[
  {
    "id": "gt-auc-vickrey-0012",
    "category": "auction_theory",
    "subcategory": "vickrey_second_price",
    "difficulty": "easy",
    "problem": "In the following Vickrey auction, 3 bidders compete for a single item:\n\nEach bidder has a private valuation for the item:\n- Bidder 1: values the item at $18\n- Bidder 2: values the item at $112\n- Bidder 3: values the item at $90\n\nThe highest bidder wins and pays the second-highest bid.\n\nDetermine the dominant strategy equilibrium, the winner, and the payment.",
    "solution": "**Dominant Strategy Analysis:**\n\nIn a second-price (Vickrey) auction, each bidder's dominant strategy is to bid their true valuation.\nThis is because:\n- Bidding above your value risks winning at a price above your value (negative surplus).\n- Bidding below your value risks losing when you could have won profitably.\n- Bidding your true value is weakly dominant regardless of others' bids. \u2713\n\n**Equilibrium Bids:**\n- Bidder 1 bids $18 (= true valuation)\n- Bidder 2 bids $112 (= true valuation)\n- Bidder 3 bids $90 (= true valuation)\n\n**Auction Outcome:**\n- Winner: Bidder 2 (highest bid: $112)\n- Payment: $90 (second-highest bid)\n- Winner's surplus: $112 - $90 = $22\n- All other bidders pay nothing and get nothing.",
    "answer": "Dominant strategy: bid true value. Winner: Bidder 2 ($112), Price: $90, Surplus: $22",
    "game_type": "sequential",
    "players": 3,
    "tags": [
      "auction",
      "vickrey",
      "second_price",
      "dominant_strategy",
      "private_value",
      "3_bidders"
    ]
  },
  {
    "id": "gt-auc-vickrey-0001",
    "category": "auction_theory",
    "subcategory": "vickrey_second_price",
    "difficulty": "easy",
    "problem": "3 bidders participate in a sealed-bid second-price auction:\n\nEach bidder has a private valuation for the item:\n- Bidder 1: values the item at $45\n- Bidder 2: values the item at $198\n- Bidder 3: values the item at $36\n\nThe highest bidder wins and pays the second-highest bid.\n\nWhat is the equilibrium bidding strategy? Determine the winner and price paid.",
    "solution": "**Dominant Strategy Analysis:**\n\nIn a second-price (Vickrey) auction, each bidder's dominant strategy is to bid their true valuation.\nThis is because:\n- Bidding above your value risks winning at a price above your value (negative surplus).\n- Bidding below your value risks losing when you could have won profitably.\n- Bidding your true value is weakly dominant regardless of others' bids. \u2713\n\n**Equilibrium Bids:**\n- Bidder 1 bids $45 (= true valuation)\n- Bidder 2 bids $198 (= true valuation)\n- Bidder 3 bids $36 (= true valuation)\n\n**Auction Outcome:**\n- Winner: Bidder 2 (highest bid: $198)\n- Payment: $45 (second-highest bid)\n- Winner's surplus: $198 - $45 = $153\n- All other bidders pay nothing and get nothing.",
    "answer": "Dominant strategy: bid true value. Winner: Bidder 2 ($198), Price: $45, Surplus: $153",
    "game_type": "sequential",
    "players": 3,
    "tags": [
      "auction",
      "vickrey",
      "second_price",
      "dominant_strategy",
      "private_value",
      "3_bidders"
    ]
  },
  {
    "id": "gt-auc-vickrey-0033",
    "category": "auction_theory",
    "subcategory": "vickrey_second_price",
    "difficulty": "easy",
    "problem": "In the following Vickrey auction, 2 bidders compete for a single item:\n\nEach bidder has a private valuation for the item:\n- Bidder 1: values the item at $159\n- Bidder 2: values the item at $22\n\nThe highest bidder wins and pays the second-highest bid.\n\nDetermine the dominant strategy equilibrium, the winner, and the payment.",
    "solution": "**Dominant Strategy Analysis:**\n\nIn a second-price (Vickrey) auction, each bidder's dominant strategy is to bid their true valuation.\nThis is because:\n- Bidding above your value risks winning at a price above your value (negative surplus).\n- Bidding below your value risks losing when you could have won profitably.\n- Bidding your true value is weakly dominant regardless of others' bids. \u2713\n\n**Equilibrium Bids:**\n- Bidder 1 bids $159 (= true valuation)\n- Bidder 2 bids $22 (= true valuation)\n\n**Auction Outcome:**\n- Winner: Bidder 1 (highest bid: $159)\n- Payment: $22 (second-highest bid)\n- Winner's surplus: $159 - $22 = $137\n- All other bidders pay nothing and get nothing.",
    "answer": "Dominant strategy: bid true value. Winner: Bidder 1 ($159), Price: $22, Surplus: $137",
    "game_type": "sequential",
    "players": 2,
    "tags": [
      "auction",
      "vickrey",
      "second_price",
      "dominant_strategy",
      "private_value",
      "2_bidders"
    ]
  },
  {
    "id": "gt-auc-fp-0049",
    "category": "auction_theory",
    "subcategory": "first_price_sealed",
    "difficulty": "medium",
    "problem": "In this first-price auction, 3 bidders have private valuations drawn from Uniform[0, 100]:\n\n3 bidders, each with a private valuation drawn independently from Uniform[0, 100].\nHighest bidder wins and pays their own bid.\nAll bidders are risk-neutral and rational.\n\nDetermine the equilibrium bid function and expected revenue.",
    "solution": "**Equilibrium Bidding Strategy:**\n\nWith 3 bidders and values drawn from Uniform[0, 100], the symmetric Bayesian Nash Equilibrium bid function is:\n\n  b(v) = ((n-1)/n) \u00d7 v = (2/3) \u00d7 v\n\n**Derivation:**\n- Each bidder shades their bid below their true value.\n- The optimal shade balances: higher bid = more likely to win, but less surplus if you win.\n- With Uniform[0,100] values, probability of winning with bid b when others bid (2/3)v:\n  P(win) = (b \u00d7 3/2)^2 / 100^2\n- Maximizing expected payoff (v - b) \u00d7 P(win) yields b*(v) = (2/3)v. \u2713\n\n**Expected Revenue:**\n- E[Revenue] = (n-1)/(n+1) \u00d7 V_max = (2/4) \u00d7 100 = 50.00\n\n**Comparison to Vickrey:**\n- By Revenue Equivalence Theorem, this equals the expected revenue from a second-price auction. \u2713",
    "answer": "BNE: b(v)=(2/3)v, E[Revenue]=50.00",
    "game_type": "simultaneous",
    "players": 3,
    "tags": [
      "auction",
      "first_price",
      "sealed_bid",
      "bayesian_nash",
      "private_value",
      "3_bidders",
      "bid_shading"
    ]
  },
  {
    "id": "gt-auc-fp-0034",
    "category": "auction_theory",
    "subcategory": "first_price_sealed",
    "difficulty": "medium",
    "problem": "Consider a first-price sealed-bid auction with 2 bidders whose values are independently drawn from Uniform[0, 200]:\n\n2 bidders, each with a private valuation drawn independently from Uniform[0, 200].\nHighest bidder wins and pays their own bid.\nAll bidders are risk-neutral and rational.\n\nFind the symmetric Bayesian Nash Equilibrium bidding strategy.",
    "solution": "**Equilibrium Bidding Strategy:**\n\nWith 2 bidders and values drawn from Uniform[0, 200], the symmetric Bayesian Nash Equilibrium bid function is:\n\n  b(v) = ((n-1)/n) \u00d7 v = (1/2) \u00d7 v\n\n**Derivation:**\n- Each bidder shades their bid below their true value.\n- The optimal shade balances: higher bid = more likely to win, but less surplus if you win.\n- With Uniform[0,200] values, probability of winning with bid b when others bid (1/2)v:\n  P(win) = (b \u00d7 2/1)^1 / 200^1\n- Maximizing expected payoff (v - b) \u00d7 P(win) yields b*(v) = (1/2)v. \u2713\n\n**Expected Revenue:**\n- E[Revenue] = (n-1)/(n+1) \u00d7 V_max = (1/3) \u00d7 200 = 66.67\n\n**Comparison to Vickrey:**\n- By Revenue Equivalence Theorem, this equals the expected revenue from a second-price auction. \u2713",
    "answer": "BNE: b(v)=(1/2)v, E[Revenue]=66.67",
    "game_type": "simultaneous",
    "players": 2,
    "tags": [
      "auction",
      "first_price",
      "sealed_bid",
      "bayesian_nash",
      "private_value",
      "2_bidders",
      "bid_shading"
    ]
  },
  {
    "id": "gt-auc-vickrey-0068",
    "category": "auction_theory",
    "subcategory": "vickrey_second_price",
    "difficulty": "medium",
    "problem": "Analyze this second-price auction:\n\nEach bidder has a private valuation for the item:\n- Bidder 1: values the item at $116\n- Bidder 2: values the item at $25\n- Bidder 3: values the item at $117\n- Bidder 4: values the item at $109\n- Bidder 5: values the item at $129\n- Bidder 6: values the item at $62\n\nThe highest bidder wins and pays the second-highest bid.\n\nFind each bidder's optimal bidding strategy and the auction outcome.",
    "solution": "**Dominant Strategy Analysis:**\n\nIn a second-price (Vickrey) auction, each bidder's dominant strategy is to bid their true valuation.\nThis is because:\n- Bidding above your value risks winning at a price above your value (negative surplus).\n- Bidding below your value risks losing when you could have won profitably.\n- Bidding your true value is weakly dominant regardless of others' bids. \u2713\n\n**Equilibrium Bids:**\n- Bidder 1 bids $116 (= true valuation)\n- Bidder 2 bids $25 (= true valuation)\n- Bidder 3 bids $117 (= true valuation)\n- Bidder 4 bids $109 (= true valuation)\n- Bidder 5 bids $129 (= true valuation)\n- Bidder 6 bids $62 (= true valuation)\n\n**Auction Outcome:**\n- Winner: Bidder 5 (highest bid: $129)\n- Payment: $117 (second-highest bid)\n- Winner's surplus: $129 - $117 = $12\n- All other bidders pay nothing and get nothing.",
    "answer": "Dominant strategy: bid true value. Winner: Bidder 5 ($129), Price: $117, Surplus: $12",
    "game_type": "sequential",
    "players": 6,
    "tags": [
      "auction",
      "vickrey",
      "second_price",
      "dominant_strategy",
      "private_value",
      "6_bidders"
    ]
  },
  {
    "id": "gt-auc-fp-0043",
    "category": "auction_theory",
    "subcategory": "first_price_sealed",
    "difficulty": "hard",
    "problem": "In this first-price auction, 10 bidders have private valuations drawn from Uniform[0, 100]:\n\n10 bidders, each with a private valuation drawn independently from Uniform[0, 100].\nHighest bidder wins and pays their own bid.\nAll bidders are risk-neutral and rational.\n\nDetermine the equilibrium bid function and expected revenue.",
    "solution": "**Equilibrium Bidding Strategy:**\n\nWith 10 bidders and values drawn from Uniform[0, 100], the symmetric Bayesian Nash Equilibrium bid function is:\n\n  b(v) = ((n-1)/n) \u00d7 v = (9/10) \u00d7 v\n\n**Derivation:**\n- Each bidder shades their bid below their true value.\n- The optimal shade balances: higher bid = more likely to win, but less surplus if you win.\n- With Uniform[0,100] values, probability of winning with bid b when others bid (9/10)v:\n  P(win) = (b \u00d7 10/9)^9 / 100^9\n- Maximizing expected payoff (v - b) \u00d7 P(win) yields b*(v) = (9/10)v. \u2713\n\n**Expected Revenue:**\n- E[Revenue] = (n-1)/(n+1) \u00d7 V_max = (9/11) \u00d7 100 = 81.82\n\n**Comparison to Vickrey:**\n- By Revenue Equivalence Theorem, this equals the expected revenue from a second-price auction. \u2713",
    "answer": "BNE: b(v)=(9/10)v, E[Revenue]=81.82",
    "game_type": "simultaneous",
    "players": 10,
    "tags": [
      "auction",
      "first_price",
      "sealed_bid",
      "bayesian_nash",
      "private_value",
      "10_bidders",
      "bid_shading"
    ]
  },
  {
    "id": "gt-auc-fp-0036",
    "category": "auction_theory",
    "subcategory": "first_price_sealed",
    "difficulty": "hard",
    "problem": "Analyze the following first-price sealed-bid auction:\n\n5 bidders, each with a private valuation drawn independently from Uniform[0, 200].\nHighest bidder wins and pays their own bid.\nAll bidders are risk-neutral and rational.\n\nSuppose the realized valuations are:\n- Bidder 1: v = 154\n- Bidder 2: v = 131\n- Bidder 3: v = 99\n- Bidder 4: v = 49\n- Bidder 5: v = 38\n\nWhat is the equilibrium bidding strategy? Compare it to truthful bidding.",
    "solution": "**Equilibrium Bidding Strategy:**\n\nWith 5 bidders and values drawn from Uniform[0, 200], the symmetric Bayesian Nash Equilibrium bid function is:\n\n  b(v) = ((n-1)/n) \u00d7 v = (4/5) \u00d7 v\n\n**Derivation:**\n- Each bidder shades their bid below their true value.\n- The optimal shade balances: higher bid = more likely to win, but less surplus if you win.\n- With Uniform[0,200] values, probability of winning with bid b when others bid (4/5)v:\n  P(win) = (b \u00d7 5/4)^4 / 200^4\n- Maximizing expected payoff (v - b) \u00d7 P(win) yields b*(v) = (4/5)v. \u2713\n\n**Example with given valuations:**\n- Bidder 1: v=154, bids (4/5)\u00d7154 = 123.2\n- Bidder 2: v=131, bids (4/5)\u00d7131 = 104.8\n- Bidder 3: v=99, bids (4/5)\u00d799 = 79.2\n- Bidder 4: v=49, bids (4/5)\u00d749 = 39.2\n- Bidder 5: v=38, bids (4/5)\u00d738 = 30.4\n- Winner: Bidder 1 (highest bid: 123.2)\n- Payment: 123.2\n\n**Expected Revenue:**\n- E[Revenue] = (n-1)/(n+1) \u00d7 V_max = (4/6) \u00d7 200 = 133.33\n\n**Comparison to Vickrey:**\n- By Revenue Equivalence Theorem, this equals the expected revenue from a second-price auction. \u2713",
    "answer": "BNE: b(v)=(4/5)v, E[Revenue]=133.33",
    "game_type": "simultaneous",
    "players": 5,
    "tags": [
      "auction",
      "first_price",
      "sealed_bid",
      "bayesian_nash",
      "private_value",
      "5_bidders",
      "bid_shading"
    ]
  },
  {
    "id": "gt-auc-curse-0017",
    "category": "auction_theory",
    "subcategory": "winners_curse",
    "difficulty": "hard",
    "problem": "In this common-value auction, bidders receive noisy signals:\n\nAn item has an unknown common value V for all bidders.\nEach of 2 bidders receives a private signal: si = V + \u03b5i, where \u03b5i ~ Uniform[-10, +10].\nThe auction is a first-price sealed-bid.\n\nAnalyze the winner's curse and find the optimal shading strategy.",
    "solution": "**Winner's Curse Analysis:**\n\nThe winner's curse arises because winning a common-value auction means you likely had\nthe most optimistic signal. If you bid naively based on your signal, you systematically overpay.\n\n**Key Insight:**\n- Your signal si is an unbiased estimate of V.\n- But conditional on WINNING (having the highest signal), your signal is biased upward.\n- Expected bias of the highest of 2 signals from Uniform[-10, +10]:\n  E[max(\u03b51,...,\u03b52)] = 10 \u00d7 (2-1)/(2+1) = 10\u00d71/3 = 3.33\n- Therefore, conditional on winning: E[V | win] \u2248 si - 3.33\n\n**Optimal Bidding Strategy:**\n- Shade bid: b(si) = si - 3.33\n- This corrects for the selection bias of winning. \u2713\n",
    "answer": "Shade amount: 3.33, Optimal bid: b(s) = s - 3.33",
    "game_type": "simultaneous",
    "players": 2,
    "tags": [
      "auction",
      "winners_curse",
      "common_value",
      "bid_shading",
      "2_bidders",
      "information_asymmetry"
    ]
  },
  {
    "id": "gt-auc-vickrey-0018",
    "category": "auction_theory",
    "subcategory": "vickrey_second_price",
    "difficulty": "medium",
    "problem": "Consider a second-price sealed-bid (Vickrey) auction with 5 bidders:\n\nEach bidder has a private valuation for the item:\n- Bidder 1: values the item at $38\n- Bidder 2: values the item at $14\n- Bidder 3: values the item at $71\n- Bidder 4: values the item at $88\n- Bidder 5: values the item at $102\n\nThe highest bidder wins and pays the second-highest bid.\n\nWhat is each bidder's dominant strategy? Who wins and what price do they pay?",
    "solution": "**Dominant Strategy Analysis:**\n\nIn a second-price (Vickrey) auction, each bidder's dominant strategy is to bid their true valuation.\nThis is because:\n- Bidding above your value risks winning at a price above your value (negative surplus).\n- Bidding below your value risks losing when you could have won profitably.\n- Bidding your true value is weakly dominant regardless of others' bids. \u2713\n\n**Equilibrium Bids:**\n- Bidder 1 bids $38 (= true valuation)\n- Bidder 2 bids $14 (= true valuation)\n- Bidder 3 bids $71 (= true valuation)\n- Bidder 4 bids $88 (= true valuation)\n- Bidder 5 bids $102 (= true valuation)\n\n**Auction Outcome:**\n- Winner: Bidder 5 (highest bid: $102)\n- Payment: $88 (second-highest bid)\n- Winner's surplus: $102 - $88 = $14\n- All other bidders pay nothing and get nothing.",
    "answer": "Dominant strategy: bid true value. Winner: Bidder 5 ($102), Price: $88, Surplus: $14",
    "game_type": "sequential",
    "players": 5,
    "tags": [
      "auction",
      "vickrey",
      "second_price",
      "dominant_strategy",
      "private_value",
      "5_bidders"
    ]
  },
  {
    "id": "gt-bayes-2type-0017",
    "category": "bayesian_game",
    "subcategory": "two_type_bayesian",
    "difficulty": "hard",
    "problem": "Analyze this Bayesian game where players have private information:\n\nPlayer 1 has a private type: Strong (probability 8/10) or Weak (probability 2/10).\nPlayer 2 does not know Player 1's type.\nBoth players choose from: {Enter, Stay Out}.\n\nPayoffs (Player 1, Player 2) if Player 1 is type Strong:\nP1 \\ P2 | Enter | Stay Out\n--- | --- | ---\nEnter | (6,4) | (2,2)\nStay Out | (-5,1) | (7,5)\n\nPayoffs if Player 1 is type Weak:\nP1 \\ P2 | Enter | Stay Out\n--- | --- | ---\nEnter | (3,-3) | (3,6)\nStay Out | (7,5) | (-5,-2)\n\nWhat is the Bayesian Nash Equilibrium?",
    "solution": "**Bayesian Nash Equilibrium Analysis:**\n\nPrior: P(Strong) = 8/10, P(Weak) = 2/10\n\n**BNE 1:**\n- Player 1 (Strong type): plays Enter\n  Check: payoff from Enter = 6 vs alternative = -5. \u2713\n- Player 1 (Weak type): plays Stay Out\n  Check: payoff from Stay Out = 7 vs alternative = 3. \u2713\n- Player 2: plays Enter\n  Expected payoff: 8/10\u00d74 + 2/10\u00d75 = 4.2\n  vs alternative Stay Out: 1.2. \u2713\n\n**BNE 2:**\n- Player 1 (Strong type): plays Stay Out\n  Check: payoff from Stay Out = 7 vs alternative = 2. \u2713\n- Player 1 (Weak type): plays Enter\n  Check: payoff from Enter = 3 vs alternative = -5. \u2713\n- Player 2: plays Stay Out\n  Expected payoff: 8/10\u00d75 + 2/10\u00d76 = 5.2\n  vs alternative Enter: 0.2. \u2713\n",
    "answer": "BNE1: P1(Strong)=Enter, P1(Weak)=Stay Out, P2=Enter; BNE2: P1(Strong)=Stay Out, P1(Weak)=Enter, P2=Stay Out",
    "game_type": "simultaneous",
    "players": 2,
    "tags": [
      "bayesian",
      "incomplete_information",
      "bayesian_nash",
      "private_type",
      "pure_strategy"
    ]
  },
  {
    "id": "gt-bayes-2type-0057",
    "category": "bayesian_game",
    "subcategory": "two_type_bayesian",
    "difficulty": "hard",
    "problem": "Players have private types in the following game:\n\nPlayer 1 has a private type: High (probability 5/10) or Low (probability 5/10).\nPlayer 2 does not know Player 1's type.\nBoth players choose from: {High Price, Low Price}.\n\nPayoffs (Player 1, Player 2) if Player 1 is type High:\nP1 \\ P2 | High Price | Low Price\n--- | --- | ---\nHigh Price | (-5,0) | (3,4)\nLow Price | (5,6) | (-5,0)\n\nPayoffs if Player 1 is type Low:\nP1 \\ P2 | High Price | Low Price\n--- | --- | ---\nHigh Price | (-1,7) | (-3,-1)\nLow Price | (-5,-3) | (1,7)\n\nSolve for the Bayesian Nash Equilibrium strategies.",
    "solution": "**Bayesian Nash Equilibrium Analysis:**\n\nPrior: P(High) = 5/10, P(Low) = 5/10\n\n**BNE 1:**\n- Player 1 (High type): plays High Price\n  Check: payoff from High Price = 3 vs alternative = -5. \u2713\n- Player 1 (Low type): plays Low Price\n  Check: payoff from Low Price = 1 vs alternative = -3. \u2713\n- Player 2: plays Low Price\n  Expected payoff: 5/10\u00d74 + 5/10\u00d77 = 5.5\n  vs alternative High Price: -1.5. \u2713\n\n**BNE 2:**\n- Player 1 (High type): plays Low Price\n  Check: payoff from Low Price = 5 vs alternative = -5. \u2713\n- Player 1 (Low type): plays High Price\n  Check: payoff from High Price = -1 vs alternative = -5. \u2713\n- Player 2: plays High Price\n  Expected payoff: 5/10\u00d76 + 5/10\u00d77 = 6.5\n  vs alternative Low Price: -0.5. \u2713\n",
    "answer": "BNE1: P1(High)=High Price, P1(Low)=Low Price, P2=Low Price; BNE2: P1(High)=Low Price, P1(Low)=High Price, P2=High Price",
    "game_type": "simultaneous",
    "players": 2,
    "tags": [
      "bayesian",
      "incomplete_information",
      "bayesian_nash",
      "private_type",
      "pure_strategy"
    ]
  },
  {
    "id": "gt-bayes-2type-0133",
    "category": "bayesian_game",
    "subcategory": "two_type_bayesian",
    "difficulty": "hard",
    "problem": "Consider the following Bayesian game with incomplete information:\n\nPlayer 1 has a private type: Cooperative (probability 4/10) or Aggressive (probability 6/10).\nPlayer 2 does not know Player 1's type.\nBoth players choose from: {Invest, Wait}.\n\nPayoffs (Player 1, Player 2) if Player 1 is type Cooperative:\nP1 \\ P2 | Invest | Wait\n--- | --- | ---\nInvest | (10,7) | (-2,2)\nWait | (10,-3) | (-5,6)\n\nPayoffs if Player 1 is type Aggressive:\nP1 \\ P2 | Invest | Wait\n--- | --- | ---\nInvest | (5,-1) | (7,8)\nWait | (6,0) | (10,-3)\n\nFind the Bayesian Nash Equilibrium.",
    "solution": "**Bayesian Nash Equilibrium Analysis:**\n\nPrior: P(Cooperative) = 4/10, P(Aggressive) = 6/10\n\n**BNE 1:**\n- Player 1 (Cooperative type): plays Invest\n  Check: payoff from Invest = 10 vs alternative = 10. \u2713\n- Player 1 (Aggressive type): plays Wait\n  Check: payoff from Wait = 6 vs alternative = 5. \u2713\n- Player 2: plays Invest\n  Expected payoff: 4/10\u00d77 + 6/10\u00d70 = 2.8\n  vs alternative Wait: -1.0. \u2713\n",
    "answer": "BNE1: P1(Cooperative)=Invest, P1(Aggressive)=Wait, P2=Invest",
    "game_type": "simultaneous",
    "players": 2,
    "tags": [
      "bayesian",
      "incomplete_information",
      "bayesian_nash",
      "private_type",
      "pure_strategy"
    ]
  },
  {
    "id": "gt-bayes-2type-0146",
    "category": "bayesian_game",
    "subcategory": "two_type_bayesian",
    "difficulty": "hard",
    "problem": "Analyze this Bayesian game where players have private information:\n\nPlayer 1 has a private type: Strong (probability 8/10) or Weak (probability 2/10).\nPlayer 2 does not know Player 1's type.\nBoth players choose from: {Cooperate, Defect}.\n\nPayoffs (Player 1, Player 2) if Player 1 is type Strong:\nP1 \\ P2 | Cooperate | Defect\n--- | --- | ---\nCooperate | (8,10) | (-4,-3)\nDefect | (3,7) | (-1,8)\n\nPayoffs if Player 1 is type Weak:\nP1 \\ P2 | Cooperate | Defect\n--- | --- | ---\nCooperate | (1,2) | (-5,7)\nDefect | (6,10) | (10,6)\n\nWhat is the Bayesian Nash Equilibrium?",
    "solution": "**Bayesian Nash Equilibrium Analysis:**\n\nPrior: P(Strong) = 8/10, P(Weak) = 2/10\n\n**BNE 1:**\n- Player 1 (Strong type): plays Cooperate\n  Check: payoff from Cooperate = 8 vs alternative = 3. \u2713\n- Player 1 (Weak type): plays Defect\n  Check: payoff from Defect = 6 vs alternative = 1. \u2713\n- Player 2: plays Cooperate\n  Expected payoff: 8/10\u00d710 + 2/10\u00d710 = 10.0\n  vs alternative Defect: -1.2. \u2713\n\n**BNE 2:**\n- Player 1 (Strong type): plays Defect\n  Check: payoff from Defect = -1 vs alternative = -4. \u2713\n- Player 1 (Weak type): plays Defect\n  Check: payoff from Defect = 10 vs alternative = -5. \u2713\n- Player 2: plays Defect\n  Expected payoff: 8/10\u00d78 + 2/10\u00d76 = 7.6\n  vs alternative Cooperate: 7.6. \u2713\n",
    "answer": "BNE1: P1(Strong)=Cooperate, P1(Weak)=Defect, P2=Cooperate; BNE2: P1(Strong)=Defect, P1(Weak)=Defect, P2=Defect",
    "game_type": "simultaneous",
    "players": 2,
    "tags": [
      "bayesian",
      "incomplete_information",
      "bayesian_nash",
      "private_type",
      "pure_strategy"
    ]
  },
  {
    "id": "gt-bayes-signal-0094",
    "category": "bayesian_game",
    "subcategory": "signaling_game",
    "difficulty": "hard",
    "problem": "Consider a signaling game:\n\nA Sender (worker) has private type: High ability (probability 0.4) or Low ability (probability 0.6).\nThe Sender can choose to Signal (e.g., get education) or Not Signal.\nThe Receiver (employer) observes the signal and offers a wage.\n\nParameters:\n- Cost of signaling for High type: 2\n- Cost of signaling for Low type: 7\n- Wage if Receiver believes High: 11\n- Wage if Receiver believes Low: 2\n- Sender payoff = wage - signal cost (if signaled), or wage (if not)\n\nFind the Perfect Bayesian Equilibrium. Is there a separating or pooling equilibrium?",
