{"paper_meta":{"paper_id":"arxiv:0704.2779","title":"0704.2779","authors":[],"year":null,"published":null,"updated":null,"venue_or_source":"arxiv-papers-shard","primary_category":"cs","secondary_categories":[],"doi":null,"license":"unknown","source_type":"pdf+shard","source_url":null,"pdf_url":null},"abstract":{"raw":"","cleaned":"","offsets":[]},"full_paper_text":{"raw_ordered_text":"The Complexity of Simple Stochastic Games\nJonas Dieckelmann\nOctober 22, 2018\nAbstract\nIn this paper we survey the computational time complexity of assorted simple stochastic game\nproblems, and we give an overview of the best known algorithms associated with each problem.\n1\nIntroduction\nA simple stochastic game G = (V,E) is a directed graph whose vertices are partitioned into four disjoint\nsets Vmax, Vmin, Vavg and Vsink. Depending on the set a vertex belongs to, it is called max, min, average\nand sink vertex, respectively. In addition, one of the vertices in V is given the property of being the start\nvertex. Vsink contains exactly two vertices, called the 1-sink and the 0-sink. The 1-sink and the 0-sink\nhave no children, while all other vertices have exactly two distinct children. Loop edges e = (i,i) are\nallowed. In the rest of this paper we assume w.l.o.g. that V = {1,...,n} where n−1 is the 0-sink and n\nis the 1-sink.\n(1/4)\n3\n6\n5\n2\n(0)\n(1)\n(1/4)\n(1/2)\n(0)\n(0)\n(0)\nmax\nmin\navg\n0-sink\n1-sink\n1\n8\n4\n7\nFigure 1: A simple stochastic game with 8 vertices. Vertex 1 is the start vertex. The numbers in paren-\ntheses denote the optimal vertex values.\nThe game is played by two players, called the max player and the min player, who have diametrically\nopposed objectives. At the start of the game, a token is placed on the start vertex. In each round, the\n1\narXiv:0704.2779v1 [cs.CC] 20 Apr 2007\n\ntoken is moved from a vertex to one of its children obeying the following rule: whenever the token is\npositioned on a max vertex, the max player decides to which child the token is moved; whenever the token\nis positioned on a min vertex, the min player decides to which child the token is moved and whenever the\ntoken is positioned on an average vertex, the token is moved with probability 1/2 to one of its children.\nAverage vertices hence model randomness in this kind of stochastic game. The game ends, when the\ntoken reaches a sink vertex. The max player wins the game, if the token reaches the 1-sink. The min\nplayer wins the game in all other cases – that is either when the token reaches the 0-sink or when he can\nforce an infinite play where the token reaches neither sink vertex.\nThe optimal value of a vertex is defined as the probability that the max player wins the game starting\nat that vertex, assuming both players employ optimal strategies (a strategy is optimal, if the probability\nof winning the game with it is greater or equal to that of any other strategy, regardless of the strategy\nchosen by the opponent). We will later see that both players of a simple stochastic game posses optimal\nstrategies, albeit not unique ones. The value of the game is defined to be the optimal value of its start\nvertex, and the optimal value vector of the game is defined to be the vector whose components are the\noptimal vertex values of the game.\nThe most intriguing question to be asked about a simple stochastic game is: what is its value? As we\nshall see later, the complexity1 of the associated function problem is polynomial-time equivalent to that\nof finding the optimal value vector of the game. The problem of computing the optimal value vector of a\nsimple stochastic game has been studied extensively from an algorithmic point of view [2,5,11,12,15,16]\n(no polynomial time algorithm has been found), but the author is not aware of any previous efforts in\nstudying its complexity. Therefore, we shall prove containment of the problem in FNP in a subsequent\nsection of this paper.\nThe question about a simple stochastic game’s value is not only intriguing, it has also pratical relevance.\nThis is because stochastic games are nowadays used as a formal tool in a variety of different application\nareas, including automated software verification and controler optimization, where the game’s value con-\nstitutes the single most crucial information. Apart from this, there exist some other motivations behind\nthe study of simple stochastic games. Most of these are related to the SSG-VALUE problem – given a\nsimple stochastic game, is its value greater than 1/2? Though Condon [4] was able to show that the\nSSG-VALUE problem is contained in NP∩coNP, despite significant efforts [2,8,9] to obtain a hardness\nresult for a specific complexity class, the problem’s exact complexity status is unknown. Thus one of\nthe motivations behind the study of simple stochastic games is the desire to find a complexity class for\nwhich SSG-VALUE is complete, so as to obtain a clue whether the problem is intractable2 or not. The\npresent consensus is that, since contained in NP ∩coNP, SSG-VALUE is very likely not NP-complete\nand may allow for more efficient algorithms than the exponential ones currently known. Condon rein-\nforces this hypothesis by stating that SSG-VALUE constitutes one of the rare combinatorial problems to\nbe contained in NP∩coNP, but for which containment also in P is an open question.\nA last motivation behind the study of simple stochastic games can be expressed as the “kill two birds\nwith one stone” factor; many computational problems, such as the generalized linear complementarity\nproblem (GLCP) and the minimum stable circuit problem for min/max/avg-circuits (STABLE-CIRCUIT),\nwere shown [8,9] to be polynomial-time reducible to SSG-VALUE – hence more efficient algorithms for\nSSG-VALUE will also yield more efficient algorithms for those other problems. The rest of this paper is\n1We will refer to the computational time complexity of a problem as the problem’s complexity.\n2Intractable computational problems are those that feature exponential time or space complexities.\n2\n\norganized as follows: in section 2, we will restate the essential definitions for simple stochastic games\nas given in Condon’s initial paper on the subject. In section 3, we will provide a more detailed view of\nsimple stochastic games which will enable us to conduct our complexity survey in section 4. The paper\nconcludes with a summary of the important points and an overview of open problems in section 5.\n2\nDefinitions\n2.1\nPlayer Strategies\nGiven a simple stochastic game, a strategy τ for the min player (or min strategy) is a subset of the\ngame’s edges such that for each min vertex i with children j and k, either (i, j) ∈τ or (i,k) ∈τ applies.\nSubstituting τ with σ and min with max in the above sentence, we obtain the analog definition for the\nmax strategy σ. Informally, a strategy denotes the player’s choice to which child the token is to be moved\nwhenever it is positioned on a vertex belonging to that player. The reason for defining player strategies\nlike this will be explained in the next section of this paper.\n2.2\nReduced Games\nGiven a simple stochastic game G = (V,E) and a strategy τ to be employed by the min player, the reduced\ngame Gτ is defined to be the sub-graph of G obtained by removing all edges from G that are not selected\nby the min strategy τ, i. e.\nGτ = (V,Eτ)\nwhere\nEτ = E \\{(i, j) ∈E : i ∈Vmin ∧(i, j) /∈τ}\nGτ can be regarded as the 1-player equivalent of G, where it is certain that the min player employs τ. In\na similar manner, the reduced games Gσ and Gτ,σ are defined as\nGσ = (V,Eσ)\nwhere\nEσ = E \\{(i, j) ∈E : i ∈Vmax ∧(i, j) /∈σ}\nGτ,σ = (V,Eτ,σ)\nwhere\nEτ,σ = E \\{(i, j) ∈E : i ∈Vmin ∪Vmax ∧(i, j) /∈τ∪σ}\nWe observe that in the reduced game Gτ,σ, the strategies of both players are fixed to τ and σ and the\nwinner is decided by a (more or less) random walk of the token on the graph of Gτ,σ.\n2.3\nVertex Values\nGiven a reduced game Gτ,σ (or alternatively a simple stochastic game G and a pair of strategies τ and σ\nto be employed by the players), the value of vertex i, vτ,σ(i), is defined to be the probability that the token\nreaches the 1-sink in a random walk on the graph of Gτ,σ, starting at vertex i. The value vector ⃗vτ,σ of\nGτ,σ is defined to be the vector whose components are the vertex values of Gτ,σ.\n3\n\n2.4\nOptimal Player Strategies and Optimal Vertex Values\nGiven a simple stochastic game G, a strategy τopt for the min player satisfying\nvτopt,σopt(i) ≤vτ,σopt(i)\n∀i,τ\nis called an optimal strategy for the min player. Similarly, a strategy σopt for the max player satisfying\nvτopt,σopt(i) ≥vτopt,σ(i)\n∀i,σ\nis called an optimal strategy for the max player. Informally, the formulas say that by employing an\noptimal strategy for G, a player assures himself the highest probability of winning G no matter what the\nstart vertex.\nThe optimal value of vertex i, v(i), is defined as\nv(i) = vτopt,σopt(i)\nwhere τopt and σopt are a pair of optimal player strategies for G. The optimal value of a vertex denotes\nthe probability that the max player wins the game starting at that vertex, assuming both players employ\noptimal strategies. The optimal value vector⃗v of G is defined to be the vector whose components are the\noptimal vertex values of G. Misleadingly, the value of a simple stochastic game is the optimal (and not\njust any) value of the start vertex. It is also important not to confuse vertex values with optimal vertex\nvalues, as their meaning is different.\n2.5\nStopping Simple Stochastic Games\nA stopping simple stochastic game is a simple stochastic game which does not permit infinite plays, i. e.\nthe token always reaches a sink vertex after a finite number of rounds, regardless of the strategies chosen\nby the players. More precisely, if for all pairs of strategies τ,σ each vertex of the reduced game Gτ,σ has\na path to a sink vertex, then G is stopping. Stopping stochastic games are also referred to as stochastic\ngames that halt with probability 1.\n3\nProperties of Simple Stochastic Games\nFrom the introduction, we can already derive some important game theoretic properties of simple stochas-\ntic games. We will use these in the elaborations to follow:\n✓determined – the optimal value vector exists and is unique.\n✓finite – the game has n states, and in each state the players have at most two actions to choose from.\n✓zero sum – in every state of the game, the win expectancy for one player is the complement of the\nwin expectancy for the opponent.\n4\n\n✓perfect information – the players act sequentially and each player is completely informed about the\nhistory and the state of the game.\n✓reachability objective – the objective of the players is to force the token to reach their respective\nsink vertex.\n✓21\n2 player – the coin-flipping ruler over average vertices (nature) is given the status of a half player.\nLet us start by discussing player strategies for simple stochastic games. The rationale of defining player\nstrategies just as we did originates from the initial work on stochastic games by Shapley [14]; he showed\nthat perfect information stopping stochastic games – and hence stopping simple stochastic games – have\na Nash equilibrium [3] in pure3 memoryless4 optimal strategies (Somla [15] points out in a footnote\nthat it is possible to extend this results to non-stopping simple stochastic games, using advanced proof\ntechniques). Because of this fact, it suffices to denote a player’s strategy by a prescription that, for each\nvertex belonging to the player, states to which child the token is to be moved; such a prescription can be\nmodeled as a subset of the game’s edges.\nFurthermore, as a direct consequence of the definition of optimal strategies, we find that a min strategy\nis optimal for a particular game G if and only if it is locally optimal (or greedy) at every min vertex of G\nwith respect to the optimal value vector. That is to say, the best strategy for the min player is to always\nmove the token from a min vertex to the child which has got the lower optimal value. The same statement\ncan be made for the max player, but certainly, the max player always moves the token from a max vertex\nto the child which posesses the higher optimal value. We conclude that a player cannot improve his\nperformance by making local concessions – unlike in chess, non-greediness will not be rewarded.\nAnother property of optimal player strategies is that they are not necessarily unique; instead, a simple\nstochastic game may posses more than one optimal strategy for a player. As a trivial example, picture\na simple stochastic game which contains a min vertex that has itself and the 0-sink as children. In this\ngame, the optimal value of the min vertex is 0 and the min player possesses at least two different optimal\nstrategies for the game – one of which contains the edge to the 0-sink and one of which contains the loop\nedge.\nWe have mentioned that player strategies are independent of the game’s history and deduce that they can\nbe fixed before the start of the game. Once both players have fixed their strategies to be τ and σ, a random\nwalk of the token on the reduced game Gτ,σ decides upon the winner. For the upcoming discussion about\nthe reduced game Gτ,σ let us w.l.o.g. assume that the vertices in Gτ,σ are labeled in such a way that the\nvertices 1,...,t are those that have a path to a sink vertex. With this in mind, we can easily verify that\n– following its definition – the value vector ⃗vτ,σ of Gτ,σ is a solution to the following system of linear\nequations: vτ,σ(n) = 1, vτ,σ(i) = 0 for t < i < n and otherwise\nvτ,σ(i) =\n(\nvτ,σ(j)\nif i is a min or max vertex with child j\n1\n2(vτ,σ(j)+vτ,σ(k))\nif i is an average vertex with children j and k\nwhich can be written as\n⃗vτ,σ = Q⃗vτ,σ +⃗b\n⇔\n(I −Q)⃗vτ,σ =⃗b\n(1)\n3a strategy is called pure if it consists of deterministic (as opposed to random) choices by the player.\n4a strategy is called memoryless if the players choice only depends on the state of the game, and not its history.