Games Miheer Dewaskar Chennai Mathematical Institute April 27, - PowerPoint PPT Presentation

Parity Games Odd Wins 1 Even Wins Even 3 Odd 5 2 6 5 1 2 6 2 3 3 6 Finite game 6 5 Every loop has max priority even Extension to infinite plays π = 1 2 1 5 6 Stack = 1 5 6 9 / 19

Parity Games Odd Wins 1 Even Wins Even 3 Odd 5 2 6 5 1 2 6 2 3 3 6 Finite game 6 5 Every loop has max priority even Extension to infinite plays π = 1 2 1 5 6 5 Stack = 1 5 6 5 9 / 19

Parity Games Odd Wins 1 Even Wins Even 3 Odd 5 2 6 5 1 2 6 2 3 3 6 Finite game 6 5 Every loop has max priority even Extension to infinite plays π = 1 2 1 5 6 5 Stack = 1 5 9 / 19

Parity Games Odd Wins 1 Even Wins Even 3 Odd 5 2 6 5 1 2 6 2 3 3 6 Finite game 6 5 Every loop has max priority even Extension to infinite plays π = 1 2 1 5 6 5 2 Stack = 1 5 2 9 / 19

Parity Games Odd Wins 1 Even Wins Even 3 Odd 5 2 6 5 1 2 6 2 3 3 6 Finite game 6 5 Every loop has max priority even Extension to infinite plays π = 1 2 1 5 6 5 2 3 Stack = 1 5 2 3 9 / 19

Parity Games Odd Wins 1 Even Wins Even 3 Odd 5 2 6 5 1 2 6 2 3 3 6 Finite game 6 5 Every loop has max priority even Extension to infinite plays π = 1 2 1 5 6 5 2 3 6 Stack = 1 5 2 3 6 9 / 19

Parity Games Odd Wins 1 Even Wins Even 3 Odd 5 2 6 5 1 2 6 2 3 3 6 Finite game 6 5 Every loop has max priority even Every eliminated cycle has Extension to infinite plays max priority even π = 1 2 1 5 6 5 2 3 6 5 Stack = 1 5 2 3 6 5 9 / 19

Parity Games Odd Wins 1 Even Wins Even 3 Odd 5 2 6 5 1 2 6 2 3 3 6 Finite game 6 5 Every loop has max priority even Every eliminated cycle has Extension to infinite plays max priority even π = 1 2 1 5 6 5 2 3 6 5 Stack = 1 5 . . . 9 / 19

Parity Games Odd Wins 1 Even Wins Even 3 Odd 5 2 6 5 1 2 6 2 3 3 6 Finite game 6 5 Every loop has max priority even Every eliminated cycle has Extension to infinite plays max priority even π = 1 2 1 5 6 5 2 3 6 5 Hence max Inf priority in π is Even Stack = 1 5 . . . 9 / 19

Parity Games Better Algorithms Marcin Jurdzinski and Jens Vöge. “A discrete strategy improvement algorithm for solving parity games”. In: Computer Aided Verification . Springer, 2000, pp. 202–215 Upper bound 1 : O � ( n / d ) d � 1 see also Friedmann, “Exponential Lower Bounds for Solving Infinitary Payoff Games and Linear Programs”. 10 / 19

Parity Games Better Algorithms Marcin Jurdzinski and Jens Vöge. “A discrete strategy improvement algorithm for solving parity games”. In: Computer Aided Verification . Springer, 2000, pp. 202–215 Upper bound 1 : O � ( n / d ) d � Marcin Jurdzinski, Mike Paterson, and Uri Zwick. “A Deterministic Subexponential Algorithm for Solving Parity Games”. In: SIAM Journal on Computing 38.4 (Jan. 2008), pp. 1519–1532 n O ( √ n ) 1 see also Friedmann, “Exponential Lower Bounds for Solving Infinitary Payoff Games and Linear Programs”. 10 / 19

Outline Finite Duration Games Infinite Duration Games Parity Games Mean Payoff Games Simple Stochastic Games 11 / 19

Mean Payoff Games Payoffs 3 b -2 a 2 -2 -1 Max c Min -1 12 / 19

Mean Payoff Games Payoffs ( ab ) ω 3 b -2 a 2 -2 -1 Max c Min -1 12 / 19

Mean Payoff Games Payoffs ( ab ) ω 3 b -2 a 2 -2 -1 Max c Min -1 12 / 19

Mean Payoff Games Payoffs ( ab ) ω 3 b -2 a 2 -2 -1 Max c Min -1 12 / 19

Mean Payoff Games Payoffs ( ab ) ω 3 b -2 a 2 -2 -1 Max c Min -1 12 / 19

Mean Payoff Games Payoffs ( ab ) ω 3 b -2 a 2 -2 -1 Max c Min -1 12 / 19

Mean Payoff Games Payoffs ( ab ) ω 3 b -2 a 2 -2 -1 Max c Min -1 12 / 19

Mean Payoff Games Payoffs ( ab ) ω a 3 b -2 a 2 -2 -1 Max c Min -1 12 / 19

Mean Payoff Games Payoffs ( ab ) ω a − 2 − − → b − 2 3 b 1 -2 a 2 -2 -1 Max c Min -1 12 / 19

Mean Payoff Games Payoffs ( ab ) ω a − 2 → b +3 − − − − → a − 2 + 3 3 b 2 -2 a 2 -2 -1 Max c Min -1 12 / 19