    "solution": "**Signaling Game Analysis:**\n\n**Separating Equilibrium** (High signals, Low doesn't):\n- High type: Signal \u2192 gets wage 11, net payoff = 11 - 2 = 9\n  vs Not Signal \u2192 gets wage 2, net payoff = 2\n  Prefer Signal? 9 > 2? Yes \u2713\n- Low type: Not Signal \u2192 gets wage 2, net payoff = 2\n  vs Signal \u2192 gets wage 11, net payoff = 11 - 7 = 4\n  Prefer Not Signal? 2 > 4? No \u2717\n- Separating equilibrium DOES NOT EXIST \u2717\n\n**Pooling Equilibrium** (Both signal or both don't):\n- If both signal: Receiver can't distinguish, offers pooling wage = 0.4\u00d711 + 0.6\u00d72 = 5.6\n  High type net: 5.6 - 2 = 3.6\n  Low type net: 5.6 - 7 = -1.4\n\n**Result:** No pure strategy PBE found with these parameters",
    "answer": "Equilibria: No pure strategy PBE found with these parameters",
    "game_type": "sequential",
    "players": 2,
    "tags": [
      "bayesian",
      "signaling",
      "incomplete_information",
      "perfect_bayesian",
      "separating",
      "pooling",
      "private_type"
    ]
  },
  {
    "id": "gt-bayes-2type-0015",
    "category": "bayesian_game",
    "subcategory": "two_type_bayesian",
    "difficulty": "hard",
    "problem": "In this game of incomplete information:\n\nPlayer 1 has a private type: Strong (probability 3/10) or Weak (probability 7/10).\nPlayer 2 does not know Player 1's type.\nBoth players choose from: {Cooperate, Defect}.\n\nPayoffs (Player 1, Player 2) if Player 1 is type Strong:\nP1 \\ P2 | Cooperate | Defect\n--- | --- | ---\nCooperate | (9,8) | (-5,-2)\nDefect | (-3,-1) | (-4,6)\n\nPayoffs if Player 1 is type Weak:\nP1 \\ P2 | Cooperate | Defect\n--- | --- | ---\nCooperate | (-1,8) | (-1,-4)\nDefect | (4,6) | (-4,6)\n\nDetermine each type's optimal strategy in the Bayesian Nash Equilibrium.",
    "solution": "**Bayesian Nash Equilibrium Analysis:**\n\nPrior: P(Strong) = 3/10, P(Weak) = 7/10\n\n**BNE 1:**\n- Player 1 (Strong type): plays Cooperate\n  Check: payoff from Cooperate = 9 vs alternative = -3. \u2713\n- Player 1 (Weak type): plays Defect\n  Check: payoff from Defect = 4 vs alternative = -1. \u2713\n- Player 2: plays Cooperate\n  Expected payoff: 3/10\u00d78 + 7/10\u00d76 = 6.6\n  vs alternative Defect: 3.6. \u2713\n",
    "answer": "BNE1: P1(Strong)=Cooperate, P1(Weak)=Defect, P2=Cooperate",
    "game_type": "simultaneous",
    "players": 2,
    "tags": [
      "bayesian",
      "incomplete_information",
      "bayesian_nash",
      "private_type",
      "pure_strategy"
    ]
  },
  {
    "id": "gt-bayes-2type-0124",
    "category": "bayesian_game",
    "subcategory": "two_type_bayesian",
    "difficulty": "hard",
    "problem": "Analyze this Bayesian game where players have private information:\n\nPlayer 1 has a private type: Informed (probability 8/10) or Uninformed (probability 2/10).\nPlayer 2 does not know Player 1's type.\nBoth players choose from: {Fight, Yield}.\n\nPayoffs (Player 1, Player 2) if Player 1 is type Informed:\nP1 \\ P2 | Fight | Yield\n--- | --- | ---\nFight | (10,0) | (-3,-5)\nYield | (2,4) | (-4,3)\n\nPayoffs if Player 1 is type Uninformed:\nP1 \\ P2 | Fight | Yield\n--- | --- | ---\nFight | (2,4) | (0,9)\nYield | (10,-2) | (-2,3)\n\nWhat is the Bayesian Nash Equilibrium?",
    "solution": "**Bayesian Nash Equilibrium Analysis:**\n\nPrior: P(Informed) = 8/10, P(Uninformed) = 2/10\n\n**BNE 1:**\n- Player 1 (Informed type): plays Fight\n  Check: payoff from Fight = 10 vs alternative = 2. \u2713\n- Player 1 (Uninformed type): plays Yield\n  Check: payoff from Yield = 10 vs alternative = 2. \u2713\n- Player 2: plays Fight\n  Expected payoff: 8/10\u00d70 + 2/10\u00d7-2 = -0.4\n  vs alternative Yield: -3.4. \u2713\n",
    "answer": "BNE1: P1(Informed)=Fight, P1(Uninformed)=Yield, P2=Fight",
    "game_type": "simultaneous",
    "players": 2,
    "tags": [
      "bayesian",
      "incomplete_information",
      "bayesian_nash",
      "private_type",
      "pure_strategy"
    ]
  },
  {
    "id": "gt-bayes-signal-0050",
    "category": "bayesian_game",
    "subcategory": "signaling_game",
    "difficulty": "hard",
    "problem": "In this signaling game with two types:\n\nA Sender (worker) has private type: High ability (probability 0.4) or Low ability (probability 0.6).\nThe Sender can choose to Signal (e.g., get education) or Not Signal.\nThe Receiver (employer) observes the signal and offers a wage.\n\nParameters:\n- Cost of signaling for High type: 3\n- Cost of signaling for Low type: 6\n- Wage if Receiver believes High: 11\n- Wage if Receiver believes Low: 4\n- Sender payoff = wage - signal cost (if signaled), or wage (if not)\n\nDetermine whether a separating equilibrium, pooling equilibrium, or both exist.",
    "solution": "**Signaling Game Analysis:**\n\n**Separating Equilibrium** (High signals, Low doesn't):\n- High type: Signal \u2192 gets wage 11, net payoff = 11 - 3 = 8\n  vs Not Signal \u2192 gets wage 4, net payoff = 4\n  Prefer Signal? 8 > 4? Yes \u2713\n- Low type: Not Signal \u2192 gets wage 4, net payoff = 4\n  vs Signal \u2192 gets wage 11, net payoff = 11 - 6 = 5\n  Prefer Not Signal? 4 > 5? No \u2717\n- Separating equilibrium DOES NOT EXIST \u2717\n\n**Pooling Equilibrium** (Both signal or both don't):\n- If both signal: Receiver can't distinguish, offers pooling wage = 0.4\u00d711 + 0.6\u00d74 = 6.8\n  High type net: 6.8 - 3 = 3.8\n  Low type net: 6.8 - 6 = 0.8\n\n**Result:** No pure strategy PBE found with these parameters",
    "answer": "Equilibria: No pure strategy PBE found with these parameters",
    "game_type": "sequential",
    "players": 2,
    "tags": [
      "bayesian",
      "signaling",
      "incomplete_information",
      "perfect_bayesian",
      "separating",
      "pooling",
      "private_type"
    ]
  },
  {
    "id": "gt-bayes-2type-0138",
    "category": "bayesian_game",
    "subcategory": "two_type_bayesian",
    "difficulty": "hard",
    "problem": "Analyze this Bayesian game where players have private information:\n\nPlayer 1 has a private type: Informed (probability 6/10) or Uninformed (probability 4/10).\nPlayer 2 does not know Player 1's type.\nBoth players choose from: {Enter, Stay Out}.\n\nPayoffs (Player 1, Player 2) if Player 1 is type Informed:\nP1 \\ P2 | Enter | Stay Out\n--- | --- | ---\nEnter | (2,8) | (5,3)\nStay Out | (-3,6) | (-2,-4)\n\nPayoffs if Player 1 is type Uninformed:\nP1 \\ P2 | Enter | Stay Out\n--- | --- | ---\nEnter | (0,2) | (-4,7)\nStay Out | (-3,9) | (4,4)\n\nWhat is the Bayesian Nash Equilibrium?",
    "solution": "**Bayesian Nash Equilibrium Analysis:**\n\nPrior: P(Informed) = 6/10, P(Uninformed) = 4/10\n\n**BNE 1:**\n- Player 1 (Informed type): plays Enter\n  Check: payoff from Enter = 2 vs alternative = -3. \u2713\n- Player 1 (Uninformed type): plays Enter\n  Check: payoff from Enter = 0 vs alternative = -3. \u2713\n- Player 2: plays Enter\n  Expected payoff: 6/10\u00d78 + 4/10\u00d72 = 5.6\n  vs alternative Stay Out: 4.6. \u2713\n",
    "answer": "BNE1: P1(Informed)=Enter, P1(Uninformed)=Enter, P2=Enter",
    "game_type": "simultaneous",
    "players": 2,
    "tags": [
      "bayesian",
      "incomplete_information",
      "bayesian_nash",
      "private_type",
      "pure_strategy"
    ]
  },
  {
    "id": "gt-bayes-signal-0065",
    "category": "bayesian_game",
    "subcategory": "signaling_game",
    "difficulty": "hard",
    "problem": "In this signaling game with two types:\n\nA Sender (worker) has private type: High ability (probability 0.4) or Low ability (probability 0.6).\nThe Sender can choose to Signal (e.g., get education) or Not Signal.\nThe Receiver (employer) observes the signal and offers a wage.\n\nParameters:\n- Cost of signaling for High type: 2\n- Cost of signaling for Low type: 5\n- Wage if Receiver believes High: 13\n- Wage if Receiver believes Low: 4\n- Sender payoff = wage - signal cost (if signaled), or wage (if not)\n\nDetermine whether a separating equilibrium, pooling equilibrium, or both exist.",
    "solution": "**Signaling Game Analysis:**\n\n**Separating Equilibrium** (High signals, Low doesn't):\n- High type: Signal \u2192 gets wage 13, net payoff = 13 - 2 = 11\n  vs Not Signal \u2192 gets wage 4, net payoff = 4\n  Prefer Signal? 11 > 4? Yes \u2713\n- Low type: Not Signal \u2192 gets wage 4, net payoff = 4\n  vs Signal \u2192 gets wage 13, net payoff = 13 - 5 = 8\n  Prefer Not Signal? 4 > 8? No \u2717\n- Separating equilibrium DOES NOT EXIST \u2717\n\n**Pooling Equilibrium** (Both signal or both don't):\n- If both signal: Receiver can't distinguish, offers pooling wage = 0.4\u00d713 + 0.6\u00d74 = 7.6\n  High type net: 7.6 - 2 = 5.6\n  Low type net: 7.6 - 5 = 2.6\n\n**Result:** No pure strategy PBE found with these parameters",
    "answer": "Equilibria: No pure strategy PBE found with these parameters",
    "game_type": "sequential",
    "players": 2,
    "tags": [
      "bayesian",
      "signaling",
      "incomplete_information",
      "perfect_bayesian",
      "separating",
      "pooling",
      "private_type"
    ]
  },
  {
    "id": "gt-coop-shapley-0118",
    "category": "cooperative_game",
    "subcategory": "shapley_value",
    "difficulty": "medium",
    "problem": "Calculate the Shapley value for each player in the following cooperative game:\n\nPlayers: {P1, P2, P3}\nCharacteristic function v(S):\n\n- v({P1}) = 9\n- v({P2}) = 19\n- v({P3}) = 17\n- v({P1, P2}) = 27\n- v({P1, P3}) = 26\n- v({P2, P3}) = 24\n- v({P1, P2, P3}) = 90",
    "solution": "**Shapley Value Computation:**\n\nThe Shapley value \u03c6\u1d62 assigns each player their average marginal contribution across all possible orderings.\n\nFormula: \u03c6\u1d62 = \u03a3 [|S|!(n-|S|-1)!/n!] \u00d7 [v(S\u222a{i}) - v(S)]\n\n**Player P1:**\n  S=\u2205: v(S\u222aP1)=9 - v(S)=0 = 9, weight=0.3333\n  S={P2}: v(S\u222aP1)=27 - v(S)=19 = 8, weight=0.1667\n  S={P3}: v(S\u222aP1)=26 - v(S)=17 = 9, weight=0.1667\n  S={P2, P3}: v(S\u222aP1)=90 - v(S)=24 = 66, weight=0.3333\n  **\u03c6(P1) = 27.8333** \u2713\n\n**Player P2:**\n  S=\u2205: v(S\u222aP2)=19 - v(S)=0 = 19, weight=0.3333\n  S={P1}: v(S\u222aP2)=27 - v(S)=9 = 18, weight=0.1667\n  S={P3}: v(S\u222aP2)=24 - v(S)=17 = 7, weight=0.1667\n  S={P1, P3}: v(S\u222aP2)=90 - v(S)=26 = 64, weight=0.3333\n  **\u03c6(P2) = 31.8333** \u2713\n\n**Player P3:**\n  S=\u2205: v(S\u222aP3)=17 - v(S)=0 = 17, weight=0.3333\n  S={P1}: v(S\u222aP3)=26 - v(S)=9 = 17, weight=0.1667\n  S={P2}: v(S\u222aP3)=24 - v(S)=19 = 5, weight=0.1667\n  S={P1, P2}: v(S\u222aP3)=90 - v(S)=27 = 63, weight=0.3333\n  **\u03c6(P3) = 30.3333** \u2713\n\n**Efficiency check:** \u03a3\u03c6\u1d62 = 27.8333 + 31.8333 + 30.3333 = 89.9999\nv(N) = 90. Match \u2713",
    "answer": "\u03c6(P1)=27.8333, \u03c6(P2)=31.8333, \u03c6(P3)=30.3333",
    "game_type": "cooperative",
    "players": 3,
    "tags": [
      "cooperative",
      "shapley_value",
      "coalition",
      "fair_division",
      "3_player",
      "characteristic_function"
    ]
  },
  {
    "id": "gt-coop-shapley-0053",
    "category": "cooperative_game",
    "subcategory": "shapley_value",
    "difficulty": "medium",
    "problem": "Calculate the Shapley value for each player in the following cooperative game:\n\nPlayers: {P1, P2, P3}\nCharacteristic function v(S):\n\n- v({P1}) = 17\n- v({P2}) = 15\n- v({P3}) = 14\n- v({P1, P2}) = 18\n- v({P1, P3}) = 40\n- v({P2, P3}) = 34\n- v({P1, P2, P3}) = 74",
    "solution": "**Shapley Value Computation:**\n\nThe Shapley value \u03c6\u1d62 assigns each player their average marginal contribution across all possible orderings.\n\nFormula: \u03c6\u1d62 = \u03a3 [|S|!(n-|S|-1)!/n!] \u00d7 [v(S\u222a{i}) - v(S)]\n\n**Player P1:**\n  S=\u2205: v(S\u222aP1)=17 - v(S)=0 = 17, weight=0.3333\n  S={P2}: v(S\u222aP1)=18 - v(S)=15 = 3, weight=0.1667\n  S={P3}: v(S\u222aP1)=40 - v(S)=14 = 26, weight=0.1667\n  S={P2, P3}: v(S\u222aP1)=74 - v(S)=34 = 40, weight=0.3333\n  **\u03c6(P1) = 23.8333** \u2713\n\n**Player P2:**\n  S=\u2205: v(S\u222aP2)=15 - v(S)=0 = 15, weight=0.3333\n  S={P1}: v(S\u222aP2)=18 - v(S)=17 = 1, weight=0.1667\n  S={P3}: v(S\u222aP2)=34 - v(S)=14 = 20, weight=0.1667\n  S={P1, P3}: v(S\u222aP2)=74 - v(S)=40 = 34, weight=0.3333\n  **\u03c6(P2) = 19.8333** \u2713\n\n**Player P3:**\n  S=\u2205: v(S\u222aP3)=14 - v(S)=0 = 14, weight=0.3333\n  S={P1}: v(S\u222aP3)=40 - v(S)=17 = 23, weight=0.1667\n  S={P2}: v(S\u222aP3)=34 - v(S)=15 = 19, weight=0.1667\n  S={P1, P2}: v(S\u222aP3)=74 - v(S)=18 = 56, weight=0.3333\n  **\u03c6(P3) = 30.3333** \u2713\n\n**Efficiency check:** \u03a3\u03c6\u1d62 = 23.8333 + 19.8333 + 30.3333 = 73.9999\nv(N) = 74. Match \u2713",
    "answer": "\u03c6(P1)=23.8333, \u03c6(P2)=19.8333, \u03c6(P3)=30.3333",
    "game_type": "cooperative",
    "players": 3,
    "tags": [
      "cooperative",
      "shapley_value",
      "coalition",
      "fair_division",
      "3_player",
      "characteristic_function"
    ]
  },
  {
    "id": "gt-coop-vote-0081",
    "category": "cooperative_game",
    "subcategory": "voting_power",
    "difficulty": "medium",
    "problem": "In this voting game:\n\nWeighted voting game [12; 15, 4, 1]\n- Quota to pass: 12\n- Total weight: 20\n- Voter 1: weight = 15\n- Voter 2: weight = 4\n- Voter 3: weight = 1\n\nDetermine each voter's power using the Shapley-Shubik index.",
    "solution": "**Shapley-Shubik Power Index:**\n\nCount how many orderings each voter is pivotal (brings total to \u2265 12).\n\n- Voter 1 (w=15): pivotal in 6/6 orderings \u2192 SSI = 1.0 (100.0%)\n- Voter 2 (w=4): pivotal in 0/6 orderings \u2192 SSI = 0.0 (0.0%)\n- Voter 3 (w=1): pivotal in 0/6 orderings \u2192 SSI = 0.0 (0.0%)\n\n**Power distribution:**\n  Voter 1: \u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588 100.0%\n  Voter 2:  0.0%\n  Voter 3:  0.0%\n\nSum of SSI: 1.0000 \u2713\nNote: Voter 1 is a **dictator**.\nNote: Voter 1 has **veto power** (nothing passes without them).\nNote: Voter 2 is a **dummy** (zero power despite having weight).\nNote: Voter 3 is a **dummy** (zero power despite having weight).",
    "answer": "SSI(V1)=1.0, SSI(V2)=0.0, SSI(V3)=0.0",
    "game_type": "cooperative",
    "players": 3,
    "tags": [
      "cooperative",
      "voting",
      "shapley_shubik",
      "power_index",
      "3_player",
      "weighted_voting"
    ]
  },
  {
    "id": "gt-coop-shapley-0005",
    "category": "cooperative_game",
    "subcategory": "shapley_value",
    "difficulty": "hard",
    "problem": "Determine a fair division using the Shapley value:\n\nPlayers: {P1, P2, P3, P4}\nCharacteristic function v(S):\n\n- v({P1}) = 2\n- v({P2}) = 7\n- v({P3}) = 6\n- v({P4}) = 4\n- v({P1, P2}) = 12\n- v({P1, P3}) = 7\n- v({P1, P4}) = 6\n- v({P2, P3}) = 15\n- v({P2, P4}) = 8\n- v({P3, P4}) = 9\n- v({P1, P2, P3}) = 35\n- v({P1, P2, P4}) = 27\n- v({P1, P3, P4}) = 34\n- v({P2, P3, P4}) = 35\n- v({P1, P2, P3, P4}) = 41\n\nWhat does each player receive?",
    "solution": "**Shapley Value Computation:**\n\nThe Shapley value \u03c6\u1d62 assigns each player their average marginal contribution across all possible orderings.\n\nFormula: \u03c6\u1d62 = \u03a3 [|S|!(n-|S|-1)!/n!] \u00d7 [v(S\u222a{i}) - v(S)]\n\n**Player P1:**\n  S=\u2205: v(S\u222aP1)=2 - v(S)=0 = 2, weight=0.2500\n  S={P2}: v(S\u222aP1)=12 - v(S)=7 = 5, weight=0.0833\n  S={P3}: v(S\u222aP1)=7 - v(S)=6 = 1, weight=0.0833\n  S={P4}: v(S\u222aP1)=6 - v(S)=4 = 2, weight=0.0833\n  S={P2, P3}: v(S\u222aP1)=35 - v(S)=15 = 20, weight=0.0833\n  S={P2, P4}: v(S\u222aP1)=27 - v(S)=8 = 19, weight=0.0833\n  S={P3, P4}: v(S\u222aP1)=34 - v(S)=9 = 25, weight=0.0833\n  S={P2, P3, P4}: v(S\u222aP1)=41 - v(S)=35 = 6, weight=0.2500\n  **\u03c6(P1) = 8.0** \u2713\n\n**Player P2:**\n  S=\u2205: v(S\u222aP2)=7 - v(S)=0 = 7, weight=0.2500\n  S={P1}: v(S\u222aP2)=12 - v(S)=2 = 10, weight=0.0833\n  S={P3}: v(S\u222aP2)=15 - v(S)=6 = 9, weight=0.0833\n  S={P4}: v(S\u222aP2)=8 - v(S)=4 = 4, weight=0.0833\n  S={P1, P3}: v(S\u222aP2)=35 - v(S)=7 = 28, weight=0.0833\n  S={P1, P4}: v(S\u222aP2)=27 - v(S)=6 = 21, weight=0.0833\n  S={P3, P4}: v(S\u222aP2)=35 - v(S)=9 = 26, weight=0.0833\n  S={P1, P3, P4}: v(S\u222aP2)=41 - v(S)=34 = 7, weight=0.2500\n  **\u03c6(P2) = 11.6667** \u2713\n\n**Player P3:**\n  S=\u2205: v(S\u222aP3)=6 - v(S)=0 = 6, weight=0.2500\n  S={P1}: v(S\u222aP3)=7 - v(S)=2 = 5, weight=0.0833\n  S={P2}: v(S\u222aP3)=15 - v(S)=7 = 8, weight=0.0833\n  S={P4}: v(S\u222aP3)=9 - v(S)=4 = 5, weight=0.0833\n  S={P1, P2}: v(S\u222aP3)=35 - v(S)=12 = 23, weight=0.0833\n  S={P1, P4}: v(S\u222aP3)=34 - v(S)=6 = 28, weight=0.0833\n  S={P2, P4}: v(S\u222aP3)=35 - v(S)=8 = 27, weight=0.0833\n  S={P1, P2, P4}: v(S\u222aP3)=41 - v(S)=27 = 14, weight=0.2500\n  **\u03c6(P3) = 13.0** \u2713\n\n**Player P4:**\n  S=\u2205: v(S\u222aP4)=4 - v(S)=0 = 4, weight=0.2500\n  S={P1}: v(S\u222aP4)=6 - v(S)=2 = 4, weight=0.0833\n  S={P2}: v(S\u222aP4)=8 - v(S)=7 = 1, weight=0.0833\n  S={P3}: v(S\u222aP4)=9 - v(S)=6 = 3, weight=0.0833\n  S={P1, P2}: v(S\u222aP4)=27 - v(S)=12 = 15, weight=0.0833\n  S={P1, P3}: v(S\u222aP4)=34 - v(S)=7 = 27, weight=0.0833\n  S={P2, P3}: v(S\u222aP4)=35 - v(S)=15 = 20, weight=0.0833\n  S={P1, P2, P3}: v(S\u222aP4)=41 - v(S)=35 = 6, weight=0.2500\n  **\u03c6(P4) = 8.3333** \u2713\n\n**Efficiency check:** \u03a3\u03c6\u1d62 = 8.0 + 11.6667 + 13.0 + 8.3333 = 41.0\nv(N) = 41. Match \u2713",
    "answer": "\u03c6(P1)=8.0, \u03c6(P2)=11.6667, \u03c6(P3)=13.0, \u03c6(P4)=8.3333",
    "game_type": "cooperative",
    "players": 4,
    "tags": [
      "cooperative",
      "shapley_value",
      "coalition",
      "fair_division",
      "4_player",
      "characteristic_function"
    ]
  },
  {
    "id": "gt-coop-shapley-0112",
    "category": "cooperative_game",
    "subcategory": "shapley_value",
    "difficulty": "hard",
    "problem": "Calculate the Shapley value for each player in the following cooperative game:\n\nPlayers: {P1, P2, P3, P4}\nCharacteristic function v(S):\n\n- v({P1}) = 7\n- v({P2}) = 5\n- v({P3}) = 5\n- v({P4}) = 7\n- v({P1, P2}) = 9\n- v({P1, P3}) = 19\n- v({P1, P4}) = 15\n- v({P2, P3}) = 6\n- v({P2, P4}) = 12\n- v({P3, P4}) = 14\n- v({P1, P2, P3}) = 27\n- v({P1, P2, P4}) = 21\n- v({P1, P3, P4}) = 27\n- v({P2, P3, P4}) = 17\n- v({P1, P2, P3, P4}) = 29",
    "solution": "**Shapley Value Computation:**\n\nThe Shapley value \u03c6\u1d62 assigns each player their average marginal contribution across all possible orderings.\n\nFormula: \u03c6\u1d62 = \u03a3 [|S|!(n-|S|-1)!/n!] \u00d7 [v(S\u222a{i}) - v(S)]\n\n**Player P1:**\n  S=\u2205: v(S\u222aP1)=7 - v(S)=0 = 7, weight=0.2500\n  S={P2}: v(S\u222aP1)=9 - v(S)=5 = 4, weight=0.0833\n  S={P3}: v(S\u222aP1)=19 - v(S)=5 = 14, weight=0.0833\n  S={P4}: v(S\u222aP1)=15 - v(S)=7 = 8, weight=0.0833\n  S={P2, P3}: v(S\u222aP1)=27 - v(S)=6 = 21, weight=0.0833\n  S={P2, P4}: v(S\u222aP1)=21 - v(S)=12 = 9, weight=0.0833\n  S={P3, P4}: v(S\u222aP1)=27 - v(S)=14 = 13, weight=0.0833\n  S={P2, P3, P4}: v(S\u222aP1)=29 - v(S)=17 = 12, weight=0.2500\n  **\u03c6(P1) = 10.5** \u2713\n\n**Player P2:**\n  S=\u2205: v(S\u222aP2)=5 - v(S)=0 = 5, weight=0.2500\n  S={P1}: v(S\u222aP2)=9 - v(S)=7 = 2, weight=0.0833\n  S={P3}: v(S\u222aP2)=6 - v(S)=5 = 1, weight=0.0833\n  S={P4}: v(S\u222aP2)=12 - v(S)=7 = 5, weight=0.0833\n  S={P1, P3}: v(S\u222aP2)=27 - v(S)=19 = 8, weight=0.0833\n  S={P1, P4}: v(S\u222aP2)=21 - v(S)=15 = 6, weight=0.0833\n  S={P3, P4}: v(S\u222aP2)=17 - v(S)=14 = 3, weight=0.0833\n  S={P1, P3, P4}: v(S\u222aP2)=29 - v(S)=27 = 2, weight=0.2500\n  **\u03c6(P2) = 3.8333** \u2713\n\n**Player P3:**\n  S=\u2205: v(S\u222aP3)=5 - v(S)=0 = 5, weight=0.2500\n  S={P1}: v(S\u222aP3)=19 - v(S)=7 = 12, weight=0.0833\n  S={P2}: v(S\u222aP3)=6 - v(S)=5 = 1, weight=0.0833\n  S={P4}: v(S\u222aP3)=14 - v(S)=7 = 7, weight=0.0833\n  S={P1, P2}: v(S\u222aP3)=27 - v(S)=9 = 18, weight=0.0833\n  S={P1, P4}: v(S\u222aP3)=27 - v(S)=15 = 12, weight=0.0833\n  S={P2, P4}: v(S\u222aP3)=17 - v(S)=12 = 5, weight=0.0833\n  S={P1, P2, P4}: v(S\u222aP3)=29 - v(S)=21 = 8, weight=0.2500\n  **\u03c6(P3) = 7.8333** \u2713\n\n**Player P4:**\n  S=\u2205: v(S\u222aP4)=7 - v(S)=0 = 7, weight=0.2500\n  S={P1}: v(S\u222aP4)=15 - v(S)=7 = 8, weight=0.0833\n  S={P2}: v(S\u222aP4)=12 - v(S)=5 = 7, weight=0.0833\n  S={P3}: v(S\u222aP4)=14 - v(S)=5 = 9, weight=0.0833\n  S={P1, P2}: v(S\u222aP4)=21 - v(S)=9 = 12, weight=0.0833\n  S={P1, P3}: v(S\u222aP4)=27 - v(S)=19 = 8, weight=0.0833\n  S={P2, P3}: v(S\u222aP4)=17 - v(S)=6 = 11, weight=0.0833\n  S={P1, P2, P3}: v(S\u222aP4)=29 - v(S)=27 = 2, weight=0.2500\n  **\u03c6(P4) = 6.8333** \u2713\n\n**Efficiency check:** \u03a3\u03c6\u1d62 = 10.5 + 3.8333 + 7.8333 + 6.8333 = 28.9999\nv(N) = 29. Match \u2713",
    "answer": "\u03c6(P1)=10.5, \u03c6(P2)=3.8333, \u03c6(P3)=7.8333, \u03c6(P4)=6.8333",
    "game_type": "cooperative",
    "players": 4,
    "tags": [
      "cooperative",
      "shapley_value",
      "coalition",
      "fair_division",
      "4_player",
      "characteristic_function"
    ]
  },
  {
    "id": "gt-coop-vote-0082",
    "category": "cooperative_game",
    "subcategory": "voting_power",
    "difficulty": "hard",
    "problem": "In this voting game:\n\nWeighted voting game [19; 14, 7, 5, 1]\n- Quota to pass: 19\n- Total weight: 27\n- Voter 1: weight = 14\n- Voter 2: weight = 7\n- Voter 3: weight = 5\n- Voter 4: weight = 1\n\nDetermine each voter's power using the Shapley-Shubik index.",
    "solution": "**Shapley-Shubik Power Index:**\n\nCount how many orderings each voter is pivotal (brings total to \u2265 19).\n\n- Voter 1 (w=14): pivotal in 16/24 orderings \u2192 SSI = 0.6667 (66.7%)\n- Voter 2 (w=7): pivotal in 4/24 orderings \u2192 SSI = 0.1667 (16.7%)\n- Voter 3 (w=5): pivotal in 4/24 orderings \u2192 SSI = 0.1667 (16.7%)\n- Voter 4 (w=1): pivotal in 0/24 orderings \u2192 SSI = 0.0 (0.0%)\n\n**Power distribution:**\n  Voter 1: \u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588 66.7%\n  Voter 2: \u2588\u2588\u2588\u2588\u2588\u2588 16.7%\n  Voter 3: \u2588\u2588\u2588\u2588\u2588\u2588 16.7%\n  Voter 4:  0.0%\n\nSum of SSI: 1.0001 \u2713\nNote: Voter 1 has **veto power** (nothing passes without them).\nNote: Voter 4 is a **dummy** (zero power despite having weight).",
    "answer": "SSI(V1)=0.6667, SSI(V2)=0.1667, SSI(V3)=0.1667, SSI(V4)=0.0",
    "game_type": "cooperative",
    "players": 4,
    "tags": [
      "cooperative",
      "voting",
      "shapley_shubik",
      "power_index",
      "4_player",
      "weighted_voting"
    ]
  },
  {
    "id": "gt-coop-vote-0020",
    "category": "cooperative_game",
    "subcategory": "voting_power",
    "difficulty": "hard",
    "problem": "Consider the following weighted voting game:\n\nWeighted voting game [24; 13, 12, 11, 5, 4]\n- Quota to pass: 24\n- Total weight: 45\n- Voter 1: weight = 13\n- Voter 2: weight = 12\n- Voter 3: weight = 11\n- Voter 4: weight = 5\n- Voter 5: weight = 4\n\nCompute the Shapley-Shubik power index for each player.",
    "solution": "**Shapley-Shubik Power Index:**\n\nCount how many orderings each voter is pivotal (brings total to \u2265 24).\n\n- Voter 1 (w=13): pivotal in 44/120 orderings \u2192 SSI = 0.3667 (36.7%)\n- Voter 2 (w=12): pivotal in 34/120 orderings \u2192 SSI = 0.2833 (28.3%)\n- Voter 3 (w=11): pivotal in 34/120 orderings \u2192 SSI = 0.2833 (28.3%)\n- Voter 4 (w=5): pivotal in 4/120 orderings \u2192 SSI = 0.0333 (3.3%)\n- Voter 5 (w=4): pivotal in 4/120 orderings \u2192 SSI = 0.0333 (3.3%)\n\n**Power distribution:**\n  Voter 1: \u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588 36.7%\n  Voter 2: \u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588 28.3%\n  Voter 3: \u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588\u2588 28.3%\n  Voter 4: \u2588 3.3%\n  Voter 5: \u2588 3.3%\n\nSum of SSI: 0.9999 \u2713",
    "answer": "SSI(V1)=0.3667, SSI(V2)=0.2833, SSI(V3)=0.2833, SSI(V4)=0.0333, SSI(V5)=0.0333",
    "game_type": "cooperative",
    "players": 5,
    "tags": [
      "cooperative",
      "voting",
      "shapley_shubik",
      "power_index",
      "5_player",
      "weighted_voting"
    ]
  },
  {
    "id": "gt-coop-core-0058",