\n5\n\nwhere Q ∈Qn×n is related to the topology of Gτ,σ as follows: Qi j = 0 if i > t and otherwise\nQij =\n(\n1\nif i is a min or max vertex with child j\n1\n2\nif i is an average vertex with child j\nand⃗b ∈Qn is defined as\nbi =\n(\n1\nif i = n\n0\notherwise\nWe will rewrite the proof given by Condon [4] that (1) has a unique solution. Let λi be the i-th eigenvalue\nof Q. The idea is to show that as m →∞, Qmn →0, from which the following chain of deductions can be\nmade\nQmn →0\n⇒\nλi ̸= 1\ni = 1...n\n⇔\ndet(Q−I) ̸= 0\n⇔\nrank(Q−I) = n\n⇔\n(1) has a unique solution\nLet us denote the upper t rows of Q by the t ×n matrix Qt, which is the 1-step transition matrix of non-\nsink vertices of Gτ,σ that have a path to a sink vertex. As a matter of fact, entry ij of Qmn\nt\n= Qt ···Qt\ndenotes the probability that the token reaches vertex j from vertex i in a random walk on the graph Gτ,σ\nin exactly mn steps. Therefore, the sum of values in the i-th row of Qmn\nt\nequals one minus the probability\nof reaching a sink vertex from i in k < mn steps. The probability of reaching a sink vertex from i in k < n\nsteps is greater than zero, as i has at least one path to a sink vertex of length no more than the maximal\ndiameter of Gτ,σ – which is n−1. Additionally, for m′ > m, the probability of reaching a sink vertex from\ni in k < m′n steps is obviously greater than the probability of reaching a sink vertex from i in k < mn\nsteps. As the values of Qmn\nt\nare all positive, it follows that as m →∞, Qmn\nt\n→0 and thus Qmn →0.\nUsing a local graph search algorithm, one is able to verify in time O(n) whether a given vertex of Gτ,σ\nhas a path to a sink vertex. Therefore, Q and⃗b can be constructed from Gτ,σ in time O(n2). By solving\n(1) with Gauss-elimination or LU-decomposition (both O(n3)), we obtain a cubic time algorithm for the\nproblem of computing the value vector of Gτ,σ.\nIn a related concern, Condon [4] showed that the vertex values vτ,σ(i) of the reduced game Gτ,σ are\nrational numbers from the set\nΩt = {p/q ∈Q : 0 ≤p ≤q ≤4t}\n(2)\nwhere t is defined as before, i. e. t is the number of non-sink vertices of Gτ,σ which have a path to a sink\nvertex. To understand why this is true, consider that, as ⃗vτ,σ is a solution to (1), the components of ⃗vτ,σ\ncan be denoted by vτ,σ(i) = Di/D where D is the determinant of the matrix I−Q and Di is the determinant\nof I −Q which has the i-th column replaced by⃗b. Since the components of I −Q and⃗b are all rational,\nboth Di and D must also be rational. Condon concludes the proof by showing that 0 ≤Di ≤D ≤4t holds.\nNote that not all values from the set Ωt can occur as vertex values in the game Gτ,σ; instead, Ωt is a\nsuperset – or approximation – of the possible vertex values of Gτ,σ. In the context of finding a reduced\ngame’s optimal vertex values, Ωt can be regarded as a search space of that problem.\n6\n\n0\n1/2\n1/16\n7/16\n1/3\n1/4\n1/2\n1\n9/16\n15/16\n2/3\n3/4\nFigure 2: Visualization of the set Ω2. The possible vertex values of a simple stochastic game with 4\nvertices are a subset of the depicted values.\nConcluding the reduced game topic, it is worth mentioning that reduced games can also be studied in the\nframework of Markov processes. If we were to assign transition probabilities of 1 to the single edges\nleaving player vertices in Gσ,τ, next to the already established transition probabilities of 1/2 assigned\nto the edges leaving average vertices, then the so modified graph Gσ,τ would formally conform to the\ndefinition of a Markov chain. In a similar fashion, the reduced games Gτ and Gσ can be transformed into\nMarkov decision processes (MDP’s).\nFollowing its definition, the optimal value vector ⃗v of a simple stochastic game G is a solution to the\nequation system\nv(i) =\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nmax(v(j),v(k))\nif i is a max vertex with children j and k\nmin(v(j),v(k))\nif i is a min vertex with children j and k\n1\n2(v(j)+v(k))\nif i is an average vertex with children j and k\n0\nif i = n−1\n1\nif i = n\nwhich can be written as\n⃗v = IG(⃗v)\n(3)\nfor IG : [0,1]n →[0,1]n as defined above. Contrary to (1), the equations in (3) are non-linear and a solution\ncan no longer be derived analytically but rather has to be computed numerically. Shapley [14] showed\nthat in the case of stopping stochastic games – and hence stopping simple stochastic games – the operator\nIG is contracting on the hypercube [0,1]n and therefore has a unique fixed point. In this case, the solution\nto (3) is the optimal value vector of the game. In the case of non-stopping simple stochastic games\nhowever, (3) is necessary, but not sufficient, for the optimal value vector of the game. For an example of\nambiguous values in a simple stochastic game that has only one connected component and is non-trivial\nobserve figure 1. In the depicted game, any value below 1/4 can be assigned uniformly to the vertices\n{2,4,6} without violating (3) – though only 0 is the correct optimal value for each of the vertices. We\nfinish this paragraph with the statement that, contrary to optimal strategies, the optimal value vector is\nunique in every simple stochastic game.\nThe last discovery about simple stochastic games which is relevant in the context of this paper is again\ndue to Condon [4]. She found out that from a simple stochastic game G, a stopping simple stochastic\ngame G′ can be constructed whose vertex values are arbitrarily close to the vertex values of G. The\nconstruction rule is as follows: For β = 1/2cn, the so-called β-stopping game G′ adopts all the vertices\n7\n\nof G but it does not adopt any edge of G. For each edge (i, j) of G, the graph G′ instead contains a path\nof m = cn average vertices which are connected to the vertices i, j and n −1 (the 0-sink) as depicted in\nfigure 3.\ne1\ne2\nem\n...\ni\nj\nn-1\nFigure 3: Construction rule for the β-stopping game G′ of G, where β = 1/2m and m = cn. The dashed\nedge is present in G, but not in G′; it is replaced by the depicted elements.\nTwo important properties of the β-stopping game G′ have been set forth by Condon:\n1. G′ can be constructed from G in time O(n2), where the constant c only has a linear effect on the\nruntime of the construction algorithm and is hidden in the O-notation.\n2. For arbitrary player strategies τ and σ, the corresponding value vectors ⃗vτ,σ of G and ⃗v′\nτ,σ of G′\nsatisfy\n|vτ,σ(i)−v′\nτ,σ(i)| ≤2n(3−c)\ni ∈V\n(4)\nWe observe that by choosing c large enough, the differences between corresponding vertex values of the\ngames G and G′ can be made arbitrarily small, though G′ can still be constructed in time polynomial in\nthe size of G. We will use this result, as well as previous results from this section, for proving the claims\nwe make in the next section.\n4\nComplexity Survey\nWe begin this section by showing that – given a simple stochastic game G – the below function problems\nhave polynomial-time equivalent complexities:\n1. What is the value of G?\n2. What is the optimal value vector of G?\n3. What are optimal player strategies of G?\nProof. “2 ≤p 1”: Let us assume that Algorithm A1 computes the value of the game G. From A1, we\nconstruct an Algorithm A2 which computes the optimal value vector of G. On input of G, A2 iterates\nover all vertices i of G. In each iteration, A2 changes the start vertex of G to be i and then performs a\nrun of A1 on G, yielding the optimal vertex value v(i). After the last iteration, A2 outputs the computed\nvalues in form of the optimal value vector of G. The runtime of A2 is dominated by the queries to A1 and\nsince A2 makes n queries to A1, A2 is efficient given A1 is.\n8\n\n“3 ≤p 2”: Suppose Algorithm A2 computes the optimal value vector of the game G. We give an informal\ndescription of an Algorithm A3 which, using A2, computes the optimal player strategies of G. Our tool\nwill be the result from the previous section that every min (max) strategy, which is locally optimal at every\nmin (max) vertex of G, is an optimal min (max) strategy of G; it therefore suffices for A3 to compute\ntwo locally optimal strategies τopt and σopt for the players. On input of G, A3 runs A2 on G, yielding the\noptimal value vector⃗v of G. For each min vertex i with children j and k, A3 adds (i, j) to the initial empty\nτ if v(j) ≤v(k), else A3 adds (i,k) to τ. Similarly, for each max vertex i with children j and k, A3 adds\n(i, j) to the initial empty σ if v(j) ≥v(k), else A3 adds (i,k) to σ. Following this construction, τopt and\nσopt are locally optimal strategies and since A3 makes one query to A2 and performs O(n) instructions,\nA3 is efficient given A2 is.\n“1 ≤p 3”: From an algorithm A3 which computes a pair of optimal player strategies of the game G, we\nconstruct an algorithm A1 which computes the value of G. On input of G, A1 runs A3 on G to obtain the\noptimal player strategies τopt and σopt of G. In time O(n), A1 then constructs the reduced game Gτopt,σopt\ncorresponding to τopt and σopt. We already argued in the previous section that the value vector (and\nhence the value) of a reduced game can be computed in time polynomial in the size of the game. Since\nA1 makes one call to A3, A1 is efficient given A3 is.\nSSG-TWOKIND (function)\nInput:\nSimple stochastic game G, lacking one vertex kind\nQuestion:\nWhat is the optimal value vector⃗v of G?\nComplexity class: FP\nAlgorithms:\nPapers discussing algorithms for SSG-TWOKIND include [1,5–7,15,16]. In the case that G lacks min ver-\ntices, SSG-TWOKIND can be expressed as the following linear optimization (or programming) problem,\nas was first shown by Derman [6]\nn\n∑\ni=1\nv(i) →min\nsubject to v(n−1) = 0, v(n) = 1 and\nv(i) ≥\n \n \n \n \n \n0\n1 ≤i ≤n\nv(j)\nif i is a max vertex with child j\n1\n2(v(j)+v(k))\nif i is an average vertex with children j and k\nSimilarly, in the case that G lacks max vertices, Derman showed that the optimal value vector of G is the\nunique solution to the linear optimization problem\nn\n∑\ni=1\nv(i) →max\n9\n\nsubject to v(n−1) = 0, v(n) = 1 and\nv(i) ≤\n \n \n \n \n \n0\n1 ≤i ≤n\nv(j)\nif i is a min vertex with child j\n1\n2(v(j)+v(k))\nif i is an average vertex with children j and k\nKhachian [10] was the first to show that linear programming problems can be solved in time polynomial\nin the bits needed to describe the problem. Since the amount of bits needed to encode any of the above\nformulas is a polynomial function of n [4], we obtain the fruit that the value vector⃗v of G can be computed\nin polynomial time.\nSo called interior point algorithms for linear programming problems, which can move through the fea-\nsible region5 instead than just along its boundary, perform best in practice. According to wikipedia,\nMehrotra’s [13] interior point algorithm is regarded as the fastest, though a worst case boundary is not\navailable. Up to date, there exists no strongly polynomial time algorithm for linear programming prob-\nlems, i.e one that is polynomial in the number of variables of the problem up to order ~4.\nIn the case that G lacks average vertices, the algorithm given in appendix A correctly computes the\noptimal value vector of G. The number of executions of the repeat loop is O(n), as D is static after at\nmost n−2 executions. It follows that the algorithm has quadratic runtime, which is a much better result\nas that obtained for the above linear programming problems.\nSSG-OVV (function)\nInput:\nSimple stochastic game G\nQuestion:\nWhat is the optimal value vector⃗v of G?\nComplexity class: FNP\nWe will first give an informal description of the proof that SSG-OVV∈FNP. Let Ωn ⊃Ωt be a superset of\nthe possible vertex values of G, as discussed in (2). The proof is based on the fact that there exists exactly\none vector ⃗z from the set Ωn\nn – namely the optimal value vector of G – which satisfies |z(i) −v′(i)| ≤\n4−2n/2 for all i ∈V, where⃗v′ is the optimal value vector of the 1/29n-stopping game G′ of G. We deduce\nthat a nondeterministic Turing-machine M for problem SSG-OVV could guess an arbitrary vector⃗z ∈Ωn\nn\nand – assuming M knows⃗v′ – argue in polynomial time that⃗z =⃗v if and only if⃗z satisfies the mentioned\nconstraint. Of course, M cannot compute⃗v′ from scratch, but since M is nondeterministic, we can happily\nlet it, next to ⃗v, also guess ⃗v′. By evaluating (3), M can easily verify whether its guess of ⃗v′ is correct.\nTherefore, letting M guess the optimal value vectors of both G and G′, M is able to conduct a polynomial\ntime verification of both guesses.\nConsidering a formal proof, we first show that the difference between two vertex values of G is either 0\n5The feasible region of a linear programming problem is the set of variable evaluations (vertices) which satisfy the con-\nstraints given in the problem.\n10\n\nor has a lower bound δ. Let α,β ∈Ωn. Then either |α−β| = 0 or\n|α−β| =\n \np\nq −p′\nq′\n =\n \npq′ −p′q\nqq′\n ≥1\nqq′ ≥\n1\n4n4n = 4−2n =: δ\nFollowing (4), we further observe that the differences between corresponding vertex values in the games\nG and the associated 1/29n-stopping game G′ are bound below δ/2, i. e. for arbitrary strategies τ and σ\n|vτ,σ(i)−v′τ,σ(i)| ≤2−6n = 4−3n < δ/2\ni ∈V\nOn input of G, a nondeterministic Turing-machine M for problem SSG-OVV first guesses one vector from\nthe set Ωn\nn to be the optimal value vector of G and one vector from the set Ωn′\nn′ to be the optimal value\nvector of G′, where n′ = 9n|E| + n. Let us denote these vectors by the tuple (⃗z,⃗s). If we define that M\naccepts (⃗z,⃗s) if⃗z = IG(⃗z),⃗s = IG′(⃗s) and |z(i)−s(i)| < δ/2 for all i ∈V, then M accepts (⃗z,⃗s) if and only\nif⃗z =⃗v∧⃗s =⃗v′. Also it should be clear that M comes to a conclusion in time polynomial in the number\nof vertices of G, as in particular M is able to construct G′ from G in time O(n2).\nProof. Instead of saying “M accepts (⃗z,⃗s)”, we will just say “M accepts”. “⇒”: If M accepts,⃗s = IG′(⃗s)\nand hence ⃗s =⃗v′. It remains to prove that if M accepts, ⃗z =⃗v holds. Suppose M accepts but⃗z ̸=⃗v. If\n⃗z ̸=⃗v, then |z(i)−v(i)| ≥δ for at least one i ∈V. Since M accepts, |z(i)−s(i)| = |z(i)−v(i)′| < δ/2 for\nall i ∈V. It follows that |v(i) −v(i)′| ≥δ/2 for at least one i ∈V which contradicts the construction of\nG′. “⇐”: Suppose⃗z =⃗v and ⃗s =⃗v′. Then obviously⃗z = IG(⃗z), ⃗s = IG′(⃗s) and |z(i) −s(i)| < δ/2 for all\ni ∈V by construction of G′. Hence M accepts.\nAlgorithms:\nAll algorithms [2,5,7,11,12,15,16] suggested to date for the problem of finding a solution to (3) have ex-\nponential time complexities. The most intuitive of those operates on the basis of the iterative update rule\n⃗vi+1 = IG(⃗vi) and is called successive approximation or value iteration algorithm. As already mentioned,\nthis algorithm is guaranteed to converge to the correct solution only if G is stopping. In her paper about\nalgorithms for simple stochastic games, Condon [5] presents a “worst case” example for the successive\napproximation algorithm in form of a special game graph, where the algorithm takes an exponential\nnumber of updates until it finds the optimal value vector.