Stochastic  -regular Games

1. Stochastic -regular Games Krishnendu Chatterjee*, Luca de Alfaro**, Thomas A. Henzinger*, Marcin Jurdzinski***, Rupak Majumdar ***** EECS, Berkeley, ** CE, UCSC, *** University of Warwick, **** CS, UCLA • Results • Notion of nonzero-sum games: objectives are not complementary • Concept of rationality in nonzero-sum game: Nash equilibrium • Existence of Nash equilibrium in stochastic games open problem • Results in nonzero-sum stochastic games[CMJ04] • Concurrent games: • Existence of -Nash equilibrium for reachability objectives, • for all >0 • Complexity of computing equilibrium values: NP • Turn-based stochastic games: • Existence of -Nash equilibrium for all Borel objectives, • for all >0 • Existence of Nash equilibrium for -regular objectives • Existence of Nash equilibrium for Borel objectives for turn-based games • Refined notion of equilibria [CHJ04] • Turn-based nonzero-sum games with adverserial external criteria • Relevant from verification perspective • Existence of unique equilibrium payoff profile • Nash equilibrium payoff profile can be several • Computability for -regular objectives • In the same complexity class as zero-sum games • Future directions of research • Relation of the refined notion of equilibria and assume-guarantee verification • Notion of bounded-rationality in concurrent games • Identify the class of objectives for which simple optimal strategies exist in • turn-based stochastic games • Study complexity of verifying several other class objectives relevant from • verification of quantitative properties • Stochastic Games • Games played on graphs with stochastic transitions • Markov decision processes • Games against nature • Turn-based games • Games against adversary • Turn-based stochastic games • Games against nature and adversary • Concurrent Games • Simultaneous games • Objectives • -regular: generalization of classical regular language to infinite strings • Specify properties like reachability, safety, fairness, liveness • Canonical representation of such objectives are • Rabin objectives • Streett objectives • Rabin-chain objectives • Simpler objectives • Reachability- Safety • Computational issues: • Maximal value with which players can win • Example of game graphs: • Results • Complexity of turn-based stochastic games[CdAH04a]: • : NP-complete for Rabin objectives • coNP-complete for Streett objectives • NP Å coNP for Rabin-chain objectives • Previous best known results 3EXPTIME. • Existence of simple optimal strategies • Hence simple controllers for stochastic reactive systems Condon’92: Turn-based stochastic games with Reachability objectives Emerson-Jutla’88: Turn-based games with Rabin objectives Game complexity Objective complexity Turn-based stochastic games with Rabin objectives • Complexity of Concurrent games[CdAH04b] • NP Å coNP for Rabin chain objectives • Previous best known 3EXPTIME • Strategy classification: • Complete the precise requirements of optimal strategies • in terms of memory and randomization • Characterize several interesting properties of optimal • strategies Pl. 1 Pl. 2 Turn-based stochastic game Pl. random deAlfaroMajumdar01: 3EXPTIME algorithm ad Complexity improvement Player 1 actions: a, b Player 2 actions: c, d ac,bd bc Complexity: NP \cap coNP Concurrent game November 18, 2004