    "category": "cooperative_game",
    "subcategory": "core_analysis",
    "difficulty": "hard",
    "problem": "Analyze the core of this coalition game:\n\nPlayers: {P1, P2, P3}\nCharacteristic function:\n- v({P1}) = 1, v({P2}) = 14, v({P3}) = 1\n- v({P1,P2}) = 29, v({P1,P3}) = 48, v({P2,P3}) = 16\n- v({P1,P2,P3}) = 48\n\nIs the core non-empty? If so, describe it.",
    "solution": "**Core Analysis:**\n\nThe core is the set of allocations (x1,x2,x3) such that:\n1. x1 + x2 + x3 = 48 (efficiency)\n2. x1 \u2265 1, x2 \u2265 14, x3 \u2265 1 (individual rationality)\n3. x1+x2 \u2265 29, x1+x3 \u2265 48, x2+x3 \u2265 16 (coalition rationality)\n\nFrom efficiency + coalition rationality:\n- x1 \u2264 48 - 16 = 32\n- x2 \u2264 48 - 48 = 0\n- x3 \u2264 48 - 29 = 19\n\nBounds: 1 \u2264 x1 \u2264 32, 14 \u2264 x2 \u2264 0, 1 \u2264 x3 \u2264 19\n\nThe core is **empty** \u2717",
    "answer": "Core empty. Bounds: 1\u2264x1\u226432, 14\u2264x2\u22640, 1\u2264x3\u226419",
    "game_type": "cooperative",
    "players": 3,
    "tags": [
      "cooperative",
      "core",
      "coalition",
      "stability",
      "3_player",
      "characteristic_function"
    ]
  },
  {
    "id": "gt-coop-core-0026",
    "category": "cooperative_game",
    "subcategory": "core_analysis",
    "difficulty": "hard",
    "problem": "Analyze the core of this coalition game:\n\nPlayers: {P1, P2, P3}\nCharacteristic function:\n- v({P1}) = 12, v({P2}) = 12, v({P3}) = 11\n- v({P1,P2}) = 21, v({P1,P3}) = 30, v({P2,P3}) = 14\n- v({P1,P2,P3}) = 46\n\nIs the core non-empty? If so, describe it.",
    "solution": "**Core Analysis:**\n\nThe core is the set of allocations (x1,x2,x3) such that:\n1. x1 + x2 + x3 = 46 (efficiency)\n2. x1 \u2265 12, x2 \u2265 12, x3 \u2265 11 (individual rationality)\n3. x1+x2 \u2265 21, x1+x3 \u2265 30, x2+x3 \u2265 14 (coalition rationality)\n\nFrom efficiency + coalition rationality:\n- x1 \u2264 46 - 14 = 32\n- x2 \u2264 46 - 30 = 16\n- x3 \u2264 46 - 21 = 25\n\nBounds: 12 \u2264 x1 \u2264 32, 12 \u2264 x2 \u2264 16, 11 \u2264 x3 \u2264 25\n\nThe core is **non-empty** \u2713\nSum of lower bounds: 35 \u2264 46 \u2713\nAll upper \u2265 lower: True \u2713",
    "answer": "Core non-empty. Bounds: 12\u2264x1\u226432, 12\u2264x2\u226416, 11\u2264x3\u226425",
    "game_type": "cooperative",
    "players": 3,
    "tags": [
      "cooperative",
      "core",
      "coalition",
      "stability",
      "3_player",
      "characteristic_function"
    ]
  },
  {
    "id": "gt-coop-shapley-0081",
    "category": "cooperative_game",
    "subcategory": "shapley_value",
    "difficulty": "medium",
    "problem": "Determine a fair division using the Shapley value:\n\nPlayers: {P1, P2, P3}\nCharacteristic function v(S):\n\n- v({P1}) = 4\n- v({P2}) = 17\n- v({P3}) = 1\n- v({P1, P2}) = 27\n- v({P1, P3}) = 7\n- v({P2, P3}) = 27\n- v({P1, P2, P3}) = 57\n\nWhat does each player receive?",
    "solution": "**Shapley Value Computation:**\n\nThe Shapley value \u03c6\u1d62 assigns each player their average marginal contribution across all possible orderings.\n\nFormula: \u03c6\u1d62 = \u03a3 [|S|!(n-|S|-1)!/n!] \u00d7 [v(S\u222a{i}) - v(S)]\n\n**Player P1:**\n  S=\u2205: v(S\u222aP1)=4 - v(S)=0 = 4, weight=0.3333\n  S={P2}: v(S\u222aP1)=27 - v(S)=17 = 10, weight=0.1667\n  S={P3}: v(S\u222aP1)=7 - v(S)=1 = 6, weight=0.1667\n  S={P2, P3}: v(S\u222aP1)=57 - v(S)=27 = 30, weight=0.3333\n  **\u03c6(P1) = 14.0** \u2713\n\n**Player P2:**\n  S=\u2205: v(S\u222aP2)=17 - v(S)=0 = 17, weight=0.3333\n  S={P1}: v(S\u222aP2)=27 - v(S)=4 = 23, weight=0.1667\n  S={P3}: v(S\u222aP2)=27 - v(S)=1 = 26, weight=0.1667\n  S={P1, P3}: v(S\u222aP2)=57 - v(S)=7 = 50, weight=0.3333\n  **\u03c6(P2) = 30.5** \u2713\n\n**Player P3:**\n  S=\u2205: v(S\u222aP3)=1 - v(S)=0 = 1, weight=0.3333\n  S={P1}: v(S\u222aP3)=7 - v(S)=4 = 3, weight=0.1667\n  S={P2}: v(S\u222aP3)=27 - v(S)=17 = 10, weight=0.1667\n  S={P1, P2}: v(S\u222aP3)=57 - v(S)=27 = 30, weight=0.3333\n  **\u03c6(P3) = 12.5** \u2713\n\n**Efficiency check:** \u03a3\u03c6\u1d62 = 14.0 + 30.5 + 12.5 = 57.0\nv(N) = 57. Match \u2713",
    "answer": "\u03c6(P1)=14.0, \u03c6(P2)=30.5, \u03c6(P3)=12.5",
    "game_type": "cooperative",
    "players": 3,
    "tags": [
      "cooperative",
      "shapley_value",
      "coalition",
      "fair_division",
      "3_player",
      "characteristic_function"
    ]
  },
  {
    "id": "gt-mech-vcg1-0055",
    "category": "mechanism_design",
    "subcategory": "vcg_single_item",
    "difficulty": "medium",
    "problem": "Consider a VCG (Vickrey-Clarke-Groves) mechanism for the following allocation problem:\n\n6 agents compete for a single indivisible item.\nEach agent has a private valuation:\n- Agent 1: v = 78\n- Agent 2: v = 85\n- Agent 3: v = 14\n- Agent 4: v = 58\n- Agent 5: v = 63\n- Agent 6: v = 83\n\nThe mechanism allocates the item to maximize total welfare and charges VCG payments.\n\nCompute the VCG payments and final allocation.",
    "solution": "**VCG Mechanism for Single Item Allocation:**\n\nStep 1: **Efficient allocation** - give item to highest-value agent.\n- Agent 2 has highest valuation (85). \u2713\n\nStep 2: **VCG Payment** - winner pays the externality imposed on others.\n- Without Agent 2: next highest (83) would get the item.\n- Others' welfare WITH Agent 2: 0 (they don't get item)\n- Others' welfare WITHOUT Agent 2: 83 (next agent gets item)\n- VCG payment = 83 - 0 = **83**\n\n**Result:**\n- Winner: Agent 2 (value 85)\n- Payment: 83\n- Surplus: 2\n- All other agents: pay 0, receive nothing.\n\n**Note:** VCG for single item reduces to Vickrey (second-price) auction. \u2713\nTruthful reporting is a dominant strategy. \u2713",
    "answer": "Winner: Agent 2 (v=85), Payment: 83, Surplus: 2",
    "game_type": "mechanism",
    "players": 6,
    "tags": [
      "mechanism_design",
      "vcg",
      "incentive_compatible",
      "dominant_strategy",
      "efficient_allocation",
      "6_agents"
    ]
  },
  {
    "id": "gt-mech-ic-0006",
    "category": "mechanism_design",
    "subcategory": "incentive_compatibility",
    "difficulty": "medium",
    "problem": "Determine whether the following mechanism is incentive compatible:\n\nMechanism: Median voter mechanism\nRule: Each agent reports preferred level. Outcome is the median of reports.\nNumber of agents: 4\nAgent valuations: v1=41, v2=23, v3=49, v4=23\n\nCheck if truthful reporting is a dominant strategy.",
    "solution": "**Incentive Compatibility Analysis of Median voter mechanism:**\n\nA mechanism is **incentive compatible (IC)** if truthful reporting is a dominant strategy \u2014 no agent benefits from misreporting regardless of others' reports.\n\n**Mechanism rule:** Each agent reports preferred level. Outcome is the median of reports.\n\n**Is it IC?** Yes \u2713\n\n**Reasoning:** The median mechanism is strategy-proof for single-peaked preferences. No agent can move the median closer to their peak by misreporting.",
    "answer": "Median voter mechanism: IC \u2713",
    "game_type": "mechanism",
    "players": 4,
    "tags": [
      "mechanism_design",
      "incentive_compatibility",
      "strategy_proof",
      "revelation_principle",
      "4_agents"
    ]
  },
  {
    "id": "gt-mech-vcg1-0026",
    "category": "mechanism_design",
    "subcategory": "vcg_single_item",
    "difficulty": "medium",
    "problem": "Apply the VCG mechanism to determine the efficient allocation and payments:\n\n4 agents compete for a single indivisible item.\nEach agent has a private valuation:\n- Agent 1: v = 66\n- Agent 2: v = 57\n- Agent 3: v = 67\n- Agent 4: v = 32\n\nThe mechanism allocates the item to maximize total welfare and charges VCG payments.",
    "solution": "**VCG Mechanism for Single Item Allocation:**\n\nStep 1: **Efficient allocation** - give item to highest-value agent.\n- Agent 3 has highest valuation (67). \u2713\n\nStep 2: **VCG Payment** - winner pays the externality imposed on others.\n- Without Agent 3: next highest (66) would get the item.\n- Others' welfare WITH Agent 3: 0 (they don't get item)\n- Others' welfare WITHOUT Agent 3: 66 (next agent gets item)\n- VCG payment = 66 - 0 = **66**\n\n**Result:**\n- Winner: Agent 3 (value 67)\n- Payment: 66\n- Surplus: 1\n- All other agents: pay 0, receive nothing.\n\n**Note:** VCG for single item reduces to Vickrey (second-price) auction. \u2713\nTruthful reporting is a dominant strategy. \u2713",
    "answer": "Winner: Agent 3 (v=67), Payment: 66, Surplus: 1",
    "game_type": "mechanism",
    "players": 4,
    "tags": [
      "mechanism_design",
      "vcg",
      "incentive_compatible",
      "dominant_strategy",
      "efficient_allocation",
      "4_agents"
    ]
  },
  {
    "id": "gt-mech-vcgm-0005",
    "category": "mechanism_design",
    "subcategory": "vcg_multi_item",
    "difficulty": "hard",
    "problem": "Consider a VCG (Vickrey-Clarke-Groves) mechanism for the following allocation problem:\n\n3 agents, 2 items (Item A, Item B).\nEach agent's valuation:\n\nAgent | Item A | Item B\n--- | --- | ---\nAgent 1 | 13 | 16\nAgent 2 | 3 | 19\nAgent 3 | 21 | 22\n\nEach item is allocated to one agent to maximize total welfare. VCG payments apply.\n\nCompute the VCG payments and final allocation.",
    "solution": "**VCG Mechanism for Multi-Item Allocation:**\n\n**Step 1: Efficient allocation (maximize total welfare):**\n- Item A: highest value is Agent 3 (21). [Agent 1=13, Agent 2=3, Agent 3=21]\n- Item B: highest value is Agent 3 (22). [Agent 1=16, Agent 2=19, Agent 3=22]\n- Total welfare: 43\n\n**Step 2: VCG payments (externality on others):**\n- Agent 1: wins nothing, payment = 0 \u2713\n- Agent 2: wins nothing, payment = 0 \u2713\n- Agent 3 (wins Item A, Item B):\n  Others' welfare with Agent 3: 0\n  Others' welfare without Agent 3: 32\n  Payment = 32 - 0 = **32** \u2713\n\n**Summary:**\n- Agent 1: gets nothing, value=0, pays=0, net=0\n- Agent 2: gets nothing, value=0, pays=0, net=0\n- Agent 3: gets ['Item A', 'Item B'], value=43, pays=32, net=11",
    "answer": "Item A\u2192Agent 3; Item B\u2192Agent 3; Payments: A1=0, A2=0, A3=32",
    "game_type": "mechanism",
    "players": 3,
    "tags": [
      "mechanism_design",
      "vcg",
      "multi_item",
      "incentive_compatible",
      "efficient_allocation",
      "3_agents"
    ]
  },
  {
    "id": "gt-mech-vcgm-0024",
    "category": "mechanism_design",
    "subcategory": "vcg_multi_item",
    "difficulty": "hard",
    "problem": "Using the Vickrey-Clarke-Groves mechanism, find the optimal allocation and each agent's payment:\n\n3 agents, 2 items (Item A, Item B).\nEach agent's valuation:\n\nAgent | Item A | Item B\n--- | --- | ---\nAgent 1 | 2 | 25\nAgent 2 | 25 | 6\nAgent 3 | 16 | 17\n\nEach item is allocated to one agent to maximize total welfare. VCG payments apply.",
    "solution": "**VCG Mechanism for Multi-Item Allocation:**\n\n**Step 1: Efficient allocation (maximize total welfare):**\n- Item A: highest value is Agent 2 (25). [Agent 1=2, Agent 2=25, Agent 3=16]\n- Item B: highest value is Agent 1 (25). [Agent 1=25, Agent 2=6, Agent 3=17]\n- Total welfare: 50\n\n**Step 2: VCG payments (externality on others):**\n- Agent 1 (wins Item B):\n  Others' welfare with Agent 1: 25\n  Others' welfare without Agent 1: 42\n  Payment = 42 - 25 = **17** \u2713\n- Agent 2 (wins Item A):\n  Others' welfare with Agent 2: 25\n  Others' welfare without Agent 2: 41\n  Payment = 41 - 25 = **16** \u2713\n- Agent 3: wins nothing, payment = 0 \u2713\n\n**Summary:**\n- Agent 1: gets ['Item B'], value=25, pays=17, net=8\n- Agent 2: gets ['Item A'], value=25, pays=16, net=9\n- Agent 3: gets nothing, value=0, pays=0, net=0",
    "answer": "Item A\u2192Agent 2; Item B\u2192Agent 1; Payments: A1=17, A2=16, A3=0",
    "game_type": "mechanism",
    "players": 3,
    "tags": [
      "mechanism_design",
      "vcg",
      "multi_item",
      "incentive_compatible",
      "efficient_allocation",
      "3_agents"
    ]
  },
  {
    "id": "gt-mech-vcgm-0006",
    "category": "mechanism_design",
    "subcategory": "vcg_multi_item",
    "difficulty": "hard",
    "problem": "Consider a VCG (Vickrey-Clarke-Groves) mechanism for the following allocation problem:\n\n3 agents, 2 items (Item A, Item B).\nEach agent's valuation:\n\nAgent | Item A | Item B\n--- | --- | ---\nAgent 1 | 5 | 5\nAgent 2 | 26 | 19\nAgent 3 | 10 | 3\n\nEach item is allocated to one agent to maximize total welfare. VCG payments apply.\n\nCompute the VCG payments and final allocation.",
    "solution": "**VCG Mechanism for Multi-Item Allocation:**\n\n**Step 1: Efficient allocation (maximize total welfare):**\n- Item A: highest value is Agent 2 (26). [Agent 1=5, Agent 2=26, Agent 3=10]\n- Item B: highest value is Agent 2 (19). [Agent 1=5, Agent 2=19, Agent 3=3]\n- Total welfare: 45\n\n**Step 2: VCG payments (externality on others):**\n- Agent 1: wins nothing, payment = 0 \u2713\n- Agent 2 (wins Item A, Item B):\n  Others' welfare with Agent 2: 0\n  Others' welfare without Agent 2: 15\n  Payment = 15 - 0 = **15** \u2713\n- Agent 3: wins nothing, payment = 0 \u2713\n\n**Summary:**\n- Agent 1: gets nothing, value=0, pays=0, net=0\n- Agent 2: gets ['Item A', 'Item B'], value=45, pays=15, net=30\n- Agent 3: gets nothing, value=0, pays=0, net=0",
    "answer": "Item A\u2192Agent 2; Item B\u2192Agent 2; Payments: A1=0, A2=15, A3=0",
    "game_type": "mechanism",
    "players": 3,
    "tags": [
      "mechanism_design",
      "vcg",
      "multi_item",
      "incentive_compatible",
      "efficient_allocation",
      "3_agents"
    ]
  },
  {
    "id": "gt-mech-vcgm-0015",
    "category": "mechanism_design",
    "subcategory": "vcg_multi_item",
    "difficulty": "hard",
    "problem": "Using the Vickrey-Clarke-Groves mechanism, find the optimal allocation and each agent's payment:\n\n3 agents, 2 items (Item A, Item B).\nEach agent's valuation:\n\nAgent | Item A | Item B\n--- | --- | ---\nAgent 1 | 30 | 9\nAgent 2 | 26 | 26\nAgent 3 | 21 | 19\n\nEach item is allocated to one agent to maximize total welfare. VCG payments apply.",
    "solution": "**VCG Mechanism for Multi-Item Allocation:**\n\n**Step 1: Efficient allocation (maximize total welfare):**\n- Item A: highest value is Agent 1 (30). [Agent 1=30, Agent 2=26, Agent 3=21]\n- Item B: highest value is Agent 2 (26). [Agent 1=9, Agent 2=26, Agent 3=19]\n- Total welfare: 56\n\n**Step 2: VCG payments (externality on others):**\n- Agent 1 (wins Item A):\n  Others' welfare with Agent 1: 26\n  Others' welfare without Agent 1: 52\n  Payment = 52 - 26 = **26** \u2713\n- Agent 2 (wins Item B):\n  Others' welfare with Agent 2: 30\n  Others' welfare without Agent 2: 49\n  Payment = 49 - 30 = **19** \u2713\n- Agent 3: wins nothing, payment = 0 \u2713\n\n**Summary:**\n- Agent 1: gets ['Item A'], value=30, pays=26, net=4\n- Agent 2: gets ['Item B'], value=26, pays=19, net=7\n- Agent 3: gets nothing, value=0, pays=0, net=0",
    "answer": "Item A\u2192Agent 1; Item B\u2192Agent 2; Payments: A1=26, A2=19, A3=0",
    "game_type": "mechanism",
    "players": 3,
    "tags": [
      "mechanism_design",
      "vcg",
      "multi_item",
      "incentive_compatible",
      "efficient_allocation",
      "3_agents"
    ]
  },
  {
    "id": "gt-mech-vcgm-0012",
    "category": "mechanism_design",
    "subcategory": "vcg_multi_item",
    "difficulty": "hard",
    "problem": "Using the Vickrey-Clarke-Groves mechanism, find the optimal allocation and each agent's payment:\n\n3 agents, 2 items (Item A, Item B).\nEach agent's valuation:\n\nAgent | Item A | Item B\n--- | --- | ---\nAgent 1 | 18 | 3\nAgent 2 | 6 | 1\nAgent 3 | 14 | 15\n\nEach item is allocated to one agent to maximize total welfare. VCG payments apply.",
    "solution": "**VCG Mechanism for Multi-Item Allocation:**\n\n**Step 1: Efficient allocation (maximize total welfare):**\n- Item A: highest value is Agent 1 (18). [Agent 1=18, Agent 2=6, Agent 3=14]\n- Item B: highest value is Agent 3 (15). [Agent 1=3, Agent 2=1, Agent 3=15]\n- Total welfare: 33\n\n**Step 2: VCG payments (externality on others):**\n- Agent 1 (wins Item A):\n  Others' welfare with Agent 1: 15\n  Others' welfare without Agent 1: 29\n  Payment = 29 - 15 = **14** \u2713\n- Agent 2: wins nothing, payment = 0 \u2713\n- Agent 3 (wins Item B):\n  Others' welfare with Agent 3: 18\n  Others' welfare without Agent 3: 21\n  Payment = 21 - 18 = **3** \u2713\n\n**Summary:**\n- Agent 1: gets ['Item A'], value=18, pays=14, net=4\n- Agent 2: gets nothing, value=0, pays=0, net=0\n- Agent 3: gets ['Item B'], value=15, pays=3, net=12",
    "answer": "Item A\u2192Agent 1; Item B\u2192Agent 3; Payments: A1=14, A2=0, A3=3",
    "game_type": "mechanism",
    "players": 3,
    "tags": [
      "mechanism_design",
      "vcg",
      "multi_item",
      "incentive_compatible",
      "efficient_allocation",
      "3_agents"
    ]
  },
  {
    "id": "gt-mech-ic-0020",
    "category": "mechanism_design",
    "subcategory": "incentive_compatibility",
    "difficulty": "medium",
    "problem": "Analyze the incentive compatibility of this mechanism:\n\nMechanism: First-price auction\nRule: Highest bidder wins, pays their bid.\nNumber of agents: 2\nAgent valuations: v1=20, v2=39\n\nWould any agent benefit from misreporting?",
    "solution": "**Incentive Compatibility Analysis of First-price auction:**\n\nA mechanism is **incentive compatible (IC)** if truthful reporting is a dominant strategy \u2014 no agent benefits from misreporting regardless of others' reports.\n\n**Mechanism rule:** Highest bidder wins, pays their bid.\n\n**Is it IC?** No \u2717\n\n**Reasoning:** Bidders have incentive to shade bids below true value to increase surplus. Truthful bidding is NOT a dominant strategy.\n\n**Example:** Agent with value 39 bids 36 instead of 39:\n- If they win at 36: surplus = 39 - 36 = 3 (better than 0 surplus from truthful bid)\n- Rational agents shade bids below true values. \u2713",
    "answer": "First-price auction: NOT IC \u2717",
    "game_type": "mechanism",
    "players": 2,
    "tags": [
      "mechanism_design",
      "incentive_compatibility",
      "strategy_proof",
      "revelation_principle",
      "2_agents"
    ]
  },
  {
    "id": "gt-mech-vcg1-0069",
    "category": "mechanism_design",
    "subcategory": "vcg_single_item",
    "difficulty": "medium",
    "problem": "Using the Vickrey-Clarke-Groves mechanism, find the optimal allocation and each agent's payment:\n\n4 agents compete for a single indivisible item.\nEach agent has a private valuation:\n- Agent 1: v = 80\n- Agent 2: v = 14\n- Agent 3: v = 93\n- Agent 4: v = 85\n\nThe mechanism allocates the item to maximize total welfare and charges VCG payments.",
    "solution": "**VCG Mechanism for Single Item Allocation:**\n\nStep 1: **Efficient allocation** - give item to highest-value agent.\n- Agent 3 has highest valuation (93). \u2713\n\nStep 2: **VCG Payment** - winner pays the externality imposed on others.\n- Without Agent 3: next highest (85) would get the item.\n- Others' welfare WITH Agent 3: 0 (they don't get item)\n- Others' welfare WITHOUT Agent 3: 85 (next agent gets item)\n- VCG payment = 85 - 0 = **85**\n\n**Result:**\n- Winner: Agent 3 (value 93)\n- Payment: 85\n- Surplus: 8\n- All other agents: pay 0, receive nothing.\n\n**Note:** VCG for single item reduces to Vickrey (second-price) auction. \u2713\nTruthful reporting is a dominant strategy. \u2713",
    "answer": "Winner: Agent 3 (v=93), Payment: 85, Surplus: 8",
    "game_type": "mechanism",
    "players": 4,
    "tags": [
      "mechanism_design",
      "vcg",
      "incentive_compatible",
      "dominant_strategy",
      "efficient_allocation",
      "4_agents"
    ]
  },
  {
    "id": "gt-2x2-classic-scaled-0023",
    "category": "normal_form_2x2",
    "subcategory": "classic_stag_hunt_scaled",
    "difficulty": "easy",
    "problem": "Given the following payoff matrix for a two-player simultaneous game:\n\nPlayer 1 \\ Player 2 | Stag | Hare\n--- | --- | ---\nStag | (12,12) | (0,9)\nHare | (9,0) | (9,9)\n\nWhat are the Nash Equilibria?",
    "solution": "This game has 2 pure strategy Nash Equilibria.\n\nTo verify (Up, Left) is a Nash Equilibrium, we check each player's incentive to deviate:\n- Player 1 currently plays Up, getting payoff 12. If Player 1 switches to Down (while Player 2 stays at Left), payoff drops to 9. Player 1 has no incentive to deviate. \u2713\n- Player 2 currently plays Left, getting payoff 12. If Player 2 switches to Right (while Player 1 stays at Up), payoff drops to 9. Player 2 has no incentive to deviate. \u2713\nTherefore (Up, Left) is a Nash Equilibrium with payoffs (12,12).\n\nTo verify (Down, Right) is a Nash Equilibrium, we check each player's incentive to deviate:\n- Player 1 currently plays Down, getting payoff 9. If Player 1 switches to Up (while Player 2 stays at Right), payoff drops to 0. Player 1 has no incentive to deviate. \u2713\n- Player 2 currently plays Right, getting payoff 9. If Player 2 switches to Left (while Player 1 stays at Down), payoff drops to 0. Player 2 has no incentive to deviate. \u2713\nTherefore (Down, Right) is a Nash Equilibrium with payoffs (9,9).\n\nMixed Strategy Nash Equilibrium:\nPlayer 1 randomizes over 2 strategies: Up with probability 0.7500, Down with probability 0.2500.\nPlayer 2 randomizes over 2 strategies: Left with probability 0.7500, Right with probability 0.2500.\nIn this equilibrium, each player's mixture makes the opponent indifferent among their active strategies.\nExpected payoffs: Player 1 gets 9.0000, Player 2 gets 9.0000.\n\nIn addition to the pure strategy equilibria, this game has 1 mixed strategy Nash Equilibrium.",
    "answer": "[(Up, Left), (Down, Right), P1=[Up:0.7500, Down:0.2500], P2=[Left:0.7500, Right:0.2500]]",
    "game_type": "simultaneous",
    "players": 2,
    "tags": [
      "2x2",
      "classic_game",
      "mixed_strategy",
      "nash_equilibrium",
      "normal_form",
      "pure_strategy",
      "scaled",
      "stag_hunt"
    ]
  },
  {
    "id": "gt-2x2-wide-0026",
    "category": "normal_form_2x2",
    "subcategory": "random_payoffs",
    "difficulty": "easy",
    "problem": "Analyze the following strategic form game and determine all Nash Equilibria:\n\nPlayer 1 \\ Player 2 | Left | Right\n--- | --- | ---\nUp | (4,0) | (10,-2)\nDown | (-8,-6) | (6,-1)",
    "solution": "This game has 1 pure strategy Nash Equilibrium.\n\nTo verify (Up, Left) is a Nash Equilibrium, we check each player's incentive to deviate:\n- Player 1 currently plays Up, getting payoff 4. If Player 1 switches to Down (while Player 2 stays at Left), payoff drops to -8. Player 1 has no incentive to deviate. \u2713\n- Player 2 currently plays Left, getting payoff 0. If Player 2 switches to Right (while Player 1 stays at Up), payoff drops to -2. Player 2 has no incentive to deviate. \u2713\nTherefore (Up, Left) is a Nash Equilibrium with payoffs (4,0).\n\nThere are no additional mixed strategy equilibria beyond the pure strategy equilibrium identified above.",
    "answer": "[(Up, Left)]",
    "game_type": "simultaneous",
    "players": 2,
    "tags": [
      "2x2",
      "dominant_strategy",
      "nash_equilibrium",
      "normal_form",
      "pure_strategy"
    ]
  },
  {
    "id": "gt-2x2-wide-0068",
    "category": "normal_form_2x2",
    "subcategory": "random_payoffs",
    "difficulty": "easy",
    "problem": "Given the following payoff matrix for a two-player simultaneous game:\n\nPlayer 1 \\ Player 2 | Left | Right\n--- | --- | ---\nUp | (5,0) | (-1,-1)\nDown | (5,-7) | (-4,7)\n\nWhat are the Nash Equilibria?",
    "solution": "This game has 1 pure strategy Nash Equilibrium.\n\nTo verify (Up, Left) is a Nash Equilibrium, we check each player's incentive to deviate:\n- Player 1 currently plays Up, getting payoff 5. If Player 1 switches to Down (while Player 2 stays at Left), payoff stays at 5. Player 1 has no incentive to deviate. \u2713\n- Player 2 currently plays Left, getting payoff 0. If Player 2 switches to Right (while Player 1 stays at Up), payoff drops to -1. Player 2 has no incentive to deviate. \u2713\nTherefore (Up, Left) is a Nash Equilibrium with payoffs (5,0).\n\nThere are no additional mixed strategy equilibria beyond the pure strategy equilibrium identified above.",