\nSo called strategy improvement algorithms try to iteratively improve an initial pair of strategies until\nconvergence. A particularly simple algorithm of this class is the one of Hoffman & Karp [11], for which\na worst case running time of O(2n/n) is established. Björklund & Vorobyov’s [2] randomized algorithm\ndating from 2005 has a worst case running time of O(2\n√\nn·log(n)), which as of the authors knowledge, is the\nbest result obtained to date.\nSSG-VALUE* (decision)\nInput:\nSimple stochastic game G, α ∈Ωn\nQuestion:\nIs the value of G > α?\n11\n\nComplexity class: NP∩coNP\nSSG-VALUE* is a straightforward extension of the SSG-VALUE Problem, which is defined by α = 1/2.\nUsing the same terminology as for the previous problem, a nondeterministic Turing-machine M for SSG-\nVALUE* first guesses one vector from the set Ωn′\nn′ to be the optimal value vector of the 1/29n-stopping\ngame G′ of G, where n′ = 9n|E|+n. If we denote this vector by⃗s and define that M accepts⃗s if⃗s = IG′(⃗s)\nand s(start) > α, then M accepts⃗s if and only if value > α∧⃗s =⃗v′. It should also be clear that M comes\nto a conclusion in time polynomial in the size of G.\nProof. “⇒”: If M accepts ⃗s, then ⃗s = IG′(⃗s) and therefore ⃗s =⃗v′. It remains to show that if M accepts\n⃗s, value > α. Suppose M accepts ⃗s but value ≤α. If value ≤α then value′ = s(start) ≤α since the\nstopping game G′ always has lower value by construction. Hence M cannot accept ⃗s. “⇐”: Suppose\nvalue > α and ⃗s =⃗v′. Then value ≥α + δ and since, by construction of G′, |value −s(start)| ≤δ/2, it\nfollows that s(start) > α. Since also ⃗s = IG′(⃗s), M accepts ⃗s. Applying a small modification to M by\nletting it accept if s(start) ≤α, M can obviously also decide the complement of SSG-VALUE* .\nAlgorithms:\nThe author is not aware of any algorithm specially tailored for SSG-VALUE*, instead, algorithms for\nthe more general SSG-OVV are used to solve SSG-VALUE*. Though theoretically, algorithms for SSG-\nVALUE* could exploit the circumstance that – depending on the topology of G – not all optimal vertex\nvalues of G would need to be computed in order to solve the problem, as we are mainly concerned with\nthe worst case behavior of such algorithms, the case where the value of G depends on its whole game\ngraph must be assumed. Therefore, the value vector of G must be computed after all.\n5\nConclusion and Open Problems\nWe have seen that the most interesting and also most difficult simple stochastic game problem, that of\ncomputing the optimal value vector, is hard to solve. However, restricting the input to simple stochastic\ngames that lack one vertex kind, we observed that the same problem becomes tractable and can be solved\nrather efficiently. Another result we obtained was that the value vector of reduced games can be computed\nin polynomial time, i.e that given a simple stochastic game and a pair of player strategies, the question\nabout the winning-probabilities of the players is efficient answerable.\nSadly, we were not able to show polynomial time equivalence of SSG-OVV and SSG-VALUE* but we\nwould still like to know what the decision equivalent of SSG-OVV is. Another major open problem is to\nshow completeness of SSG-VALUE (or SSG-OVV) for a specific complexity class, thereby specifying the\nproblem’s exact complexity. In a last word, Condon expressed the possibility that finding an algorithm\nthat seperates simple stochastic games with low value (α < 0,25) from those with high value (α > 0,75)\nmight be of significance in solving the master problem: SSG-VALUE ∈P?\n12\n\nAppendix A\nAlgorithm 1 optimalValueVector(G)\ninput simple stochastic game G = (V,E)\noutput optimal value vector⃗v of G\nrequire Vavg = /0\nbegin\nD = {n−1,n},⃗v =⃗0\nrepeat\nfor i ∈V \\D do\nif i is a max vertex with a 1-valued child in D then\nD = D∪{i}, v(i) = 1\nelse if i is a max vertex with two 0-valued children in D then\nD = D∪{i}, v(i) = 0\nelse if i is a min vertex with a 0-valued child in D then\nD = D∪{i}, v(i) = 0\nelse if i is a min vertex with two 1-valued children in D then\nD = D∪{i}, v(i) = 1\nend if\nend for\nuntil D is static\nreturn⃗v\nend\nReferences\n[1] Daniel Anderson. Improved combinatorial algorithms for mean payoff games. Master’s thesis,\nUniversity of Upsala, 2006.\n[2] Henrik Björklund and Sergei G. Vorobyov. Combinatorial structure and randomized subexponential\nalgorithms for infinite games. Theoretical Computer Science, 349(3):347–360, 2005.\n[3] Chatterjee, Majumdar, and Jurdzinski. On nash equilibria in stochastic games. In CSL: 18th Work-\nshop on Computer Science Logic. LNCS, Springer-Verlag, 2004.\n[4] A. Condon. The complexity of stochastic games. Information and Computation, 96:203–224, 1992.\n[5] A. Condon. On algorithms for simple stochastic games. DIMACS Series in Discrete Mathematics\nand Theoretical Computer Science, 13:51–71, 1993.\n[6] Cyrus Derman. Finite State Markovian Decision Processes, volume 67 of Mathematics in Science\nand Engineering. Academic Press, New York, NY, 1970.\n[7] Jerzy A. Filar, T. A. Schultz, F. Thuijsman, and O. J. Vrieze. Nonlinear programming and stationary\nequilibria in stochastic games. Math. Program, 50:227–237, 1991.\n13\n\n[8] Gartner and Rust. Simple stochastic games and P-matrix generalized linear complementarity prob-\nlems. FCT: Fundamentals (or Foundations) of Computation Theory, 15, 2005.\n[9] Brendan Juba. On the hardness of simple stochastic games. Master’s thesis, Massachusetts Institute\nof Technology, 2006.\n[10] L. G. Khachian. A polynomial algorithm in linear programming. Soviet Mathematics, 20(191–194),\n1979.\n[11] V. S. Anil Kumar and R. Tripathi. Algorithmic results in simple stochastic games. Technical Report\nTR855, University of Rochester, Computer Science Department, 2004.\n[12] Walter Ludwig. A subexponential randomized algorithm for the simple stochastic game problem.\nInf. Comput., 117(1):151–155, 1995.\n[13] S. Mehrotra. On finding a vertex solution using interior point methods. Linear Algebra and its\nApplications, 152:233–254, 1991.\n[14] L. S. Shalpey. Stochastic games. Proceedings of the National Academy of Sciences, 39:1095–1100,\n1953.\n[15] Rafal Somla. New algorithms for solving simple stochastic games. Electr. Notes Theor. Comput.\nSci, 119(1):51–65, 2005.\n[16] Csaba Szepesvári and Michael L. Littman. Generalized Markov decision processes: Dynamic-\nprogramming and reinforcement-learning algorithms. Technical Report CS-96-11, Brown Univer-\nsity, Providence, RI, 1996.\n14","paragraphs":[{"paragraph_id":"p1","order":1,"text":"The Complexity of Simple Stochastic Games\nJonas Dieckelmann\nOctober 22, 2018\nAbstract\nIn this paper we survey the computational time complexity of assorted simple stochastic game\nproblems, and we give an overview of the best known algorithms associated with each problem.\n1\nIntroduction\nA simple stochastic game G = (V,E) is a directed graph whose vertices are partitioned into four disjoint\nsets Vmax, Vmin, Vavg and Vsink. Depending on the set a vertex belongs to, it is called max, min, average\nand sink vertex, respectively. In addition, one of the vertices in V is given the property of being the start\nvertex. Vsink contains exactly two vertices, called the 1-sink and the 0-sink. The 1-sink and the 0-sink\nhave no children, while all other vertices have exactly two distinct children. Loop edges e = (i,i) are\nallowed. In the rest of this paper we assume w.l.o.g. that V = {1,...,n} where n−1 is the 0-sink and n\nis the 1-sink.\n(1/4)\n3\n6\n5\n2\n(0)\n(1)\n(1/4)\n(1/2)\n(0)\n(0)\n(0)\nmax\nmin\navg\n0-sink\n1-sink\n1\n8\n4\n7\nFigure 1: A simple stochastic game with 8 vertices. Vertex 1 is the start vertex. The numbers in paren-\ntheses denote the optimal vertex values.\nThe game is played by two players, called the max player and the min player, who have diametrically\nopposed objectives. At the start of the game, a token is placed on the start vertex. In each round, the\n1\narXiv:0704.2779v1 [cs.CC] 20 Apr 2007"},{"paragraph_id":"p2","order":2,"text":"token is moved from a vertex to one of its children obeying the following rule: whenever the token is\npositioned on a max vertex, the max player decides to which child the token is moved; whenever the token\nis positioned on a min vertex, the min player decides to which child the token is moved and whenever the\ntoken is positioned on an average vertex, the token is moved with probability 1/2 to one of its children.\nAverage vertices hence model randomness in this kind of stochastic game. The game ends, when the\ntoken reaches a sink vertex. The max player wins the game, if the token reaches the 1-sink. The min\nplayer wins the game in all other cases – that is either when the token reaches the 0-sink or when he can\nforce an infinite play where the token reaches neither sink vertex.\nThe optimal value of a vertex is defined as the probability that the max player wins the game starting\nat that vertex, assuming both players employ optimal strategies (a strategy is optimal, if the probability\nof winning the game with it is greater or equal to that of any other strategy, regardless of the strategy\nchosen by the opponent). We will later see that both players of a simple stochastic game posses optimal\nstrategies, albeit not unique ones. The value of the game is defined to be the optimal value of its start\nvertex, and the optimal value vector of the game is defined to be the vector whose components are the\noptimal vertex values of the game.\nThe most intriguing question to be asked about a simple stochastic game is: what is its value? As we\nshall see later, the complexity1 of the associated function problem is polynomial-time equivalent to that\nof finding the optimal value vector of the game. The problem of computing the optimal value vector of a\nsimple stochastic game has been studied extensively from an algorithmic point of view [2,5,11,12,15,16]\n(no polynomial time algorithm has been found), but the author is not aware of any previous efforts in\nstudying its complexity. Therefore, we shall prove containment of the problem in FNP in a subsequent\nsection of this paper.\nThe question about a simple stochastic game’s value is not only intriguing, it has also pratical relevance.\nThis is because stochastic games are nowadays used as a formal tool in a variety of different application\nareas, including automated software verification and controler optimization, where the game’s value con-\nstitutes the single most crucial information. Apart from this, there exist some other motivations behind\nthe study of simple stochastic games. Most of these are related to the SSG-VALUE problem – given a\nsimple stochastic game, is its value greater than 1/2? Though Condon [4] was able to show that the\nSSG-VALUE problem is contained in NP∩coNP, despite significant efforts [2,8,9] to obtain a hardness\nresult for a specific complexity class, the problem’s exact complexity status is unknown. Thus one of\nthe motivations behind the study of simple stochastic games is the desire to find a complexity class for\nwhich SSG-VALUE is complete, so as to obtain a clue whether the problem is intractable2 or not. The\npresent consensus is that, since contained in NP ∩coNP, SSG-VALUE is very likely not NP-complete\nand may allow for more efficient algorithms than the exponential ones currently known. Condon rein-\nforces this hypothesis by stating that SSG-VALUE constitutes one of the rare combinatorial problems to\nbe contained in NP∩coNP, but for which containment also in P is an open question.\nA last motivation behind the study of simple stochastic games can be expressed as the “kill two birds\nwith one stone” factor; many computational problems, such as the generalized linear complementarity\nproblem (GLCP) and the minimum stable circuit problem for min/max/avg-circuits (STABLE-CIRCUIT),\nwere shown [8,9] to be polynomial-time reducible to SSG-VALUE – hence more efficient algorithms for\nSSG-VALUE will also yield more efficient algorithms for those other problems. The rest of this paper is\n1We will refer to the computational time complexity of a problem as the problem’s complexity.\n2Intractable computational problems are those that feature exponential time or space complexities.\n2"},{"paragraph_id":"p3","order":3,"text":"organized as follows: in section 2, we will restate the essential definitions for simple stochastic games\nas given in Condon’s initial paper on the subject. In section 3, we will provide a more detailed view of\nsimple stochastic games which will enable us to conduct our complexity survey in section 4. The paper\nconcludes with a summary of the important points and an overview of open problems in section 5.\n2\nDefinitions\n2.1\nPlayer Strategies\nGiven a simple stochastic game, a strategy τ for the min player (or min strategy) is a subset of the\ngame’s edges such that for each min vertex i with children j and k, either (i, j) ∈τ or (i,k) ∈τ applies.\nSubstituting τ with σ and min with max in the above sentence, we obtain the analog definition for the\nmax strategy σ. Informally, a strategy denotes the player’s choice to which child the token is to be moved\nwhenever it is positioned on a vertex belonging to that player. The reason for defining player strategies\nlike this will be explained in the next section of this paper.\n2.2\nReduced Games\nGiven a simple stochastic game G = (V,E) and a strategy τ to be employed by the min player, the reduced\ngame Gτ is defined to be the sub-graph of G obtained by removing all edges from G that are not selected\nby the min strategy τ, i. e.\nGτ = (V,Eτ)\nwhere\nEτ = E \\{(i, j) ∈E : i ∈Vmin ∧(i, j) /∈τ}\nGτ can be regarded as the 1-player equivalent of G, where it is certain that the min player employs τ. In\na similar manner, the reduced games Gσ and Gτ,σ are defined as\nGσ = (V,Eσ)\nwhere\nEσ = E \\{(i, j) ∈E : i ∈Vmax ∧(i, j) /∈σ}\nGτ,σ = (V,Eτ,σ)\nwhere\nEτ,σ = E \\{(i, j) ∈E : i ∈Vmin ∪Vmax ∧(i, j) /∈τ∪σ}\nWe observe that in the reduced game Gτ,σ, the strategies of both players are fixed to τ and σ and the\nwinner is decided by a (more or less) random walk of the token on the graph of Gτ,σ.\n2.3\nVertex Values\nGiven a reduced game Gτ,σ (or alternatively a simple stochastic game G and a pair of strategies τ and σ\nto be employed by the players), the value of vertex i, vτ,σ(i), is defined to be the probability that the token\nreaches the 1-sink in a random walk on the graph of Gτ,σ, starting at vertex i. The value vector ⃗vτ,σ of\nGτ,σ is defined to be the vector whose components are the vertex values of Gτ,σ.