Boolean Games Revisited

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Boolean games revisited

Elise Bonzon Marie-Christine Lagasquie-Schiex J´erˆome Lang

Bruno Zanuttini

{bonzon,lagasq,lang}@irit.fr [email protected]

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Introduction Boolean games Nash equilibria Dominated strategies Conclusion 1 Introduction 2 Boolean games 3 Nash equilibria 4 Dominated strategies 5 Conclusion

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1 Introduction 2 Boolean games

3 Nash equilibria 4 Dominated strategies 5 Conclusion

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Introduction Boolean games Nash equilibria Dominated strategies Conclusion

Introduction

Boolean games as introduced in Harrenstein, Van der Hoek, Meyer, Witteveen (2001, 2004)

2-players games with p binary decision variables

Each decision variable is controlled by only one player Zero-sum games

Static games

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2-players games with p binary decision variables Each decision variable is controlled by only one player

Zero-sum games Static games

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Introduction

2-players games with p binary decision variables Each decision variable is controlled by only one player Zero-sum games

Static games

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2-players games with p binary decision variables Each decision variable is controlled by only one player Zero-sum games

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n prisoners (denoted by 1, . . . , n).

The same proposal is made to each of them:

“Either you cover your accomplices (Ci, i = 1, . . . , n) or you

denounce them (¬Ci, i = 1, . . . , n).”

Denouncing makes you freed while your partners will be sent to prison (except those who denounced you as well; these ones will be freed as well),

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Introduction

Boolean games

Nash equilibria Dominated strategies Conclusion

Boolean n-players version of prisoners’ dilemma

Normal form for n = 3:

3 : C3 H H H H H 1 2 C2 C2 C1 (1, 1, 1) (0, 1, 0) C1 (1, 0, 0) (1, 1, 0) 3 : C3 H H H H H 1 2 C2 C2 C1 (0, 0, 1) (0, 1, 1) C1 (1, 0, 1) (1, 1, 1)

n prisoners : n-dimension matrix, therefore 2nn-tuples must be specified.

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Normal form for n = 3: 3 : C3 H H H H H 1 2 C2 C2 C1 (1, 1, 1) (0, 1, 0) C1 (1, 0, 0) (1, 1, 0) 3 : C3 H H H H H 1 2 C2 C2 C1 (0, 0, 1) (0, 1, 1) C1 (1, 0, 1) (1, 1, 1)

n prisoners : n-dimension matrix, therefore 2nn-tuples must be

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Introduction

Boolean games

Boolean n-players version of prisoners’ dilemma

Expressed much more compactly by Boolean game G = (A, V , π, Φ): A = {1, . . . , n},

V = {C1, . . . , Cn},

∀i ∈ {1, . . . , n}, πi = {Ci}, and ϕi = (C1∧ C2∧ . . . Cn) ∨ ¬Ci.

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Expressed much more compactly by Boolean game G = (A, V , π, Φ): A = {1, . . . , n},

V = {C1, . . . , Cn},

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Introduction

Boolean games

Boolean n-players version of prisoners’ dilemma

Expressed much more compactly by Boolean game G = (A, V , π, Φ):

A = {1, . . . , n},

V = {C1, . . . , Cn},

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A = {1, . . . , n}, V = {C1, . . . , Cn},

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Introduction

Boolean games

Boolean n-players version of prisoners’ dilemma

A = {1, . . . , n}, V = {C1, . . . , Cn},

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A = {1, . . . , n}, V = {C1, . . . , Cn},

∀i ∈ {1, . . . , n}, πi = {Ci}, and

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Introduction

Boolean games

Boolean n-players version of prisoners’ dilemma

∀i , i has 2 possible strategies: si1 = {Ci} and si2 = {Ci}

the strategy Ci is a winning strategy for i .