    "answer": "[(Up, Left)]",
    "game_type": "simultaneous",
    "players": 2,
    "tags": [
      "2x2",
      "dominant_strategy",
      "nash_equilibrium",
      "normal_form",
      "pure_strategy"
    ]
  },
  {
    "id": "gt-2x2-pure-ne-0110",
    "category": "normal_form_2x2",
    "subcategory": "pure_nash_equilibrium",
    "difficulty": "medium",
    "problem": "In the game represented by the payoff matrix below, find every Nash Equilibrium:\n\nPlayer 1 \\ Player 2 | Left | Right\n--- | --- | ---\nUp | (3,-1) | (-1,-1)\nDown | (-3,-3) | (-1,5)",
    "solution": "This game has 3 pure strategy Nash Equilibria.\n\nTo verify (Up, Left) is a Nash Equilibrium, we check each player's incentive to deviate:\n- Player 1 currently plays Up, getting payoff 3. If Player 1 switches to Down (while Player 2 stays at Left), payoff drops to -3. Player 1 has no incentive to deviate. \u2713\n- Player 2 currently plays Left, getting payoff -1. If Player 2 switches to Right (while Player 1 stays at Up), payoff stays at -1. Player 2 has no incentive to deviate. \u2713\nTherefore (Up, Left) is a Nash Equilibrium with payoffs (3,-1).\n\nTo verify (Up, Right) is a Nash Equilibrium, we check each player's incentive to deviate:\n- Player 1 currently plays Up, getting payoff -1. If Player 1 switches to Down (while Player 2 stays at Right), payoff stays at -1. Player 1 has no incentive to deviate. \u2713\n- Player 2 currently plays Right, getting payoff -1. If Player 2 switches to Left (while Player 1 stays at Up), payoff stays at -1. Player 2 has no incentive to deviate. \u2713\nTherefore (Up, Right) is a Nash Equilibrium with payoffs (-1,-1).\n\nTo verify (Down, Right) is a Nash Equilibrium, we check each player's incentive to deviate:\n- Player 1 currently plays Down, getting payoff -1. If Player 1 switches to Up (while Player 2 stays at Right), payoff stays at -1. Player 1 has no incentive to deviate. \u2713\n- Player 2 currently plays Right, getting payoff 5. If Player 2 switches to Left (while Player 1 stays at Down), payoff drops to -3. Player 2 has no incentive to deviate. \u2713\nTherefore (Down, Right) is a Nash Equilibrium with payoffs (-1,5).\n\nThere are no additional mixed strategy equilibria beyond the pure strategy equilibria identified above.",
    "answer": "[(Up, Left), (Up, Right), (Down, Right)]",
    "game_type": "simultaneous",
    "players": 2,
    "tags": [
      "2x2",
      "nash_equilibrium",
      "normal_form",
      "pure_strategy"
    ]
  },
  {
    "id": "gt-2x2-mixed-ne-0060",
    "category": "normal_form_2x2",
    "subcategory": "mixed_nash_equilibrium",
    "difficulty": "medium",
    "problem": "In the following game, players must randomize. Find the mixed strategy Nash Equilibrium and expected payoffs:\n\nPlayer 1 \\ Player 2 | Left | Right\n--- | --- | ---\nUp | (1,-4) | (2,2)\nDown | (-3,2) | (3,-2)",
    "solution": "There are no pure strategy Nash Equilibria.\n\nMixed Strategy Nash Equilibrium:\nPlayer 1 randomizes over 2 strategies: Up with probability 0.4000, Down with probability 0.6000.\nPlayer 2 randomizes over 2 strategies: Left with probability 0.2000, Right with probability 0.8000.\nIn this equilibrium, each player's mixture makes the opponent indifferent among their active strategies.\nExpected payoffs: Player 1 gets 1.8000, Player 2 gets -0.4000.",
    "answer": "[P1=[Up:0.4000, Down:0.6000], P2=[Left:0.2000, Right:0.8000]]",
    "game_type": "simultaneous",
    "players": 2,
    "tags": [
      "2x2",
      "mixed_strategy",
      "nash_equilibrium",
      "normal_form"
    ]
  },
  {
    "id": "gt-2x2-pure-ne-0066",
    "category": "normal_form_2x2",
    "subcategory": "pure_nash_equilibrium",
    "difficulty": "medium",
    "problem": "Given the following payoff matrix for a two-player simultaneous game:\n\nPlayer 1 \\ Player 2 | Left | Right\n--- | --- | ---\nUp | (1,4) | (3,4)\nDown | (1,1) | (-1,4)\n\nWhat are the Nash Equilibria?",
    "solution": "This game has 2 pure strategy Nash Equilibria.\n\nTo verify (Up, Left) is a Nash Equilibrium, we check each player's incentive to deviate:\n- Player 1 currently plays Up, getting payoff 1. If Player 1 switches to Down (while Player 2 stays at Left), payoff stays at 1. Player 1 has no incentive to deviate. \u2713\n- Player 2 currently plays Left, getting payoff 4. If Player 2 switches to Right (while Player 1 stays at Up), payoff stays at 4. Player 2 has no incentive to deviate. \u2713\nTherefore (Up, Left) is a Nash Equilibrium with payoffs (1,4).\n\nTo verify (Up, Right) is a Nash Equilibrium, we check each player's incentive to deviate:\n- Player 1 currently plays Up, getting payoff 3. If Player 1 switches to Down (while Player 2 stays at Right), payoff drops to -1. Player 1 has no incentive to deviate. \u2713\n- Player 2 currently plays Right, getting payoff 4. If Player 2 switches to Left (while Player 1 stays at Up), payoff stays at 4. Player 2 has no incentive to deviate. \u2713\nTherefore (Up, Right) is a Nash Equilibrium with payoffs (3,4).\n\nThere are no additional mixed strategy equilibria beyond the pure strategy equilibria identified above.",
    "answer": "[(Up, Left), (Up, Right)]",
    "game_type": "simultaneous",
    "players": 2,
    "tags": [
      "2x2",
      "nash_equilibrium",
      "normal_form",
      "pure_strategy"
    ]
  },
  {
    "id": "gt-2x2-dom-strat-0037",
    "category": "normal_form_2x2",
    "subcategory": "dominant_strategy",
    "difficulty": "easy",
    "problem": "Consider the game below:\n\nPlayer 1 \\ Player 2 | Left | Right\n--- | --- | ---\nUp | (-4,5) | (-2,3)\nDown | (2,1) | (1,-5)\n\nIdentify any strictly dominant or dominated strategies for each player.",
    "solution": "To identify dominant strategies, we compare each player's payoffs across all opponent strategies:\n\nPlayer 1's analysis:\n- Strategy Up: payoffs [-4, -2] against ['Left', 'Right']\n- Strategy Down: payoffs [2, 1] against ['Left', 'Right']\nStrategy Down yields a strictly higher payoff than every other strategy in every column.\nTherefore, Down is a strictly dominant strategy for Player 1. \u2713\nStrategy Up is strictly dominated (another strategy always gives higher payoff). \u2717\n\nPlayer 2's analysis:\n- Strategy Left: payoffs [5, 1] against ['Up', 'Down']\n- Strategy Right: payoffs [3, -5] against ['Up', 'Down']\nStrategy Left yields a strictly higher payoff than every other strategy in every row.\nTherefore, Left is a strictly dominant strategy for Player 2. \u2713\nStrategy Right is strictly dominated. \u2717",
    "answer": "{'p1_dominant': [1], 'p2_dominant': [0]}",
    "game_type": "simultaneous",
    "players": 2,
    "tags": [
      "2x2",
      "dominant_strategy",
      "normal_form"
    ]
  },
  {
    "id": "gt-2x2-mixed-ne-0004",
    "category": "normal_form_2x2",
    "subcategory": "mixed_nash_equilibrium",
    "difficulty": "medium",
    "problem": "In the game represented by the payoff matrix below, find every Nash Equilibrium:\n\nPlayer 1 \\ Player 2 | Left | Right\n--- | --- | ---\nUp | (0,4) | (4,0)\nDown | (1,-1) | (-1,5)",
    "solution": "There are no pure strategy Nash Equilibria.\n\nMixed Strategy Nash Equilibrium:\nPlayer 1 randomizes over 2 strategies: Up with probability 0.6000, Down with probability 0.4000.\nPlayer 2 randomizes over 2 strategies: Left with probability 0.8333, Right with probability 0.1667.\nIn this equilibrium, each player's mixture makes the opponent indifferent among their active strategies.\nExpected payoffs: Player 1 gets 0.6667, Player 2 gets 2.0000.",
    "answer": "[P1=[Up:0.6000, Down:0.4000], P2=[Left:0.8333, Right:0.1667]]",
    "game_type": "simultaneous",
    "players": 2,
    "tags": [
      "2x2",
      "mixed_strategy",
      "nash_equilibrium",
      "normal_form"
    ]
  },
  {
    "id": "gt-2x2-extra-0001",
    "category": "normal_form_2x2",
    "subcategory": "random_extra",
    "difficulty": "easy",
    "problem": "Examine the payoff matrix and identify all pure strategy Nash Equilibria:\n\nPlayer 1 \\ Player 2 | Left | Right\n--- | --- | ---\nUp | (4,7) | (-1,1)\nDown | (-4,0) | (1,6)",
    "solution": "This game has 2 pure strategy Nash Equilibria.\n\nTo verify (Up, Left) is a Nash Equilibrium, we check each player's incentive to deviate:\n- Player 1 currently plays Up, getting payoff 4. If Player 1 switches to Down (while Player 2 stays at Left), payoff drops to -4. Player 1 has no incentive to deviate. \u2713\n- Player 2 currently plays Left, getting payoff 7. If Player 2 switches to Right (while Player 1 stays at Up), payoff drops to 1. Player 2 has no incentive to deviate. \u2713\nTherefore (Up, Left) is a Nash Equilibrium with payoffs (4,7).\n\nTo verify (Down, Right) is a Nash Equilibrium, we check each player's incentive to deviate:\n- Player 1 currently plays Down, getting payoff 1. If Player 1 switches to Up (while Player 2 stays at Right), payoff drops to -1. Player 1 has no incentive to deviate. \u2713\n- Player 2 currently plays Right, getting payoff 6. If Player 2 switches to Left (while Player 1 stays at Down), payoff drops to 0. Player 2 has no incentive to deviate. \u2713\nTherefore (Down, Right) is a Nash Equilibrium with payoffs (1,6).\n\nMixed Strategy Nash Equilibrium:\nPlayer 1 randomizes over 2 strategies: Up with probability 0.5000, Down with probability 0.5000.\nPlayer 2 randomizes over 2 strategies: Left with probability 0.2000, Right with probability 0.8000.\nIn this equilibrium, each player's mixture makes the opponent indifferent among their active strategies.\nExpected payoffs: Player 1 gets 0.0000, Player 2 gets 3.5000.\n\nIn addition to the pure strategy equilibria, this game has 1 mixed strategy Nash Equilibrium.",
    "answer": "[(Up, Left), (Down, Right), P1=[Up:0.5000, Down:0.5000], P2=[Left:0.2000, Right:0.8000]]",
    "game_type": "simultaneous",
    "players": 2,
    "tags": [
      "2x2",
      "dominant_strategy",
      "mixed_strategy",
      "nash_equilibrium",
      "normal_form",
      "pure_strategy"
    ]
  },
  {
    "id": "gt-2x2-dom-strat-0076",
    "category": "normal_form_2x2",
    "subcategory": "dominant_strategy",
    "difficulty": "easy",
    "problem": "Analyze the following game for dominant strategies:\n\nPlayer 1 \\ Player 2 | Left | Right\n--- | --- | ---\nUp | (-3,4) | (-5,-2)\nDown | (1,0) | (0,0)\n\nDoes either player have a dominant strategy? If so, what is it?",
    "solution": "To identify dominant strategies, we compare each player's payoffs across all opponent strategies:\n\nPlayer 1's analysis:\n- Strategy Up: payoffs [-3, -5] against ['Left', 'Right']\n- Strategy Down: payoffs [1, 0] against ['Left', 'Right']\nStrategy Down yields a strictly higher payoff than every other strategy in every column.\nTherefore, Down is a strictly dominant strategy for Player 1. \u2713\nStrategy Up is strictly dominated (another strategy always gives higher payoff). \u2717\n\nPlayer 2's analysis:\n- Strategy Left: payoffs [4, 0] against ['Up', 'Down']\n- Strategy Right: payoffs [-2, 0] against ['Up', 'Down']\nStrategy Left yields a payoff at least as high as every other strategy in every row.\nTherefore, Left is a weakly dominant strategy for Player 2. \u2713\nStrategy Right is weakly dominated. \u2717",
    "answer": "{'p1_dominant': [1], 'p2_dominant': []}",
    "game_type": "simultaneous",
    "players": 2,
    "tags": [
      "2x2",
      "dominant_strategy",
      "normal_form"
    ]
  },
  {
    "id": "gt-nxm-3x3-dom-0009",
    "category": "normal_form_3x3",
    "subcategory": "3x3_dominant",
    "difficulty": "medium",
    "problem": "Analyze the following game for dominant strategies:\n\nPlayer 1 \\ Player 2 | Left | Center | Right\n--- | --- | --- | ---\nTop | (-2,5) | (-3,3) | (0,-6)\nMiddle | (-5,-2) | (4,-3) | (6,-2)\nBottom | (1,2) | (2,5) | (6,-4)\n\nDoes either player have a dominant strategy? If so, what is it?",
    "solution": "To identify dominant strategies, we compare each player's payoffs across all opponent strategies:\n\nPlayer 1's analysis:\n- Strategy Top: payoffs [-2, -3, 0] against ['Left', 'Center', 'Right']\n- Strategy Middle: payoffs [-5, 4, 6] against ['Left', 'Center', 'Right']\n- Strategy Bottom: payoffs [1, 2, 6] against ['Left', 'Center', 'Right']\nNo single strategy dominates all others across every column. Player 1 has no dominant strategy.\nStrategy Top is strictly dominated (another strategy always gives higher payoff). \u2717\n\nPlayer 2's analysis:\n- Strategy Left: payoffs [5, -2, 2] against ['Top', 'Middle', 'Bottom']\n- Strategy Center: payoffs [3, -3, 5] against ['Top', 'Middle', 'Bottom']\n- Strategy Right: payoffs [-6, -2, -4] against ['Top', 'Middle', 'Bottom']\nNo single strategy dominates all others across every row. Player 2 has no dominant strategy.\nStrategy Right is weakly dominated. \u2717",
    "answer": "{'p1': {'strictly_dominant': [], 'weakly_dominant': [], 'strictly_dominated': [0], 'weakly_dominated': []}, 'p2': {'strictly_dominant': [], 'weakly_dominant': [], 'strictly_dominated': [], 'weakly_dominated': [2]}}",
    "game_type": "simultaneous",
    "players": 2,
    "tags": [
      "3x3",
      "dominant_strategy",
      "normal_form"
    ]
  },
  {
    "id": "gt-nxm-3x3-mixed-0033",
    "category": "normal_form_3x3",
    "subcategory": "3x3_mixed_ne",
    "difficulty": "medium",
    "problem": "Analyze the following 3-by-3 game and determine all Nash Equilibria:\n\nPlayer 1 \\ Player 2 | Left | Center | Right\n--- | --- | --- | ---\nTop | (0,5) | (4,1) | (-4,-5)\nMiddle | (-6,5) | (-1,-6) | (1,-6)\nBottom | (-1,-3) | (-2,-1) | (4,6)",
    "solution": "This game has 2 pure strategy Nash Equilibria.\n\nTo verify (Top, Left) is a Nash Equilibrium, we check each player's incentive to deviate:\n- Player 1 currently plays Top, getting payoff 0. If Player 1 switches to Middle (while Player 2 stays at Left), payoff drops to -6. Player 1 has no incentive to deviate. \u2713\n- Player 1 currently plays Top, getting payoff 0. If Player 1 switches to Bottom (while Player 2 stays at Left), payoff drops to -1. Player 1 has no incentive to deviate. \u2713\n- Player 2 currently plays Left, getting payoff 5. If Player 2 switches to Center (while Player 1 stays at Top), payoff drops to 1. Player 2 has no incentive to deviate. \u2713\n- Player 2 currently plays Left, getting payoff 5. If Player 2 switches to Right (while Player 1 stays at Top), payoff drops to -5. Player 2 has no incentive to deviate. \u2713\nTherefore (Top, Left) is a Nash Equilibrium with payoffs (0,5).\n\nTo verify (Bottom, Right) is a Nash Equilibrium, we check each player's incentive to deviate:\n- Player 1 currently plays Bottom, getting payoff 4. If Player 1 switches to Top (while Player 2 stays at Right), payoff drops to -4. Player 1 has no incentive to deviate. \u2713\n- Player 1 currently plays Bottom, getting payoff 4. If Player 1 switches to Middle (while Player 2 stays at Right), payoff drops to 1. Player 1 has no incentive to deviate. \u2713\n- Player 2 currently plays Right, getting payoff 6. If Player 2 switches to Left (while Player 1 stays at Bottom), payoff drops to -3. Player 2 has no incentive to deviate. \u2713\n- Player 2 currently plays Right, getting payoff 6. If Player 2 switches to Center (while Player 1 stays at Bottom), payoff drops to -1. Player 2 has no incentive to deviate. \u2713\nTherefore (Bottom, Right) is a Nash Equilibrium with payoffs (4,6).\n\nMixed Strategy Nash Equilibrium:\nPlayer 1 randomizes over 2 strategies: Top with probability 0.4737, Bottom with probability 0.5263.\nPlayer 2 randomizes over 2 strategies: Left with probability 0.8889, Right with probability 0.1111.\nIn this equilibrium, each player's mixture makes the opponent indifferent among their active strategies.\nExpected payoffs: Player 1 gets -0.4444, Player 2 gets 0.7895.\n\nIn addition to the pure strategy equilibria, this game has 1 mixed strategy Nash Equilibrium.",
    "answer": "[(Top, Left), (Bottom, Right), P1=[Top:0.4737, Bottom:0.5263], P2=[Left:0.8889, Right:0.1111]]",
    "game_type": "simultaneous",
    "players": 2,
    "tags": [
      "3x3",
      "mixed_strategy",
      "nash_equilibrium",
      "normal_form",
      "pure_strategy"
    ]
  },
  {
    "id": "gt-nxm-3x3-iesds-0038",
    "category": "normal_form_3x3",
    "subcategory": "3x3_iesds",
    "difficulty": "medium",
    "problem": "Use iterated dominance to simplify the following game:\n\nPlayer 1 \\ Player 2 | Left | Center | Right\n--- | --- | --- | ---\nTop | (-1,4) | (-4,0) | (2,-3)\nMiddle | (2,-1) | (-5,-5) | (-2,2)\nBottom | (3,5) | (-1,0) | (5,-4)\n\nEliminate strictly dominated strategies step by step.",
    "solution": "Applying Iterated Elimination of Strictly Dominated Strategies (IESDS):\n\nStep 1: Consider Player 1's strategies.\n  Compare Top vs Bottom: for every column strategy Player 2 might play,\n  Bottom gives Player 1 a strictly higher payoff than Top.\n  Therefore Top is strictly dominated by Bottom. Eliminate Top. \u2717\n  Remaining strategies: P1=['Middle', 'Bottom'], P2=['Left', 'Center', 'Right']\n\nStep 2: Consider Player 1's strategies.\n  Compare Middle vs Bottom: for every column strategy Player 2 might play,\n  Bottom gives Player 1 a strictly higher payoff than Middle.\n  Therefore Middle is strictly dominated by Bottom. Eliminate Middle. \u2717\n  Remaining strategies: P1=['Bottom'], P2=['Left', 'Center', 'Right']\n\nStep 3: Consider Player 2's strategies.\n  Compare Center vs Left: for every row strategy Player 1 might play,\n  Left gives Player 2 a strictly higher payoff than Center.\n  Therefore Center is strictly dominated by Left. Eliminate Center. \u2717\n  Remaining strategies: P1=['Bottom'], P2=['Left', 'Right']\n\nStep 4: Consider Player 2's strategies.\n  Compare Right vs Left: for every row strategy Player 1 might play,\n  Left gives Player 2 a strictly higher payoff than Right.\n  Therefore Right is strictly dominated by Left. Eliminate Right. \u2717\n  Remaining strategies: P1=['Bottom'], P2=['Left']\n\nAfter 4 elimination steps, the unique surviving strategy profile is (Bottom, Left).\nThe resulting payoffs are (3, 5). \u2713",
    "answer": "(Bottom, Left)",
    "game_type": "simultaneous",
    "players": 2,
    "tags": [
      "3x3",
      "dominated_strategy",
      "iesds",
      "normal_form"
    ]
  },
  {
    "id": "gt-nxm-3x3-multi-0013",
    "category": "normal_form_3x3",
    "subcategory": "3x3_multi_eq",
    "difficulty": "hard",
    "problem": "The following 3x3 game has multiple equilibria. Find ALL Nash Equilibria:\n\nPlayer 1 \\ Player 2 | Left | Center | Right\n--- | --- | --- | ---\nTop | (-2,-5) | (0,5) | (1,4)\nMiddle | (-2,4) | (4,-1) | (0,4)\nBottom | (3,4) | (0,-3) | (-3,2)",
    "solution": "This game has 3 Nash Equilibria (1 pure, 2 mixed).\n\nThis game has 1 pure strategy Nash Equilibrium.\n\nTo verify (Bottom, Left) is a Nash Equilibrium, we check each player's incentive to deviate:\n- Player 1 currently plays Bottom, getting payoff 3. If Player 1 switches to Top (while Player 2 stays at Left), payoff drops to -2. Player 1 has no incentive to deviate. \u2713\n- Player 1 currently plays Bottom, getting payoff 3. If Player 1 switches to Middle (while Player 2 stays at Left), payoff drops to -2. Player 1 has no incentive to deviate. \u2713\n- Player 2 currently plays Left, getting payoff 4. If Player 2 switches to Center (while Player 1 stays at Bottom), payoff drops to -3. Player 2 has no incentive to deviate. \u2713\n- Player 2 currently plays Left, getting payoff 4. If Player 2 switches to Right (while Player 1 stays at Bottom), payoff drops to 2. Player 2 has no incentive to deviate. \u2713\nTherefore (Bottom, Left) is a Nash Equilibrium with payoffs (3,4).\n\nMixed Strategy Nash Equilibrium:\nPlayer 1 randomizes over 2 strategies: Top with probability 0.8333, Middle with probability 0.1667.\nPlayer 2 randomizes over 2 strategies: Center with probability 0.2000, Right with probability 0.8000.\nIn this equilibrium, each player's mixture makes the opponent indifferent among their active strategies.\nExpected payoffs: Player 1 gets 0.8000, Player 2 gets 4.0000.\n\nMixed Strategy Nash Equilibrium:\nPlayer 1 randomizes over 2 strategies: Top with probability 0.1818, Bottom with probability 0.8182.\nPlayer 2 randomizes over 2 strategies: Left with probability 0.4444, Right with probability 0.5556.\nIn this equilibrium, each player's mixture makes the opponent indifferent among their active strategies.\nExpected payoffs: Player 1 gets -0.3333, Player 2 gets 2.3636.\n\nIn addition to the pure strategy equilibrium, this game has 2 mixed strategy Nash Equilibria.",
    "answer": "[(Bottom, Left), P1=[Top:0.8333, Middle:0.1667], P2=[Center:0.2000, Right:0.8000], P1=[Top:0.1818, Bottom:0.8182], P2=[Left:0.4444, Right:0.5556]]",
    "game_type": "simultaneous",
    "players": 2,
    "tags": [
      "3x3",
      "mixed_strategy",
      "multiple_equilibria",
      "nash_equilibrium",
      "normal_form",
      "pure_strategy"
    ]
  },
  {
    "id": "gt-nxm-3x3-multi-0005",
    "category": "normal_form_3x3",
    "subcategory": "3x3_multi_eq",
    "difficulty": "hard",
    "problem": "The following 3x3 game has multiple equilibria. Find ALL Nash Equilibria:\n\nPlayer 1 \\ Player 2 | Left | Center | Right\n--- | --- | --- | ---\nTop | (-3,-1) | (1,2) | (2,2)\nMiddle | (-4,-4) | (-3,0) | (-2,-2)\nBottom | (5,3) | (-1,4) | (-3,-4)",
    "solution": "This game has 2 Nash Equilibria (2 pure, 0 mixed).\n\nThis game has 2 pure strategy Nash Equilibria.\n\nTo verify (Top, Center) is a Nash Equilibrium, we check each player's incentive to deviate:\n- Player 1 currently plays Top, getting payoff 1. If Player 1 switches to Middle (while Player 2 stays at Center), payoff drops to -3. Player 1 has no incentive to deviate. \u2713\n- Player 1 currently plays Top, getting payoff 1. If Player 1 switches to Bottom (while Player 2 stays at Center), payoff drops to -1. Player 1 has no incentive to deviate. \u2713\n- Player 2 currently plays Center, getting payoff 2. If Player 2 switches to Left (while Player 1 stays at Top), payoff drops to -1. Player 2 has no incentive to deviate. \u2713\n- Player 2 currently plays Center, getting payoff 2. If Player 2 switches to Right (while Player 1 stays at Top), payoff stays at 2. Player 2 has no incentive to deviate. \u2713\nTherefore (Top, Center) is a Nash Equilibrium with payoffs (1,2).\n\nTo verify (Top, Right) is a Nash Equilibrium, we check each player's incentive to deviate:\n- Player 1 currently plays Top, getting payoff 2. If Player 1 switches to Middle (while Player 2 stays at Right), payoff drops to -2. Player 1 has no incentive to deviate. \u2713\n- Player 1 currently plays Top, getting payoff 2. If Player 1 switches to Bottom (while Player 2 stays at Right), payoff drops to -3. Player 1 has no incentive to deviate. \u2713\n- Player 2 currently plays Right, getting payoff 2. If Player 2 switches to Left (while Player 1 stays at Top), payoff drops to -1. Player 2 has no incentive to deviate. \u2713\n- Player 2 currently plays Right, getting payoff 2. If Player 2 switches to Center (while Player 1 stays at Top), payoff stays at 2. Player 2 has no incentive to deviate. \u2713\nTherefore (Top, Right) is a Nash Equilibrium with payoffs (2,2).\n\nThere are no additional mixed strategy equilibria beyond the pure strategy equilibria identified above.",
    "answer": "[(Top, Center), (Top, Right)]",
    "game_type": "simultaneous",
    "players": 2,
    "tags": [
      "3x3",
      "multiple_equilibria",
      "nash_equilibrium",
      "normal_form",
      "pure_strategy"
    ]
  },
  {
    "id": "gt-nxm-3x3-multi-0003",
    "category": "normal_form_3x3",
    "subcategory": "3x3_multi_eq",
    "difficulty": "hard",
    "problem": "The following 3x3 game has multiple equilibria. Find ALL Nash Equilibria:\n\nPlayer 1 \\ Player 2 | Left | Center | Right\n--- | --- | --- | ---\nTop | (5,-1) | (-5,-3) | (-2,-4)\nMiddle | (4,-3) | (5,3) | (0,-3)\nBottom | (0,-3) | (4,-5) | (0,-4)",