\n3"},{"paragraph_id":"p4","order":4,"text":"2.4\nOptimal Player Strategies and Optimal Vertex Values\nGiven a simple stochastic game G, a strategy τopt for the min player satisfying\nvτopt,σopt(i) ≤vτ,σopt(i)\n∀i,τ\nis called an optimal strategy for the min player. Similarly, a strategy σopt for the max player satisfying\nvτopt,σopt(i) ≥vτopt,σ(i)\n∀i,σ\nis called an optimal strategy for the max player. Informally, the formulas say that by employing an\noptimal strategy for G, a player assures himself the highest probability of winning G no matter what the\nstart vertex.\nThe optimal value of vertex i, v(i), is defined as\nv(i) = vτopt,σopt(i)\nwhere τopt and σopt are a pair of optimal player strategies for G. The optimal value of a vertex denotes\nthe probability that the max player wins the game starting at that vertex, assuming both players employ\noptimal strategies. The optimal value vector⃗v of G is defined to be the vector whose components are the\noptimal vertex values of G. Misleadingly, the value of a simple stochastic game is the optimal (and not\njust any) value of the start vertex. It is also important not to confuse vertex values with optimal vertex\nvalues, as their meaning is different.\n2.5\nStopping Simple Stochastic Games\nA stopping simple stochastic game is a simple stochastic game which does not permit infinite plays, i. e.\nthe token always reaches a sink vertex after a finite number of rounds, regardless of the strategies chosen\nby the players. More precisely, if for all pairs of strategies τ,σ each vertex of the reduced game Gτ,σ has\na path to a sink vertex, then G is stopping. Stopping stochastic games are also referred to as stochastic\ngames that halt with probability 1.\n3\nProperties of Simple Stochastic Games\nFrom the introduction, we can already derive some important game theoretic properties of simple stochas-\ntic games. We will use these in the elaborations to follow:\n✓determined – the optimal value vector exists and is unique.\n✓finite – the game has n states, and in each state the players have at most two actions to choose from.\n✓zero sum – in every state of the game, the win expectancy for one player is the complement of the\nwin expectancy for the opponent.\n4"},{"paragraph_id":"p5","order":5,"text":"✓perfect information – the players act sequentially and each player is completely informed about the\nhistory and the state of the game.\n✓reachability objective – the objective of the players is to force the token to reach their respective\nsink vertex.\n✓21\n2 player – the coin-flipping ruler over average vertices (nature) is given the status of a half player.\nLet us start by discussing player strategies for simple stochastic games. The rationale of defining player\nstrategies just as we did originates from the initial work on stochastic games by Shapley [14]; he showed\nthat perfect information stopping stochastic games – and hence stopping simple stochastic games – have\na Nash equilibrium [3] in pure3 memoryless4 optimal strategies (Somla [15] points out in a footnote\nthat it is possible to extend this results to non-stopping simple stochastic games, using advanced proof\ntechniques). Because of this fact, it suffices to denote a player’s strategy by a prescription that, for each\nvertex belonging to the player, states to which child the token is to be moved; such a prescription can be\nmodeled as a subset of the game’s edges.\nFurthermore, as a direct consequence of the definition of optimal strategies, we find that a min strategy\nis optimal for a particular game G if and only if it is locally optimal (or greedy) at every min vertex of G\nwith respect to the optimal value vector. That is to say, the best strategy for the min player is to always\nmove the token from a min vertex to the child which has got the lower optimal value. The same statement\ncan be made for the max player, but certainly, the max player always moves the token from a max vertex\nto the child which posesses the higher optimal value. We conclude that a player cannot improve his\nperformance by making local concessions – unlike in chess, non-greediness will not be rewarded.\nAnother property of optimal player strategies is that they are not necessarily unique; instead, a simple\nstochastic game may posses more than one optimal strategy for a player. As a trivial example, picture\na simple stochastic game which contains a min vertex that has itself and the 0-sink as children. In this\ngame, the optimal value of the min vertex is 0 and the min player possesses at least two different optimal\nstrategies for the game – one of which contains the edge to the 0-sink and one of which contains the loop\nedge.\nWe have mentioned that player strategies are independent of the game’s history and deduce that they can\nbe fixed before the start of the game. Once both players have fixed their strategies to be τ and σ, a random\nwalk of the token on the reduced game Gτ,σ decides upon the winner. For the upcoming discussion about\nthe reduced game Gτ,σ let us w.l.o.g. assume that the vertices in Gτ,σ are labeled in such a way that the\nvertices 1,...,t are those that have a path to a sink vertex. With this in mind, we can easily verify that\n– following its definition – the value vector ⃗vτ,σ of Gτ,σ is a solution to the following system of linear\nequations: vτ,σ(n) = 1, vτ,σ(i) = 0 for t < i < n and otherwise\nvτ,σ(i) =\n(\nvτ,σ(j)\nif i is a min or max vertex with child j\n1\n2(vτ,σ(j)+vτ,σ(k))\nif i is an average vertex with children j and k\nwhich can be written as\n⃗vτ,σ = Q⃗vτ,σ +⃗b\n⇔\n(I −Q)⃗vτ,σ =⃗b\n(1)\n3a strategy is called pure if it consists of deterministic (as opposed to random) choices by the player.\n4a strategy is called memoryless if the players choice only depends on the state of the game, and not its history.\n5"},{"paragraph_id":"p6","order":6,"text":"where Q ∈Qn×n is related to the topology of Gτ,σ as follows: Qi j = 0 if i > t and otherwise\nQij =\n(\n1\nif i is a min or max vertex with child j\n1\n2\nif i is an average vertex with child j\nand⃗b ∈Qn is defined as\nbi =\n(\n1\nif i = n\n0\notherwise\nWe will rewrite the proof given by Condon [4] that (1) has a unique solution. Let λi be the i-th eigenvalue\nof Q. The idea is to show that as m →∞, Qmn →0, from which the following chain of deductions can be\nmade\nQmn →0\n⇒\nλi ̸= 1\ni = 1...n\n⇔\ndet(Q−I) ̸= 0\n⇔\nrank(Q−I) = n\n⇔\n(1) has a unique solution\nLet us denote the upper t rows of Q by the t ×n matrix Qt, which is the 1-step transition matrix of non-\nsink vertices of Gτ,σ that have a path to a sink vertex. As a matter of fact, entry ij of Qmn\nt\n= Qt ···Qt\ndenotes the probability that the token reaches vertex j from vertex i in a random walk on the graph Gτ,σ\nin exactly mn steps. Therefore, the sum of values in the i-th row of Qmn\nt\nequals one minus the probability\nof reaching a sink vertex from i in k < mn steps. The probability of reaching a sink vertex from i in k < n\nsteps is greater than zero, as i has at least one path to a sink vertex of length no more than the maximal\ndiameter of Gτ,σ – which is n−1. Additionally, for m′ > m, the probability of reaching a sink vertex from\ni in k < m′n steps is obviously greater than the probability of reaching a sink vertex from i in k < mn\nsteps. As the values of Qmn\nt\nare all positive, it follows that as m →∞, Qmn\nt\n→0 and thus Qmn →0.\nUsing a local graph search algorithm, one is able to verify in time O(n) whether a given vertex of Gτ,σ\nhas a path to a sink vertex. Therefore, Q and⃗b can be constructed from Gτ,σ in time O(n2). By solving\n(1) with Gauss-elimination or LU-decomposition (both O(n3)), we obtain a cubic time algorithm for the\nproblem of computing the value vector of Gτ,σ.\nIn a related concern, Condon [4] showed that the vertex values vτ,σ(i) of the reduced game Gτ,σ are\nrational numbers from the set\nΩt = {p/q ∈Q : 0 ≤p ≤q ≤4t}\n(2)\nwhere t is defined as before, i. e. t is the number of non-sink vertices of Gτ,σ which have a path to a sink\nvertex. To understand why this is true, consider that, as ⃗vτ,σ is a solution to (1), the components of ⃗vτ,σ\ncan be denoted by vτ,σ(i) = Di/D where D is the determinant of the matrix I−Q and Di is the determinant\nof I −Q which has the i-th column replaced by⃗b. Since the components of I −Q and⃗b are all rational,\nboth Di and D must also be rational. Condon concludes the proof by showing that 0 ≤Di ≤D ≤4t holds.\nNote that not all values from the set Ωt can occur as vertex values in the game Gτ,σ; instead, Ωt is a\nsuperset – or approximation – of the possible vertex values of Gτ,σ. In the context of finding a reduced\ngame’s optimal vertex values, Ωt can be regarded as a search space of that problem.\n6"},{"paragraph_id":"p7","order":7,"text":"0\n1/2\n1/16\n7/16\n1/3\n1/4\n1/2\n1\n9/16\n15/16\n2/3\n3/4\nFigure 2: Visualization of the set Ω2. The possible vertex values of a simple stochastic game with 4\nvertices are a subset of the depicted values.\nConcluding the reduced game topic, it is worth mentioning that reduced games can also be studied in the\nframework of Markov processes. If we were to assign transition probabilities of 1 to the single edges\nleaving player vertices in Gσ,τ, next to the already established transition probabilities of 1/2 assigned\nto the edges leaving average vertices, then the so modified graph Gσ,τ would formally conform to the\ndefinition of a Markov chain. In a similar fashion, the reduced games Gτ and Gσ can be transformed into\nMarkov decision processes (MDP’s).\nFollowing its definition, the optimal value vector ⃗v of a simple stochastic game G is a solution to the\nequation system\nv(i) ="},{"paragraph_id":"p8","order":8,"text":"max(v(j),v(k))\nif i is a max vertex with children j and k\nmin(v(j),v(k))\nif i is a min vertex with children j and k\n1\n2(v(j)+v(k))\nif i is an average vertex with children j and k\n0\nif i = n−1\n1\nif i = n\nwhich can be written as\n⃗v = IG(⃗v)\n(3)\nfor IG : [0,1]n →[0,1]n as defined above. Contrary to (1), the equations in (3) are non-linear and a solution\ncan no longer be derived analytically but rather has to be computed numerically. Shapley [14] showed\nthat in the case of stopping stochastic games – and hence stopping simple stochastic games – the operator\nIG is contracting on the hypercube [0,1]n and therefore has a unique fixed point. In this case, the solution\nto (3) is the optimal value vector of the game. In the case of non-stopping simple stochastic games\nhowever, (3) is necessary, but not sufficient, for the optimal value vector of the game. For an example of\nambiguous values in a simple stochastic game that has only one connected component and is non-trivial\nobserve figure 1. In the depicted game, any value below 1/4 can be assigned uniformly to the vertices\n{2,4,6} without violating (3) – though only 0 is the correct optimal value for each of the vertices. We\nfinish this paragraph with the statement that, contrary to optimal strategies, the optimal value vector is\nunique in every simple stochastic game.\nThe last discovery about simple stochastic games which is relevant in the context of this paper is again\ndue to Condon [4]. She found out that from a simple stochastic game G, a stopping simple stochastic\ngame G′ can be constructed whose vertex values are arbitrarily close to the vertex values of G. The\nconstruction rule is as follows: For β = 1/2cn, the so-called β-stopping game G′ adopts all the vertices\n7"},{"paragraph_id":"p9","order":9,"text":"of G but it does not adopt any edge of G. For each edge (i, j) of G, the graph G′ instead contains a path\nof m = cn average vertices which are connected to the vertices i, j and n −1 (the 0-sink) as depicted in\nfigure 3.\ne1\ne2\nem\n...\ni\nj\nn-1\nFigure 3: Construction rule for the β-stopping game G′ of G, where β = 1/2m and m = cn. The dashed\nedge is present in G, but not in G′; it is replaced by the depicted elements.\nTwo important properties of the β-stopping game G′ have been set forth by Condon:\n1. G′ can be constructed from G in time O(n2), where the constant c only has a linear effect on the\nruntime of the construction algorithm and is hidden in the O-notation.\n2. For arbitrary player strategies τ and σ, the corresponding value vectors ⃗vτ,σ of G and ⃗v′\nτ,σ of G′\nsatisfy\n|vτ,σ(i)−v′\nτ,σ(i)| ≤2n(3−c)\ni ∈V\n(4)\nWe observe that by choosing c large enough, the differences between corresponding vertex values of the\ngames G and G′ can be made arbitrarily small, though G′ can still be constructed in time polynomial in\nthe size of G. We will use this result, as well as previous results from this section, for proving the claims\nwe make in the next section.\n4\nComplexity Survey\nWe begin this section by showing that – given a simple stochastic game G – the below function problems\nhave polynomial-time equivalent complexities:\n1. What is the value of G?\n2. What is the optimal value vector of G?\n3. What are optimal player strategies of G?\nProof. “2 ≤p 1”: Let us assume that Algorithm A1 computes the value of the game G. From A1, we\nconstruct an Algorithm A2 which computes the optimal value vector of G. On input of G, A2 iterates\nover all vertices i of G. In each iteration, A2 changes the start vertex of G to be i and then performs a\nrun of A1 on G, yielding the optimal vertex value v(i). After the last iteration, A2 outputs the computed\nvalues in form of the optimal value vector of G. The runtime of A2 is dominated by the queries to A1 and\nsince A2 makes n queries to A1, A2 is efficient given A1 is.\n8"},{"paragraph_id":"p10","order":10,"text":"“3 ≤p 2”: Suppose Algorithm A2 computes the optimal value vector of the game G. We give an informal\ndescription of an Algorithm A3 which, using A2, computes the optimal player strategies of G. Our tool\nwill be the result from the previous section that every min (max) strategy, which is locally optimal at every\nmin (max) vertex of G, is an optimal min (max) strategy of G; it therefore suffices for A3 to compute\ntwo locally optimal strategies τopt and σopt for the players. On input of G, A3 runs A2 on G, yielding the\noptimal value vector⃗v of G. For each min vertex i with children j and k, A3 adds (i, j) to the initial empty\nτ if v(j) ≤v(k), else A3 adds (i,k) to τ. Similarly, for each max vertex i with children j and k, A3 adds\n(i, j) to the initial empty σ if v(j) ≥v(k), else A3 adds (i,k) to σ. Following this construction, τopt and\nσopt are locally optimal strategies and since A3 makes one query to A2 and performs O(n) instructions,\nA3 is efficient given A2 is.