The Complexity of Admissibility in -Regular Games R. Brenguier - PowerPoint PPT Presentation

The Complexity of Admissibility in ω -Regular Games R. Brenguier J.-F. Raskin M. Sassolas Romain Brenguier (ULB) Admissibility Thursday, July 17, 2014 1 / 23

Multiplayer non-zero-sum games Romain Brenguier (ULB) Admissibility Thursday, July 17, 2014 2 / 23

Models of rationality Nash equilibria � no player has interest in deviating. Regret minimization � players prefer moves that would induce less regret had they known the other players strategy. Elimination of dominated strategies � players eliminate “bad” strategies Romain Brenguier (ULB) Admissibility Thursday, July 17, 2014 3 / 23

Iterative elimination of dominated strategies σ is dominated by σ ′ (wrt S ): for all strategies of the other players (in S ), if σ wins, then σ ′ wins. and for some strategy of the other players (in S ), σ loses while σ ′ wins. c v 1 :-) a v 0 e b v 2 :-( f Should player � play a or b ? Romain Brenguier (ULB) Admissibility Thursday, July 17, 2014 4 / 23

Iterative elimination of dominated strategies σ is dominated by σ ′ (wrt S ): for all strategies of the other players (in S ), if σ wins, then σ ′ wins. and for some strategy of the other players (in S ), σ loses while σ ′ wins. c v 1 :-) a d v 0 b v 2 :-( f Romain Brenguier (ULB) Admissibility Thursday, July 17, 2014 4 / 23

Iterative elimination of dominated strategies σ is dominated by σ ′ (wrt S ): for all strategies of the other players (in S ), if σ wins, then σ ′ wins. and for some strategy of the other players (in S ), σ loses while σ ′ wins. c v 1 :-) a d v 0 e b v 2 :-( f Romain Brenguier (ULB) Admissibility Thursday, July 17, 2014 4 / 23

Iterative elimination of dominated strategies σ is dominated by σ ′ (wrt S ): for all strategies of the other players (in S ), if σ wins, then σ ′ wins. and for some strategy of the other players (in S ), σ loses while σ ′ wins. , , a a , , b a 1 2 3 b a , , a b b , , a b a , , b b a , , b , , Romain Brenguier (ULB) Admissibility Thursday, July 17, 2014 5 / 23

Iterative elimination of dominated strategies σ is dominated by σ ′ (wrt S ): for all strategies of the other players (in S ), if σ wins, then σ ′ wins. and for some strategy of the other players (in S ), σ loses while σ ′ wins. , , a a , , b a 1 2 3 b a , , a b b , , a b a , , b b a , , b , , Romain Brenguier (ULB) Admissibility Thursday, July 17, 2014 5 / 23

Iterative elimination of dominated strategies σ is dominated by σ ′ (wrt S ): for all strategies of the other players (in S ), if σ wins, then σ ′ wins. and for some strategy of the other players (in S ), σ loses while σ ′ wins. , , a a , , b a 1 2 3 b a , , a b b , , a b a , , b b a , , b , , Romain Brenguier (ULB) Admissibility Thursday, July 17, 2014 5 / 23

Iterative elimination of dominated strategies σ is dominated by σ ′ (wrt S ): for all strategies of the other players (in S ), if σ wins, then σ ′ wins. and for some strategy of the other players (in S ), σ loses while σ ′ wins. , , , , b a 1 2 3 b a , , a b b , , a b a , , b a , , , , Romain Brenguier (ULB) Admissibility Thursday, July 17, 2014 5 / 23

Iterative elimination of dominated strategies σ is dominated by σ ′ (wrt S ): for all strategies of the other players (in S ), if σ wins, then σ ′ wins. and for some strategy of the other players (in S ), σ loses while σ ′ wins. , , , , b a 1 2 3 b a , , a b b , , a b a , , b a , , , , Romain Brenguier (ULB) Admissibility Thursday, July 17, 2014 5 / 23

Iterative elimination of dominated strategies σ is dominated by σ ′ (wrt S ): for all strategies of the other players (in S ), if σ wins, then σ ′ wins. and for some strategy of the other players (in S ), σ loses while σ ′ wins. , , , , b a 1 2 3 b a , , a b b , , a a , , b a , , , , Romain Brenguier (ULB) Admissibility Thursday, July 17, 2014 5 / 23

Our setting Turn based games on graphs. Muller objectives: ρ ∈ Win i iff Inf ( ρ ) ∈ F . Weak Muller objectives: ρ ∈ Win i iff Occ ( ρ ) ∈ F . Romain Brenguier (ULB) Admissibility Thursday, July 17, 2014 6 / 23

Our setting Turn based games on graphs. Muller objectives: ρ ∈ Win i iff Inf ( ρ ) ∈ F . Weak Muller objectives: ρ ∈ Win i iff Occ ( ρ ) ∈ F . Dominance: σ ′ i ≻ S n σ i if σ ′ i dominates σ i w.r.t S n . Iterative admissibility: S 0 i = S i (all strategies) S n +1 � ∃ σ ′ = S n � i ∈ S n i , σ ′ i \ � i ≻ S n σ i � σ i . i Set of iteratively admissible strategies: S ∗ = � n ∈ N S n Romain Brenguier (ULB) Admissibility Thursday, July 17, 2014 6 / 23