    "solution": "This game has 3 Nash Equilibria (2 pure, 1 mixed).\n\nThis game has 2 pure strategy Nash Equilibria.\n\nTo verify (Top, Left) is a Nash Equilibrium, we check each player's incentive to deviate:\n- Player 1 currently plays Top, getting payoff 5. If Player 1 switches to Middle (while Player 2 stays at Left), payoff drops to 4. Player 1 has no incentive to deviate. \u2713\n- Player 1 currently plays Top, getting payoff 5. If Player 1 switches to Bottom (while Player 2 stays at Left), payoff drops to 0. Player 1 has no incentive to deviate. \u2713\n- Player 2 currently plays Left, getting payoff -1. If Player 2 switches to Center (while Player 1 stays at Top), payoff drops to -3. Player 2 has no incentive to deviate. \u2713\n- Player 2 currently plays Left, getting payoff -1. If Player 2 switches to Right (while Player 1 stays at Top), payoff drops to -4. Player 2 has no incentive to deviate. \u2713\nTherefore (Top, Left) is a Nash Equilibrium with payoffs (5,-1).\n\nTo verify (Middle, Center) is a Nash Equilibrium, we check each player's incentive to deviate:\n- Player 1 currently plays Middle, getting payoff 5. If Player 1 switches to Top (while Player 2 stays at Center), payoff drops to -5. Player 1 has no incentive to deviate. \u2713\n- Player 1 currently plays Middle, getting payoff 5. If Player 1 switches to Bottom (while Player 2 stays at Center), payoff drops to 4. Player 1 has no incentive to deviate. \u2713\n- Player 2 currently plays Center, getting payoff 3. If Player 2 switches to Left (while Player 1 stays at Middle), payoff drops to -3. Player 2 has no incentive to deviate. \u2713\n- Player 2 currently plays Center, getting payoff 3. If Player 2 switches to Right (while Player 1 stays at Middle), payoff drops to -3. Player 2 has no incentive to deviate. \u2713\nTherefore (Middle, Center) is a Nash Equilibrium with payoffs (5,3).\n\nMixed Strategy Nash Equilibrium:\nPlayer 1 randomizes over 2 strategies: Top with probability 0.7500, Middle with probability 0.2500.\nPlayer 2 randomizes over 2 strategies: Left with probability 0.9091, Center with probability 0.0909.\nIn this equilibrium, each player's mixture makes the opponent indifferent among their active strategies.\nExpected payoffs: Player 1 gets 4.0909, Player 2 gets -1.5000.\n\nIn addition to the pure strategy equilibria, this game has 1 mixed strategy Nash Equilibrium.",
    "answer": "[(Top, Left), (Middle, Center), P1=[Top:0.7500, Middle:0.2500], P2=[Left:0.9091, Center:0.0909]]",
    "game_type": "simultaneous",
    "players": 2,
    "tags": [
      "3x3",
      "mixed_strategy",
      "multiple_equilibria",
      "nash_equilibrium",
      "normal_form",
      "pure_strategy"
    ]
  },
  {
    "id": "gt-nxm-3x3-iesds-0024",
    "category": "normal_form_3x3",
    "subcategory": "3x3_iesds",
    "difficulty": "medium",
    "problem": "Apply Iterated Elimination of Strictly Dominated Strategies (IESDS) to the following game:\n\nPlayer 1 \\ Player 2 | Left | Center | Right\n--- | --- | --- | ---\nTop | (0,-2) | (2,0) | (2,2)\nMiddle | (5,0) | (-2,-3) | (-5,3)\nBottom | (1,-3) | (0,-2) | (-1,-1)\n\nShow each elimination step and find the surviving strategy profile.",
    "solution": "Applying Iterated Elimination of Strictly Dominated Strategies (IESDS):\n\nStep 1: Consider Player 2's strategies.\n  Compare Left vs Right: for every row strategy Player 1 might play,\n  Right gives Player 2 a strictly higher payoff than Left.\n  Therefore Left is strictly dominated by Right. Eliminate Left. \u2717\n  Remaining strategies: P1=['Top', 'Middle', 'Bottom'], P2=['Center', 'Right']\n\nStep 2: Consider Player 1's strategies.\n  Compare Middle vs Top: for every column strategy Player 2 might play,\n  Top gives Player 1 a strictly higher payoff than Middle.\n  Therefore Middle is strictly dominated by Top. Eliminate Middle. \u2717\n  Remaining strategies: P1=['Top', 'Bottom'], P2=['Center', 'Right']\n\nStep 3: Consider Player 1's strategies.\n  Compare Bottom vs Top: for every column strategy Player 2 might play,\n  Top gives Player 1 a strictly higher payoff than Bottom.\n  Therefore Bottom is strictly dominated by Top. Eliminate Bottom. \u2717\n  Remaining strategies: P1=['Top'], P2=['Center', 'Right']\n\nStep 4: Consider Player 2's strategies.\n  Compare Center vs Right: for every row strategy Player 1 might play,\n  Right gives Player 2 a strictly higher payoff than Center.\n  Therefore Center is strictly dominated by Right. Eliminate Center. \u2717\n  Remaining strategies: P1=['Top'], P2=['Right']\n\nAfter 4 elimination steps, the unique surviving strategy profile is (Top, Right).\nThe resulting payoffs are (2, 2). \u2713",
    "answer": "(Top, Right)",
    "game_type": "simultaneous",
    "players": 2,
    "tags": [
      "3x3",
      "dominated_strategy",
      "iesds",
      "normal_form"
    ]
  },
  {
    "id": "gt-nxm-3x3-pure-0059",
    "category": "normal_form_3x3",
    "subcategory": "3x3_pure_ne",
    "difficulty": "medium",
    "problem": "Consider the following 3x3 strategic form game:\n\nPlayer 1 \\ Player 2 | Left | Center | Right\n--- | --- | --- | ---\nTop | (-1,-8) | (-7,7) | (-1,0)\nMiddle | (4,4) | (4,2) | (-8,4)\nBottom | (-3,1) | (7,-5) | (-1,-3)\n\nFind all Nash Equilibria (pure and mixed).",
    "solution": "This game has 1 pure strategy Nash Equilibrium.\n\nTo verify (Middle, Left) is a Nash Equilibrium, we check each player's incentive to deviate:\n- Player 1 currently plays Middle, getting payoff 4. If Player 1 switches to Top (while Player 2 stays at Left), payoff drops to -1. Player 1 has no incentive to deviate. \u2713\n- Player 1 currently plays Middle, getting payoff 4. If Player 1 switches to Bottom (while Player 2 stays at Left), payoff drops to -3. Player 1 has no incentive to deviate. \u2713\n- Player 2 currently plays Left, getting payoff 4. If Player 2 switches to Center (while Player 1 stays at Middle), payoff drops to 2. Player 2 has no incentive to deviate. \u2713\n- Player 2 currently plays Left, getting payoff 4. If Player 2 switches to Right (while Player 1 stays at Middle), payoff stays at 4. Player 2 has no incentive to deviate. \u2713\nTherefore (Middle, Left) is a Nash Equilibrium with payoffs (4,4).\n\nMixed Strategy Nash Equilibrium:\nPlayer 1 randomizes over 3 strategies: Top with probability 0.2222, Middle with probability 0.3333, Bottom with probability 0.4444.\nPlayer 2 randomizes over 3 strategies: Left with probability 0.4804, Center with probability 0.0686, Right with probability 0.4510.\nIn this equilibrium, each player's mixture makes the opponent indifferent among their active strategies.\nExpected payoffs: Player 1 gets -1.4118, Player 2 gets -0.0000.\n\nIn addition to the pure strategy equilibrium, this game has 1 mixed strategy Nash Equilibrium.",
    "answer": "[(Middle, Left), P1=[Top:0.2222, Middle:0.3333, Bottom:0.4444], P2=[Left:0.4804, Center:0.0686, Right:0.4510]]",
    "game_type": "simultaneous",
    "players": 2,
    "tags": [
      "3x3",
      "mixed_strategy",
      "nash_equilibrium",
      "normal_form",
      "pure_strategy"
    ]
  },
  {
    "id": "gt-nxm-3x3-dom-0014",
    "category": "normal_form_3x3",
    "subcategory": "3x3_dominant",
    "difficulty": "medium",
    "problem": "Consider the game below:\n\nPlayer 1 \\ Player 2 | Left | Center | Right\n--- | --- | --- | ---\nTop | (3,3) | (4,5) | (-1,0)\nMiddle | (-6,-1) | (-6,-5) | (1,-4)\nBottom | (4,1) | (-1,-3) | (6,6)\n\nIdentify any strictly dominant or dominated strategies for each player.",
    "solution": "To identify dominant strategies, we compare each player's payoffs across all opponent strategies:\n\nPlayer 1's analysis:\n- Strategy Top: payoffs [3, 4, -1] against ['Left', 'Center', 'Right']\n- Strategy Middle: payoffs [-6, -6, 1] against ['Left', 'Center', 'Right']\n- Strategy Bottom: payoffs [4, -1, 6] against ['Left', 'Center', 'Right']\nNo single strategy dominates all others across every column. Player 1 has no dominant strategy.\nStrategy Middle is strictly dominated (another strategy always gives higher payoff). \u2717\n\nPlayer 2's analysis:\n- Strategy Left: payoffs [3, -1, 1] against ['Top', 'Middle', 'Bottom']\n- Strategy Center: payoffs [5, -5, -3] against ['Top', 'Middle', 'Bottom']\n- Strategy Right: payoffs [0, -4, 6] against ['Top', 'Middle', 'Bottom']\nNo single strategy dominates all others across every row. Player 2 has no dominant strategy.",
    "answer": "{'p1': {'strictly_dominant': [], 'weakly_dominant': [], 'strictly_dominated': [1], 'weakly_dominated': []}, 'p2': {'strictly_dominant': [], 'weakly_dominant': [], 'strictly_dominated': [], 'weakly_dominated': []}}",
    "game_type": "simultaneous",
    "players": 2,
    "tags": [
      "3x3",
      "dominant_strategy",
      "normal_form"
    ]
  },
  {
    "id": "gt-nxm-3x3-mixed-0015",
    "category": "normal_form_3x3",
    "subcategory": "3x3_mixed_ne",
    "difficulty": "medium",
    "problem": "Analyze the following 3-by-3 game and determine all Nash Equilibria:\n\nPlayer 1 \\ Player 2 | Left | Center | Right\n--- | --- | --- | ---\nTop | (3,3) | (-5,-3) | (2,2)\nMiddle | (2,-1) | (-2,6) | (4,-4)\nBottom | (-6,-1) | (0,6) | (3,3)",
    "solution": "This game has 2 pure strategy Nash Equilibria.\n\nTo verify (Top, Left) is a Nash Equilibrium, we check each player's incentive to deviate:\n- Player 1 currently plays Top, getting payoff 3. If Player 1 switches to Middle (while Player 2 stays at Left), payoff drops to 2. Player 1 has no incentive to deviate. \u2713\n- Player 1 currently plays Top, getting payoff 3. If Player 1 switches to Bottom (while Player 2 stays at Left), payoff drops to -6. Player 1 has no incentive to deviate. \u2713\n- Player 2 currently plays Left, getting payoff 3. If Player 2 switches to Center (while Player 1 stays at Top), payoff drops to -3. Player 2 has no incentive to deviate. \u2713\n- Player 2 currently plays Left, getting payoff 3. If Player 2 switches to Right (while Player 1 stays at Top), payoff drops to 2. Player 2 has no incentive to deviate. \u2713\nTherefore (Top, Left) is a Nash Equilibrium with payoffs (3,3).\n\nTo verify (Bottom, Center) is a Nash Equilibrium, we check each player's incentive to deviate:\n- Player 1 currently plays Bottom, getting payoff 0. If Player 1 switches to Top (while Player 2 stays at Center), payoff drops to -5. Player 1 has no incentive to deviate. \u2713\n- Player 1 currently plays Bottom, getting payoff 0. If Player 1 switches to Middle (while Player 2 stays at Center), payoff drops to -2. Player 1 has no incentive to deviate. \u2713\n- Player 2 currently plays Center, getting payoff 6. If Player 2 switches to Left (while Player 1 stays at Bottom), payoff drops to -1. Player 2 has no incentive to deviate. \u2713\n- Player 2 currently plays Center, getting payoff 6. If Player 2 switches to Right (while Player 1 stays at Bottom), payoff drops to 3. Player 2 has no incentive to deviate. \u2713\nTherefore (Bottom, Center) is a Nash Equilibrium with payoffs (0,6).\n\nMixed Strategy Nash Equilibrium:\nPlayer 1 randomizes over 2 strategies: Top with probability 0.5385, Middle with probability 0.4615.\nPlayer 2 randomizes over 2 strategies: Left with probability 0.7500, Center with probability 0.2500.\nIn this equilibrium, each player's mixture makes the opponent indifferent among their active strategies.\nExpected payoffs: Player 1 gets 1.0000, Player 2 gets 1.1538.\n\nIn addition to the pure strategy equilibria, this game has 1 mixed strategy Nash Equilibrium.",
    "answer": "[(Top, Left), (Bottom, Center), P1=[Top:0.5385, Middle:0.4615], P2=[Left:0.7500, Center:0.2500]]",
    "game_type": "simultaneous",
    "players": 2,
    "tags": [
      "3x3",
      "mixed_strategy",
      "nash_equilibrium",
      "normal_form",
      "pure_strategy"
    ]
  },
  {
    "id": "gt-nxm-3x4-iesds-0011",
    "category": "normal_form_3x4",
    "subcategory": "3x4_iesds",
    "difficulty": "medium",
    "problem": "Use iterated dominance to simplify the following game:\n\nPlayer 1 \\ Player 2 | C1 | C2 | C3 | C4\n--- | --- | --- | --- | ---\nTop | (-1,-2) | (-5,1) | (-4,5) | (5,0)\nMiddle | (-2,3) | (0,-4) | (5,5) | (1,1)\nBottom | (-5,5) | (2,-1) | (-5,1) | (3,-5)\n\nEliminate strictly dominated strategies step by step.",
    "solution": "Applying Iterated Elimination of Strictly Dominated Strategies (IESDS):\n\nStep 1: Consider Player 2's strategies.\n  Compare C2 vs C3: for every row strategy Player 1 might play,\n  C3 gives Player 2 a strictly higher payoff than C2.\n  Therefore C2 is strictly dominated by C3. Eliminate C2. \u2717\n  Remaining strategies: P1=['Top', 'Middle', 'Bottom'], P2=['C1', 'C3', 'C4']\n\nStep 2: Consider Player 1's strategies.\n  Compare Bottom vs Top: for every column strategy Player 2 might play,\n  Top gives Player 1 a strictly higher payoff than Bottom.\n  Therefore Bottom is strictly dominated by Top. Eliminate Bottom. \u2717\n  Remaining strategies: P1=['Top', 'Middle'], P2=['C1', 'C3', 'C4']\n\nStep 3: Consider Player 2's strategies.\n  Compare C1 vs C3: for every row strategy Player 1 might play,\n  C3 gives Player 2 a strictly higher payoff than C1.\n  Therefore C1 is strictly dominated by C3. Eliminate C1. \u2717\n  Remaining strategies: P1=['Top', 'Middle'], P2=['C3', 'C4']\n\nStep 4: Consider Player 2's strategies.\n  Compare C4 vs C3: for every row strategy Player 1 might play,\n  C3 gives Player 2 a strictly higher payoff than C4.\n  Therefore C4 is strictly dominated by C3. Eliminate C4. \u2717\n  Remaining strategies: P1=['Top', 'Middle'], P2=['C3']\n\nStep 5: Consider Player 1's strategies.\n  Compare Top vs Middle: for every column strategy Player 2 might play,\n  Middle gives Player 1 a strictly higher payoff than Top.\n  Therefore Top is strictly dominated by Middle. Eliminate Top. \u2717\n  Remaining strategies: P1=['Middle'], P2=['C3']\n\nAfter 5 elimination steps, the unique surviving strategy profile is (Middle, C3).\nThe resulting payoffs are (5, 5). \u2713",
    "answer": "(Middle, C3)",
    "game_type": "simultaneous",
    "players": 2,
    "tags": [
      "3x4",
      "dominated_strategy",
      "iesds",
      "normal_form"
    ]
  },
  {
    "id": "gt-nxm-3x4-iesds-0030",
    "category": "normal_form_3x4",
    "subcategory": "3x4_iesds",
    "difficulty": "medium",
    "problem": "Use iterated dominance to simplify the following game:\n\nPlayer 1 \\ Player 2 | C1 | C2 | C3 | C4\n--- | --- | --- | --- | ---\nTop | (0,-3) | (-5,-4) | (-2,0) | (-2,-1)\nMiddle | (3,2) | (3,-5) | (0,5) | (2,4)\nBottom | (4,-3) | (-2,1) | (4,4) | (2,-2)\n\nEliminate strictly dominated strategies step by step.",
    "solution": "Applying Iterated Elimination of Strictly Dominated Strategies (IESDS):\n\nStep 1: Consider Player 1's strategies.\n  Compare Top vs Middle: for every column strategy Player 2 might play,\n  Middle gives Player 1 a strictly higher payoff than Top.\n  Therefore Top is strictly dominated by Middle. Eliminate Top. \u2717\n  Remaining strategies: P1=['Middle', 'Bottom'], P2=['C1', 'C2', 'C3', 'C4']\n\nStep 2: Consider Player 2's strategies.\n  Compare C1 vs C3: for every row strategy Player 1 might play,\n  C3 gives Player 2 a strictly higher payoff than C1.\n  Therefore C1 is strictly dominated by C3. Eliminate C1. \u2717\n  Remaining strategies: P1=['Middle', 'Bottom'], P2=['C2', 'C3', 'C4']\n\nStep 3: Consider Player 2's strategies.\n  Compare C2 vs C3: for every row strategy Player 1 might play,\n  C3 gives Player 2 a strictly higher payoff than C2.\n  Therefore C2 is strictly dominated by C3. Eliminate C2. \u2717\n  Remaining strategies: P1=['Middle', 'Bottom'], P2=['C3', 'C4']\n\nStep 4: Consider Player 2's strategies.\n  Compare C4 vs C3: for every row strategy Player 1 might play,\n  C3 gives Player 2 a strictly higher payoff than C4.\n  Therefore C4 is strictly dominated by C3. Eliminate C4. \u2717\n  Remaining strategies: P1=['Middle', 'Bottom'], P2=['C3']\n\nStep 5: Consider Player 1's strategies.\n  Compare Middle vs Bottom: for every column strategy Player 2 might play,\n  Bottom gives Player 1 a strictly higher payoff than Middle.\n  Therefore Middle is strictly dominated by Bottom. Eliminate Middle. \u2717\n  Remaining strategies: P1=['Bottom'], P2=['C3']\n\nAfter 5 elimination steps, the unique surviving strategy profile is (Bottom, C3).\nThe resulting payoffs are (4, 4). \u2713",
    "answer": "(Bottom, C3)",
    "game_type": "simultaneous",
    "players": 2,
    "tags": [
      "3x4",
      "dominated_strategy",
      "iesds",
      "normal_form"
    ]
  },
  {
    "id": "gt-nxm-3x4-iesds-0013",
    "category": "normal_form_3x4",
    "subcategory": "3x4_iesds",
    "difficulty": "medium",
    "problem": "In the game below, apply IESDS to find the solution:\n\nPlayer 1 \\ Player 2 | C1 | C2 | C3 | C4\n--- | --- | --- | --- | ---\nTop | (3,-5) | (-5,2) | (2,4) | (3,-5)\nMiddle | (3,-5) | (2,-3) | (-2,2) | (1,2)\nBottom | (-5,1) | (1,-1) | (0,2) | (-1,-4)",
    "solution": "Applying Iterated Elimination of Strictly Dominated Strategies (IESDS):\n\nStep 1: Consider Player 2's strategies.\n  Compare C1 vs C3: for every row strategy Player 1 might play,\n  C3 gives Player 2 a strictly higher payoff than C1.\n  Therefore C1 is strictly dominated by C3. Eliminate C1. \u2717\n  Remaining strategies: P1=['Top', 'Middle', 'Bottom'], P2=['C2', 'C3', 'C4']\n\nStep 2: Consider Player 2's strategies.\n  Compare C2 vs C3: for every row strategy Player 1 might play,\n  C3 gives Player 2 a strictly higher payoff than C2.\n  Therefore C2 is strictly dominated by C3. Eliminate C2. \u2717\n  Remaining strategies: P1=['Top', 'Middle', 'Bottom'], P2=['C3', 'C4']\n\nStep 3: Consider Player 1's strategies.\n  Compare Middle vs Top: for every column strategy Player 2 might play,\n  Top gives Player 1 a strictly higher payoff than Middle.\n  Therefore Middle is strictly dominated by Top. Eliminate Middle. \u2717\n  Remaining strategies: P1=['Top', 'Bottom'], P2=['C3', 'C4']\n\nStep 4: Consider Player 1's strategies.\n  Compare Bottom vs Top: for every column strategy Player 2 might play,\n  Top gives Player 1 a strictly higher payoff than Bottom.\n  Therefore Bottom is strictly dominated by Top. Eliminate Bottom. \u2717\n  Remaining strategies: P1=['Top'], P2=['C3', 'C4']\n\nStep 5: Consider Player 2's strategies.\n  Compare C4 vs C3: for every row strategy Player 1 might play,\n  C3 gives Player 2 a strictly higher payoff than C4.\n  Therefore C4 is strictly dominated by C3. Eliminate C4. \u2717\n  Remaining strategies: P1=['Top'], P2=['C3']\n\nAfter 5 elimination steps, the unique surviving strategy profile is (Top, C3).\nThe resulting payoffs are (2, 4). \u2713",
    "answer": "(Top, C3)",
    "game_type": "simultaneous",
    "players": 2,
    "tags": [
      "3x4",
      "dominated_strategy",
      "iesds",
      "normal_form"
    ]
  },
  {
    "id": "gt-nxm-3x4-mixed-0038",
    "category": "normal_form_3x4",
    "subcategory": "3x4_mixed_ne",
    "difficulty": "hard",
    "problem": "Analyze the following 3-by-4 game and determine all Nash Equilibria:\n\nPlayer 1 \\ Player 2 | C1 | C2 | C3 | C4\n--- | --- | --- | --- | ---\nTop | (-6,1) | (5,2) | (-1,-3) | (-4,-1)\nMiddle | (4,-2) | (2,-2) | (-2,-3) | (6,-5)\nBottom | (-1,5) | (4,-1) | (2,2) | (2,5)",
    "solution": "This game has 2 pure strategy Nash Equilibria.\n\nTo verify (Top, C2) is a Nash Equilibrium, we check each player's incentive to deviate:\n- Player 1 currently plays Top, getting payoff 5. If Player 1 switches to Middle (while Player 2 stays at C2), payoff drops to 2. Player 1 has no incentive to deviate. \u2713\n- Player 1 currently plays Top, getting payoff 5. If Player 1 switches to Bottom (while Player 2 stays at C2), payoff drops to 4. Player 1 has no incentive to deviate. \u2713\n- Player 2 currently plays C2, getting payoff 2. If Player 2 switches to C1 (while Player 1 stays at Top), payoff drops to 1. Player 2 has no incentive to deviate. \u2713\n- Player 2 currently plays C2, getting payoff 2. If Player 2 switches to C3 (while Player 1 stays at Top), payoff drops to -3. Player 2 has no incentive to deviate. \u2713\n- Player 2 currently plays C2, getting payoff 2. If Player 2 switches to C4 (while Player 1 stays at Top), payoff drops to -1. Player 2 has no incentive to deviate. \u2713\nTherefore (Top, C2) is a Nash Equilibrium with payoffs (5,2).\n\nTo verify (Middle, C1) is a Nash Equilibrium, we check each player's incentive to deviate:\n- Player 1 currently plays Middle, getting payoff 4. If Player 1 switches to Top (while Player 2 stays at C1), payoff drops to -6. Player 1 has no incentive to deviate. \u2713\n- Player 1 currently plays Middle, getting payoff 4. If Player 1 switches to Bottom (while Player 2 stays at C1), payoff drops to -1. Player 1 has no incentive to deviate. \u2713\n- Player 2 currently plays C1, getting payoff -2. If Player 2 switches to C2 (while Player 1 stays at Middle), payoff stays at -2. Player 2 has no incentive to deviate. \u2713\n- Player 2 currently plays C1, getting payoff -2. If Player 2 switches to C3 (while Player 1 stays at Middle), payoff drops to -3. Player 2 has no incentive to deviate. \u2713\n- Player 2 currently plays C1, getting payoff -2. If Player 2 switches to C4 (while Player 1 stays at Middle), payoff drops to -5. Player 2 has no incentive to deviate. \u2713\nTherefore (Middle, C1) is a Nash Equilibrium with payoffs (4,-2).\n\nMixed Strategy Nash Equilibrium:\nPlayer 1 randomizes over 2 strategies: Top with probability 0.8571, Bottom with probability 0.1429.\nPlayer 2 randomizes over 2 strategies: C1 with probability 0.1667, C2 with probability 0.8333.\nIn this equilibrium, each player's mixture makes the opponent indifferent among their active strategies.\nExpected payoffs: Player 1 gets 3.1667, Player 2 gets 1.5714.\n\nIn addition to the pure strategy equilibria, this game has 1 mixed strategy Nash Equilibrium.",
    "answer": "[(Top, C2), (Middle, C1), P1=[Top:0.8571, Bottom:0.1429], P2=[C1:0.1667, C2:0.8333]]",
    "game_type": "simultaneous",
    "players": 2,
    "tags": [
      "3x4",
      "mixed_strategy",
      "nash_equilibrium",
      "normal_form",
      "pure_strategy"
    ]
  },
  {
    "id": "gt-nxm-3x4-mixed-0012",
    "category": "normal_form_3x4",
    "subcategory": "3x4_mixed_ne",
    "difficulty": "hard",
    "problem": "Consider the following 3x4 strategic form game:\n\nPlayer 1 \\ Player 2 | C1 | C2 | C3 | C4\n--- | --- | --- | --- | ---\nTop | (-5,-1) | (0,2) | (3,3) | (3,6)\nMiddle | (-5,4) | (-6,5) | (-2,-6) | (5,-2)\nBottom | (6,2) | (-6,2) | (-6,-6) | (-6,1)\n\nFind all Nash Equilibria (pure and mixed).",
    "solution": "This game has 1 pure strategy Nash Equilibrium.\n\nTo verify (Bottom, C1) is a Nash Equilibrium, we check each player's incentive to deviate:\n- Player 1 currently plays Bottom, getting payoff 6. If Player 1 switches to Top (while Player 2 stays at C1), payoff drops to -5. Player 1 has no incentive to deviate. \u2713\n- Player 1 currently plays Bottom, getting payoff 6. If Player 1 switches to Middle (while Player 2 stays at C1), payoff drops to -5. Player 1 has no incentive to deviate. \u2713\n- Player 2 currently plays C1, getting payoff 2. If Player 2 switches to C2 (while Player 1 stays at Bottom), payoff stays at 2. Player 2 has no incentive to deviate. \u2713\n- Player 2 currently plays C1, getting payoff 2. If Player 2 switches to C3 (while Player 1 stays at Bottom), payoff drops to -6. Player 2 has no incentive to deviate. \u2713\n- Player 2 currently plays C1, getting payoff 2. If Player 2 switches to C4 (while Player 1 stays at Bottom), payoff drops to 1. Player 2 has no incentive to deviate. \u2713\nTherefore (Bottom, C1) is a Nash Equilibrium with payoffs (6,2).\n\nMixed Strategy Nash Equilibrium:\nPlayer 1 randomizes over 2 strategies: Top with probability 0.6364, Middle with probability 0.3636.\nPlayer 2 randomizes over 2 strategies: C2 with probability 0.2500, C4 with probability 0.7500.\nIn this equilibrium, each player's mixture makes the opponent indifferent among their active strategies.\nExpected payoffs: Player 1 gets 2.2500, Player 2 gets 3.0909.\n\nIn addition to the pure strategy equilibrium, this game has 1 mixed strategy Nash Equilibrium.",