\n“1 ≤p 3”: From an algorithm A3 which computes a pair of optimal player strategies of the game G, we\nconstruct an algorithm A1 which computes the value of G. On input of G, A1 runs A3 on G to obtain the\noptimal player strategies τopt and σopt of G. In time O(n), A1 then constructs the reduced game Gτopt,σopt\ncorresponding to τopt and σopt. We already argued in the previous section that the value vector (and\nhence the value) of a reduced game can be computed in time polynomial in the size of the game. Since\nA1 makes one call to A3, A1 is efficient given A3 is.\nSSG-TWOKIND (function)\nInput:\nSimple stochastic game G, lacking one vertex kind\nQuestion:\nWhat is the optimal value vector⃗v of G?\nComplexity class: FP\nAlgorithms:\nPapers discussing algorithms for SSG-TWOKIND include [1,5–7,15,16]. In the case that G lacks min ver-\ntices, SSG-TWOKIND can be expressed as the following linear optimization (or programming) problem,\nas was first shown by Derman [6]\nn\n∑\ni=1\nv(i) →min\nsubject to v(n−1) = 0, v(n) = 1 and\nv(i) ≥"},{"paragraph_id":"p11","order":11,"text":"0\n1 ≤i ≤n\nv(j)\nif i is a max vertex with child j\n1\n2(v(j)+v(k))\nif i is an average vertex with children j and k\nSimilarly, in the case that G lacks max vertices, Derman showed that the optimal value vector of G is the\nunique solution to the linear optimization problem\nn\n∑\ni=1\nv(i) →max\n9"},{"paragraph_id":"p12","order":12,"text":"subject to v(n−1) = 0, v(n) = 1 and\nv(i) ≤"},{"paragraph_id":"p13","order":13,"text":"0\n1 ≤i ≤n\nv(j)\nif i is a min vertex with child j\n1\n2(v(j)+v(k))\nif i is an average vertex with children j and k\nKhachian [10] was the first to show that linear programming problems can be solved in time polynomial\nin the bits needed to describe the problem. Since the amount of bits needed to encode any of the above\nformulas is a polynomial function of n [4], we obtain the fruit that the value vector⃗v of G can be computed\nin polynomial time.\nSo called interior point algorithms for linear programming problems, which can move through the fea-\nsible region5 instead than just along its boundary, perform best in practice. According to wikipedia,\nMehrotra’s [13] interior point algorithm is regarded as the fastest, though a worst case boundary is not\navailable. Up to date, there exists no strongly polynomial time algorithm for linear programming prob-\nlems, i.e one that is polynomial in the number of variables of the problem up to order ~4.\nIn the case that G lacks average vertices, the algorithm given in appendix A correctly computes the\noptimal value vector of G. The number of executions of the repeat loop is O(n), as D is static after at\nmost n−2 executions. It follows that the algorithm has quadratic runtime, which is a much better result\nas that obtained for the above linear programming problems.\nSSG-OVV (function)\nInput:\nSimple stochastic game G\nQuestion:\nWhat is the optimal value vector⃗v of G?\nComplexity class: FNP\nWe will first give an informal description of the proof that SSG-OVV∈FNP. Let Ωn ⊃Ωt be a superset of\nthe possible vertex values of G, as discussed in (2). The proof is based on the fact that there exists exactly\none vector ⃗z from the set Ωn\nn – namely the optimal value vector of G – which satisfies |z(i) −v′(i)| ≤\n4−2n/2 for all i ∈V, where⃗v′ is the optimal value vector of the 1/29n-stopping game G′ of G. We deduce\nthat a nondeterministic Turing-machine M for problem SSG-OVV could guess an arbitrary vector⃗z ∈Ωn\nn\nand – assuming M knows⃗v′ – argue in polynomial time that⃗z =⃗v if and only if⃗z satisfies the mentioned\nconstraint. Of course, M cannot compute⃗v′ from scratch, but since M is nondeterministic, we can happily\nlet it, next to ⃗v, also guess ⃗v′. By evaluating (3), M can easily verify whether its guess of ⃗v′ is correct.\nTherefore, letting M guess the optimal value vectors of both G and G′, M is able to conduct a polynomial\ntime verification of both guesses.\nConsidering a formal proof, we first show that the difference between two vertex values of G is either 0\n5The feasible region of a linear programming problem is the set of variable evaluations (vertices) which satisfy the con-\nstraints given in the problem.\n10"},{"paragraph_id":"p14","order":14,"text":"or has a lower bound δ. Let α,β ∈Ωn. Then either |α−β| = 0 or\n|α−β| ="},{"paragraph_id":"p15","order":15,"text":"p\nq −p′\nq′\n ="},{"paragraph_id":"p16","order":16,"text":"pq′ −p′q\nqq′\n ≥1\nqq′ ≥\n1\n4n4n = 4−2n =: δ\nFollowing (4), we further observe that the differences between corresponding vertex values in the games\nG and the associated 1/29n-stopping game G′ are bound below δ/2, i. e. for arbitrary strategies τ and σ\n|vτ,σ(i)−v′τ,σ(i)| ≤2−6n = 4−3n < δ/2\ni ∈V\nOn input of G, a nondeterministic Turing-machine M for problem SSG-OVV first guesses one vector from\nthe set Ωn\nn to be the optimal value vector of G and one vector from the set Ωn′\nn′ to be the optimal value\nvector of G′, where n′ = 9n|E| + n. Let us denote these vectors by the tuple (⃗z,⃗s). If we define that M\naccepts (⃗z,⃗s) if⃗z = IG(⃗z),⃗s = IG′(⃗s) and |z(i)−s(i)| < δ/2 for all i ∈V, then M accepts (⃗z,⃗s) if and only\nif⃗z =⃗v∧⃗s =⃗v′. Also it should be clear that M comes to a conclusion in time polynomial in the number\nof vertices of G, as in particular M is able to construct G′ from G in time O(n2).\nProof. Instead of saying “M accepts (⃗z,⃗s)”, we will just say “M accepts”. “⇒”: If M accepts,⃗s = IG′(⃗s)\nand hence ⃗s =⃗v′. It remains to prove that if M accepts, ⃗z =⃗v holds. Suppose M accepts but⃗z ̸=⃗v. If\n⃗z ̸=⃗v, then |z(i)−v(i)| ≥δ for at least one i ∈V. Since M accepts, |z(i)−s(i)| = |z(i)−v(i)′| < δ/2 for\nall i ∈V. It follows that |v(i) −v(i)′| ≥δ/2 for at least one i ∈V which contradicts the construction of\nG′. “⇐”: Suppose⃗z =⃗v and ⃗s =⃗v′. Then obviously⃗z = IG(⃗z), ⃗s = IG′(⃗s) and |z(i) −s(i)| < δ/2 for all\ni ∈V by construction of G′. Hence M accepts.\nAlgorithms:\nAll algorithms [2,5,7,11,12,15,16] suggested to date for the problem of finding a solution to (3) have ex-\nponential time complexities. The most intuitive of those operates on the basis of the iterative update rule\n⃗vi+1 = IG(⃗vi) and is called successive approximation or value iteration algorithm. As already mentioned,\nthis algorithm is guaranteed to converge to the correct solution only if G is stopping. In her paper about\nalgorithms for simple stochastic games, Condon [5] presents a “worst case” example for the successive\napproximation algorithm in form of a special game graph, where the algorithm takes an exponential\nnumber of updates until it finds the optimal value vector.\nSo called strategy improvement algorithms try to iteratively improve an initial pair of strategies until\nconvergence. A particularly simple algorithm of this class is the one of Hoffman & Karp [11], for which\na worst case running time of O(2n/n) is established. Björklund & Vorobyov’s [2] randomized algorithm\ndating from 2005 has a worst case running time of O(2\n√\nn·log(n)), which as of the authors knowledge, is the\nbest result obtained to date.\nSSG-VALUE* (decision)\nInput:\nSimple stochastic game G, α ∈Ωn\nQuestion:\nIs the value of G > α?\n11"},{"paragraph_id":"p17","order":17,"text":"Complexity class: NP∩coNP\nSSG-VALUE* is a straightforward extension of the SSG-VALUE Problem, which is defined by α = 1/2.\nUsing the same terminology as for the previous problem, a nondeterministic Turing-machine M for SSG-\nVALUE* first guesses one vector from the set Ωn′\nn′ to be the optimal value vector of the 1/29n-stopping\ngame G′ of G, where n′ = 9n|E|+n. If we denote this vector by⃗s and define that M accepts⃗s if⃗s = IG′(⃗s)\nand s(start) > α, then M accepts⃗s if and only if value > α∧⃗s =⃗v′. It should also be clear that M comes\nto a conclusion in time polynomial in the size of G.\nProof. “⇒”: If M accepts ⃗s, then ⃗s = IG′(⃗s) and therefore ⃗s =⃗v′. It remains to show that if M accepts\n⃗s, value > α. Suppose M accepts ⃗s but value ≤α. If value ≤α then value′ = s(start) ≤α since the\nstopping game G′ always has lower value by construction. Hence M cannot accept ⃗s. “⇐”: Suppose\nvalue > α and ⃗s =⃗v′. Then value ≥α + δ and since, by construction of G′, |value −s(start)| ≤δ/2, it\nfollows that s(start) > α. Since also ⃗s = IG′(⃗s), M accepts ⃗s. Applying a small modification to M by\nletting it accept if s(start) ≤α, M can obviously also decide the complement of SSG-VALUE* .\nAlgorithms:\nThe author is not aware of any algorithm specially tailored for SSG-VALUE*, instead, algorithms for\nthe more general SSG-OVV are used to solve SSG-VALUE*. Though theoretically, algorithms for SSG-\nVALUE* could exploit the circumstance that – depending on the topology of G – not all optimal vertex\nvalues of G would need to be computed in order to solve the problem, as we are mainly concerned with\nthe worst case behavior of such algorithms, the case where the value of G depends on its whole game\ngraph must be assumed. Therefore, the value vector of G must be computed after all.\n5\nConclusion and Open Problems\nWe have seen that the most interesting and also most difficult simple stochastic game problem, that of\ncomputing the optimal value vector, is hard to solve. However, restricting the input to simple stochastic\ngames that lack one vertex kind, we observed that the same problem becomes tractable and can be solved\nrather efficiently. Another result we obtained was that the value vector of reduced games can be computed\nin polynomial time, i.e that given a simple stochastic game and a pair of player strategies, the question\nabout the winning-probabilities of the players is efficient answerable.\nSadly, we were not able to show polynomial time equivalence of SSG-OVV and SSG-VALUE* but we\nwould still like to know what the decision equivalent of SSG-OVV is. Another major open problem is to\nshow completeness of SSG-VALUE (or SSG-OVV) for a specific complexity class, thereby specifying the\nproblem’s exact complexity. In a last word, Condon expressed the possibility that finding an algorithm\nthat seperates simple stochastic games with low value (α < 0,25) from those with high value (α > 0,75)\nmight be of significance in solving the master problem: SSG-VALUE ∈P?\n12"},{"paragraph_id":"p18","order":18,"text":"Appendix A\nAlgorithm 1 optimalValueVector(G)\ninput simple stochastic game G = (V,E)\noutput optimal value vector⃗v of G\nrequire Vavg = /0\nbegin\nD = {n−1,n},⃗v =⃗0\nrepeat\nfor i ∈V \\D do\nif i is a max vertex with a 1-valued child in D then\nD = D∪{i}, v(i) = 1\nelse if i is a max vertex with two 0-valued children in D then\nD = D∪{i}, v(i) = 0\nelse if i is a min vertex with a 0-valued child in D then\nD = D∪{i}, v(i) = 0\nelse if i is a min vertex with two 1-valued children in D then\nD = D∪{i}, v(i) = 1\nend if\nend for\nuntil D is static\nreturn⃗v\nend\nReferences\n[1] Daniel Anderson. Improved combinatorial algorithms for mean payoff games. Master’s thesis,\nUniversity of Upsala, 2006.\n[2] Henrik Björklund and Sergei G. Vorobyov. Combinatorial structure and randomized subexponential\nalgorithms for infinite games. Theoretical Computer Science, 349(3):347–360, 2005.\n[3] Chatterjee, Majumdar, and Jurdzinski. On nash equilibria in stochastic games. In CSL: 18th Work-\nshop on Computer Science Logic. LNCS, Springer-Verlag, 2004.\n[4] A. Condon. The complexity of stochastic games. Information and Computation, 96:203–224, 1992.\n[5] A. Condon. On algorithms for simple stochastic games. DIMACS Series in Discrete Mathematics\nand Theoretical Computer Science, 13:51–71, 1993.\n[6] Cyrus Derman. Finite State Markovian Decision Processes, volume 67 of Mathematics in Science\nand Engineering. Academic Press, New York, NY, 1970.\n[7] Jerzy A. Filar, T. A. Schultz, F. Thuijsman, and O. J. Vrieze. Nonlinear programming and stationary\nequilibria in stochastic games. Math. Program, 50:227–237, 1991.\n13"},{"paragraph_id":"p19","order":19,"text":"[8] Gartner and Rust. Simple stochastic games and P-matrix generalized linear complementarity prob-\nlems. FCT: Fundamentals (or Foundations) of Computation Theory, 15, 2005.\n[9] Brendan Juba. On the hardness of simple stochastic games. Master’s thesis, Massachusetts Institute\nof Technology, 2006.\n[10] L. G. Khachian. A polynomial algorithm in linear programming. Soviet Mathematics, 20(191–194),\n1979.\n[11] V. S. Anil Kumar and R. Tripathi. Algorithmic results in simple stochastic games. Technical Report\nTR855, University of Rochester, Computer Science Department, 2004.\n[12] Walter Ludwig. A subexponential randomized algorithm for the simple stochastic game problem.\nInf. Comput., 117(1):151–155, 1995.\n[13] S. Mehrotra. On finding a vertex solution using interior point methods. Linear Algebra and its\nApplications, 152:233–254, 1991.\n[14] L. S. Shalpey. Stochastic games. Proceedings of the National Academy of Sciences, 39:1095–1100,\n1953.\n[15] Rafal Somla. New algorithms for solving simple stochastic games. Electr. Notes Theor. Comput.\nSci, 119(1):51–65, 2005.\n[16] Csaba Szepesvári and Michael L. Littman. Generalized Markov decision processes: Dynamic-\nprogramming and reinforcement-learning algorithms. Technical Report CS-96-11, Brown Univer-\nsity, Providence, RI, 1996.\n14"}],"pages":[{"page":1,"text":"The Complexity of Simple Stochastic Games\nJonas Dieckelmann\nOctober 22, 2018\nAbstract\nIn this paper we survey the computational time complexity of assorted simple stochastic game\nproblems, and we give an overview of the best known algorithms associated with each problem.\n1\nIntroduction\nA simple stochastic game G = (V,E) is a directed graph whose vertices are partitioned into four disjoint\nsets Vmax, Vmin, Vavg and Vsink. Depending on the set a vertex belongs to, it is called max, min, average\nand sink vertex, respectively. In addition, one of the vertices in V is given the property of being the start\nvertex. Vsink contains exactly two vertices, called the 1-sink and the 0-sink. The 1-sink and the 0-sink\nhave no children, while all other vertices have exactly two distinct children. Loop edges e = (i,i) are\nallowed. In the rest of this paper we assume w.l.o.g. that V = {1,...,n} where n−1 is the 0-sink and n\nis the 1-sink.\n(1/4)\n3\n6\n5\n2\n(0)\n(1)\n(1/4)\n(1/2)\n(0)\n(0)\n(0)\nmax\nmin\navg\n0-sink\n1-sink\n1\n8\n4\n7\nFigure 1: A simple stochastic game with 8 vertices. Vertex 1 is the start vertex. The numbers in paren-\ntheses denote the optimal vertex values.