Mean Payoff Games Payoffs ( ab ) ω a − 2 → b +3 → a − 2 − − − − − − → b − 2 + 3 − 2 3 b 3 -2 a 2 -2 -1 Max c Min -1 12 / 19

Mean Payoff Games Payoffs ( ab ) ω a − 2 → b +3 → a − 2 → b +3 − − − − − − − − → a − 2 + 3 − 2 + 3 3 b 4 -2 a 2 -2 -1 Max c Min -1 12 / 19

Mean Payoff Games Payoffs ( ab ) ω a − 2 → b +3 → a − 2 → b +3 → a − 2 − − − − − − − − − − → b − 2 + 3 − 2 + 3 − 2 3 b 5 -2 a 2 -2 -1 Max c Min -1 12 / 19

Mean Payoff Games Payoffs ( ab ) ω a − 2 → b +3 → a − 2 → b +3 → a − 2 → b +3 − − − − − − − − − − − − → a − 2 + 3 − 2 + 3 − 2 + 3 3 b 6 -2 a 2 -2 -1 Max c Min -1 12 / 19

Mean Payoff Games Payoffs ( ab ) ω Min pays 1 2 units to Max a − 2 → b +3 → a − 2 → b +3 → a − 2 → b +3 − − − − − − − − − − − − → a . . . − 2 + 3 − 2 + 3 − 2 + 3 → 1 3 ∼ n ( − 2+3) b 2 n 6 2 -2 a 2 -2 -1 Max c Min -1 12 / 19

Mean Payoff Games Payoffs ( ab ) ω Min pays 1 2 units to Max a − 2 → b +3 → a − 2 → b +3 → a − 2 → b +3 − − − − − − − − − − − − → a . . . − 2 + 3 − 2 + 3 − 2 + 3 → 1 3 ∼ n ( − 2+3) b 2 n 6 2 -2 a 2 ( acb ) ω -2 -1 Max c Min -1 12 / 19

Mean Payoff Games Payoffs ( ab ) ω Min pays 1 2 units to Max a − 2 → b +3 → a − 2 → b +3 → a − 2 → b +3 − − − − − − − − − − − − → a . . . − 2 + 3 − 2 + 3 − 2 + 3 → 1 3 ∼ n ( − 2+3) b 2 n 6 2 -2 a 2 ( acb ) ω -2 -1 Max c Min -1 12 / 19

Mean Payoff Games Payoffs ( ab ) ω Min pays 1 2 units to Max a − 2 → b +3 → a − 2 → b +3 → a − 2 → b +3 − − − − − − − − − − − − → a . . . − 2 + 3 − 2 + 3 − 2 + 3 → 1 3 ∼ n ( − 2+3) b 2 n 6 2 -2 a 2 ( acb ) ω -2 -1 Max c Min -1 12 / 19

Mean Payoff Games Payoffs ( ab ) ω Min pays 1 2 units to Max a − 2 → b +3 → a − 2 → b +3 → a − 2 → b +3 − − − − − − − − − − − − → a . . . − 2 + 3 − 2 + 3 − 2 + 3 → 1 3 ∼ n ( − 2+3) b 2 n 6 2 -2 a 2 ( acb ) ω -2 -1 Max c Min -1 12 / 19

Mean Payoff Games Payoffs ( ab ) ω Min pays 1 2 units to Max a − 2 → b +3 → a − 2 → b +3 → a − 2 → b +3 − − − − − − − − − − − − → a . . . − 2 + 3 − 2 + 3 − 2 + 3 → 1 3 ∼ n ( − 2+3) b 2 n 6 2 -2 a 2 ( acb ) ω -2 -1 Max c Min -1 12 / 19

Mean Payoff Games Payoffs ( ab ) ω Min pays 1 2 units to Max a − 2 → b +3 → a − 2 → b +3 → a − 2 → b +3 − − − − − − − − − − − − → a . . . − 2 + 3 − 2 + 3 − 2 + 3 → 1 3 ∼ n ( − 2+3) b 2 n 6 2 -2 a 2 ( acb ) ω -2 -1 Max c Min -1 12 / 19

Mean Payoff Games Payoffs ( ab ) ω Min pays 1 2 units to Max a − 2 → b +3 → a − 2 → b +3 → a − 2 → b +3 − − − − − − − − − − − − → a . . . − 2 + 3 − 2 + 3 − 2 + 3 → 1 3 ∼ n ( − 2+3) b 2 n 6 2 -2 a 2 ( acb ) ω -2 -1 Max c Min -1 12 / 19

Mean Payoff Games Payoffs ( ab ) ω Min pays 1 2 units to Max a − 2 → b +3 → a − 2 → b +3 → a − 2 → b +3 − − − − − − − − − − − − → a . . . − 2 + 3 − 2 + 3 − 2 + 3 → 1 3 ∼ n ( − 2+3) b 2 n 6 2 -2 a 2 ( acb ) ω -2 -1 Max c Min -1 12 / 19

Mean Payoff Games Payoffs ( ab ) ω Min pays 1 2 units to Max a − 2 → b +3 → a − 2 → b +3 → a − 2 → b +3 − − − − − − − − − − − − → a . . . − 2 + 3 − 2 + 3 − 2 + 3 → 1 3 ∼ n ( − 2+3) b 2 n 6 2 -2 a 2 ( acb ) ω -2 -1 Max c Min a -1 12 / 19