    "answer": "[(Bottom, C1), P1=[Top:0.6364, Middle:0.3636], P2=[C2:0.2500, C4:0.7500]]",
    "game_type": "simultaneous",
    "players": 2,
    "tags": [
      "3x4",
      "mixed_strategy",
      "nash_equilibrium",
      "normal_form",
      "pure_strategy"
    ]
  },
  {
    "id": "gt-nxm-3x4-multi-0007",
    "category": "normal_form_3x4",
    "subcategory": "3x4_multi_eq",
    "difficulty": "hard",
    "problem": "The following 3x4 game has multiple equilibria. Find ALL Nash Equilibria:\n\nPlayer 1 \\ Player 2 | C1 | C2 | C3 | C4\n--- | --- | --- | --- | ---\nTop | (-3,5) | (2,3) | (4,0) | (-5,2)\nMiddle | (-5,3) | (4,3) | (2,2) | (3,4)\nBottom | (4,2) | (0,-3) | (-1,-3) | (-4,-4)",
    "solution": "This game has 2 Nash Equilibria (2 pure, 0 mixed).\n\nThis game has 2 pure strategy Nash Equilibria.\n\nTo verify (Middle, C4) is a Nash Equilibrium, we check each player's incentive to deviate:\n- Player 1 currently plays Middle, getting payoff 3. If Player 1 switches to Top (while Player 2 stays at C4), payoff drops to -5. Player 1 has no incentive to deviate. \u2713\n- Player 1 currently plays Middle, getting payoff 3. If Player 1 switches to Bottom (while Player 2 stays at C4), payoff drops to -4. Player 1 has no incentive to deviate. \u2713\n- Player 2 currently plays C4, getting payoff 4. If Player 2 switches to C1 (while Player 1 stays at Middle), payoff drops to 3. Player 2 has no incentive to deviate. \u2713\n- Player 2 currently plays C4, getting payoff 4. If Player 2 switches to C2 (while Player 1 stays at Middle), payoff drops to 3. Player 2 has no incentive to deviate. \u2713\n- Player 2 currently plays C4, getting payoff 4. If Player 2 switches to C3 (while Player 1 stays at Middle), payoff drops to 2. Player 2 has no incentive to deviate. \u2713\nTherefore (Middle, C4) is a Nash Equilibrium with payoffs (3,4).\n\nTo verify (Bottom, C1) is a Nash Equilibrium, we check each player's incentive to deviate:\n- Player 1 currently plays Bottom, getting payoff 4. If Player 1 switches to Top (while Player 2 stays at C1), payoff drops to -3. Player 1 has no incentive to deviate. \u2713\n- Player 1 currently plays Bottom, getting payoff 4. If Player 1 switches to Middle (while Player 2 stays at C1), payoff drops to -5. Player 1 has no incentive to deviate. \u2713\n- Player 2 currently plays C1, getting payoff 2. If Player 2 switches to C2 (while Player 1 stays at Bottom), payoff drops to -3. Player 2 has no incentive to deviate. \u2713\n- Player 2 currently plays C1, getting payoff 2. If Player 2 switches to C3 (while Player 1 stays at Bottom), payoff drops to -3. Player 2 has no incentive to deviate. \u2713\n- Player 2 currently plays C1, getting payoff 2. If Player 2 switches to C4 (while Player 1 stays at Bottom), payoff drops to -4. Player 2 has no incentive to deviate. \u2713\nTherefore (Bottom, C1) is a Nash Equilibrium with payoffs (4,2).\n\nThere are no additional mixed strategy equilibria beyond the pure strategy equilibria identified above.",
    "answer": "[(Middle, C4), (Bottom, C1)]",
    "game_type": "simultaneous",
    "players": 2,
    "tags": [
      "3x4",
      "multiple_equilibria",
      "nash_equilibrium",
      "normal_form",
      "pure_strategy"
    ]
  },
  {
    "id": "gt-nxm-3x4-iesds-0018",
    "category": "normal_form_3x4",
    "subcategory": "3x4_iesds",
    "difficulty": "medium",
    "problem": "Apply Iterated Elimination of Strictly Dominated Strategies (IESDS) to the following game:\n\nPlayer 1 \\ Player 2 | C1 | C2 | C3 | C4\n--- | --- | --- | --- | ---\nTop | (-2,2) | (-5,5) | (-5,-2) | (-5,1)\nMiddle | (1,-2) | (-5,-5) | (0,0) | (-4,-1)\nBottom | (0,3) | (0,3) | (5,5) | (-3,2)\n\nShow each elimination step and find the surviving strategy profile.",
    "solution": "Applying Iterated Elimination of Strictly Dominated Strategies (IESDS):\n\nStep 1: Consider Player 1's strategies.\n  Compare Top vs Bottom: for every column strategy Player 2 might play,\n  Bottom gives Player 1 a strictly higher payoff than Top.\n  Therefore Top is strictly dominated by Bottom. Eliminate Top. \u2717\n  Remaining strategies: P1=['Middle', 'Bottom'], P2=['C1', 'C2', 'C3', 'C4']\n\nStep 2: Consider Player 2's strategies.\n  Compare C1 vs C3: for every row strategy Player 1 might play,\n  C3 gives Player 2 a strictly higher payoff than C1.\n  Therefore C1 is strictly dominated by C3. Eliminate C1. \u2717\n  Remaining strategies: P1=['Middle', 'Bottom'], P2=['C2', 'C3', 'C4']\n\nStep 3: Consider Player 1's strategies.\n  Compare Middle vs Bottom: for every column strategy Player 2 might play,\n  Bottom gives Player 1 a strictly higher payoff than Middle.\n  Therefore Middle is strictly dominated by Bottom. Eliminate Middle. \u2717\n  Remaining strategies: P1=['Bottom'], P2=['C2', 'C3', 'C4']\n\nStep 4: Consider Player 2's strategies.\n  Compare C2 vs C3: for every row strategy Player 1 might play,\n  C3 gives Player 2 a strictly higher payoff than C2.\n  Therefore C2 is strictly dominated by C3. Eliminate C2. \u2717\n  Remaining strategies: P1=['Bottom'], P2=['C3', 'C4']\n\nStep 5: Consider Player 2's strategies.\n  Compare C4 vs C3: for every row strategy Player 1 might play,\n  C3 gives Player 2 a strictly higher payoff than C4.\n  Therefore C4 is strictly dominated by C3. Eliminate C4. \u2717\n  Remaining strategies: P1=['Bottom'], P2=['C3']\n\nAfter 5 elimination steps, the unique surviving strategy profile is (Bottom, C3).\nThe resulting payoffs are (5, 5). \u2713",
    "answer": "(Bottom, C3)",
    "game_type": "simultaneous",
    "players": 2,
    "tags": [
      "3x4",
      "dominated_strategy",
      "iesds",
      "normal_form"
    ]
  },
  {
    "id": "gt-nxm-3x4-dom-0030",
    "category": "normal_form_3x4",
    "subcategory": "3x4_dominant",
    "difficulty": "medium",
    "problem": "Analyze the following game for dominant strategies:\n\nPlayer 1 \\ Player 2 | C1 | C2 | C3 | C4\n--- | --- | --- | --- | ---\nTop | (4,2) | (2,5) | (1,-3) | (6,-3)\nMiddle | (3,-2) | (-1,1) | (4,4) | (3,-2)\nBottom | (4,-3) | (3,1) | (-6,4) | (0,-2)\n\nDoes either player have a dominant strategy? If so, what is it?",
    "solution": "To identify dominant strategies, we compare each player's payoffs across all opponent strategies:\n\nPlayer 1's analysis:\n- Strategy Top: payoffs [4, 2, 1, 6] against ['C1', 'C2', 'C3', 'C4']\n- Strategy Middle: payoffs [3, -1, 4, 3] against ['C1', 'C2', 'C3', 'C4']\n- Strategy Bottom: payoffs [4, 3, -6, 0] against ['C1', 'C2', 'C3', 'C4']\nNo single strategy dominates all others across every column. Player 1 has no dominant strategy.\n\nPlayer 2's analysis:\n- Strategy C1: payoffs [2, -2, -3] against ['Top', 'Middle', 'Bottom']\n- Strategy C2: payoffs [5, 1, 1] against ['Top', 'Middle', 'Bottom']\n- Strategy C3: payoffs [-3, 4, 4] against ['Top', 'Middle', 'Bottom']\n- Strategy C4: payoffs [-3, -2, -2] against ['Top', 'Middle', 'Bottom']\nNo single strategy dominates all others across every row. Player 2 has no dominant strategy.\nStrategy C1 is strictly dominated. \u2717\nStrategy C4 is strictly dominated. \u2717",
    "answer": "{'p1': {'strictly_dominant': [], 'weakly_dominant': [], 'strictly_dominated': [], 'weakly_dominated': []}, 'p2': {'strictly_dominant': [], 'weakly_dominant': [], 'strictly_dominated': [0, 3], 'weakly_dominated': []}}",
    "game_type": "simultaneous",
    "players": 2,
    "tags": [
      "3x4",
      "dominant_strategy",
      "normal_form"
    ]
  },
  {
    "id": "gt-nxm-3x4-mixed-0035",
    "category": "normal_form_3x4",
    "subcategory": "3x4_mixed_ne",
    "difficulty": "hard",
    "problem": "Players 1 and 2 play the following 3x4 simultaneous game. Find all equilibria:\n\nPlayer 1 \\ Player 2 | C1 | C2 | C3 | C4\n--- | --- | --- | --- | ---\nTop | (5,6) | (-1,1) | (-6,4) | (-5,-1)\nMiddle | (-2,4) | (-5,-4) | (3,-6) | (-6,-5)\nBottom | (-1,-2) | (1,6) | (1,3) | (6,-2)",
    "solution": "This game has 2 pure strategy Nash Equilibria.\n\nTo verify (Top, C1) is a Nash Equilibrium, we check each player's incentive to deviate:\n- Player 1 currently plays Top, getting payoff 5. If Player 1 switches to Middle (while Player 2 stays at C1), payoff drops to -2. Player 1 has no incentive to deviate. \u2713\n- Player 1 currently plays Top, getting payoff 5. If Player 1 switches to Bottom (while Player 2 stays at C1), payoff drops to -1. Player 1 has no incentive to deviate. \u2713\n- Player 2 currently plays C1, getting payoff 6. If Player 2 switches to C2 (while Player 1 stays at Top), payoff drops to 1. Player 2 has no incentive to deviate. \u2713\n- Player 2 currently plays C1, getting payoff 6. If Player 2 switches to C3 (while Player 1 stays at Top), payoff drops to 4. Player 2 has no incentive to deviate. \u2713\n- Player 2 currently plays C1, getting payoff 6. If Player 2 switches to C4 (while Player 1 stays at Top), payoff drops to -1. Player 2 has no incentive to deviate. \u2713\nTherefore (Top, C1) is a Nash Equilibrium with payoffs (5,6).\n\nTo verify (Bottom, C2) is a Nash Equilibrium, we check each player's incentive to deviate:\n- Player 1 currently plays Bottom, getting payoff 1. If Player 1 switches to Top (while Player 2 stays at C2), payoff drops to -1. Player 1 has no incentive to deviate. \u2713\n- Player 1 currently plays Bottom, getting payoff 1. If Player 1 switches to Middle (while Player 2 stays at C2), payoff drops to -5. Player 1 has no incentive to deviate. \u2713\n- Player 2 currently plays C2, getting payoff 6. If Player 2 switches to C1 (while Player 1 stays at Bottom), payoff drops to -2. Player 2 has no incentive to deviate. \u2713\n- Player 2 currently plays C2, getting payoff 6. If Player 2 switches to C3 (while Player 1 stays at Bottom), payoff drops to 3. Player 2 has no incentive to deviate. \u2713\n- Player 2 currently plays C2, getting payoff 6. If Player 2 switches to C4 (while Player 1 stays at Bottom), payoff drops to -2. Player 2 has no incentive to deviate. \u2713\nTherefore (Bottom, C2) is a Nash Equilibrium with payoffs (1,6).\n\nMixed Strategy Nash Equilibrium:\nPlayer 1 randomizes over 3 strategies: Top with probability 0.4819, Middle with probability 0.1084, Bottom with probability 0.4096.\nPlayer 2 randomizes over 3 strategies: C1 with probability 0.5169, C2 with probability 0.0562, C3 with probability 0.4270.\nIn this equilibrium, each player's mixture makes the opponent indifferent among their active strategies.\nExpected payoffs: Player 1 gets -0.0337, Player 2 gets 2.5060.\n\nIn addition to the pure strategy equilibria, this game has 1 mixed strategy Nash Equilibrium.",
    "answer": "[(Top, C1), (Bottom, C2), P1=[Top:0.4819, Middle:0.1084, Bottom:0.4096], P2=[C1:0.5169, C2:0.0562, C3:0.4270]]",
    "game_type": "simultaneous",
    "players": 2,
    "tags": [
      "3x4",
      "mixed_strategy",
      "nash_equilibrium",
      "normal_form",
      "pure_strategy"
    ]
  },
  {
    "id": "gt-nxm-3x4-pure-0042",
    "category": "normal_form_3x4",
    "subcategory": "3x4_pure_ne",
    "difficulty": "hard",
    "problem": "Players 1 and 2 play the following 3x4 simultaneous game. Find all equilibria:\n\nPlayer 1 \\ Player 2 | C1 | C2 | C3 | C4\n--- | --- | --- | --- | ---\nTop | (4,-7) | (1,8) | (5,0) | (0,3)\nMiddle | (-2,-6) | (-6,2) | (-3,-8) | (-7,4)\nBottom | (-3,6) | (-6,-4) | (7,8) | (-1,-3)",
    "solution": "This game has 2 pure strategy Nash Equilibria.\n\nTo verify (Top, C2) is a Nash Equilibrium, we check each player's incentive to deviate:\n- Player 1 currently plays Top, getting payoff 1. If Player 1 switches to Middle (while Player 2 stays at C2), payoff drops to -6. Player 1 has no incentive to deviate. \u2713\n- Player 1 currently plays Top, getting payoff 1. If Player 1 switches to Bottom (while Player 2 stays at C2), payoff drops to -6. Player 1 has no incentive to deviate. \u2713\n- Player 2 currently plays C2, getting payoff 8. If Player 2 switches to C1 (while Player 1 stays at Top), payoff drops to -7. Player 2 has no incentive to deviate. \u2713\n- Player 2 currently plays C2, getting payoff 8. If Player 2 switches to C3 (while Player 1 stays at Top), payoff drops to 0. Player 2 has no incentive to deviate. \u2713\n- Player 2 currently plays C2, getting payoff 8. If Player 2 switches to C4 (while Player 1 stays at Top), payoff drops to 3. Player 2 has no incentive to deviate. \u2713\nTherefore (Top, C2) is a Nash Equilibrium with payoffs (1,8).\n\nTo verify (Bottom, C3) is a Nash Equilibrium, we check each player's incentive to deviate:\n- Player 1 currently plays Bottom, getting payoff 7. If Player 1 switches to Top (while Player 2 stays at C3), payoff drops to 5. Player 1 has no incentive to deviate. \u2713\n- Player 1 currently plays Bottom, getting payoff 7. If Player 1 switches to Middle (while Player 2 stays at C3), payoff drops to -3. Player 1 has no incentive to deviate. \u2713\n- Player 2 currently plays C3, getting payoff 8. If Player 2 switches to C1 (while Player 1 stays at Bottom), payoff drops to 6. Player 2 has no incentive to deviate. \u2713\n- Player 2 currently plays C3, getting payoff 8. If Player 2 switches to C2 (while Player 1 stays at Bottom), payoff drops to -4. Player 2 has no incentive to deviate. \u2713\n- Player 2 currently plays C3, getting payoff 8. If Player 2 switches to C4 (while Player 1 stays at Bottom), payoff drops to -3. Player 2 has no incentive to deviate. \u2713\nTherefore (Bottom, C3) is a Nash Equilibrium with payoffs (7,8).\n\nMixed Strategy Nash Equilibrium:\nPlayer 1 randomizes over 2 strategies: Top with probability 0.6000, Bottom with probability 0.4000.\nPlayer 2 randomizes over 2 strategies: C2 with probability 0.2222, C3 with probability 0.7778.\nIn this equilibrium, each player's mixture makes the opponent indifferent among their active strategies.\nExpected payoffs: Player 1 gets 4.1111, Player 2 gets 3.2000.\n\nIn addition to the pure strategy equilibria, this game has 1 mixed strategy Nash Equilibrium.",
    "answer": "[(Top, C2), (Bottom, C3), P1=[Top:0.6000, Bottom:0.4000], P2=[C2:0.2222, C3:0.7778]]",
    "game_type": "simultaneous",
    "players": 2,
    "tags": [
      "3x4",
      "mixed_strategy",
      "nash_equilibrium",
      "normal_form",
      "pure_strategy"
    ]
  },
  {
    "id": "gt-nxm-4x4-dom-0012",
    "category": "normal_form_4x4",
    "subcategory": "4x4_dominant",
    "difficulty": "medium",
    "problem": "For each player in the following game, determine if they have a dominant strategy:\n\nPlayer 1 \\ Player 2 | C1 | C2 | C3 | C4\n--- | --- | --- | --- | ---\nR1 | (-1,3) | (-5,3) | (-4,1) | (-4,-5)\nR2 | (-3,6) | (-6,-6) | (6,-1) | (-1,-5)\nR3 | (4,4) | (-1,1) | (-3,6) | (1,-2)\nR4 | (-2,-5) | (3,4) | (-3,5) | (0,-2)",
    "solution": "To identify dominant strategies, we compare each player's payoffs across all opponent strategies:\n\nPlayer 1's analysis:\n- Strategy R1: payoffs [-1, -5, -4, -4] against ['C1', 'C2', 'C3', 'C4']\n- Strategy R2: payoffs [-3, -6, 6, -1] against ['C1', 'C2', 'C3', 'C4']\n- Strategy R3: payoffs [4, -1, -3, 1] against ['C1', 'C2', 'C3', 'C4']\n- Strategy R4: payoffs [-2, 3, -3, 0] against ['C1', 'C2', 'C3', 'C4']\nNo single strategy dominates all others across every column. Player 1 has no dominant strategy.\nStrategy R1 is strictly dominated (another strategy always gives higher payoff). \u2717\n\nPlayer 2's analysis:\n- Strategy C1: payoffs [3, 6, 4, -5] against ['R1', 'R2', 'R3', 'R4']\n- Strategy C2: payoffs [3, -6, 1, 4] against ['R1', 'R2', 'R3', 'R4']\n- Strategy C3: payoffs [1, -1, 6, 5] against ['R1', 'R2', 'R3', 'R4']\n- Strategy C4: payoffs [-5, -5, -2, -2] against ['R1', 'R2', 'R3', 'R4']\nNo single strategy dominates all others across every row. Player 2 has no dominant strategy.\nStrategy C4 is strictly dominated. \u2717",
    "answer": "{'p1': {'strictly_dominant': [], 'weakly_dominant': [], 'strictly_dominated': [0], 'weakly_dominated': []}, 'p2': {'strictly_dominant': [], 'weakly_dominant': [], 'strictly_dominated': [3], 'weakly_dominated': []}}",
    "game_type": "simultaneous",
    "players": 2,
    "tags": [
      "4x4",
      "dominant_strategy",
      "normal_form"
    ]
  },
  {
    "id": "gt-nxm-4x4-dom-0007",
    "category": "normal_form_4x4",
    "subcategory": "4x4_dominant",
    "difficulty": "medium",
    "problem": "Consider the game below:\n\nPlayer 1 \\ Player 2 | C1 | C2 | C3 | C4\n--- | --- | --- | --- | ---\nR1 | (0,1) | (0,5) | (2,-2) | (-6,-3)\nR2 | (1,-2) | (5,-2) | (6,6) | (4,3)\nR3 | (1,-4) | (0,-3) | (5,-3) | (-1,1)\nR4 | (-6,-2) | (4,-1) | (3,3) | (4,-5)\n\nIdentify any strictly dominant or dominated strategies for each player.",
    "solution": "To identify dominant strategies, we compare each player's payoffs across all opponent strategies:\n\nPlayer 1's analysis:\n- Strategy R1: payoffs [0, 0, 2, -6] against ['C1', 'C2', 'C3', 'C4']\n- Strategy R2: payoffs [1, 5, 6, 4] against ['C1', 'C2', 'C3', 'C4']\n- Strategy R3: payoffs [1, 0, 5, -1] against ['C1', 'C2', 'C3', 'C4']\n- Strategy R4: payoffs [-6, 4, 3, 4] against ['C1', 'C2', 'C3', 'C4']\nStrategy R2 yields a payoff at least as high as every other strategy in every column (and strictly higher in at least one).\nTherefore, R2 is a weakly dominant strategy for Player 1. \u2713\nStrategy R1 is strictly dominated (another strategy always gives higher payoff). \u2717\nStrategy R3 is weakly dominated. \u2717\nStrategy R4 is weakly dominated. \u2717\n\nPlayer 2's analysis:\n- Strategy C1: payoffs [1, -2, -4, -2] against ['R1', 'R2', 'R3', 'R4']\n- Strategy C2: payoffs [5, -2, -3, -1] against ['R1', 'R2', 'R3', 'R4']\n- Strategy C3: payoffs [-2, 6, -3, 3] against ['R1', 'R2', 'R3', 'R4']\n- Strategy C4: payoffs [-3, 3, 1, -5] against ['R1', 'R2', 'R3', 'R4']\nNo single strategy dominates all others across every row. Player 2 has no dominant strategy.\nStrategy C1 is weakly dominated. \u2717",
    "answer": "{'p1': {'strictly_dominant': [], 'weakly_dominant': [1], 'strictly_dominated': [0], 'weakly_dominated': [2, 3]}, 'p2': {'strictly_dominant': [], 'weakly_dominant': [], 'strictly_dominated': [], 'weakly_dominated': [0]}}",
    "game_type": "simultaneous",
    "players": 2,
    "tags": [
      "4x4",
      "dominant_strategy",
      "normal_form"
    ]
  },
  {
    "id": "gt-nxm-4x4-dom-0022",
    "category": "normal_form_4x4",
    "subcategory": "4x4_dominant",
    "difficulty": "medium",
    "problem": "Consider the game below:\n\nPlayer 1 \\ Player 2 | C1 | C2 | C3 | C4\n--- | --- | --- | --- | ---\nR1 | (2,-4) | (-4,-6) | (-3,-6) | (3,-2)\nR2 | (-2,3) | (0,3) | (-2,-5) | (2,-5)\nR3 | (-1,5) | (-4,-3) | (-1,3) | (-1,4)\nR4 | (5,5) | (3,5) | (-6,0) | (0,1)\n\nIdentify any strictly dominant or dominated strategies for each player.",
    "solution": "To identify dominant strategies, we compare each player's payoffs across all opponent strategies:\n\nPlayer 1's analysis:\n- Strategy R1: payoffs [2, -4, -3, 3] against ['C1', 'C2', 'C3', 'C4']\n- Strategy R2: payoffs [-2, 0, -2, 2] against ['C1', 'C2', 'C3', 'C4']\n- Strategy R3: payoffs [-1, -4, -1, -1] against ['C1', 'C2', 'C3', 'C4']\n- Strategy R4: payoffs [5, 3, -6, 0] against ['C1', 'C2', 'C3', 'C4']\nNo single strategy dominates all others across every column. Player 1 has no dominant strategy.\n\nPlayer 2's analysis:\n- Strategy C1: payoffs [-4, 3, 5, 5] against ['R1', 'R2', 'R3', 'R4']\n- Strategy C2: payoffs [-6, 3, -3, 5] against ['R1', 'R2', 'R3', 'R4']\n- Strategy C3: payoffs [-6, -5, 3, 0] against ['R1', 'R2', 'R3', 'R4']\n- Strategy C4: payoffs [-2, -5, 4, 1] against ['R1', 'R2', 'R3', 'R4']\nNo single strategy dominates all others across every row. Player 2 has no dominant strategy.\nStrategy C3 is strictly dominated. \u2717\nStrategy C2 is weakly dominated. \u2717",
    "answer": "{'p1': {'strictly_dominant': [], 'weakly_dominant': [], 'strictly_dominated': [], 'weakly_dominated': []}, 'p2': {'strictly_dominant': [], 'weakly_dominant': [], 'strictly_dominated': [2], 'weakly_dominated': [1]}}",
    "game_type": "simultaneous",
    "players": 2,
    "tags": [
      "4x4",
      "dominant_strategy",
      "normal_form"
    ]
  },
  {
    "id": "gt-nxm-4x4-mixed-0009",
    "category": "normal_form_4x4",
    "subcategory": "4x4_mixed_ne",
    "difficulty": "hard",
    "problem": "Consider the following 4x4 strategic form game:\n\nPlayer 1 \\ Player 2 | C1 | C2 | C3 | C4\n--- | --- | --- | --- | ---\nR1 | (-1,-6) | (-6,1) | (0,4) | (4,6)\nR2 | (-3,4) | (-2,-3) | (-1,-5) | (4,4)\nR3 | (-6,-1) | (2,-5) | (3,-2) | (-3,-5)\nR4 | (1,-4) | (-2,4) | (4,-1) | (1,-6)\n\nFind all Nash Equilibria (pure and mixed).",
    "solution": "This game has 2 pure strategy Nash Equilibria.\n\nTo verify (R1, C4) is a Nash Equilibrium, we check each player's incentive to deviate:\n- Player 1 currently plays R1, getting payoff 4. If Player 1 switches to R2 (while Player 2 stays at C4), payoff stays at 4. Player 1 has no incentive to deviate. \u2713\n- Player 1 currently plays R1, getting payoff 4. If Player 1 switches to R3 (while Player 2 stays at C4), payoff drops to -3. Player 1 has no incentive to deviate. \u2713\n- Player 1 currently plays R1, getting payoff 4. If Player 1 switches to R4 (while Player 2 stays at C4), payoff drops to 1. Player 1 has no incentive to deviate. \u2713\n- Player 2 currently plays C4, getting payoff 6. If Player 2 switches to C1 (while Player 1 stays at R1), payoff drops to -6. Player 2 has no incentive to deviate. \u2713\n- Player 2 currently plays C4, getting payoff 6. If Player 2 switches to C2 (while Player 1 stays at R1), payoff drops to 1. Player 2 has no incentive to deviate. \u2713\n- Player 2 currently plays C4, getting payoff 6. If Player 2 switches to C3 (while Player 1 stays at R1), payoff drops to 4. Player 2 has no incentive to deviate. \u2713\nTherefore (R1, C4) is a Nash Equilibrium with payoffs (4,6).\n\nTo verify (R2, C4) is a Nash Equilibrium, we check each player's incentive to deviate:\n- Player 1 currently plays R2, getting payoff 4. If Player 1 switches to R1 (while Player 2 stays at C4), payoff stays at 4. Player 1 has no incentive to deviate. \u2713\n- Player 1 currently plays R2, getting payoff 4. If Player 1 switches to R3 (while Player 2 stays at C4), payoff drops to -3. Player 1 has no incentive to deviate. \u2713\n- Player 1 currently plays R2, getting payoff 4. If Player 1 switches to R4 (while Player 2 stays at C4), payoff drops to 1. Player 1 has no incentive to deviate. \u2713\n- Player 2 currently plays C4, getting payoff 4. If Player 2 switches to C1 (while Player 1 stays at R2), payoff stays at 4. Player 2 has no incentive to deviate. \u2713\n- Player 2 currently plays C4, getting payoff 4. If Player 2 switches to C2 (while Player 1 stays at R2), payoff drops to -3. Player 2 has no incentive to deviate. \u2713\n- Player 2 currently plays C4, getting payoff 4. If Player 2 switches to C3 (while Player 1 stays at R2), payoff drops to -5. Player 2 has no incentive to deviate. \u2713\nTherefore (R2, C4) is a Nash Equilibrium with payoffs (4,4).\n\nMixed Strategy Nash Equilibrium:\nPlayer 1 randomizes over 3 strategies: R1 with probability 0.0673, R2 with probability 0.5288, R4 with probability 0.4038.\nPlayer 2 randomizes over 3 strategies: C1 with probability 0.3529, C2 with probability 0.1765, C4 with probability 0.4706.\nIn this equilibrium, each player's mixture makes the opponent indifferent among their active strategies.\nExpected payoffs: Player 1 gets 0.4706, Player 2 gets 0.0962.\n\nIn addition to the pure strategy equilibria, this game has 1 mixed strategy Nash Equilibrium.",
    "answer": "[(R1, C4), (R2, C4), P1=[R1:0.0673, R2:0.5288, R4:0.4038], P2=[C1:0.3529, C2:0.1765, C4:0.4706]]",
    "game_type": "simultaneous",
    "players": 2,
    "tags": [
      "4x4",
      "mixed_strategy",
      "nash_equilibrium",
      "normal_form",
      "pure_strategy"
