1. 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 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  -regular Games Krishnendu Chatterjee*, Luca de Alfaro**, Thomas A. Henzinger*, Marcin Jurdzinski***, Rupak Majumdar **** * EECS, Berkeley, ** CE, UCSC, *** University of Warwick, **** CS, UCLA November 18, 2004 Pl. 1 Pl. 2 Pl. random Turn-based stochastic game ac,bd bc Player 1 actions: a, b Player 2 actions: c, d ad Concurrent game Emerson-Jutla’88: Turn-based games with Rabin objectives Condon’92: Turn-based stochastic games with Reachability objectives Turn-based stochastic games with Rabin objectives Game complexity Objective complexity 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 deAlfaroMajumdar01: 3EXPTIME algorithm Complexity: NP \cap coNP Complexity improvement