Strategy Complexity of Concurrent Safety Games

© Krishnendu Chatterjee, Rasmus Ibsen-Jensen, and Kristoffer Arnsfelt Hansen; licensed under Creative Commons License CC-BY. We consider two player, zero-sum, finite-state concurrent reachability games, played for an infinite number of rounds, where in every round, each player simultaneously and independently of the other players chooses an action, whereafter the successor state is determined by a probability distribution given by the current state and the chosen actions. Player 1 wins iff a designated goal state is eventually visited. We are interested in the complexity of stationary strategies measured by their patience, which is defined as the inverse of the smallest non-zero probability employed. Our main results are as follows: We show that: (i) the optimal bound on the patience of optimal and -optimal strategies, for both players is doubly exponential; and (ii) even in games with a single non-absorbing state exponential (in the number of actions) patience is necessary.

[1]  Thomas A. Henzinger,et al.  The Control of Synchronous Systems , 2000, CONCUR.

[2]  P. Ramadge,et al.  Supervisory control of a class of discrete event processes , 1987 .

[3]  Thomas A. Henzinger,et al.  Alternating-time temporal logic , 1997, Proceedings 38th Annual Symposium on Foundations of Computer Science.

[4]  Nicolas Vieille,et al.  Computing uniformly optimal strategies in two-player stochastic games , 2009 .

[5]  Krishnendu Chatterjee,et al.  Qualitative concurrent parity games , 2011, TOCL.

[6]  Peter Bro Miltersen,et al.  Solving simple stochastic games with few coin toss positions , 2012, ESA.

[7]  Stef Tijs,et al.  Fictitious play applied to sequences of games and discounted stochastic games , 1982 .

[8]  Kristoffer Arnsfelt Hansen,et al.  Strategy Complexity of Concurrent Stochastic Games with Safety and Reachability Objectives , 2015, ArXiv.

[9]  Thomas A. Henzinger,et al.  Concurrent reachability games , 2007, Theor. Comput. Sci..

[10]  Peter Bro Miltersen,et al.  A near-optimal strategy for a heads-up no-limit Texas Hold'em poker tournament , 2007, AAMAS '07.

[11]  M. Dufwenberg.  Game theory. , 2011, Wiley interdisciplinary reviews. Cognitive science.

[12]  Amir Pnueli,et al.  On the synthesis of a reactive module , 1989, POPL '89.

[13]  L. Shapley,et al.  Stochastic Games* , 1953, Proceedings of the National Academy of Sciences.

[14]  Krishnendu Chatterjee,et al.  Concurrent games with tail objectives , 2006, Theor. Comput. Sci..

[15]  Thomas A. Henzinger,et al.  The Control of Synchronous Systems, Part II , 2001, CONCUR.

[16]  Kousha Etessami,et al.  Recursive Concurrent Stochastic Games , 2008, Log. Methods Comput. Sci..

[17]  T. Parthasarathy.  Discounted and positive stochastic games , 1971 .

[18]  Aranyak Mehta,et al.  Playing large games using simple strategies , 2003, EC '03.

[19]  J. Filar,et al.  Competitive Markov Decision Processes , 1996 .

[20]  E. Rowland.  Theory of Games and Economic Behavior , 1946, Nature.

[21]  C. J. Himmelberg,et al.  Existence of p-equilibrium and optimal stationary strategies in stochastic games , 1976 .

[22]  Kristoffer Arnsfelt Hansen,et al.  Winning Concurrent Reachability Games Requires Doubly-Exponential Patience , 2009, 2009 24th Annual IEEE Symposium on Logic In Computer Science.

[23]  Kristoffer Arnsfelt Hansen,et al.  The Complexity of Solving Reachability Games Using Value and Strategy Iteration , 2013, Theory of Computing Systems.

[24]  Krishnendu Chatterjee,et al.  The Complexity of Ergodic Mean-payoff Games , 2014, ICALP.

[25]  O. J. Vrieze,et al.  On equilibria in repeated games with absorbing states , 1989 .