8 strategy profiles for G

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the strategy Ci is a winning strategy for i . 8 strategy profiles for G

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Introduction

Boolean games

Boolean n-players version of prisoners’ dilemma

8 strategy profiles for G

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s−i denotes the projection of S on A \ {i }

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Introduction Boolean games

Nash equilibria

Dominated strategies Conclusion

Boolean n-players version of prisoners’ dilemma

S = {C1C2C3}. s−1= (C2, C3) ; s−2= (C1, C3) ; s−3= (C1, C2)

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Nash equilibria

Boolean n-players version of prisoners’ dilemma

A pure-strategy Nash equilibrium (PNE) is a strategy profile such as each player’s strategy is an optimal response to other players’ strategies. S = {s1, . . . , sn} is a PNE iff

∀i ∈ {1, . . . , n}, ∀s0 i ∈ 2

πi_{, u}

i(S ) ≥ ui(s−i, si0).

2 pure-strategy Nash equilibria: C1C2C3and C1C2C3

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∀i ∈ {1, . . . , n}, ∀s0 i ∈ 2

πi_{, u}

i(S ) ≥ ui(s−i, si0).

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Nash equilibria

Boolean n-players version of prisoners’ dilemma

∀i ∈ {1, . . . , n}, ∀s0 i ∈ 2

πi_{, u}

i(S ) ≥ ui(s−i, si0).

2 pure-strategy Nash equilibria: C1C2C3and C1C2C3

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Characterization

A strategy profile S is a pure-strategy Nash equilibrium for a Boolean-game G iff for all i , either

S |= ϕi

or s−i |= ¬ϕi

Example : G = (A, V , π, φ) with A = {1, 2}, V = {a, b}, π1= {a},

π2= {b}, ϕ1= a ∧ ¬b, ϕ2= ¬a ∧ b.

G has 2 PNE:

S = ab. We have S |= ϕ1and (s−2= a) |= ¬ϕ2.

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Characterization

∃i : ϕi = projection of ϕi on variables of π−i.

∃i : ϕi: obtained by forgetting in ϕi all variables controlled by i . Example : G = (A, V , π, φ) with A = {1, 2}, V = {a, b}, π1= {a}, π2= {b}, ϕ1= a ∧ ¬b, ϕ2= ¬a ∧ b.

∃1 : ϕ1= (> ∧ ¬b) ∨ (⊥ ∧ ¬b) = ¬b ∃2 : ϕ2= (¬a ∧ >) ∨ (¬a ∧ ⊥) = ¬a

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∃i : ϕi: obtained by forgetting in ϕi all variables controlled by i .

Example : G = (A, V , π, φ) with A = {1, 2}, V = {a, b}, π1= {a}, π2= {b}, ϕ1= a ∧ ¬b, ϕ2= ¬a ∧ b.

∃1 : ϕ1= (> ∧ ¬b) ∨ (⊥ ∧ ¬b) = ¬b

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Characterization

π2= {b}, ϕ1= a ∧ ¬b, ϕ2= ¬a ∧ b.

∃1 : ϕ1= (> ∧ ¬b) ∨ (⊥ ∧ ¬b) = ¬b ∃2 : ϕ2= (¬a ∧ >) ∨ (¬a ∧ ⊥) = ¬a

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π2= {b}, ϕ1= a ∧ ¬b, ϕ2= ¬a ∧ b.

∃1 : ϕ1= (> ∧ ¬b) ∨ (⊥ ∧ ¬b) = ¬b ∃2 : ϕ2= (¬a ∧ >) ∨ (¬a ∧ ⊥) = ¬a

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Characterization

π2= {b}, ϕ1= a ∧ ¬b, ϕ2= ¬a ∧ b.

∃1 : ϕ1= (> ∧ ¬b) ∨ (⊥ ∧ ¬b) = ¬b

∃2 : ϕ2= (¬a ∧ >) ∨ (¬a ∧ ⊥) = ¬a

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Simplification of the characterization

S is a pure-strategy Nash equilibrium for G if and only if

S |=^

i

(ϕi∨ (¬∃i : ϕi))

π2= {b}, ϕ1= a ∧ ¬b, ϕ2= ¬a ∧ b.

Recall that: ∃1 : ϕ1= ¬b and ∃2 : ϕ2= ¬a.

G has 2 PNE:

S = ab. We have S |= ϕ1∧ (¬∃2 : ϕ2).

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Complexity

Nash equilibrium

Deciding whether there is a pure-strategy Nash equilibrium in a Boolean

game is Σp₂-complete. Completeness holds even under the restriction to

two-players zero-sum games.

