Boolean games revisited
Elise Bonzon Marie-Christine Lagasquie-Schiex J´erˆome Lang
Bruno Zanuttini
{bonzon,lagasq,lang}@irit.fr zanutti@info.unicaen.fr
Introduction Boolean games Nash equilibria Dominated strategies Conclusion 1 Introduction 2 Boolean games 3 Nash equilibria 4 Dominated strategies 5 Conclusion
1 Introduction 2 Boolean games
3 Nash equilibria 4 Dominated strategies 5 Conclusion
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
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
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
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
Introduction Boolean games Nash equilibria Dominated strategies Conclusion 1 Introduction 2 Boolean games 3 Nash equilibria 4 Dominated strategies 5 Conclusion
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),
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.
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
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)
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.
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)
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.
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)
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.
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)
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.
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)
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.
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)
Expressed much more compactly by Boolean game G = (A, V , π, Φ):
A = {1, . . . , n}, V = {C1, . . . , Cn},
∀i ∈ {1, . . . , n}, πi = {Ci}, and
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)
∀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
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)
∀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
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)
∀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
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)
∀i , i has 2 possible strategies: si1 = {Ci} and si2 = {Ci}
the strategy Ci is a winning strategy for i .
Introduction Boolean games Nash equilibria Dominated strategies Conclusion 1 Introduction 2 Boolean games 3 Nash equilibria 4 Dominated strategies 5 Conclusion
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)
s−i denotes the projection of S on A \ {i }
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)
s−i denotes the projection of S on A \ {i }
S = {C1C2C3}. s−1= (C2, C3) ; s−2= (C1, C3) ; s−3= (C1, C2)
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)
s−i denotes the projection of S on A \ {i }
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)
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
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)
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).
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)
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
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.
Introduction Boolean games Nash equilibria Dominated strategies Conclusion
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
∃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
Introduction Boolean games Nash equilibria Dominated strategies Conclusion
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
∃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
Introduction Boolean games Nash equilibria Dominated strategies Conclusion
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
Simplification of the characterization
S is a pure-strategy Nash equilibrium for G if and only if
S |=^
i
(ϕi∨ (¬∃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.
Recall that: ∃1 : ϕ1= ¬b and ∃2 : ϕ2= ¬a.
G has 2 PNE:
S = ab. We have S |= ϕ1∧ (¬∃2 : ϕ2).
Introduction Boolean games Nash equilibria Dominated strategies Conclusion
Complexity
Nash equilibriumDeciding 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.
1 Introduction 2 Boolean games
3 Nash equilibria
4 Dominated strategies 5 Conclusion
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)
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
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)
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).
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)
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
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)
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).
Introduction Boolean games Nash equilibria Dominated strategies Conclusion
Characterization
Strict dominanceStrategy si strictly dominates strategy si0 if and only if:
si |= (¬∃ − i : ¬ϕi) and
si0 |= (¬∃ − i : ϕi).
Example : G = (A, V , π, φ) with A = {1, 2}, V = {a, b}, π1= {a},
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.
Weak dominance
Strategy si weakly dominates strategy si0 if and only if:
(ϕi)s0
i |= (ϕi)si and (ϕi)si 6|= (ϕi)si0.
Example : G = (A, V , π, φ) with A = {1, 2}, V = {a, b}, π1= {a},
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|= ⊥
Introduction Boolean games Nash equilibria Dominated strategies Conclusion
Complexity
Nash equilibriumDeciding whether a given strategy si0 is weakly dominated is
Σp2-complete.
Hardness holds even if ϕi is restricted to be in DNF.
1 Introduction 2 Boolean games
3 Nash equilibria 4 Dominated strategies
Introduction Boolean games Nash equilibria Dominated strategies Conclusion
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;
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).