| --- |
| license: cc-by-4.0 |
| pretty_name: Nash Equilibria of 2x2 Normal-Form Games |
| tags: |
| - game-theory |
| - nash-equilibrium |
| - normal-form-games |
| - combinatorics |
| - synthetic |
| size_categories: |
| - 100M<n<1B |
| configs: |
| - config_name: default |
| data_files: |
| - split: payoffs_0_10 |
| path: matrices_2x2_0_to_10.parquet |
| - split: payoffs_0_5 |
| path: matrices_2x2_0_to_5.parquet |
| - split: payoffs_0_3 |
| path: matrices_2x2_0_to_3.parquet |
| - split: payoffs_0_2 |
| path: matrices_2x2_0_to_2.parquet |
| --- |
| |
| # Nash Equilibria of 2×2 Normal-Form Games |
|
|
| An **exhaustive enumeration** of every two-player 2×2 normal-form (bimatrix) game with |
| non-negative integer payoffs in a fixed range, each annotated with its pure- and |
| mixed-strategy Nash equilibria and a set of game-theoretic classifications. |
|
|
| Because the payoff space is enumerated completely, the largest split contains **all |
| `11^8 = 214,358,881`** games with payoffs in `0..10` — not a sample. |
|
|
| ## Splits |
|
|
| Each split is the complete enumeration for a payoff range. Payoffs `0..k` is a strict |
| subset of payoffs `0..10` (identical rows), so the smaller splits are convenience |
| subsets of `payoffs_0_10`. |
|
|
| | split | payoff range | games (rows) | |
| |---|---|--:| |
| | `payoffs_0_10` | 0–10 | 214,358,881 | |
| | `payoffs_0_5` | 0–5 | 1,679,616 | |
| | `payoffs_0_3` | 0–3 | 65,536 | |
| | `payoffs_0_2` | 0–2 | 6,561 | |
|
|
| ```python |
| from datasets import load_dataset |
| ds = load_dataset("AlexDunstan/nash-equilibria-matrices", split="payoffs_0_10") |
| ``` |
|
|
| ## Game encoding |
|
|
| A 2×2 game has two players (p1, p2) choosing a row / column. Each of the 4 cells holds a |
| payoff pair. The 8 payoff columns are named `r{row}c{col}_p{player}` (row-major, 0-indexed): |
|
|
| | | col 0 | col 1 | |
| |---|---|---| |
| | **row 0** | `r0c0_p1`, `r0c0_p2` | `r0c1_p1`, `r0c1_p2` | |
| | **row 1** | `r1c0_p1`, `r1c0_p2` | `r1c1_p1`, `r1c1_p2` | |
|
|
| ## Columns (32) |
|
|
| **Payoffs (8)** — `r0c0_p1 … r1c1_p2`: `int8`, the integer payoff to each player in each cell. |
|
|
| **Equilibria** |
| - `num_equilibria` (`int8`): number of pure-strategy Nash equilibria. |
| - `equilibrium_positions` (`string`): list of `(row, col)` pure-NE cells, e.g. `"[(0, 0), (1, 1)]"`. |
| - `category` (`string`): `Solved` if ≥1 pure NE, else `Unsolved`. |
|
|
| **Structural flags** (`bool`) |
| - `p1_has_dominant`, `p2_has_dominant`, `both_dominant`: dominant-strategy existence. |
| - `is_zero_sum`, `is_symmetric`: structural properties of the payoff matrix. |
| - `has_pareto_dom_ne`, `all_ne_pareto_eff`: Pareto properties of the equilibria. |
|
|
| **Welfare** |
| - `max_welfare` (`int16`): maximum total payoff (`p1+p2`) over all cells. |
| - `ne_welfare` (`string`): total welfare at each NE, e.g. `"[0, 2]"`. |
| - `welfare_loss` (`int16`): efficiency gap between `max_welfare` and the equilibria. |
|
|
| **Mixed-strategy equilibrium** (populated for 2×2 games with no pure NE; `null`/empty otherwise) |
| - `mixed_exists` (`bool`). |
| - `mixed_p`, `mixed_q` (`double`): equilibrium mixing probabilities. |
| - `mixed_payoff_p1`, `mixed_payoff_p2` (`double`): expected payoffs under the mixed NE. |
|
|
| **Payoff asymmetry at equilibria** |
| - `ne_p1_payoffs`, `ne_p2_payoffs`, `ne_payoff_diffs` (`string`): per-NE payoff lists and differences. |
| - `ne_has_equal_payoffs` (`bool`). |
| - `ne_mean_abs_diff` (`double`): mean absolute payoff difference across the equilibria. |
|
|
| **Classification** |
| - `game_type` (`string`): a categorical label. Distribution over `payoffs_0_10`: |
|
|
| | game_type | rows | |
| |---|--:| |
| | Harmony | 56,150,160 | |
| | Dominant (P1 only) | 51,967,234 | |
| | Dominant (P2 only) | 51,967,234 | |
| | Other | 18,295,200 | |
| | No Equilibrium | 18,283,848 | |
| | Deadlock | 11,809,512 | |
| | Prisoner's Dilemma | 5,814,336 | |
| | Zero-Sum | 65,307 | |
| | Coordination | 6,050 | |
| |
| > **List columns are stored as strings** (Python `repr` of a list), for exact |
| > round-trip fidelity. Parse them with `ast.literal_eval`: |
| > ```python |
| > import ast |
| > positions = ast.literal_eval(row["equilibrium_positions"]) |
| > ``` |
|
|
| ## Provenance |
|
|
| Games were generated by exhaustive Cartesian enumeration of the payoff space, then |
| enriched with equilibrium detection and classification computed deterministically from |
| the payoff matrix (no sampling, no randomness). Payoffs are compact integer types and |
| repetitive categorical/list columns are dictionary-encoded, so the full 214M-row split is |
| ~172 MB of Parquet. |
|
|
| ## License |
|
|
| CC-BY-4.0 — free to use with attribution. |
|
|
