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

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The Complexity of Admissibility in ω-Regular Games

R. Brenguier

J.-F. Raskin

M. Sassolas

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Multiplayer non-zero-sum games

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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

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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. v0 v1 v2 :-) :-( a b c e f Should player play a or b?

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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. v0 v1 v2 :-) :-( a b c d f

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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. v0 v1 v2 :-) :-( a b c d e f

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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. 1 2 3 a b , , , , , , , , , , , , , , a b a b a b a b a b a b a b

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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. 1 2 3 a b , , , , , , , , , , , , , , a b a b a b a b a b a b a b

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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. 1 2 3 a b , , , , , , , , , , , , , , a b a b a b a b a b a b a b

Romain Brenguier (ULB) Admissibility Thursday, July 17, 2014 5 / 23

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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. 1 2 3 a b , , , , , , , , , , , , , , a b b a b a b a b a

Romain Brenguier (ULB) Admissibility Thursday, July 17, 2014 5 / 23

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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. 1 2 3 a b , , , , , , , , , , , , , , a b b a b a b a b a

Romain Brenguier (ULB) Admissibility Thursday, July 17, 2014 5 / 23

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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. 1 2 3 a b , , , , , , , , , , , , , , a b a b a b a b a

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Our setting

Turn based games on graphs. Muller objectives: ρ ∈ Wini iff Inf(ρ) ∈ F. Weak Muller objectives: ρ ∈ Wini iff Occ(ρ) ∈ F.

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Our setting

Turn based games on graphs. Muller objectives: ρ ∈ Wini iff Inf(ρ) ∈ F. Weak Muller objectives: ρ ∈ Wini iff Occ(ρ) ∈ F. Dominance: σ′

i ≻Sn σi if σ′ i dominates σi w.r.t Sn.

Iterative admissibility: S0

i = Si

(all strategies) Sn+1

i

= Sn

i \

σi
∃σ′

i ∈ Sn i , σ′ i ≻Sn σi

.

Set of iteratively admissible strategies: S∗ =

n∈N Sn

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Our setting

Turn based games on graphs. Muller objectives: ρ ∈ Wini iff Inf(ρ) ∈ F. Weak Muller objectives: ρ ∈ Wini iff Occ(ρ) ∈ F. Dominance: σ′

i ≻Sn σi if σ′ i dominates σi w.r.t Sn.

Iterative admissibility: S0

i = Si

(all strategies) Sn+1

i

= Sn

i \

σi
∃σ′

i ∈ Sn i , σ′ i ≻Sn σi

.

Set of iteratively admissible strategies: S∗ =

n∈N Sn

“Admissibility in Infinite Games” [Berwanger, STACS’07]: S∗ is well defined and is reached after a finite number of iterations.

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Values [Berwanger, STACS’07]

:-)

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Values [Berwanger, STACS’07]

:-) Winning Losing

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Values [Berwanger, STACS’07]

:-) Winning Losing

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Values [Berwanger, STACS’07]

:-) Winning Surely Losing Potentially Losing

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Values [Berwanger, STACS’07]

:-) Winning Surely Losing Potentially Losing Val = 0 Val = −1 Val = 1 no strategy profile σP in Sn such that h · outcome(σP) winning for player i = ⇒ Valn

i (h) = −1;

∃σi ∈ Sn

i such that ∀σ−i ∈ Sn −i, h · outcome(σP) winning for player i

= ⇒ Valn

i (h) = 1;

therwise Valn

i (h) = 0;

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Outline

1

Introduction

2

Setting

3

Simple Safety

4

Muller objectives

5

Conclusion

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Outline

1

Introduction

2

Setting

3

Simple Safety

4

Muller objectives

5

Conclusion

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Simple Safety

Safety objective: avoid Bad states Simple safety: Bad states are absorbing v4 v3 v2 v6 v7 v1 v0 v5 Bad Bad

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A local notion of dominance

In simple safety games the rule to never decrease one’s own value is sufficient for admissibility. 0∗1ω 0∗ − 1ω 0ω

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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 Valn

i (s) := 1;

else if there is no winning run for player i from s in G \ T n−1 then Valn

i (s) := −1;

else Valn

i (s) := 0;

forall the i ∈ P do T n

i := T n−1 i

∪ {(s, s′) ∈ E | s ∈ Vi ∧ Valn

i (s) > Valn i (s′)}

n := n + 1 until ∀i ∈ P. T n

i = T n−1 i

;

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