\nThe game is played by two players, called the max player and the min player, who have diametrically\nopposed objectives. At the start of the game, a token is placed on the start vertex. In each round, the\n1\narXiv:0704.2779v1 [cs.CC] 20 Apr 2007"},{"page":2,"text":"token is moved from a vertex to one of its children obeying the following rule: whenever the token is\npositioned on a max vertex, the max player decides to which child the token is moved; whenever the token\nis positioned on a min vertex, the min player decides to which child the token is moved and whenever the\ntoken is positioned on an average vertex, the token is moved with probability 1/2 to one of its children.\nAverage vertices hence model randomness in this kind of stochastic game. The game ends, when the\ntoken reaches a sink vertex. The max player wins the game, if the token reaches the 1-sink. The min\nplayer wins the game in all other cases – that is either when the token reaches the 0-sink or when he can\nforce an infinite play where the token reaches neither sink vertex.\nThe optimal value of a vertex is defined as the probability that the max player wins the game starting\nat that vertex, assuming both players employ optimal strategies (a strategy is optimal, if the probability\nof winning the game with it is greater or equal to that of any other strategy, regardless of the strategy\nchosen by the opponent). We will later see that both players of a simple stochastic game posses optimal\nstrategies, albeit not unique ones. The value of the game is defined to be the optimal value of its start\nvertex, and the optimal value vector of the game is defined to be the vector whose components are the\noptimal vertex values of the game.\nThe most intriguing question to be asked about a simple stochastic game is: what is its value? As we\nshall see later, the complexity1 of the associated function problem is polynomial-time equivalent to that\nof finding the optimal value vector of the game. The problem of computing the optimal value vector of a\nsimple stochastic game has been studied extensively from an algorithmic point of view [2,5,11,12,15,16]\n(no polynomial time algorithm has been found), but the author is not aware of any previous efforts in\nstudying its complexity. Therefore, we shall prove containment of the problem in FNP in a subsequent\nsection of this paper.\nThe question about a simple stochastic game’s value is not only intriguing, it has also pratical relevance.\nThis is because stochastic games are nowadays used as a formal tool in a variety of different application\nareas, including automated software verification and controler optimization, where the game’s value con-\nstitutes the single most crucial information. Apart from this, there exist some other motivations behind\nthe study of simple stochastic games. Most of these are related to the SSG-VALUE problem – given a\nsimple stochastic game, is its value greater than 1/2? Though Condon [4] was able to show that the\nSSG-VALUE problem is contained in NP∩coNP, despite significant efforts [2,8,9] to obtain a hardness\nresult for a specific complexity class, the problem’s exact complexity status is unknown. Thus one of\nthe motivations behind the study of simple stochastic games is the desire to find a complexity class for\nwhich SSG-VALUE is complete, so as to obtain a clue whether the problem is intractable2 or not. The\npresent consensus is that, since contained in NP ∩coNP, SSG-VALUE is very likely not NP-complete\nand may allow for more efficient algorithms than the exponential ones currently known. Condon rein-\nforces this hypothesis by stating that SSG-VALUE constitutes one of the rare combinatorial problems to\nbe contained in NP∩coNP, but for which containment also in P is an open question.\nA last motivation behind the study of simple stochastic games can be expressed as the “kill two birds\nwith one stone” factor; many computational problems, such as the generalized linear complementarity\nproblem (GLCP) and the minimum stable circuit problem for min/max/avg-circuits (STABLE-CIRCUIT),\nwere shown [8,9] to be polynomial-time reducible to SSG-VALUE – hence more efficient algorithms for\nSSG-VALUE will also yield more efficient algorithms for those other problems. The rest of this paper is\n1We will refer to the computational time complexity of a problem as the problem’s complexity.\n2Intractable computational problems are those that feature exponential time or space complexities.\n2"},{"page":3,"text":"organized as follows: in section 2, we will restate the essential definitions for simple stochastic games\nas given in Condon’s initial paper on the subject. In section 3, we will provide a more detailed view of\nsimple stochastic games which will enable us to conduct our complexity survey in section 4. The paper\nconcludes with a summary of the important points and an overview of open problems in section 5.\n2\nDefinitions\n2.1\nPlayer Strategies\nGiven a simple stochastic game, a strategy τ for the min player (or min strategy) is a subset of the\ngame’s edges such that for each min vertex i with children j and k, either (i, j) ∈τ or (i,k) ∈τ applies.\nSubstituting τ with σ and min with max in the above sentence, we obtain the analog definition for the\nmax strategy σ. Informally, a strategy denotes the player’s choice to which child the token is to be moved\nwhenever it is positioned on a vertex belonging to that player. The reason for defining player strategies\nlike this will be explained in the next section of this paper.\n2.2\nReduced Games\nGiven a simple stochastic game G = (V,E) and a strategy τ to be employed by the min player, the reduced\ngame Gτ is defined to be the sub-graph of G obtained by removing all edges from G that are not selected\nby the min strategy τ, i. e.\nGτ = (V,Eτ)\nwhere\nEτ = E \\{(i, j) ∈E : i ∈Vmin ∧(i, j) /∈τ}\nGτ can be regarded as the 1-player equivalent of G, where it is certain that the min player employs τ. In\na similar manner, the reduced games Gσ and Gτ,σ are defined as\nGσ = (V,Eσ)\nwhere\nEσ = E \\{(i, j) ∈E : i ∈Vmax ∧(i, j) /∈σ}\nGτ,σ = (V,Eτ,σ)\nwhere\nEτ,σ = E \\{(i, j) ∈E : i ∈Vmin ∪Vmax ∧(i, j) /∈τ∪σ}\nWe observe that in the reduced game Gτ,σ, the strategies of both players are fixed to τ and σ and the\nwinner is decided by a (more or less) random walk of the token on the graph of Gτ,σ.\n2.3\nVertex Values\nGiven a reduced game Gτ,σ (or alternatively a simple stochastic game G and a pair of strategies τ and σ\nto be employed by the players), the value of vertex i, vτ,σ(i), is defined to be the probability that the token\nreaches the 1-sink in a random walk on the graph of Gτ,σ, starting at vertex i. The value vector ⃗vτ,σ of\nGτ,σ is defined to be the vector whose components are the vertex values of Gτ,σ.\n3"},{"page":4,"text":"2.4\nOptimal Player Strategies and Optimal Vertex Values\nGiven a simple stochastic game G, a strategy τopt for the min player satisfying\nvτopt,σopt(i) ≤vτ,σopt(i)\n∀i,τ\nis called an optimal strategy for the min player. Similarly, a strategy σopt for the max player satisfying\nvτopt,σopt(i) ≥vτopt,σ(i)\n∀i,σ\nis called an optimal strategy for the max player. Informally, the formulas say that by employing an\noptimal strategy for G, a player assures himself the highest probability of winning G no matter what the\nstart vertex.\nThe optimal value of vertex i, v(i), is defined as\nv(i) = vτopt,σopt(i)\nwhere τopt and σopt are a pair of optimal player strategies for G. The optimal value of a vertex denotes\nthe probability that the max player wins the game starting at that vertex, assuming both players employ\noptimal strategies. The optimal value vector⃗v of G is defined to be the vector whose components are the\noptimal vertex values of G. Misleadingly, the value of a simple stochastic game is the optimal (and not\njust any) value of the start vertex. It is also important not to confuse vertex values with optimal vertex\nvalues, as their meaning is different.\n2.5\nStopping Simple Stochastic Games\nA stopping simple stochastic game is a simple stochastic game which does not permit infinite plays, i. e.\nthe token always reaches a sink vertex after a finite number of rounds, regardless of the strategies chosen\nby the players. More precisely, if for all pairs of strategies τ,σ each vertex of the reduced game Gτ,σ has\na path to a sink vertex, then G is stopping. Stopping stochastic games are also referred to as stochastic\ngames that halt with probability 1.\n3\nProperties of Simple Stochastic Games\nFrom the introduction, we can already derive some important game theoretic properties of simple stochas-\ntic games. We will use these in the elaborations to follow:\n✓determined – the optimal value vector exists and is unique.\n✓finite – the game has n states, and in each state the players have at most two actions to choose from.\n✓zero sum – in every state of the game, the win expectancy for one player is the complement of the\nwin expectancy for the opponent.\n4"},{"page":5,"text":"✓perfect information – the players act sequentially and each player is completely informed about the\nhistory and the state of the game.\n✓reachability objective – the objective of the players is to force the token to reach their respective\nsink vertex.\n✓21\n2 player – the coin-flipping ruler over average vertices (nature) is given the status of a half player.\nLet us start by discussing player strategies for simple stochastic games. The rationale of defining player\nstrategies just as we did originates from the initial work on stochastic games by Shapley [14]; he showed\nthat perfect information stopping stochastic games – and hence stopping simple stochastic games – have\na Nash equilibrium [3] in pure3 memoryless4 optimal strategies (Somla [15] points out in a footnote\nthat it is possible to extend this results to non-stopping simple stochastic games, using advanced proof\ntechniques). Because of this fact, it suffices to denote a player’s strategy by a prescription that, for each\nvertex belonging to the player, states to which child the token is to be moved; such a prescription can be\nmodeled as a subset of the game’s edges.\nFurthermore, as a direct consequence of the definition of optimal strategies, we find that a min strategy\nis optimal for a particular game G if and only if it is locally optimal (or greedy) at every min vertex of G\nwith respect to the optimal value vector. That is to say, the best strategy for the min player is to always\nmove the token from a min vertex to the child which has got the lower optimal value. The same statement\ncan be made for the max player, but certainly, the max player always moves the token from a max vertex\nto the child which posesses the higher optimal value. We conclude that a player cannot improve his\nperformance by making local concessions – unlike in chess, non-greediness will not be rewarded.\nAnother property of optimal player strategies is that they are not necessarily unique; instead, a simple\nstochastic game may posses more than one optimal strategy for a player. As a trivial example, picture\na simple stochastic game which contains a min vertex that has itself and the 0-sink as children. In this\ngame, the optimal value of the min vertex is 0 and the min player possesses at least two different optimal\nstrategies for the game – one of which contains the edge to the 0-sink and one of which contains the loop\nedge.\nWe have mentioned that player strategies are independent of the game’s history and deduce that they can\nbe fixed before the start of the game. Once both players have fixed their strategies to be τ and σ, a random\nwalk of the token on the reduced game Gτ,σ decides upon the winner. For the upcoming discussion about\nthe reduced game Gτ,σ let us w.l.o.g. assume that the vertices in Gτ,σ are labeled in such a way that the\nvertices 1,...,t are those that have a path to a sink vertex. With this in mind, we can easily verify that\n– following its definition – the value vector ⃗vτ,σ of Gτ,σ is a solution to the following system of linear\nequations: vτ,σ(n) = 1, vτ,σ(i) = 0 for t < i < n and otherwise\nvτ,σ(i) =\n(\nvτ,σ(j)\nif i is a min or max vertex with child j\n1\n2(vτ,σ(j)+vτ,σ(k))\nif i is an average vertex with children j and k\nwhich can be written as\n⃗vτ,σ = Q⃗vτ,σ +⃗b\n⇔\n(I −Q)⃗vτ,σ =⃗b\n(1)\n3a strategy is called pure if it consists of deterministic (as opposed to random) choices by the player.\n4a strategy is called memoryless if the players choice only depends on the state of the game, and not its history.\n5"},{"page":6,"text":"where Q ∈Qn×n is related to the topology of Gτ,σ as follows: Qi j = 0 if i > t and otherwise\nQij =\n(\n1\nif i is a min or max vertex with child j\n1\n2\nif i is an average vertex with child j\nand⃗b ∈Qn is defined as\nbi =\n(\n1\nif i = n\n0\notherwise\nWe will rewrite the proof given by Condon [4] that (1) has a unique solution. Let λi be the i-th eigenvalue\nof Q. The idea is to show that as m →∞, Qmn →0, from which the following chain of deductions can be\nmade\nQmn →0\n⇒\nλi ̸= 1\ni = 1...n\n⇔\ndet(Q−I) ̸= 0\n⇔\nrank(Q−I) = n\n⇔\n(1) has a unique solution\nLet us denote the upper t rows of Q by the t ×n matrix Qt, which is the 1-step transition matrix of non-\nsink vertices of Gτ,σ that have a path to a sink vertex. As a matter of fact, entry ij of Qmn\nt\n= Qt ···Qt\ndenotes the probability that the token reaches vertex j from vertex i in a random walk on the graph Gτ,σ\nin exactly mn steps. Therefore, the sum of values in the i-th row of Qmn\nt\nequals one minus the probability\nof reaching a sink vertex from i in k < mn steps. The probability of reaching a sink vertex from i in k < n\nsteps is greater than zero, as i has at least one path to a sink vertex of length no more than the maximal\ndiameter of Gτ,σ – which is n−1. Additionally, for m′ > m, the probability of reaching a sink vertex from\ni in k < m′n steps is obviously greater than the probability of reaching a sink vertex from i in k < mn\nsteps. As the values of Qmn\nt\nare all positive, it follows that as m →∞, Qmn\nt\n→0 and thus Qmn →0.\nUsing a local graph search algorithm, one is able to verify in time O(n) whether a given vertex of Gτ,σ\nhas a path to a sink vertex. Therefore, Q and⃗b can be constructed from Gτ,σ in time O(n2). By solving\n(1) with Gauss-elimination or LU-decomposition (both O(n3)), we obtain a cubic time algorithm for the\nproblem of computing the value vector of Gτ,σ.\nIn a related concern, Condon [4] showed that the vertex values vτ,σ(i) of the reduced game Gτ,σ are\nrational numbers from the set\nΩt = {p/q ∈Q : 0 ≤p ≤q ≤4t}\n(2)\nwhere t is defined as before, i. e. t is the number of non-sink vertices of Gτ,σ which have a path to a sink\nvertex. To understand why this is true, consider that, as ⃗vτ,σ is a solution to (1), the components of ⃗vτ,σ\ncan be denoted by vτ,σ(i) = Di/D where D is the determinant of the matrix I−Q and Di is the determinant\nof I −Q which has the i-th column replaced by⃗b. Since the components of I −Q and⃗b are all rational,\nboth Di and D must also be rational. Condon concludes the proof by showing that 0 ≤Di ≤D ≤4t holds.