Our setting Turn based games on graphs. Muller objectives: ρ ∈ Win i iff Inf ( ρ ) ∈ F . Weak Muller objectives: ρ ∈ Win i iff Occ ( ρ ) ∈ F . Dominance: σ ′ i ≻ S n σ i if σ ′ i dominates σ i w.r.t S n . Iterative admissibility: S 0 i = S i (all strategies) S n +1 � ∃ σ ′ = S n � i ∈ S n i , σ ′ i \ � i ≻ S n σ i � σ i . i Set of iteratively admissible strategies: S ∗ = � n ∈ N S n “Admissibility in Infinite Games” [Berwanger, STACS’07]: S ∗ is well defined and is reached after a finite number of iterations. Romain Brenguier (ULB) Admissibility Thursday, July 17, 2014 6 / 23

Values [Berwanger, STACS’07] :-) Romain Brenguier (ULB) Admissibility Thursday, July 17, 2014 7 / 23

Values [Berwanger, STACS’07] :-) Winning Losing Romain Brenguier (ULB) Admissibility Thursday, July 17, 2014 7 / 23

Values [Berwanger, STACS’07] :-) Winning Losing Romain Brenguier (ULB) Admissibility Thursday, July 17, 2014 7 / 23

Values [Berwanger, STACS’07] :-) Winning Potentially Surely Losing Losing Romain Brenguier (ULB) Admissibility Thursday, July 17, 2014 7 / 23

Values [Berwanger, STACS’07] :-) Winning Val = 1 Potentially Surely Losing Losing Val = − 1 Val = 0 no strategy profile σ P in S n such that h · outcome ( σ P ) winning for ⇒ Val n player i = i ( h ) = − 1; ∃ σ i ∈ S n i such that ∀ σ − i ∈ S n − i , h · outcome ( σ P ) winning for player i ⇒ Val n = i ( h ) = 1; otherwise Val n i ( h ) = 0; Romain Brenguier (ULB) Admissibility Thursday, July 17, 2014 7 / 23

Outline Introduction 1 Setting 2 Simple Safety 3 Muller objectives 4 Conclusion 5 Romain Brenguier (ULB) Admissibility Thursday, July 17, 2014 8 / 23

Outline Introduction 1 Setting 2 Simple Safety 3 Muller objectives 4 Conclusion 5 Romain Brenguier (ULB) Admissibility Thursday, July 17, 2014 9 / 23

Simple Safety Safety objective: avoid Bad states Simple safety: Bad states are absorbing Bad � v 0 v 1 v 2 v 3 v 4 v 5 v 6 v 7 Bad � Romain Brenguier (ULB) Admissibility Thursday, July 17, 2014 10 / 23

A local notion of dominance In simple safety games the rule to never decrease one’s own value is sufficient for admissibility. 0 ∗ 1 ω 0 ∗ − 1 ω 0 ω Romain Brenguier (ULB) Admissibility Thursday, July 17, 2014 11 / 23

Algorithm n := 0 ; T − 1 := ∅ i repeat forall the s ∈ V do if there is a winning strategy for player i from s in G \ T n − 1 then Val n i ( s ) := 1 ; else if there is no winning run for player i from s in G \ T n − 1 then Val n i ( s ) := − 1 ; else Val n i ( s ) := 0 ; forall the i ∈ P do i := T n − 1 T n ∪ { ( s , s ′ ) ∈ E | s ∈ V i ∧ Val n i ( s ) > Val n i ( s ′ ) } i n := n + 1 i = T n − 1 until ∀ i ∈ P . T n ; i Romain Brenguier (ULB) Admissibility Thursday, July 17, 2014 12 / 23

The algorithm in action Bad � v 0 v 1 v 2 v 3 v 4 v 5 v 6 v 7 Bad � Romain Brenguier (ULB) Admissibility Thursday, July 17, 2014 13 / 23

The algorithm in action Bad � v 0 v 1 v 2 v 3 v 4 v 5 v 6 v 7 Bad � Which states have value − 1 for player � ? Which states have value 1 for player � ? Romain Brenguier (ULB) Admissibility Thursday, July 17, 2014 13 / 23

The algorithm in action Bad � 0 , 0 − 1 , 0 0 , 0 − 1 , 0 − 1 , 0 v 0 v 1 v 2 v 3 v 4 Val � , Val � v 5 v 6 v 7 − 1 , 0 − 1 , − 1 − 1 , 0 Bad � Romain Brenguier (ULB) Admissibility Thursday, July 17, 2014 13 / 23

The algorithm in action Bad � − 1 , 0 0 , 0 − 1 , 0 − 1 , 0 0 , 0 Val � , Val � v 0 v 1 v 2 v 3 v 4 − 1 , 0 v 5 v 6 − 1 , − 1 v 7 − 1 , 0 Bad � On which transition would a player decrease its own value? Romain Brenguier (ULB) Admissibility Thursday, July 17, 2014 13 / 23

The algorithm in action Bad � 0 , 0 − 1 , 0 0 , 0 − 1 , 0 − 1 , 0 v 0 v 1 v 2 v 3 v 4 Val � , Val � v 5 v 6 v 7 − 1 , 0 − 1 , − 1 − 1 , 0 Bad � Romain Brenguier (ULB) Admissibility Thursday, July 17, 2014 13 / 23

The algorithm in action Bad � v 0 v 1 v 2 v 3 v 4 v 5 v 6 v 7 Bad � Romain Brenguier (ULB) Admissibility Thursday, July 17, 2014 13 / 23

The algorithm in action Bad � v 0 v 1 v 2 v 3 v 4 v 5 v 6 v 7 Bad � Romain Brenguier (ULB) Admissibility Thursday, July 17, 2014 13 / 23

The algorithm in action Bad � − 1 , 0 0 , 1 − 1 , 0 − 1 , 0 v 0 v 1 v 2 v 3 v 4 0 , 1 − 1 , 0 v 5 v 6 v 7 − 1 , − 1 − 1 , 0 Bad � Romain Brenguier (ULB) Admissibility Thursday, July 17, 2014 13 / 23