    ]
  },
  {
    "id": "gt-nxm-4x4-iesds-0030",
    "category": "normal_form_4x4",
    "subcategory": "4x4_iesds",
    "difficulty": "hard",
    "problem": "Apply Iterated Elimination of Strictly Dominated Strategies (IESDS) to the following game:\n\nPlayer 1 \\ Player 2 | C1 | C2 | C3 | C4\n--- | --- | --- | --- | ---\nR1 | (-3,0) | (1,1) | (3,3) | (-5,0)\nR2 | (0,4) | (5,-5) | (-4,2) | (-1,-4)\nR3 | (1,-3) | (1,-1) | (4,0) | (1,1)\nR4 | (-2,-2) | (0,-3) | (5,3) | (4,-2)\n\nShow each elimination step and find the surviving strategy profile.",
    "solution": "Applying Iterated Elimination of Strictly Dominated Strategies (IESDS):\n\nStep 1: Consider Player 2's strategies.\n  Compare C2 vs C3: for every row strategy Player 1 might play,\n  C3 gives Player 2 a strictly higher payoff than C2.\n  Therefore C2 is strictly dominated by C3. Eliminate C2. \u2717\n  Remaining strategies: P1=['R1', 'R2', 'R3', 'R4'], P2=['C1', 'C3', 'C4']\n\nStep 2: Consider Player 1's strategies.\n  Compare R1 vs R3: for every column strategy Player 2 might play,\n  R3 gives Player 1 a strictly higher payoff than R1.\n  Therefore R1 is strictly dominated by R3. Eliminate R1. \u2717\n  Remaining strategies: P1=['R2', 'R3', 'R4'], P2=['C1', 'C3', 'C4']\n\nStep 3: Consider Player 1's strategies.\n  Compare R2 vs R3: for every column strategy Player 2 might play,\n  R3 gives Player 1 a strictly higher payoff than R2.\n  Therefore R2 is strictly dominated by R3. Eliminate R2. \u2717\n  Remaining strategies: P1=['R3', 'R4'], P2=['C1', 'C3', 'C4']\n\nStep 4: Consider Player 2's strategies.\n  Compare C1 vs C3: for every row strategy Player 1 might play,\n  C3 gives Player 2 a strictly higher payoff than C1.\n  Therefore C1 is strictly dominated by C3. Eliminate C1. \u2717\n  Remaining strategies: P1=['R3', 'R4'], P2=['C3', 'C4']\n\nStep 5: Consider Player 1's strategies.\n  Compare R3 vs R4: for every column strategy Player 2 might play,\n  R4 gives Player 1 a strictly higher payoff than R3.\n  Therefore R3 is strictly dominated by R4. Eliminate R3. \u2717\n  Remaining strategies: P1=['R4'], P2=['C3', 'C4']\n\nStep 6: Consider Player 2's strategies.\n  Compare C4 vs C3: for every row strategy Player 1 might play,\n  C3 gives Player 2 a strictly higher payoff than C4.\n  Therefore C4 is strictly dominated by C3. Eliminate C4. \u2717\n  Remaining strategies: P1=['R4'], P2=['C3']\n\nAfter 6 elimination steps, the unique surviving strategy profile is (R4, C3).\nThe resulting payoffs are (5, 3). \u2713",
    "answer": "(R4, C3)",
    "game_type": "simultaneous",
    "players": 2,
    "tags": [
      "4x4",
      "dominated_strategy",
      "iesds",
      "normal_form"
    ]
  },
  {
    "id": "gt-nxm-4x4-iesds-0025",
    "category": "normal_form_4x4",
    "subcategory": "4x4_iesds",
    "difficulty": "hard",
    "problem": "Use iterated dominance to simplify the following game:\n\nPlayer 1 \\ Player 2 | C1 | C2 | C3 | C4\n--- | --- | --- | --- | ---\nR1 | (1,0) | (1,-3) | (4,-1) | (-4,-4)\nR2 | (1,-2) | (-3,5) | (2,-3) | (2,-4)\nR3 | (3,0) | (5,5) | (-2,-3) | (4,-2)\nR4 | (-3,0) | (3,-1) | (3,-2) | (-4,-2)\n\nEliminate strictly dominated strategies step by step.",
    "solution": "Applying Iterated Elimination of Strictly Dominated Strategies (IESDS):\n\nStep 1: Consider Player 2's strategies.\n  Compare C3 vs C1: for every row strategy Player 1 might play,\n  C1 gives Player 2 a strictly higher payoff than C3.\n  Therefore C3 is strictly dominated by C1. Eliminate C3. \u2717\n  Remaining strategies: P1=['R1', 'R2', 'R3', 'R4'], P2=['C1', 'C2', 'C4']\n\nStep 2: Consider Player 1's strategies.\n  Compare R1 vs R3: for every column strategy Player 2 might play,\n  R3 gives Player 1 a strictly higher payoff than R1.\n  Therefore R1 is strictly dominated by R3. Eliminate R1. \u2717\n  Remaining strategies: P1=['R2', 'R3', 'R4'], P2=['C1', 'C2', 'C4']\n\nStep 3: Consider Player 1's strategies.\n  Compare R2 vs R3: for every column strategy Player 2 might play,\n  R3 gives Player 1 a strictly higher payoff than R2.\n  Therefore R2 is strictly dominated by R3. Eliminate R2. \u2717\n  Remaining strategies: P1=['R3', 'R4'], P2=['C1', 'C2', 'C4']\n\nStep 4: Consider Player 1's strategies.\n  Compare R4 vs R3: for every column strategy Player 2 might play,\n  R3 gives Player 1 a strictly higher payoff than R4.\n  Therefore R4 is strictly dominated by R3. Eliminate R4. \u2717\n  Remaining strategies: P1=['R3'], P2=['C1', 'C2', 'C4']\n\nStep 5: Consider Player 2's strategies.\n  Compare C1 vs C2: for every row strategy Player 1 might play,\n  C2 gives Player 2 a strictly higher payoff than C1.\n  Therefore C1 is strictly dominated by C2. Eliminate C1. \u2717\n  Remaining strategies: P1=['R3'], P2=['C2', 'C4']\n\nStep 6: Consider Player 2's strategies.\n  Compare C4 vs C2: for every row strategy Player 1 might play,\n  C2 gives Player 2 a strictly higher payoff than C4.\n  Therefore C4 is strictly dominated by C2. Eliminate C4. \u2717\n  Remaining strategies: P1=['R3'], P2=['C2']\n\nAfter 6 elimination steps, the unique surviving strategy profile is (R3, C2).\nThe resulting payoffs are (5, 5). \u2713",
    "answer": "(R3, C2)",
    "game_type": "simultaneous",
    "players": 2,
    "tags": [
      "4x4",
      "dominated_strategy",
      "iesds",
      "normal_form"
    ]
  },
  {
    "id": "gt-nxm-4x4-iesds-0017",
    "category": "normal_form_4x4",
    "subcategory": "4x4_iesds",
    "difficulty": "hard",
    "problem": "Use iterated dominance to simplify the following game:\n\nPlayer 1 \\ Player 2 | C1 | C2 | C3 | C4\n--- | --- | --- | --- | ---\nR1 | (-5,5) | (3,4) | (-1,3) | (1,-3)\nR2 | (0,-3) | (-2,0) | (-5,-3) | (-2,3)\nR3 | (4,5) | (4,2) | (2,-2) | (5,0)\nR4 | (-5,2) | (-4,-2) | (2,-1) | (-4,1)\n\nEliminate strictly dominated strategies step by step.",
    "solution": "Applying Iterated Elimination of Strictly Dominated Strategies (IESDS):\n\nStep 1: Consider Player 1's strategies.\n  Compare R1 vs R3: for every column strategy Player 2 might play,\n  R3 gives Player 1 a strictly higher payoff than R1.\n  Therefore R1 is strictly dominated by R3. Eliminate R1. \u2717\n  Remaining strategies: P1=['R2', 'R3', 'R4'], P2=['C1', 'C2', 'C3', 'C4']\n\nStep 2: Consider Player 1's strategies.\n  Compare R2 vs R3: for every column strategy Player 2 might play,\n  R3 gives Player 1 a strictly higher payoff than R2.\n  Therefore R2 is strictly dominated by R3. Eliminate R2. \u2717\n  Remaining strategies: P1=['R3', 'R4'], P2=['C1', 'C2', 'C3', 'C4']\n\nStep 3: Consider Player 2's strategies.\n  Compare C2 vs C1: for every row strategy Player 1 might play,\n  C1 gives Player 2 a strictly higher payoff than C2.\n  Therefore C2 is strictly dominated by C1. Eliminate C2. \u2717\n  Remaining strategies: P1=['R3', 'R4'], P2=['C1', 'C3', 'C4']\n\nStep 4: Consider Player 2's strategies.\n  Compare C3 vs C1: for every row strategy Player 1 might play,\n  C1 gives Player 2 a strictly higher payoff than C3.\n  Therefore C3 is strictly dominated by C1. Eliminate C3. \u2717\n  Remaining strategies: P1=['R3', 'R4'], P2=['C1', 'C4']\n\nStep 5: Consider Player 1's strategies.\n  Compare R4 vs R3: for every column strategy Player 2 might play,\n  R3 gives Player 1 a strictly higher payoff than R4.\n  Therefore R4 is strictly dominated by R3. Eliminate R4. \u2717\n  Remaining strategies: P1=['R3'], P2=['C1', 'C4']\n\nStep 6: Consider Player 2's strategies.\n  Compare C4 vs C1: for every row strategy Player 1 might play,\n  C1 gives Player 2 a strictly higher payoff than C4.\n  Therefore C4 is strictly dominated by C1. Eliminate C4. \u2717\n  Remaining strategies: P1=['R3'], P2=['C1']\n\nAfter 6 elimination steps, the unique surviving strategy profile is (R3, C1).\nThe resulting payoffs are (4, 5). \u2713",
    "answer": "(R3, C1)",
    "game_type": "simultaneous",
    "players": 2,
    "tags": [
      "4x4",
      "dominated_strategy",
      "iesds",
      "normal_form"
    ]
  },
  {
    "id": "gt-nxm-4x4-pure-0019",
    "category": "normal_form_4x4",
    "subcategory": "4x4_pure_ne",
    "difficulty": "hard",
    "problem": "Consider the following 4x4 strategic form game:\n\nPlayer 1 \\ Player 2 | C1 | C2 | C3 | C4\n--- | --- | --- | --- | ---\nR1 | (3,4) | (3,5) | (-3,-5) | (-2,-2)\nR2 | (-6,-1) | (7,4) | (7,-3) | (-2,-3)\nR3 | (-5,-6) | (-1,-6) | (4,4) | (4,-7)\nR4 | (8,7) | (8,-3) | (-4,-1) | (-3,-1)\n\nFind all Nash Equilibria (pure and mixed).",
    "solution": "This game has 1 pure strategy Nash Equilibrium.\n\nTo verify (R4, C1) is a Nash Equilibrium, we check each player's incentive to deviate:\n- Player 1 currently plays R4, getting payoff 8. If Player 1 switches to R1 (while Player 2 stays at C1), payoff drops to 3. Player 1 has no incentive to deviate. \u2713\n- Player 1 currently plays R4, getting payoff 8. If Player 1 switches to R2 (while Player 2 stays at C1), payoff drops to -6. Player 1 has no incentive to deviate. \u2713\n- Player 1 currently plays R4, getting payoff 8. If Player 1 switches to R3 (while Player 2 stays at C1), payoff drops to -5. Player 1 has no incentive to deviate. \u2713\n- Player 2 currently plays C1, getting payoff 7. If Player 2 switches to C2 (while Player 1 stays at R4), payoff drops to -3. Player 2 has no incentive to deviate. \u2713\n- Player 2 currently plays C1, getting payoff 7. If Player 2 switches to C3 (while Player 1 stays at R4), payoff drops to -1. Player 2 has no incentive to deviate. \u2713\n- Player 2 currently plays C1, getting payoff 7. If Player 2 switches to C4 (while Player 1 stays at R4), payoff drops to -1. Player 2 has no incentive to deviate. \u2713\nTherefore (R4, C1) is a Nash Equilibrium with payoffs (8,7).\n\nThere are no additional mixed strategy equilibria beyond the pure strategy equilibrium identified above.",
    "answer": "[(R4, C1)]",
    "game_type": "simultaneous",
    "players": 2,
    "tags": [
      "4x4",
      "nash_equilibrium",
      "normal_form",
      "pure_strategy"
    ]
  },
  {
    "id": "gt-nxm-4x4-iesds-0007",
    "category": "normal_form_4x4",
    "subcategory": "4x4_iesds",
    "difficulty": "hard",
    "problem": "In the game below, apply IESDS to find the solution:\n\nPlayer 1 \\ Player 2 | C1 | C2 | C3 | C4\n--- | --- | --- | --- | ---\nR1 | (-1,5) | (3,-1) | (-1,-3) | (1,4)\nR2 | (-5,-4) | (3,-5) | (-5,-4) | (-2,-2)\nR3 | (1,0) | (5,-1) | (3,-2) | (-5,2)\nR4 | (5,1) | (0,-2) | (5,4) | (4,-1)",
    "solution": "Applying Iterated Elimination of Strictly Dominated Strategies (IESDS):\n\nStep 1: Consider Player 2's strategies.\n  Compare C2 vs C1: for every row strategy Player 1 might play,\n  C1 gives Player 2 a strictly higher payoff than C2.\n  Therefore C2 is strictly dominated by C1. Eliminate C2. \u2717\n  Remaining strategies: P1=['R1', 'R2', 'R3', 'R4'], P2=['C1', 'C3', 'C4']\n\nStep 2: Consider Player 1's strategies.\n  Compare R1 vs R4: for every column strategy Player 2 might play,\n  R4 gives Player 1 a strictly higher payoff than R1.\n  Therefore R1 is strictly dominated by R4. Eliminate R1. \u2717\n  Remaining strategies: P1=['R2', 'R3', 'R4'], P2=['C1', 'C3', 'C4']\n\nStep 3: Consider Player 1's strategies.\n  Compare R2 vs R4: for every column strategy Player 2 might play,\n  R4 gives Player 1 a strictly higher payoff than R2.\n  Therefore R2 is strictly dominated by R4. Eliminate R2. \u2717\n  Remaining strategies: P1=['R3', 'R4'], P2=['C1', 'C3', 'C4']\n\nStep 4: Consider Player 1's strategies.\n  Compare R3 vs R4: for every column strategy Player 2 might play,\n  R4 gives Player 1 a strictly higher payoff than R3.\n  Therefore R3 is strictly dominated by R4. Eliminate R3. \u2717\n  Remaining strategies: P1=['R4'], P2=['C1', 'C3', 'C4']\n\nStep 5: Consider Player 2's strategies.\n  Compare C1 vs C3: for every row strategy Player 1 might play,\n  C3 gives Player 2 a strictly higher payoff than C1.\n  Therefore C1 is strictly dominated by C3. Eliminate C1. \u2717\n  Remaining strategies: P1=['R4'], P2=['C3', 'C4']\n\nStep 6: Consider Player 2's strategies.\n  Compare C4 vs C3: for every row strategy Player 1 might play,\n  C3 gives Player 2 a strictly higher payoff than C4.\n  Therefore C4 is strictly dominated by C3. Eliminate C4. \u2717\n  Remaining strategies: P1=['R4'], P2=['C3']\n\nAfter 6 elimination steps, the unique surviving strategy profile is (R4, C3).\nThe resulting payoffs are (5, 4). \u2713",
    "answer": "(R4, C3)",
    "game_type": "simultaneous",
    "players": 2,
    "tags": [
      "4x4",
      "dominated_strategy",
      "iesds",
      "normal_form"
    ]
  },
  {
    "id": "gt-nxm-4x4-iesds-0014",
    "category": "normal_form_4x4",
    "subcategory": "4x4_iesds",
    "difficulty": "hard",
    "problem": "In the game below, apply IESDS to find the solution:\n\nPlayer 1 \\ Player 2 | C1 | C2 | C3 | C4\n--- | --- | --- | --- | ---\nR1 | (2,0) | (-5,0) | (-3,-3) | (-5,-4)\nR2 | (4,3) | (3,2) | (1,-2) | (2,0)\nR3 | (3,-3) | (-3,5) | (4,2) | (5,-3)\nR4 | (-4,-4) | (0,1) | (-1,-3) | (-2,3)",
    "solution": "Applying Iterated Elimination of Strictly Dominated Strategies (IESDS):\n\nStep 1: Consider Player 1's strategies.\n  Compare R1 vs R2: for every column strategy Player 2 might play,\n  R2 gives Player 1 a strictly higher payoff than R1.\n  Therefore R1 is strictly dominated by R2. Eliminate R1. \u2717\n  Remaining strategies: P1=['R2', 'R3', 'R4'], P2=['C1', 'C2', 'C3', 'C4']\n\nStep 2: Consider Player 1's strategies.\n  Compare R4 vs R2: for every column strategy Player 2 might play,\n  R2 gives Player 1 a strictly higher payoff than R4.\n  Therefore R4 is strictly dominated by R2. Eliminate R4. \u2717\n  Remaining strategies: P1=['R2', 'R3'], P2=['C1', 'C2', 'C3', 'C4']\n\nStep 3: Consider Player 2's strategies.\n  Compare C3 vs C2: for every row strategy Player 1 might play,\n  C2 gives Player 2 a strictly higher payoff than C3.\n  Therefore C3 is strictly dominated by C2. Eliminate C3. \u2717\n  Remaining strategies: P1=['R2', 'R3'], P2=['C1', 'C2', 'C4']\n\nStep 4: Consider Player 2's strategies.\n  Compare C4 vs C2: for every row strategy Player 1 might play,\n  C2 gives Player 2 a strictly higher payoff than C4.\n  Therefore C4 is strictly dominated by C2. Eliminate C4. \u2717\n  Remaining strategies: P1=['R2', 'R3'], P2=['C1', 'C2']\n\nStep 5: Consider Player 1's strategies.\n  Compare R3 vs R2: for every column strategy Player 2 might play,\n  R2 gives Player 1 a strictly higher payoff than R3.\n  Therefore R3 is strictly dominated by R2. Eliminate R3. \u2717\n  Remaining strategies: P1=['R2'], P2=['C1', 'C2']\n\nStep 6: Consider Player 2's strategies.\n  Compare C2 vs C1: for every row strategy Player 1 might play,\n  C1 gives Player 2 a strictly higher payoff than C2.\n  Therefore C2 is strictly dominated by C1. Eliminate C2. \u2717\n  Remaining strategies: P1=['R2'], P2=['C1']\n\nAfter 6 elimination steps, the unique surviving strategy profile is (R2, C1).\nThe resulting payoffs are (4, 3). \u2713",
    "answer": "(R2, C1)",
    "game_type": "simultaneous",
    "players": 2,
    "tags": [
      "4x4",
      "dominated_strategy",
      "iesds",
      "normal_form"
    ]
  },
  {
    "id": "gt-seq-ultim-0010",
    "category": "sequential_game",
    "subcategory": "ultimatum_bargaining",
    "difficulty": "easy",
    "problem": "Analyze this ultimatum bargaining game using backward induction:\n\nA Proposer and a Responder divide $12.\nThe Proposer offers the Responder an amount from: {$0, $1, $2, $3, $4, $5, $6, $7, $8, $9, $10, $11, $12}.\nThe Proposer keeps the remainder.\nThe Responder can Accept or Reject.\n- If Accepted: Proposer gets $12 minus offer, Responder gets the offer.\n- If Rejected: both get $0.\n\nWhat offer does the proposer make in the SPE?",
    "solution": "Solving by backward induction:\n\n**Stage 2 (Responder's decision):**\n- For any offer x \u2265 $0: Accept gives Responder $x, Reject gives $0.\n- Responder accepts any offer \u2265 $0 (weakly dominant). \u2713\n\n**Stage 1 (Proposer's decision):**\n- Proposer knows Responder will accept any offer.\n- Proposer maximizes own payoff: $12 - offer.\n- Optimal offer: $0 (Proposer keeps $12). \u2713\n\n**Subgame Perfect Equilibrium:**\n- Proposer offers $0\n- Responder accepts (any offer \u2265 $0)\n- Payoffs: Proposer=$12, Responder=$0\n\n**Note:** The SPE prediction contrasts with experimental evidence where offers around 40-50% are common and low offers are frequently rejected, showing bounded rationality and fairness concerns.",
    "answer": "SPE: Offer=$0, Proposer gets $12, Responder gets $0",
    "game_type": "sequential",
    "players": 2,
    "tags": [
      "sequential",
      "backward_induction",
      "subgame_perfect",
      "ultimatum",
      "bargaining",
      "pure_strategy"
    ]
  },
  {
    "id": "gt-seq-ultim-0057",
    "category": "sequential_game",
    "subcategory": "ultimatum_bargaining",
    "difficulty": "easy",
    "problem": "In this bargaining game, the proposer makes a take-it-or-leave-it offer:\n\nA Proposer and a Responder divide $100.\nThe Proposer offers the Responder an amount from: {$0, $5, $10, $15, $20, $25, $30, $35, $40, $45, $50, $55, $60, $65, $70, $75, $80, $85, $90, $95, $100}.\nThe Proposer keeps the remainder.\nThe Responder can Accept or Reject.\n- If Accepted: Proposer gets $100 minus offer, Responder gets the offer.\n- If Rejected: both get $0.\n\nDetermine the SPE.",
    "solution": "Solving by backward induction:\n\n**Stage 2 (Responder's decision):**\n- For any offer x \u2265 $0: Accept gives Responder $x, Reject gives $0.\n- Responder accepts any offer \u2265 $0 (weakly dominant). \u2713\n\n**Stage 1 (Proposer's decision):**\n- Proposer knows Responder will accept any offer.\n- Proposer maximizes own payoff: $100 - offer.\n- Optimal offer: $0 (Proposer keeps $100). \u2713\n\n**Subgame Perfect Equilibrium:**\n- Proposer offers $0\n- Responder accepts (any offer \u2265 $0)\n- Payoffs: Proposer=$100, Responder=$0\n\n**Note:** The SPE prediction contrasts with experimental evidence where offers around 40-50% are common and low offers are frequently rejected, showing bounded rationality and fairness concerns.",
    "answer": "SPE: Offer=$0, Proposer gets $100, Responder gets $0",
    "game_type": "sequential",
    "players": 2,
    "tags": [
      "sequential",
      "backward_induction",
      "subgame_perfect",
      "ultimatum",
      "bargaining",
      "pure_strategy"
    ]
  },
  {
    "id": "gt-seq-ultim-0020",
    "category": "sequential_game",
    "subcategory": "ultimatum_bargaining",
    "difficulty": "easy",
    "problem": "Consider an ultimatum game:\n\nA Proposer and a Responder divide $8.\nThe Proposer offers the Responder an amount from: {$0, $1, $2, $3, $4, $5, $6, $7, $8}.\nThe Proposer keeps the remainder.\nThe Responder can Accept or Reject.\n- If Accepted: Proposer gets $8 minus offer, Responder gets the offer.\n- If Rejected: both get $0.\n\nFind the Subgame Perfect Equilibrium.",
    "solution": "Solving by backward induction:\n\n**Stage 2 (Responder's decision):**\n- For any offer x \u2265 $0: Accept gives Responder $x, Reject gives $0.\n- Responder accepts any offer \u2265 $0 (weakly dominant). \u2713\n\n**Stage 1 (Proposer's decision):**\n- Proposer knows Responder will accept any offer.\n- Proposer maximizes own payoff: $8 - offer.\n- Optimal offer: $0 (Proposer keeps $8). \u2713\n\n**Subgame Perfect Equilibrium:**\n- Proposer offers $0\n- Responder accepts (any offer \u2265 $0)\n- Payoffs: Proposer=$8, Responder=$0\n\n**Note:** The SPE prediction contrasts with experimental evidence where offers around 40-50% are common and low offers are frequently rejected, showing bounded rationality and fairness concerns.",
    "answer": "SPE: Offer=$0, Proposer gets $8, Responder gets $0",
    "game_type": "sequential",
    "players": 2,
    "tags": [
      "sequential",
      "backward_induction",
      "subgame_perfect",
      "ultimatum",
      "bargaining",
      "pure_strategy"
    ]
  },
  {
    "id": "gt-seq-simple-0045",
    "category": "sequential_game",
    "subcategory": "simple_2player",
    "difficulty": "medium",
    "problem": "Player 1 moves first in the following game, and Player 2 observes Player 1's choice before deciding:\n\nPlayer 1 chooses from: {A, B, C}\nIf Player 1 chooses A, Player 2 chooses from {X, Y, Z}:\n  (A, X) \u2192 payoffs (0, 5) (Player 1, Player 2)\n  (A, Y) \u2192 payoffs (8, 10) (Player 1, Player 2)\n  (A, Z) \u2192 payoffs (4, 7) (Player 1, Player 2)\nIf Player 1 chooses B, Player 2 chooses from {X, Y, Z}:\n  (B, X) \u2192 payoffs (-4, 9) (Player 1, Player 2)\n  (B, Y) \u2192 payoffs (-3, 5) (Player 1, Player 2)\n  (B, Z) \u2192 payoffs (3, 5) (Player 1, Player 2)\nIf Player 1 chooses C, Player 2 chooses from {X, Y}:\n  (C, X) \u2192 payoffs (-2, 7) (Player 1, Player 2)\n  (C, Y) \u2192 payoffs (-5, 9) (Player 1, Player 2)\n\nSolve for the Subgame Perfect Equilibrium.",
    "solution": "Solving by backward induction:\n\n**Stage 2 (Player 2's optimal responses):**\n- If Player 1 chose A: Player 2 compares payoffs [X\u21925, Y\u219210, Z\u21927]. Best response: Y (payoff 10). \u2713\n- If Player 1 chose B: Player 2 compares payoffs [X\u21929, Y\u21925, Z\u21925]. Best response: X (payoff 9). \u2713\n- If Player 1 chose C: Player 2 compares payoffs [X\u21927, Y\u21929]. Best response: Y (payoff 9). \u2713\n\n**Stage 1 (Player 1 anticipates Player 2's responses):**\n- A \u2192 P2 plays Y \u2192 payoffs (8, 10). P1 gets 8. \u2713 OPTIMAL\n- B \u2192 P2 plays X \u2192 payoffs (-4, 9). P1 gets -4. \n- C \u2192 P2 plays Y \u2192 payoffs (-5, 9). P1 gets -5. \n\n**Subgame Perfect Equilibrium:**\n- Player 1 plays A\n- Player 2's strategy: [Y after A, X after B, Y after C]\n- Equilibrium payoffs: (8, 10)",
    "answer": "SPE: P1=A, P2=[Y after A, X after B, Y after C], Payoffs=(8, 10)",
    "game_type": "sequential",
    "players": 2,
    "tags": [
      "sequential",
      "backward_induction",
      "subgame_perfect",
      "extensive_form",
      "3_actions",
      "pure_strategy"
    ]
  },
  {
    "id": "gt-seq-3stage-0074",
    "category": "sequential_game",
    "subcategory": "three_stage",
    "difficulty": "medium",
    "problem": "Analyze this multi-stage game:\n\nStage 1: Player 1 chooses from {A, B}\nStage 2: Player 2 observes Stage 1 and chooses from {X, Y}\nStage 3: Player 1 observes Stages 1-2 and chooses from {L, R}\n\nPayoffs (Player 1, Player 2):\n\n  (A, X, L) \u2192 (-5, -4)\n  (A, X, R) \u2192 (-2, 8)\n  (A, Y, L) \u2192 (9, 7)\n  (A, Y, R) \u2192 (-2, 3)\n  (B, X, L) \u2192 (10, -1)\n  (B, X, R) \u2192 (1, -5)\n  (B, Y, L) \u2192 (4, 8)\n  (B, Y, R) \u2192 (-2, 3)\n\nUse backward induction to determine the optimal strategy at each decision point.",
    "solution": "Solving by backward induction:\n\n**Stage 3 (Player 1's optimal responses):**\n- After (A,X): compares [L\u2192-5, R\u2192-2]. Best: R (payoff -2). \u2713\n- After (A,Y): compares [L\u21929, R\u2192-2]. Best: L (payoff 9). \u2713\n- After (B,X): compares [L\u219210, R\u21921]. Best: L (payoff 10). \u2713\n- After (B,Y): compares [L\u21924, R\u2192-2]. Best: L (payoff 4). \u2713\n\n**Stage 2 (Player 2's optimal responses):**\n- After (A): compares [X\u21928, Y\u21927]. Best: X (payoff 8). \u2713\n- After (B): compares [X\u2192-1, Y\u21928]. Best: Y (payoff 8). \u2713\n\n**Stage 1 (Player 1's choice):**\n- A \u2192 sequence (A,X,R) \u2192 payoffs (-2, 8). Gets -2. \n- B \u2192 sequence (B,Y,L) \u2192 payoffs (4, 8). Gets 4. \u2713 OPTIMAL\n\n**Subgame Perfect Equilibrium:**\n- Equilibrium path: (B, Y, L)\n- Equilibrium payoffs: (4, 8)",
    "answer": "SPE path: (B, Y, L), Payoffs: (4, 8)",
    "game_type": "sequential",
    "players": 2,
    "tags": [
      "sequential",
      "backward_induction",
      "subgame_perfect",
      "extensive_form",
      "three_stage",
      "2_player",
      "pure_strategy"
    ]
  },
  {
    "id": "gt-seq-3stage-0027",
    "category": "sequential_game",
    "subcategory": "three_stage",
    "difficulty": "medium",