\nNote that not all values from the set Ωt can occur as vertex values in the game Gτ,σ; instead, Ωt is a\nsuperset – or approximation – of the possible vertex values of Gτ,σ. In the context of finding a reduced\ngame’s optimal vertex values, Ωt can be regarded as a search space of that problem.\n6"},{"page":7,"text":"0\n1/2\n1/16\n7/16\n1/3\n1/4\n1/2\n1\n9/16\n15/16\n2/3\n3/4\nFigure 2: Visualization of the set Ω2. The possible vertex values of a simple stochastic game with 4\nvertices are a subset of the depicted values.\nConcluding the reduced game topic, it is worth mentioning that reduced games can also be studied in the\nframework of Markov processes. If we were to assign transition probabilities of 1 to the single edges\nleaving player vertices in Gσ,τ, next to the already established transition probabilities of 1/2 assigned\nto the edges leaving average vertices, then the so modified graph Gσ,τ would formally conform to the\ndefinition of a Markov chain. In a similar fashion, the reduced games Gτ and Gσ can be transformed into\nMarkov decision processes (MDP’s).\nFollowing its definition, the optimal value vector ⃗v of a simple stochastic game G is a solution to the\nequation system\nv(i) =\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nmax(v(j),v(k))\nif i is a max vertex with children j and k\nmin(v(j),v(k))\nif i is a min vertex with children j and k\n1\n2(v(j)+v(k))\nif i is an average vertex with children j and k\n0\nif i = n−1\n1\nif i = n\nwhich can be written as\n⃗v = IG(⃗v)\n(3)\nfor IG : [0,1]n →[0,1]n as defined above. Contrary to (1), the equations in (3) are non-linear and a solution\ncan no longer be derived analytically but rather has to be computed numerically. Shapley [14] showed\nthat in the case of stopping stochastic games – and hence stopping simple stochastic games – the operator\nIG is contracting on the hypercube [0,1]n and therefore has a unique fixed point. In this case, the solution\nto (3) is the optimal value vector of the game. In the case of non-stopping simple stochastic games\nhowever, (3) is necessary, but not sufficient, for the optimal value vector of the game. For an example of\nambiguous values in a simple stochastic game that has only one connected component and is non-trivial\nobserve figure 1. In the depicted game, any value below 1/4 can be assigned uniformly to the vertices\n{2,4,6} without violating (3) – though only 0 is the correct optimal value for each of the vertices. We\nfinish this paragraph with the statement that, contrary to optimal strategies, the optimal value vector is\nunique in every simple stochastic game.\nThe last discovery about simple stochastic games which is relevant in the context of this paper is again\ndue to Condon [4]. She found out that from a simple stochastic game G, a stopping simple stochastic\ngame G′ can be constructed whose vertex values are arbitrarily close to the vertex values of G. The\nconstruction rule is as follows: For β = 1/2cn, the so-called β-stopping game G′ adopts all the vertices\n7"},{"page":8,"text":"of G but it does not adopt any edge of G. For each edge (i, j) of G, the graph G′ instead contains a path\nof m = cn average vertices which are connected to the vertices i, j and n −1 (the 0-sink) as depicted in\nfigure 3.\ne1\ne2\nem\n...\ni\nj\nn-1\nFigure 3: Construction rule for the β-stopping game G′ of G, where β = 1/2m and m = cn. The dashed\nedge is present in G, but not in G′; it is replaced by the depicted elements.\nTwo important properties of the β-stopping game G′ have been set forth by Condon:\n1. G′ can be constructed from G in time O(n2), where the constant c only has a linear effect on the\nruntime of the construction algorithm and is hidden in the O-notation.\n2. For arbitrary player strategies τ and σ, the corresponding value vectors ⃗vτ,σ of G and ⃗v′\nτ,σ of G′\nsatisfy\n|vτ,σ(i)−v′\nτ,σ(i)| ≤2n(3−c)\ni ∈V\n(4)\nWe observe that by choosing c large enough, the differences between corresponding vertex values of the\ngames G and G′ can be made arbitrarily small, though G′ can still be constructed in time polynomial in\nthe size of G. We will use this result, as well as previous results from this section, for proving the claims\nwe make in the next section.\n4\nComplexity Survey\nWe begin this section by showing that – given a simple stochastic game G – the below function problems\nhave polynomial-time equivalent complexities:\n1. What is the value of G?\n2. What is the optimal value vector of G?\n3. What are optimal player strategies of G?\nProof. “2 ≤p 1”: Let us assume that Algorithm A1 computes the value of the game G. From A1, we\nconstruct an Algorithm A2 which computes the optimal value vector of G. On input of G, A2 iterates\nover all vertices i of G. In each iteration, A2 changes the start vertex of G to be i and then performs a\nrun of A1 on G, yielding the optimal vertex value v(i). After the last iteration, A2 outputs the computed\nvalues in form of the optimal value vector of G. The runtime of A2 is dominated by the queries to A1 and\nsince A2 makes n queries to A1, A2 is efficient given A1 is.\n8"},{"page":9,"text":"“3 ≤p 2”: Suppose Algorithm A2 computes the optimal value vector of the game G. We give an informal\ndescription of an Algorithm A3 which, using A2, computes the optimal player strategies of G. Our tool\nwill be the result from the previous section that every min (max) strategy, which is locally optimal at every\nmin (max) vertex of G, is an optimal min (max) strategy of G; it therefore suffices for A3 to compute\ntwo locally optimal strategies τopt and σopt for the players. On input of G, A3 runs A2 on G, yielding the\noptimal value vector⃗v of G. For each min vertex i with children j and k, A3 adds (i, j) to the initial empty\nτ if v(j) ≤v(k), else A3 adds (i,k) to τ. Similarly, for each max vertex i with children j and k, A3 adds\n(i, j) to the initial empty σ if v(j) ≥v(k), else A3 adds (i,k) to σ. Following this construction, τopt and\nσopt are locally optimal strategies and since A3 makes one query to A2 and performs O(n) instructions,\nA3 is efficient given A2 is.\n“1 ≤p 3”: From an algorithm A3 which computes a pair of optimal player strategies of the game G, we\nconstruct an algorithm A1 which computes the value of G. On input of G, A1 runs A3 on G to obtain the\noptimal player strategies τopt and σopt of G. In time O(n), A1 then constructs the reduced game Gτopt,σopt\ncorresponding to τopt and σopt. We already argued in the previous section that the value vector (and\nhence the value) of a reduced game can be computed in time polynomial in the size of the game. Since\nA1 makes one call to A3, A1 is efficient given A3 is.\nSSG-TWOKIND (function)\nInput:\nSimple stochastic game G, lacking one vertex kind\nQuestion:\nWhat is the optimal value vector⃗v of G?\nComplexity class: FP\nAlgorithms:\nPapers discussing algorithms for SSG-TWOKIND include [1,5–7,15,16]. In the case that G lacks min ver-\ntices, SSG-TWOKIND can be expressed as the following linear optimization (or programming) problem,\nas was first shown by Derman [6]\nn\n∑\ni=1\nv(i) →min\nsubject to v(n−1) = 0, v(n) = 1 and\nv(i) ≥\n \n \n \n \n \n0\n1 ≤i ≤n\nv(j)\nif i is a max vertex with child j\n1\n2(v(j)+v(k))\nif i is an average vertex with children j and k\nSimilarly, in the case that G lacks max vertices, Derman showed that the optimal value vector of G is the\nunique solution to the linear optimization problem\nn\n∑\ni=1\nv(i) →max\n9"},{"page":10,"text":"subject to v(n−1) = 0, v(n) = 1 and\nv(i) ≤\n \n \n \n \n \n0\n1 ≤i ≤n\nv(j)\nif i is a min vertex with child j\n1\n2(v(j)+v(k))\nif i is an average vertex with children j and k\nKhachian [10] was the first to show that linear programming problems can be solved in time polynomial\nin the bits needed to describe the problem. Since the amount of bits needed to encode any of the above\nformulas is a polynomial function of n [4], we obtain the fruit that the value vector⃗v of G can be computed\nin polynomial time.\nSo called interior point algorithms for linear programming problems, which can move through the fea-\nsible region5 instead than just along its boundary, perform best in practice. According to wikipedia,\nMehrotra’s [13] interior point algorithm is regarded as the fastest, though a worst case boundary is not\navailable. Up to date, there exists no strongly polynomial time algorithm for linear programming prob-\nlems, i.e one that is polynomial in the number of variables of the problem up to order ~4.\nIn the case that G lacks average vertices, the algorithm given in appendix A correctly computes the\noptimal value vector of G. The number of executions of the repeat loop is O(n), as D is static after at\nmost n−2 executions. It follows that the algorithm has quadratic runtime, which is a much better result\nas that obtained for the above linear programming problems.\nSSG-OVV (function)\nInput:\nSimple stochastic game G\nQuestion:\nWhat is the optimal value vector⃗v of G?\nComplexity class: FNP\nWe will first give an informal description of the proof that SSG-OVV∈FNP. Let Ωn ⊃Ωt be a superset of\nthe possible vertex values of G, as discussed in (2). The proof is based on the fact that there exists exactly\none vector ⃗z from the set Ωn\nn – namely the optimal value vector of G – which satisfies |z(i) −v′(i)| ≤\n4−2n/2 for all i ∈V, where⃗v′ is the optimal value vector of the 1/29n-stopping game G′ of G. We deduce\nthat a nondeterministic Turing-machine M for problem SSG-OVV could guess an arbitrary vector⃗z ∈Ωn\nn\nand – assuming M knows⃗v′ – argue in polynomial time that⃗z =⃗v if and only if⃗z satisfies the mentioned\nconstraint. Of course, M cannot compute⃗v′ from scratch, but since M is nondeterministic, we can happily\nlet it, next to ⃗v, also guess ⃗v′. By evaluating (3), M can easily verify whether its guess of ⃗v′ is correct.\nTherefore, letting M guess the optimal value vectors of both G and G′, M is able to conduct a polynomial\ntime verification of both guesses.\nConsidering a formal proof, we first show that the difference between two vertex values of G is either 0\n5The feasible region of a linear programming problem is the set of variable evaluations (vertices) which satisfy the con-\nstraints given in the problem.\n10"},{"page":11,"text":"or has a lower bound δ. Let α,β ∈Ωn. Then either |α−β| = 0 or\n|α−β| =\n \np\nq −p′\nq′\n =\n \npq′ −p′q\nqq′\n ≥1\nqq′ ≥\n1\n4n4n = 4−2n =: δ\nFollowing (4), we further observe that the differences between corresponding vertex values in the games\nG and the associated 1/29n-stopping game G′ are bound below δ/2, i. e. for arbitrary strategies τ and σ\n|vτ,σ(i)−v′τ,σ(i)| ≤2−6n = 4−3n < δ/2\ni ∈V\nOn input of G, a nondeterministic Turing-machine M for problem SSG-OVV first guesses one vector from\nthe set Ωn\nn to be the optimal value vector of G and one vector from the set Ωn′\nn′ to be the optimal value\nvector of G′, where n′ = 9n|E| + n. Let us denote these vectors by the tuple (⃗z,⃗s). If we define that M\naccepts (⃗z,⃗s) if⃗z = IG(⃗z),⃗s = IG′(⃗s) and |z(i)−s(i)| < δ/2 for all i ∈V, then M accepts (⃗z,⃗s) if and only\nif⃗z =⃗v∧⃗s =⃗v′. Also it should be clear that M comes to a conclusion in time polynomial in the number\nof vertices of G, as in particular M is able to construct G′ from G in time O(n2).\nProof. Instead of saying “M accepts (⃗z,⃗s)”, we will just say “M accepts”. “⇒”: If M accepts,⃗s = IG′(⃗s)\nand hence ⃗s =⃗v′. It remains to prove that if M accepts, ⃗z =⃗v holds. Suppose M accepts but⃗z ̸=⃗v. If\n⃗z ̸=⃗v, then |z(i)−v(i)| ≥δ for at least one i ∈V. Since M accepts, |z(i)−s(i)| = |z(i)−v(i)′| < δ/2 for\nall i ∈V. It follows that |v(i) −v(i)′| ≥δ/2 for at least one i ∈V which contradicts the construction of\nG′. “⇐”: Suppose⃗z =⃗v and ⃗s =⃗v′. Then obviously⃗z = IG(⃗z), ⃗s = IG′(⃗s) and |z(i) −s(i)| < δ/2 for all\ni ∈V by construction of G′. Hence M accepts.\nAlgorithms:\nAll algorithms [2,5,7,11,12,15,16] suggested to date for the problem of finding a solution to (3) have ex-\nponential time complexities. The most intuitive of those operates on the basis of the iterative update rule\n⃗vi+1 = IG(⃗vi) and is called successive approximation or value iteration algorithm. As already mentioned,\nthis algorithm is guaranteed to converge to the correct solution only if G is stopping. In her paper about\nalgorithms for simple stochastic games, Condon [5] presents a “worst case” example for the successive\napproximation algorithm in form of a special game graph, where the algorithm takes an exponential\nnumber of updates until it finds the optimal value vector.\nSo called strategy improvement algorithms try to iteratively improve an initial pair of strategies until\nconvergence. A particularly simple algorithm of this class is the one of Hoffman & Karp [11], for which\na worst case running time of O(2n/n) is established. Björklund & Vorobyov’s [2] randomized algorithm\ndating from 2005 has a worst case running time of O(2\n√\nn·log(n)), which as of the authors knowledge, is the\nbest result obtained to date.\nSSG-VALUE* (decision)\nInput:\nSimple stochastic game G, α ∈Ωn\nQuestion:\nIs the value of G > α?\n11"},{"page":12,"text":"Complexity class: NP∩coNP\nSSG-VALUE* is a straightforward extension of the SSG-VALUE Problem, which is defined by α = 1/2.\nUsing the same terminology as for the previous problem, a nondeterministic Turing-machine M for SSG-\nVALUE* first guesses one vector from the set Ωn′\nn′ to be the optimal value vector of the 1/29n-stopping\ngame G′ of G, where n′ = 9n|E|+n. If we denote this vector by⃗s and define that M accepts⃗s if⃗s = IG′(⃗s)\nand s(start) > α, then M accepts⃗s if and only if value > α∧⃗s =⃗v′. It should also be clear that M comes\nto a conclusion in time polynomial in the size of G.\nProof. “⇒”: If M accepts ⃗s, then ⃗s = IG′(⃗s) and therefore ⃗s =⃗v′. It remains to show that if M accepts\n⃗s, value > α. Suppose M accepts ⃗s but value ≤α. If value ≤α then value′ = s(start) ≤α since the\nstopping game G′ always has lower value by construction. Hence M cannot accept ⃗s. “⇐”: Suppose\nvalue > α and ⃗s =⃗v′. Then value ≥α + δ and since, by construction of G′, |value −s(start)| ≤δ/2, it\nfollows that s(start) > α. Since also ⃗s = IG′(⃗s), M accepts ⃗s. Applying a small modification to M by\nletting it accept if s(start) ≤α, M can obviously also decide the complement of SSG-VALUE* .