Deterministic Graphical Games Revisited

Andersson, Daniel; Hansen, Kristoffer Arnsfelt; Miltersen, Peter Bro; Sørensen, Troels Bjerre

doi:10.1093/logcom/exq001

Cited by 15 publications

(29 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The corresponding positional structures, game forms, and games are called Deterministic Graphical ones. They were introduced by Washburn [32] in 1990 and since then are frequent in the literature; see, for example, [32], [10], [4], [6,Section 12], [5], [1], [2], [7], [19]. Only two-person zero-sum DG games were considered in [32], [10], [2], while our main results are related to the two-person, but not necessarily zero-sum, DGMS games.…”

Section: Modeling Positional Games By Digraphsmentioning

confidence: 99%

“…Consider a a two-person zero-sum DGMS game modeled by a digraph G. The proof follows the same line as the proof of Proposition 2 until we start to analyze a component G j = (V j , E j ) from which every move comes to a terminal loop. Obviously, this is a BD subgame and, according to [2], it has a subgame perfect saddle point than can be determined in "almost linear" time (|E j | + |V j | + |A j | log |A j |), where A j is the set of terminal loops reachable from V j . These strategies defined on V j will be extended by backward induction to the whole vertex-set V .…”

Section: Proof Of Propositionmentioning

confidence: 99%

See 1 more Smart Citation

Backward induction in presence of cycles

Gurvich

2018

Journal of Logic and Computation

View full text Add to dashboard Cite

For the classical backward induction algorithm, the input is an arbitrary n-person positional game with perfect information modeled by a finite acyclic directed graph (digraph) and the output is a profile (x1, . . . , xn) of pure positional strategies that form some special subgame perfect Nash equilibrium. We extend this algorithm to work with digraphs that may have directed cycles. Each digraph admits a unique partition into strongly connected components, which will be treated as the outcomes of the game. Such a game will be called a deterministic graphical multistage(DGMS) game. If we identify the outcomes corresponding to all strongly connected components, except terminal positions, we obtain the so-called deterministic graphical(DG) games, which are frequent in the literature. The outcomes of a DG game are all terminal positions and one special outcome c that is assigned to all infinite plays. We modify the backward induction procedure to adapt it for the DGMS games. However, by doing so, we lose two important properties: the modified algorithm always outputs a Nash equilibrium (NE) only when n = 2 and, even in this case, this NE may be not subgame perfect.(Yet, in the zero-sum case it is.) The lack of these two properties is not a fault of the algorithm, just (subgame perfect) Nash equilibria in pure positional strategies may fail to exist in the considered game.

show abstract

Section: Modeling Positional Games By Digraphsmentioning

confidence: 99%

Section: Proof Of Propositionmentioning

confidence: 99%

Backward induction in presence of cycles

Gurvich

2018

Journal of Logic and Computation

View full text Add to dashboard Cite

show abstract

“…Compute an optimal strategy x for Player 1 in G using linear-time retrograde analysis [1]. Let x be the interpretation of x as a strategy in G obtained by restricting x to the vertices of Player 1 in G. By construction, from any starting vertex v, Player 1 can ensure reaching the terminal in G if and only if x ensures a non-zero probability of reaching a terminal in G. Let v be any vertex with positive value in G. Player 1 has a safe strategy that ensures a non-zero probability of reaching a terminal from v, specifically, the optimal strategy.…”

Section: Proof Of Theoremmentioning

confidence: 99%

The Complexity of Solving Stochastic Games on Graphs

Andersson

Miltersen

2009

Lecture Notes in Computer Science

View full text Add to dashboard Cite

Abstract. We consider some well-known families of two-player zero-sum perfect-information stochastic games played on finite directed graphs. Generalizing and unifying results of Liggett and Lippman, Zwick and Paterson, and Chatterjee and Henzinger, we show that the following tasks are polynomial-time (Turing) equivalent.-Solving stochastic parity games, -Solving simple stochastic games, -Solving stochastic terminal-payoff games with payoffs and probabilities given in unary, -Solving stochastic terminal-payoff games with payoffs and probabilities given in binary, -Solving stochastic mean-payoff games with rewards and probabilities given in unary, -Solving stochastic mean-payoff games with rewards and probabilities given in binary, -Solving stochastic discounted-payoff games with discount factor, rewards and probabilities given in binary. It is unknown whether these tasks can be performed in polynomial time. In the above list, "solving" may mean either quantitatively solving a game (computing the values of its positions) or strategically solving a game (computing an optimal strategy for each player). In particular, these two tasks are polynomial-time equivalent for all the games listed above. We also consider a more refined notion of equivalence between quantitatively and strategically solving a game. We exhibit a linear time algorithm that given a simple stochastic game or a terminal-payoff game and the values of all positions of that game, computes a pair of optimal strategies. Consequently, for any restriction one may put on the simple stochastic game model, quantitatively solving is polynomial-time equivalent to strategically solving the resulting class of games.

show abstract

“…The f -regions and f -strategies can be expressed in terms of deterministic games as they do not depend on what happens once a random vertex is reached. We can thus use the results of [AHMS08] to compute them in time O(|E| log * (|V |). In order to compute the f -values, we build a Markov chain M f designed to mimic the behaviour of G when the players use their f -strategies.…”

Section: Permutation Strategiesmentioning

confidence: 99%

Solving Simple Stochastic Games with Few Random Vertices

Gimbert¹,

Horn²

2009

Logical Methods in Computer Science

View full text Add to dashboard Cite

Abstract. Simple stochastic games are two-player zero-sum stochastic games with turnbased moves, perfect information, and reachability winning conditions.We present two new algorithms computing the values of simple stochastic games. Both of them rely on the existence of optimal permutation strategies, a class of positional strategies derived from permutations of the random vertices. The "permutation-enumeration" algorithm performs an exhaustive search among these strategies, while the "permutationimprovement" algorithm is based on successive improvements,à la Hoffman-Karp.Our algorithms improve previously known algorithms in several aspects. First they run in polynomial time when the number of random vertices is fixed, so the problem of solving simple stochastic games is fixed-parameter tractable when the parameter is the number of random vertices. Furthermore, our algorithms do not require the input game to be transformed into a stopping game. Finally, the permutation-enumeration algorithm does not use linear programming, while the permutation-improvement algorithm may run in polynomial time.

show abstract

Deterministic Graphical Games Revisited

Cited by 15 publications

References 20 publications

Backward induction in presence of cycles

Backward induction in presence of cycles

The Complexity of Solving Stochastic Games on Graphs

Solving Simple Stochastic Games with Few Random Vertices

Contact Info

Product

Resources

About