    "problem": "In this game, three decisions are made sequentially:\n\nStage 1: Player 1 chooses from {A, B}\nStage 2: Player 2 observes Stage 1 and chooses from {X, Y}\nStage 3: Player 1 observes Stages 1-2 and chooses from {L, R}\n\nPayoffs (Player 1, Player 2):\n\n  (A, X, L) \u2192 (10, -2)\n  (A, X, R) \u2192 (4, 8)\n  (A, Y, L) \u2192 (-3, -2)\n  (A, Y, R) \u2192 (-1, 6)\n  (B, X, L) \u2192 (4, 5)\n  (B, X, R) \u2192 (9, 1)\n  (B, Y, L) \u2192 (10, 6)\n  (B, Y, R) \u2192 (10, -2)\n\nFind the SPE by working backwards from the final stage.",
    "solution": "Solving by backward induction:\n\n**Stage 3 (Player 1's optimal responses):**\n- After (A,X): compares [L\u219210, R\u21924]. Best: L (payoff 10). \u2713\n- After (A,Y): compares [L\u2192-3, R\u2192-1]. Best: R (payoff -1). \u2713\n- After (B,X): compares [L\u21924, R\u21929]. Best: R (payoff 9). \u2713\n- After (B,Y): compares [L\u219210, R\u219210]. Best: L (payoff 10). \u2713\n\n**Stage 2 (Player 2's optimal responses):**\n- After (A): compares [X\u2192-2, Y\u21926]. Best: Y (payoff 6). \u2713\n- After (B): compares [X\u21921, Y\u21926]. Best: Y (payoff 6). \u2713\n\n**Stage 1 (Player 1's choice):**\n- A \u2192 sequence (A,Y,R) \u2192 payoffs (-1, 6). Gets -1. \n- B \u2192 sequence (B,Y,L) \u2192 payoffs (10, 6). Gets 10. \u2713 OPTIMAL\n\n**Subgame Perfect Equilibrium:**\n- Equilibrium path: (B, Y, L)\n- Equilibrium payoffs: (10, 6)",
    "answer": "SPE path: (B, Y, L), Payoffs: (10, 6)",
    "game_type": "sequential",
    "players": 2,
    "tags": [
      "sequential",
      "backward_induction",
      "subgame_perfect",
      "extensive_form",
      "three_stage",
      "2_player",
      "pure_strategy"
    ]
  },
  {
    "id": "gt-seq-3stage-0059",
    "category": "sequential_game",
    "subcategory": "three_stage",
    "difficulty": "hard",
    "problem": "Consider the following 3-stage sequential game:\n\nStage 1: Player 1 chooses from {A, B}\nStage 2: Player 2 observes Stage 1 and chooses from {X, Y}\nStage 3: Player 3 observes Stages 1-2 and chooses from {L, R}\n\nPayoffs (Player 1, Player 2, Player 3):\n\n  (A, X, L) \u2192 (-5, 8, 0)\n  (A, X, R) \u2192 (8, 0, -4)\n  (A, Y, L) \u2192 (-2, 5, 1)\n  (A, Y, R) \u2192 (1, 8, 3)\n  (B, X, L) \u2192 (4, 4, 2)\n  (B, X, R) \u2192 (-2, -4, 7)\n  (B, Y, L) \u2192 (10, -1, -4)\n  (B, Y, R) \u2192 (6, -5, 8)\n\nSolve for the Subgame Perfect Equilibrium using backward induction.",
    "solution": "Solving by backward induction:\n\n**Stage 3 (Player 3's optimal responses):**\n- After (A,X): compares [L\u21920, R\u2192-4]. Best: L (payoff 0). \u2713\n- After (A,Y): compares [L\u21921, R\u21923]. Best: R (payoff 3). \u2713\n- After (B,X): compares [L\u21922, R\u21927]. Best: R (payoff 7). \u2713\n- After (B,Y): compares [L\u2192-4, R\u21928]. Best: R (payoff 8). \u2713\n\n**Stage 2 (Player 2's optimal responses):**\n- After (A): compares [X\u21928, Y\u21928]. Best: X (payoff 8). \u2713\n- After (B): compares [X\u2192-4, Y\u2192-5]. Best: X (payoff -4). \u2713\n\n**Stage 1 (Player 1's choice):**\n- A \u2192 sequence (A,X,L) \u2192 payoffs (-5, 8, 0). Gets -5. \n- B \u2192 sequence (B,X,R) \u2192 payoffs (-2, -4, 7). Gets -2. \u2713 OPTIMAL\n\n**Subgame Perfect Equilibrium:**\n- Equilibrium path: (B, X, R)\n- Equilibrium payoffs: (-2, -4, 7)",
    "answer": "SPE path: (B, X, R), Payoffs: (-2, -4, 7)",
    "game_type": "sequential",
    "players": 3,
    "tags": [
      "sequential",
      "backward_induction",
      "subgame_perfect",
      "extensive_form",
      "three_stage",
      "3_player",
      "pure_strategy"
    ]
  },
  {
    "id": "gt-seq-stack-0055",
    "category": "sequential_game",
    "subcategory": "stackelberg_competition",
    "difficulty": "hard",
    "problem": "Consider a Stackelberg duopoly:\n\nMarket demand: P = 24 - 2(q1 + q2)\nFirm 1 (Leader) marginal cost: c1 = 0\nFirm 2 (Follower) marginal cost: c2 = 0\nFirm 1 chooses quantity q1 first. Firm 2 observes q1 and then chooses q2.\n\nFind the Subgame Perfect Equilibrium quantities, prices, and profits.",
    "solution": "Solving by backward induction:\n\n**Stage 2 (Follower's best response):**\n- Firm 2 maximizes profit: \u03c02 = (P - c2) \u00d7 q2 = (24 - 2(q1 + q2) - 0) \u00d7 q2\n- FOC: 24 - 2\u00d7q1 - 4\u00d7q2 - 0 = 0\n- Best response: q2(q1) = (24 - 0 - 2\u00d7q1) / 4\n\n**Stage 1 (Leader's optimization):**\n- Firm 1 substitutes Firm 2's BR into its profit:\n- \u03c01 = (24/2 + 0/2 - 2\u00d7q1/2 - 0) \u00d7 q1\n- FOC: 24/2 + 0/2 - 2\u00d7q1 - 0 = 0\n- q1* = (24 + 0 - 0) / 4 = 6\n\n**Follower's quantity:**\n- q2* = (24 - 0 - 2\u00d76) / 4 = 3\n\n**Market outcome:**\n- Total output: Q = 6 + 3 = 9\n- Market price: P = 24 - 2\u00d79 = 6\n- Leader profit: \u03c01 = (6 - 0) \u00d7 6 = 36\n- Follower profit: \u03c02 = (6 - 0) \u00d7 3 = 18\n\n**Leader advantage:** Firm 1 produces more (6 vs 3) and earns higher profit (36 vs 18) due to first-mover advantage. \u2713",
    "answer": "q1*=6, q2*=3, P*=6, \u03c01=36, \u03c02=18",
    "game_type": "sequential",
    "players": 2,
    "tags": [
      "sequential",
      "backward_induction",
      "subgame_perfect",
      "stackelberg",
      "duopoly",
      "cournot",
      "industrial_organization",
      "pure_strategy"
    ]
  },
  {
    "id": "gt-seq-stack-0073",
    "category": "sequential_game",
    "subcategory": "stackelberg_competition",
    "difficulty": "hard",
    "problem": "In a Stackelberg competition, Firm 1 (leader) chooses quantity first, then Firm 2 (follower) observes and chooses:\n\nMarket demand: P = 36 - 3(q1 + q2)\nFirm 1 (Leader) marginal cost: c1 = 0\nFirm 2 (Follower) marginal cost: c2 = 0\nFirm 1 chooses quantity q1 first. Firm 2 observes q1 and then chooses q2.\n\nSolve for the SPE.",
    "solution": "Solving by backward induction:\n\n**Stage 2 (Follower's best response):**\n- Firm 2 maximizes profit: \u03c02 = (P - c2) \u00d7 q2 = (36 - 3(q1 + q2) - 0) \u00d7 q2\n- FOC: 36 - 3\u00d7q1 - 6\u00d7q2 - 0 = 0\n- Best response: q2(q1) = (36 - 0 - 3\u00d7q1) / 6\n\n**Stage 1 (Leader's optimization):**\n- Firm 1 substitutes Firm 2's BR into its profit:\n- \u03c01 = (36/2 + 0/2 - 3\u00d7q1/2 - 0) \u00d7 q1\n- FOC: 36/2 + 0/2 - 3\u00d7q1 - 0 = 0\n- q1* = (36 + 0 - 0) / 6 = 6\n\n**Follower's quantity:**\n- q2* = (36 - 0 - 3\u00d76) / 6 = 3\n\n**Market outcome:**\n- Total output: Q = 6 + 3 = 9\n- Market price: P = 36 - 3\u00d79 = 9\n- Leader profit: \u03c01 = (9 - 0) \u00d7 6 = 54\n- Follower profit: \u03c02 = (9 - 0) \u00d7 3 = 27\n\n**Leader advantage:** Firm 1 produces more (6 vs 3) and earns higher profit (54 vs 27) due to first-mover advantage. \u2713",
    "answer": "q1*=6, q2*=3, P*=9, \u03c01=54, \u03c02=27",
    "game_type": "sequential",
    "players": 2,
    "tags": [
      "sequential",
      "backward_induction",
      "subgame_perfect",
      "stackelberg",
      "duopoly",
      "cournot",
      "industrial_organization",
      "pure_strategy"
    ]
  },
  {
    "id": "gt-seq-ultim-0009",
    "category": "sequential_game",
    "subcategory": "ultimatum_bargaining",
    "difficulty": "easy",
    "problem": "In this bargaining game, the proposer makes a take-it-or-leave-it offer:\n\nA Proposer and a Responder divide $8.\nThe Proposer offers the Responder an amount from: {$0, $1, $2, $3, $4, $5, $6, $7, $8}.\nThe Proposer keeps the remainder.\nThe Responder can Accept or Reject.\n- If Accepted: Proposer gets $8 minus offer, Responder gets the offer.\n- If Rejected: both get $0.\n\nDetermine the SPE.",
    "solution": "Solving by backward induction:\n\n**Stage 2 (Responder's decision):**\n- For any offer x \u2265 $0: Accept gives Responder $x, Reject gives $0.\n- Responder accepts any offer \u2265 $0 (weakly dominant). \u2713\n\n**Stage 1 (Proposer's decision):**\n- Proposer knows Responder will accept any offer.\n- Proposer maximizes own payoff: $8 - offer.\n- Optimal offer: $0 (Proposer keeps $8). \u2713\n\n**Subgame Perfect Equilibrium:**\n- Proposer offers $0\n- Responder accepts (any offer \u2265 $0)\n- Payoffs: Proposer=$8, Responder=$0\n\n**Note:** The SPE prediction contrasts with experimental evidence where offers around 40-50% are common and low offers are frequently rejected, showing bounded rationality and fairness concerns.",
    "answer": "SPE: Offer=$0, Proposer gets $8, Responder gets $0",
    "game_type": "sequential",
    "players": 2,
    "tags": [
      "sequential",
      "backward_induction",
      "subgame_perfect",
      "ultimatum",
      "bargaining",
      "pure_strategy"
    ]
  },
  {
    "id": "gt-zs-2x2-saddle-0015",
    "category": "zero_sum",
    "subcategory": "saddle_point",
    "difficulty": "easy",
    "problem": "Analyze this zero-sum game for saddle points:\n\nPlayer 1 \\ Player 2 | Left | Right\n--- | --- | ---\nUp | -8 | -6\nDown | -4 | 6\n\nWhat is the optimal pure strategy for each player?",
    "solution": "Analyzing this zero-sum game for saddle points:\n\nStep 1: For each cell, check if it is both the minimum of its row and the maximum of its column:\n  (Up, Left): value=-8, row_min=-8, col_max=-4 \u2717\n  (Down, Left): value=-4, row_min=-4, col_max=-4 \u2190 Saddle point! \u2713\n  (Down, Right): value=6, row_min=-4, col_max=6 \u2717\n\nSaddle point(s) found: (Down, Left) with value -4.\nThe maximin value (Player 1) is -4 (strategy: Down).\nThe minimax value (Player 2) is -4 (strategy: Left).\nSince maximin = minimax = -4, the value of the game is -4. \u2713\nPlayer 1's optimal strategy: Down. Player 2's optimal strategy: Left.",
    "answer": "Saddle points: [(Down, Left)], Game value: -4",
    "game_type": "simultaneous",
    "players": 2,
    "tags": [
      "2x2",
      "minimax",
      "pure_strategy",
      "saddle_point",
      "zero_sum"
    ]
  },
  {
    "id": "gt-zs-2x2-value-0007",
    "category": "zero_sum",
    "subcategory": "game_value",
    "difficulty": "easy",
    "problem": "What is the value of this zero-sum game? Justify your answer:\n\nPlayer 1 \\ Player 2 | Left | Right\n--- | --- | ---\nUp | 3 | -5\nDown | 3 | 4",
    "solution": "Applying the minimax theorem to this zero-sum game:\n\nStep 1: Compute row minima (worst case for Player 1 per strategy):\n  Up: min[3, -5] = -5\n  Down: min[3, 4] = 3\n  Maximin value = max of row minima = 3 (Player 1 plays Down)\n\nStep 2: Compute column maxima (worst case for Player 2 per strategy):\n  Left: max[3, 3] = 3\n  Right: max[-5, 4] = 4\n  Minimax value = min of column maxima = 3 (Player 2 plays Left)\n\nStep 3: Maximin (3) = Minimax (3). A saddle point exists! \u2713\n  Saddle point(s): (Down, Left) with game value 3.\n  Player 1's optimal pure strategy: Down\n  Player 2's optimal pure strategy: Left",
    "answer": "Game value: 3.0",
    "game_type": "simultaneous",
    "players": 2,
    "tags": [
      "2x2",
      "game_value",
      "minimax",
      "pure_strategy",
      "zero_sum"
    ]
  },
  {
    "id": "gt-zs-2x2-minimax-0001",
    "category": "zero_sum",
    "subcategory": "minimax",
    "difficulty": "easy",
    "problem": "Consider the following zero-sum game (payoffs for Player 1):\n\nPlayer 1 \\ Player 2 | Left | Right\n--- | --- | ---\nUp | 8 | 2\nDown | -7 | 7\n\nCompute the maximin and minimax values. Does a saddle point exist?",
    "solution": "Applying the minimax theorem to this zero-sum game:\n\nStep 1: Compute row minima (worst case for Player 1 per strategy):\n  Up: min[8, 2] = 2\n  Down: min[-7, 7] = -7\n  Maximin value = max of row minima = 2 (Player 1 plays Up)\n\nStep 2: Compute column maxima (worst case for Player 2 per strategy):\n  Left: max[8, -7] = 8\n  Right: max[2, 7] = 7\n  Minimax value = min of column maxima = 7 (Player 2 plays Right)\n\nStep 3: Maximin (2) \u2260 Minimax (7). No saddle point exists.\n  Players must use mixed strategies to find the optimal solution.\n\nStep 4: Computing the optimal mixed strategy solution:\n  Player 1's optimal mixed strategy: Up with probability 0.7000, Down with probability 0.3000.\n  Player 2's optimal mixed strategy: Left with probability 0.2500, Right with probability 0.7500.\n  The value of the game is 3.5000. \u2713\n  At this equilibrium, each player's mixture makes the opponent indifferent among their active strategies.",
    "answer": "Maximin: 2, Minimax: 7, Saddle: No, Mixed solution: P1=[0.7, 0.3], P2=[0.25, 0.75], Value=3.5000",
    "game_type": "simultaneous",
    "players": 2,
    "tags": [
      "2x2",
      "minimax",
      "mixed_strategy",
      "zero_sum"
    ]
  },
  {
    "id": "gt-zs-3x4-value-0012",
    "category": "zero_sum",
    "subcategory": "game_value",
    "difficulty": "medium",
    "problem": "What is the value of this zero-sum game? Justify your answer:\n\nPlayer 1 \\ Player 2 | C1 | C2 | C3 | C4\n--- | --- | --- | --- | ---\nTop | -8 | 5 | -5 | 3\nMiddle | -3 | 2 | 3 | 1\nBottom | -2 | -2 | 2 | -5",
    "solution": "Applying the minimax theorem to this zero-sum game:\n\nStep 1: Compute row minima (worst case for Player 1 per strategy):\n  Top: min[-8, 5, -5, 3] = -8\n  Middle: min[-3, 2, 3, 1] = -3\n  Bottom: min[-2, -2, 2, -5] = -5\n  Maximin value = max of row minima = -3 (Player 1 plays Middle)\n\nStep 2: Compute column maxima (worst case for Player 2 per strategy):\n  C1: max[-8, -3, -2] = -2\n  C2: max[5, 2, -2] = 5\n  C3: max[-5, 3, 2] = 3\n  C4: max[3, 1, -5] = 3\n  Minimax value = min of column maxima = -2 (Player 2 plays C1)\n\nStep 3: Maximin (-3) \u2260 Minimax (-2). No saddle point exists.\n  Players must use mixed strategies to find the optimal solution.\n\nStep 4: Computing the optimal mixed strategy solution:\n  Player 1's optimal mixed strategy: Middle with probability 0.4286, Bottom with probability 0.5714.\n  Player 2's optimal mixed strategy: C1 with probability 0.8571, C4 with probability 0.1429.\n  The value of the game is -2.4286. \u2713\n  At this equilibrium, each player's mixture makes the opponent indifferent among their active strategies.",
    "answer": "Game value: -2.428571, P1=[0.0, 0.428571, 0.571429], P2=[0.857143, 0.0, 0.0, 0.142857]",
    "game_type": "simultaneous",
    "players": 2,
    "tags": [
      "3x4",
      "game_value",
      "minimax",
      "mixed_strategy",
      "zero_sum"
    ]
  },
  {
    "id": "gt-zs-3x4-value-0014",
    "category": "zero_sum",
    "subcategory": "game_value",
    "difficulty": "medium",
    "problem": "Find the value of the following zero-sum game:\n\nPlayer 1 \\ Player 2 | C1 | C2 | C3 | C4\n--- | --- | --- | --- | ---\nTop | 3 | -8 | -8 | 5\nMiddle | -5 | -4 | 2 | 4\nBottom | 7 | 4 | -7 | 7\n\nSpecify whether pure or mixed strategies are required.",
    "solution": "Applying the minimax theorem to this zero-sum game:\n\nStep 1: Compute row minima (worst case for Player 1 per strategy):\n  Top: min[3, -8, -8, 5] = -8\n  Middle: min[-5, -4, 2, 4] = -5\n  Bottom: min[7, 4, -7, 7] = -7\n  Maximin value = max of row minima = -5 (Player 1 plays Middle)\n\nStep 2: Compute column maxima (worst case for Player 2 per strategy):\n  C1: max[3, -5, 7] = 7\n  C2: max[-8, -4, 4] = 4\n  C3: max[-8, 2, -7] = 2\n  C4: max[5, 4, 7] = 7\n  Minimax value = min of column maxima = 2 (Player 2 plays C3)\n\nStep 3: Maximin (-5) \u2260 Minimax (2). No saddle point exists.\n  Players must use mixed strategies to find the optimal solution.\n\nStep 4: Computing the optimal mixed strategy solution:\n  Player 1's optimal mixed strategy: Middle with probability 0.6471, Bottom with probability 0.3529.\n  Player 2's optimal mixed strategy: C2 with probability 0.5294, C3 with probability 0.4706.\n  The value of the game is -1.1765. \u2713\n  At this equilibrium, each player's mixture makes the opponent indifferent among their active strategies.",
    "answer": "Game value: -1.176471, P1=[0.0, 0.647059, 0.352941], P2=[0.0, 0.529412, 0.470588, 0.0]",
    "game_type": "simultaneous",
    "players": 2,
    "tags": [
      "3x4",
      "game_value",
      "minimax",
      "mixed_strategy",
      "zero_sum"
    ]
  },
  {
    "id": "gt-zs-3x3-saddle-0015",
    "category": "zero_sum",
    "subcategory": "saddle_point",
    "difficulty": "medium",
    "problem": "In the zero-sum game below, identify any saddle points and compute the game value:\n\nPlayer 1 \\ Player 2 | Left | Center | Right\n--- | --- | --- | ---\nTop | 6 | 8 | 2\nMiddle | 8 | 4 | -8\nBottom | -7 | 0 | -6",
    "solution": "Analyzing this zero-sum game for saddle points:\n\nStep 1: For each cell, check if it is both the minimum of its row and the maximum of its column:\n  (Top, Center): value=8, row_min=2, col_max=8 \u2717\n  (Top, Right): value=2, row_min=2, col_max=2 \u2190 Saddle point! \u2713\n  (Middle, Left): value=8, row_min=-8, col_max=8 \u2717\n  (Middle, Right): value=-8, row_min=-8, col_max=2 \u2717\n  (Bottom, Left): value=-7, row_min=-7, col_max=8 \u2717\n\nSaddle point(s) found: (Top, Right) with value 2.\nThe maximin value (Player 1) is 2 (strategy: Top).\nThe minimax value (Player 2) is 2 (strategy: Right).\nSince maximin = minimax = 2, the value of the game is 2. \u2713\nPlayer 1's optimal strategy: Top. Player 2's optimal strategy: Right.",
    "answer": "Saddle points: [(Top, Right)], Game value: 2",
    "game_type": "simultaneous",
    "players": 2,
    "tags": [
      "3x3",
      "minimax",
      "pure_strategy",
      "saddle_point",
      "zero_sum"
    ]
  },
  {
    "id": "gt-zs-3x3-mixed-0015",
    "category": "zero_sum",
    "subcategory": "mixed_strategy_minimax",
    "difficulty": "hard",
    "problem": "The following zero-sum game has no saddle point:\n\nPlayer 1 \\ Player 2 | Left | Center | Right\n--- | --- | --- | ---\nTop | -3 | 2 | -4\nMiddle | 6 | 0 | 5\nBottom | -4 | 0 | 0\n\nFind the optimal mixed strategy for each player and the value of the game.",
    "solution": "Applying the minimax theorem to this zero-sum game:\n\nStep 1: Compute row minima (worst case for Player 1 per strategy):\n  Top: min[-3, 2, -4] = -4\n  Middle: min[6, 0, 5] = 0\n  Bottom: min[-4, 0, 0] = -4\n  Maximin value = max of row minima = 0 (Player 1 plays Middle)\n\nStep 2: Compute column maxima (worst case for Player 2 per strategy):\n  Left: max[-3, 6, -4] = 6\n  Center: max[2, 0, 0] = 2\n  Right: max[-4, 5, 0] = 5\n  Minimax value = min of column maxima = 2 (Player 2 plays Center)\n\nStep 3: Maximin (0) \u2260 Minimax (2). No saddle point exists.\n  Players must use mixed strategies to find the optimal solution.\n\nStep 4: Computing the optimal mixed strategy solution:\n  Player 1's optimal mixed strategy: Top with probability 0.4545, Middle with probability 0.5455.\n  Player 2's optimal mixed strategy: Center with probability 0.8182, Right with probability 0.1818.\n  The value of the game is 0.9091. \u2713\n  At this equilibrium, each player's mixture makes the opponent indifferent among their active strategies.",
    "answer": "P1=[0.4545, 0.5455, 0.0000], P2=[0.0000, 0.8182, 0.1818], Value=0.9091",
    "game_type": "simultaneous",
    "players": 2,
    "tags": [
      "3x3",
      "minimax",
      "mixed_strategy",
      "zero_sum"
    ]
  },
  {
    "id": "gt-zs-3x3-mixed-0003",
    "category": "zero_sum",
    "subcategory": "mixed_strategy_minimax",
    "difficulty": "hard",
    "problem": "Solve the following zero-sum game using mixed strategies:\n\nPlayer 1 \\ Player 2 | Left | Center | Right\n--- | --- | --- | ---\nTop | -2 | -3 | 6\nMiddle | 2 | -4 | 0\nBottom | 4 | 1 | -1\n\nDetermine the optimal randomization and the expected game value.",
    "solution": "Applying the minimax theorem to this zero-sum game:\n\nStep 1: Compute row minima (worst case for Player 1 per strategy):\n  Top: min[-2, -3, 6] = -3\n  Middle: min[2, -4, 0] = -4\n  Bottom: min[4, 1, -1] = -1\n  Maximin value = max of row minima = -1 (Player 1 plays Bottom)\n\nStep 2: Compute column maxima (worst case for Player 2 per strategy):\n  Left: max[-2, 2, 4] = 4\n  Center: max[-3, -4, 1] = 1\n  Right: max[6, 0, -1] = 6\n  Minimax value = min of column maxima = 1 (Player 2 plays Center)\n\nStep 3: Maximin (-1) \u2260 Minimax (1). No saddle point exists.\n  Players must use mixed strategies to find the optimal solution.\n\nStep 4: Computing the optimal mixed strategy solution:\n  Player 1's optimal mixed strategy: Top with probability 0.1818, Bottom with probability 0.8182.\n  Player 2's optimal mixed strategy: Center with probability 0.6364, Right with probability 0.3636.\n  The value of the game is 0.2727. \u2713\n  At this equilibrium, each player's mixture makes the opponent indifferent among their active strategies.",
    "answer": "P1=[0.1818, 0.0000, 0.8182], P2=[0.0000, 0.6364, 0.3636], Value=0.2727",
    "game_type": "simultaneous",
    "players": 2,
    "tags": [
      "3x3",
      "minimax",
      "mixed_strategy",
      "zero_sum"
    ]
  },
  {
    "id": "gt-zs-3x3-mixed-0021",
    "category": "zero_sum",
    "subcategory": "mixed_strategy_minimax",
    "difficulty": "hard",
    "problem": "Solve the following zero-sum game using mixed strategies:\n\nPlayer 1 \\ Player 2 | Left | Center | Right\n--- | --- | --- | ---\nTop | 3 | 5 | 1\nMiddle | -2 | 3 | 5\nBottom | -2 | -2 | -6\n\nDetermine the optimal randomization and the expected game value.",
    "solution": "Applying the minimax theorem to this zero-sum game:\n\nStep 1: Compute row minima (worst case for Player 1 per strategy):\n  Top: min[3, 5, 1] = 1\n  Middle: min[-2, 3, 5] = -2\n  Bottom: min[-2, -2, -6] = -6\n  Maximin value = max of row minima = 1 (Player 1 plays Top)\n\nStep 2: Compute column maxima (worst case for Player 2 per strategy):\n  Left: max[3, -2, -2] = 3\n  Center: max[5, 3, -2] = 5\n  Right: max[1, 5, -6] = 5\n  Minimax value = min of column maxima = 3 (Player 2 plays Left)\n\nStep 3: Maximin (1) \u2260 Minimax (3). No saddle point exists.\n  Players must use mixed strategies to find the optimal solution.\n\nStep 4: Computing the optimal mixed strategy solution:\n  Player 1's optimal mixed strategy: Top with probability 0.7778, Middle with probability 0.2222.\n  Player 2's optimal mixed strategy: Left with probability 0.4444, Right with probability 0.5556.\n  The value of the game is 1.8889. \u2713\n  At this equilibrium, each player's mixture makes the opponent indifferent among their active strategies.",
    "answer": "P1=[0.7778, 0.2222, 0.0000], P2=[0.4444, 0.0000, 0.5556], Value=1.8889",
    "game_type": "simultaneous",
    "players": 2,
    "tags": [
      "3x3",
      "minimax",
      "mixed_strategy",
      "zero_sum"
    ]
  },
  {
    "id": "gt-zs-3x3-mixed-0010",
    "category": "zero_sum",
    "subcategory": "mixed_strategy_minimax",
    "difficulty": "hard",
    "problem": "The following zero-sum game has no saddle point:\n\nPlayer 1 \\ Player 2 | Left | Center | Right\n--- | --- | --- | ---\nTop | -4 | 3 | 0\nMiddle | 3 | -4 | -3\nBottom | -5 | -4 | 2\n\nFind the optimal mixed strategy for each player and the value of the game.",
    "solution": "Applying the minimax theorem to this zero-sum game:\n\nStep 1: Compute row minima (worst case for Player 1 per strategy):\n  Top: min[-4, 3, 0] = -4\n  Middle: min[3, -4, -3] = -4\n  Bottom: min[-5, -4, 2] = -5\n  Maximin value = max of row minima = -4 (Player 1 plays Top)\n\nStep 2: Compute column maxima (worst case for Player 2 per strategy):\n  Left: max[-4, 3, -5] = 3\n  Center: max[3, -4, -4] = 3\n  Right: max[0, -3, 2] = 2\n  Minimax value = min of column maxima = 2 (Player 2 plays Right)\n\nStep 3: Maximin (-4) \u2260 Minimax (2). No saddle point exists.\n  Players must use mixed strategies to find the optimal solution.\n\nStep 4: Computing the optimal mixed strategy solution:\n  Player 1's optimal mixed strategy: Top with probability 0.4216, Middle with probability 0.4412, Bottom with probability 0.1373.\n  Player 2's optimal mixed strategy: Left with probability 0.3431, Center with probability 0.1078, Right with probability 0.5490.\n  The value of the game is -1.0490. \u2713\n  At this equilibrium, each player's mixture makes the opponent indifferent among their active strategies.",
    "answer": "P1=[0.4216, 0.4412, 0.1373], P2=[0.3431, 0.1078, 0.5490], Value=-1.0490",
    "game_type": "simultaneous",
    "players": 2,
    "tags": [
      "3x3",
      "minimax",
      "mixed_strategy",
      "zero_sum"
    ]
  }
]