Stochastic ?-regular Games

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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 gamesCMJ04
• Concurrent games
• Existence of ?-Nash equilibrium for reachability
  objectives,
• for all ?gt0
• Complexity of computing equilibrium values NP
• Turn-based stochastic games
• Existence of ?-Nash equilibrium for all Borel
  objectives,
• for all ?gt0
• Existence of Nash equilibrium for ?-regular
  objectives
• Existence of Nash equilibrium for Borel
  objectives for turn-based games

• 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

• Results
• Complexity of turn-based stochastic
  gamesCdAH04a
• 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

Condon92 Turn-based stochastic games
with Reachability objectives
Emerson-Jutla88 Turn-based games with Rabin
objectives
Game complexity
Objective complexity
Turn-based stochastic games with Rabin objectives

• Complexity of Concurrent gamesCdAH04b
• 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
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