\nAlgorithms:\nThe author is not aware of any algorithm specially tailored for SSG-VALUE*, instead, algorithms for\nthe more general SSG-OVV are used to solve SSG-VALUE*. Though theoretically, algorithms for SSG-\nVALUE* could exploit the circumstance that – depending on the topology of G – not all optimal vertex\nvalues of G would need to be computed in order to solve the problem, as we are mainly concerned with\nthe worst case behavior of such algorithms, the case where the value of G depends on its whole game\ngraph must be assumed. Therefore, the value vector of G must be computed after all.\n5\nConclusion and Open Problems\nWe have seen that the most interesting and also most difficult simple stochastic game problem, that of\ncomputing the optimal value vector, is hard to solve. However, restricting the input to simple stochastic\ngames that lack one vertex kind, we observed that the same problem becomes tractable and can be solved\nrather efficiently. Another result we obtained was that the value vector of reduced games can be computed\nin polynomial time, i.e that given a simple stochastic game and a pair of player strategies, the question\nabout the winning-probabilities of the players is efficient answerable.\nSadly, we were not able to show polynomial time equivalence of SSG-OVV and SSG-VALUE* but we\nwould still like to know what the decision equivalent of SSG-OVV is. Another major open problem is to\nshow completeness of SSG-VALUE (or SSG-OVV) for a specific complexity class, thereby specifying the\nproblem’s exact complexity. In a last word, Condon expressed the possibility that finding an algorithm\nthat seperates simple stochastic games with low value (α < 0,25) from those with high value (α > 0,75)\nmight be of significance in solving the master problem: SSG-VALUE ∈P?\n12"},{"page":13,"text":"Appendix A\nAlgorithm 1 optimalValueVector(G)\ninput simple stochastic game G = (V,E)\noutput optimal value vector⃗v of G\nrequire Vavg = /0\nbegin\nD = {n−1,n},⃗v =⃗0\nrepeat\nfor i ∈V \\D do\nif i is a max vertex with a 1-valued child in D then\nD = D∪{i}, v(i) = 1\nelse if i is a max vertex with two 0-valued children in D then\nD = D∪{i}, v(i) = 0\nelse if i is a min vertex with a 0-valued child in D then\nD = D∪{i}, v(i) = 0\nelse if i is a min vertex with two 1-valued children in D then\nD = D∪{i}, v(i) = 1\nend if\nend for\nuntil D is static\nreturn⃗v\nend\nReferences\n[1] Daniel Anderson. Improved combinatorial algorithms for mean payoff games. Master’s thesis,\nUniversity of Upsala, 2006.\n[2] Henrik Björklund and Sergei G. Vorobyov. Combinatorial structure and randomized subexponential\nalgorithms for infinite games. Theoretical Computer Science, 349(3):347–360, 2005.\n[3] Chatterjee, Majumdar, and Jurdzinski. On nash equilibria in stochastic games. In CSL: 18th Work-\nshop on Computer Science Logic. LNCS, Springer-Verlag, 2004.\n[4] A. Condon. The complexity of stochastic games. Information and Computation, 96:203–224, 1992.\n[5] A. Condon. On algorithms for simple stochastic games. DIMACS Series in Discrete Mathematics\nand Theoretical Computer Science, 13:51–71, 1993.\n[6] Cyrus Derman. Finite State Markovian Decision Processes, volume 67 of Mathematics in Science\nand Engineering. Academic Press, New York, NY, 1970.\n[7] Jerzy A. Filar, T. A. Schultz, F. Thuijsman, and O. J. Vrieze. Nonlinear programming and stationary\nequilibria in stochastic games. Math. Program, 50:227–237, 1991.\n13"},{"page":14,"text":"[8] Gartner and Rust. Simple stochastic games and P-matrix generalized linear complementarity prob-\nlems. FCT: Fundamentals (or Foundations) of Computation Theory, 15, 2005.\n[9] Brendan Juba. On the hardness of simple stochastic games. Master’s thesis, Massachusetts Institute\nof Technology, 2006.\n[10] L. G. Khachian. A polynomial algorithm in linear programming. Soviet Mathematics, 20(191–194),\n1979.\n[11] V. S. Anil Kumar and R. Tripathi. Algorithmic results in simple stochastic games. Technical Report\nTR855, University of Rochester, Computer Science Department, 2004.\n[12] Walter Ludwig. A subexponential randomized algorithm for the simple stochastic game problem.\nInf. Comput., 117(1):151–155, 1995.\n[13] S. Mehrotra. On finding a vertex solution using interior point methods. Linear Algebra and its\nApplications, 152:233–254, 1991.\n[14] L. S. Shalpey. Stochastic games. Proceedings of the National Academy of Sciences, 39:1095–1100,\n1953.\n[15] Rafal Somla. New algorithms for solving simple stochastic games. Electr. Notes Theor. Comput.\nSci, 119(1):51–65, 2005.\n[16] Csaba Szepesvári and Michael L. Littman. Generalized Markov decision processes: Dynamic-\nprogramming and reinforcement-learning algorithms. Technical Report CS-96-11, Brown Univer-\nsity, Providence, RI, 1996.\n14"}]},"section_tree":[],"assets":{"figures":[],"tables":[],"images":[]},"math_expressions":{"inline":[],"block":[{"equation_id":"eq1","equation_number":null,"raw_text":"A simple stochastic game G = (V,E) is a directed graph whose vertices are partitioned into four disjoint","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq2","equation_number":null,"raw_text":"have no children, while all other vertices have exactly two distinct children. Loop edges e = (i,i) are","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq3","equation_number":null,"raw_text":"allowed. In the rest of this paper we assume w.l.o.g. that V = {1,...,n} where n−1 is the 0-sink and n","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq4","equation_number":null,"raw_text":"Given a simple stochastic game G = (V,E) and a strategy τ to be employed by the min player, the reduced","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq5","equation_number":null,"raw_text":"Gτ = (V,Eτ)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq6","equation_number":null,"raw_text":"Eτ = E \\{(i, j) ∈E : i ∈Vmin ∧(i, j) /∈τ}","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq7","equation_number":null,"raw_text":"Gσ = (V,Eσ)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq8","equation_number":null,"raw_text":"Eσ = E \\{(i, j) ∈E : i ∈Vmax ∧(i, j) /∈σ}","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq9","equation_number":null,"raw_text":"Gτ,σ = (V,Eτ,σ)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq10","equation_number":null,"raw_text":"Eτ,σ = E \\{(i, j) ∈E : i ∈Vmin ∪Vmax ∧(i, j) /∈τ∪σ}","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq11","equation_number":null,"raw_text":"v(i) = vτopt,σopt(i)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq12","equation_number":null,"raw_text":"equations: vτ,σ(n) = 1, vτ,σ(i) = 0 for t < i < n and otherwise","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq13","equation_number":null,"raw_text":"vτ,σ(i) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq14","equation_number":null,"raw_text":"⃗vτ,σ = Q⃗vτ,σ +⃗b","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq15","equation_number":null,"raw_text":"(I −Q)⃗vτ,σ =⃗b","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq16","equation_number":null,"raw_text":"where Q ∈Qn×n is related to the topology of Gτ,σ as follows: Qi j = 0 if i > t and otherwise","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq17","equation_number":null,"raw_text":"Qij =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq18","equation_number":null,"raw_text":"bi =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq19","equation_number":null,"raw_text":"if i = n","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq20","equation_number":null,"raw_text":"λi ̸= 1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq21","equation_number":null,"raw_text":"i = 1...n","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq22","equation_number":null,"raw_text":"det(Q−I) ̸= 0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq23","equation_number":null,"raw_text":"rank(Q−I) = n","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq24","equation_number":null,"raw_text":"= Qt ···Qt","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq25","equation_number":null,"raw_text":"Ωt = {p/q ∈Q : 0 ≤p ≤q ≤4t}","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq26","equation_number":null,"raw_text":"can be denoted by vτ,σ(i) = Di/D where D is the determinant of the matrix I−Q and Di is the determinant","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq27","equation_number":null,"raw_text":"v(i) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq28","equation_number":null,"raw_text":"if i = n−1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq29","equation_number":null,"raw_text":"if i = n","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq30","equation_number":null,"raw_text":"⃗v = IG(⃗v)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq31","equation_number":null,"raw_text":"construction rule is as follows: For β = 1/2cn, the so-called β-stopping game G′ adopts all the vertices","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq32","equation_number":null,"raw_text":"of m = cn average vertices which are connected to the vertices i, j and n −1 (the 0-sink) as depicted in","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq33","equation_number":null,"raw_text":"Figure 3: Construction rule for the β-stopping game G′ of G, where β = 1/2m and m = cn. The dashed","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq34","equation_number":null,"raw_text":"i=1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq35","equation_number":null,"raw_text":"subject to v(n−1) = 0, v(n) = 1 and","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq36","equation_number":null,"raw_text":"i=1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq37","equation_number":null,"raw_text":"subject to v(n−1) = 0, v(n) = 1 and","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq38","equation_number":null,"raw_text":"and – assuming M knows⃗v′ – argue in polynomial time that⃗z =⃗v if and only if⃗z satisfies the mentioned","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq39","equation_number":null,"raw_text":"or has a lower bound δ. Let α,β ∈Ωn. Then either |α−β| = 0 or","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq40","equation_number":null,"raw_text":"4n4n = 4−2n =: δ","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq41","equation_number":null,"raw_text":"|vτ,σ(i)−v′τ,σ(i)| ≤2−6n = 4−3n < δ/2","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq42","equation_number":null,"raw_text":"vector of G′, where n′ = 9n|E| + n. Let us denote these vectors by the tuple (⃗z,⃗s). If we define that M","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq43","equation_number":null,"raw_text":"accepts (⃗z,⃗s) if⃗z = IG(⃗z),⃗s = IG′(⃗s) and |z(i)−s(i)| < δ/2 for all i ∈V, then M accepts (⃗z,⃗s) if and only","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq44","equation_number":null,"raw_text":"if⃗z =⃗v∧⃗s =⃗v′. Also it should be clear that M comes to a conclusion in time polynomial in the number","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq45","equation_number":null,"raw_text":"Proof. Instead of saying “M accepts (⃗z,⃗s)”, we will just say “M accepts”. “⇒”: If M accepts,⃗s = IG′(⃗s)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq46","equation_number":null,"raw_text":"and hence ⃗s =⃗v′. It remains to prove that if M accepts, ⃗z =⃗v holds. Suppose M accepts but⃗z ̸=⃗v. If","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq47","equation_number":null,"raw_text":"⃗z ̸=⃗v, then |z(i)−v(i)| ≥δ for at least one i ∈V. Since M accepts, |z(i)−s(i)| = |z(i)−v(i)′| < δ/2 for","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq48","equation_number":null,"raw_text":"G′. “⇐”: Suppose⃗z =⃗v and ⃗s =⃗v′. Then obviously⃗z = IG(⃗z), ⃗s = IG′(⃗s) and |z(i) −s(i)| < δ/2 for all","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq49","equation_number":null,"raw_text":"⃗vi+1 = IG(⃗vi) and is called successive approximation or value iteration algorithm. As already mentioned,","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq50","equation_number":null,"raw_text":"SSG-VALUE* is a straightforward extension of the SSG-VALUE Problem, which is defined by α = 1/2.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq51","equation_number":null,"raw_text":"game G′ of G, where n′ = 9n|E|+n. If we denote this vector by⃗s and define that M accepts⃗s if⃗s = IG′(⃗s)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq52","equation_number":null,"raw_text":"and s(start) > α, then M accepts⃗s if and only if value > α∧⃗s =⃗v′. It should also be clear that M comes","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq53","equation_number":null,"raw_text":"Proof. “⇒”: If M accepts ⃗s, then ⃗s = IG′(⃗s) and therefore ⃗s =⃗v′. It remains to show that if M accepts","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq54","equation_number":null,"raw_text":"⃗s, value > α. Suppose M accepts ⃗s but value ≤α. If value ≤α then value′ = s(start) ≤α since the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq55","equation_number":null,"raw_text":"value > α and ⃗s =⃗v′. Then value ≥α + δ and since, by construction of G′, |value −s(start)| ≤δ/2, it","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq56","equation_number":null,"raw_text":"follows that s(start) > α. Since also ⃗s = IG′(⃗s), M accepts ⃗s. Applying a small modification to M by","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq57","equation_number":null,"raw_text":"input simple stochastic game G = (V,E)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq58","equation_number":null,"raw_text":"require Vavg = /0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq59","equation_number":null,"raw_text":"D = {n−1,n},⃗v =⃗0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq60","equation_number":null,"raw_text":"D = D∪{i}, v(i) = 1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq61","equation_number":null,"raw_text":"D = D∪{i}, v(i) = 0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq62","equation_number":null,"raw_text":"D = D∪{i}, v(i) = 0","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq63","equation_number":null,"raw_text":"D = D∪{i}, v(i) = 1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"}],"equation_refs":[]},"symbol_table":[],"references":{"in_text_markers":[],"bibliography":[]},"claims":[],"flow_graph":{"nodes":[],"edges":[]},"quality":{"text_char_count":34782,"parse_confidence":0.5,"equation_parse_rate_proxy":1.0,"citation_resolution_rate_proxy":0.0,"structure_coverage_proxy":0.0,"asset_coverage_proxy":0.0,"accepted_for_training":true}}