Goals in DNF

Let G be a Boolean game. If every ϕi is in DNF, then deciding whether

there is a pure-strategy Nash equilibrium is NP-complete.

Completeness holds even if both we restrict the number of players to 2 and one player controls only one variable.

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3 Nash equilibria

4 Dominated strategies 5 Conclusion

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Introduction Boolean games Nash equilibria

Dominated strategies

Conclusion

Boolean n-players version of prisoners’ dilemma

A strategy si for player i strictly dominates another strategy si0 if it does strictly better than it against all possible combinations of other players’ strategies: ∀s−i ∈ 2π−i, ui(si0, s−i) < ui(si, s−i).

si weakly dominates si0 if it does at least as well against all possible combinations of other players’ strategies, and strictly better against at least one: ∀s−i ∈ 2π−i, ui(si0, s−i) ≤ ui(si, s−i) and ∃s−i ∈ 2π−i s.t. ui(si0, s−i) < ui(si, s−i).

Elimination of dominated strategies: C1C2C3

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A strategy si for player i strictly dominates another strategy si0 if it

does strictly better than it against all possible combinations of other players’ strategies: ∀s−i ∈ 2π−i, ui(si0, s−i) < ui(si, s−i).

si weakly dominates si0 if it does at least as well against all possible combinations of other players’ strategies, and strictly better against at least one: ∀s−i ∈ 2π−i, ui(si0, s−i) ≤ ui(si, s−i) and ∃s−i ∈ 2π−i s.t. ui(si0, s−i) < ui(si, s−i).

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Introduction Boolean games Nash equilibria

Dominated strategies

Conclusion

Boolean n-players version of prisoners’ dilemma

si weakly dominates si0 if it does at least as well against all possible

combinations of other players’ strategies, and strictly better against at least one: ∀s−i ∈ 2π−i, ui(si0, s−i) ≤ ui(si, s−i) and ∃s−i ∈ 2π−i

s.t. ui(si0, s−i) < ui(si, s−i).

Elimination of dominated strategies: C1C2C3

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si weakly dominates si0 if it does at least as well against all possible

combinations of other players’ strategies, and strictly better against at least one: ∀s−i ∈ 2π−i, ui(si0, s−i) ≤ ui(si, s−i) and ∃s−i ∈ 2π−i

s.t. ui(si0, s−i) < ui(si, s−i).

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Characterization

Strict dominance

Strategy si strictly dominates strategy si0 if and only if:

si |= (¬∃ − i : ¬ϕi) and

s_i0 |= (¬∃ − i : ϕi).

π2= {b}, ϕ1= a, ϕ2= ¬a ∧ b.

Strategy s1= a strictly dominates s10 = a. We compute: ∃ − 1 : ϕ1= a

and ∃ − 1 : ¬ϕ1= ¬a, and we have:

s1|= ¬(¬a).

s₁0 |= ¬a.

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Weak dominance

Strategy si weakly dominates strategy si0 if and only if:

(ϕi)s0

i |= (ϕi)si and (ϕi)si 6|= (ϕi)s_i0.

π2= {b}, ϕ1= a, ϕ2= ¬a ∧ b.

Strategy s2= b strictly dominates s20 = b. We compute : (ϕ2)s0

2= ⊥,

(ϕ2)s2 = ¬a, and we have:

⊥ |= ¬a ¬a 6|= ⊥

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Complexity

Nash equilibrium

Deciding whether a given strategy s_i0 is weakly dominated is

Σp₂-complete.

Hardness holds even if ϕi is restricted to be in DNF.

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3 Nash equilibria 4 Dominated strategies

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Conclusion

Extension of Harrenstein and al.’s Boolean games: Arbitrary number of players;

Non zero-sum games;

Characterization of Nash equilibria and dominated strategies; Computational complexity of the related problems;

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The companion paper (PRICAI’06) considers extended Boolean games with ordinal preferences represented by prioritized goals and CP-nets with binary variables;

Computing mixed strategy Nash equilibria for Boolean games; Defining and studying dynamic Boolean games (with complete or incomplete information).