Modal Logic for
Mixed Strategies
in Games
Games
Strategies
Expected utility
Nash equilibrium
Game logics
Syntax and semantics
Proof system
Completeness and
provability
1/19
Modal Logic for Mixed Strategies in Games
Joshua Sack and Wiebe van der Hoek
August 14, 2014
Modal Logic for
Mixed Strategies
in Games
Game
Definition (Game)
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Strategies
Expected utility
Nash equilibrium
A (strategic form) game is a tuple
G = (Ag , {Πi }i∈Ag , {ui }i∈Ag ),
Game logics
Syntax and semantics
Proof system
Completeness and
provability
2/19
G where
Ag - finite set of players,
Πi - finite set of pure strategies for agent i ∈ Ag , and
ui : Π → Q - utility function, assigning agent i’s payoff
to each pure strategy profile,
Definition (Pure profile)
A pure strategy profile is a tuple
(π1 , . . . , πn ),
such that each πi ∈ Πi . Let Π be all pure strategy profiles.
Modal Logic for
Mixed Strategies
in Games
Mixed strategies
Definition (Mixed strategy)
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Expected utility
Nash equilibrium
Πi is agent i’s set of pure strategies. A mixed strategy is a
probability mass function
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Syntax and semantics
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Completeness and
provability
3/19
σi : Πi → [0, 1],
Let Σi be all mixed strategies for player i.
Definition (Mixed profile)
A mixed (strategy) profile tuple is a tuple (σ1 , . . . , σn ) for
each player i ∈ Ag .
Definition (Mixed strategy function)
A mixed strategy function is a function σ : Π → [0, 1], where
Π is the set of pure strategy profiles.
Modal Logic for
Mixed Strategies
in Games
Games
Strategies
Expected utility
Nash equilibrium
Game logics
Syntax and semantics
Proof system
Completeness and
provability
4/19
Correlated profiles
There exist mixed strategy functions that are not equivalent
to mixed profiles: correlated strategies.
Example
σ
Ha
Ta
Hb
0.2
0.2
Tb
0.2
0.4
Here, whether a chooses Ha with probability 1/2 or
probability 1/3 depends on b’s strategy.
We will call a mixed strategy function a mixed profile only if
the mixed strategy function is equivalent to a mixed profile
(and are hence uncorrelated).
Modal Logic for
Mixed Strategies
in Games
Expected utility
Games
Strategies
Expected utility
Nash equilibrium
Game logics
Syntax and semantics
Proof system
Completeness and
provability
The utility function ui : Π → R can be extended from pure
to mixed strategy profiles by
X
ui (σ) =
σ(π)ui (π).
π∈Π
5/19
Modal Logic for
Mixed Strategies
in Games
Nash equilibrium
Games
Strategies
Expected utility
Nash equilibrium
Game logics
Given a mixed strategy profile σ and a mixed strategy ρi ,
denote by (ρi , σ−i ) the strategy profile
Syntax and semantics
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Completeness and
provability
6/19
(σ1 , . . . , σi−1 , ρi , σi+1 , . . . , σn ).
Definition
A Nash equilibrium is a (mixed or pure) strategy profile σ
such that for any agent i and any strategy ρi
ui (σ) ≥ ui (ρi , σ−i ).
Modal Logic for
Mixed Strategies
in Games
Games
Strategies
Expected utility
Nash equilibrium
Game logics
Syntax and semantics
Proof system
Completeness and
provability
Nash equilibria in matching pennies
Ag = {a, b}
Πi = {Hi , Ti } (heads and tails of player i’s coin)
ui : Π → {−1, 1} is given by the following chart:
Ha
Ta
Hb
+1, −1
−1, +1
Tb
−1, +1
+1, −1
Pure strategy Nash equilibria: none
Mixed strategy Nash equilibria: one
each player plays 1/2 for both their strategies.
[Corresponding mixed strategy function: uniform
probability function (each of the 4 pure profiles gets
probability 14 )]
7/19
Modal Logic for
Mixed Strategies
in Games
Games
Strategies
Expected utility
Nash equilibrium
Game logics
Syntax and semantics
Proof system
Completeness and
provability
Background on some game logics
To give a sense of the diversity of game logics:
Marc Pauly’s dissertation, “Logic for social software”,
ILLC, University of Amsterdam, 2001.
defines “Game logic” and “Coalition logic” with formulas
describing what certain (groups of) agents can enforce.
R. Alur, T. Henzinger, O. Kupferman. Alternating-Time
temporal logic. Journal of the ACM. 2002.
describes powers of coalitions over time, using concurrent
game models.
J. Halpern: Substantive Rationality and Backward
Induction. Games and Economic Behavior, 37:425-435,
2001.
gives a logic for a fixed game that is defined on Kripke
models whose states are labelled with (pure) strategy profiles.
8/19
Modal Logic for
Mixed Strategies
in Games
Language
Given a game G, let
Games
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Syntax and semantics
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Completeness and
provability
9/19
t ::= aP(π) | t + t
ϕ ::= t ≥ a | ¬ϕ | ϕ ∧ ϕ | [G ]ϕ | [@i ]ϕ | [Ai ]ϕ | [=i ]ϕ
where a ∈ Q, π ∈ Π, i ∈ Ag , and G ⊆ Ag .
e ⊆ Σ - a set of mixed profiles (as functions). Formulas
Fix Σ
e (relations restricted to Σ
e × Σ).
e
evaluated (only) on σ, τ ∈ Σ
Pn
Pn
σ k=1 qk P(π k ) ≥ r iff
k=1 qk σ(πk ) ≥ r
σ [G ]ϕ
iff τ ϕ whenever
for each i ∈ G , σi = τi
σ [@i ]ϕ
iff τ ϕ whenever ui (σ) < ui (τ )
σ [Ai ]ϕ
iff τ ϕ whenever ui (σ) > ui (τ )
σ [=i ]ϕ
iff τ ϕ whenever ui (σ) = ui (τ )
J. Sack & W. van der Hoek. Modal Logic for Mixed
Strategies. Studia Logica, 2014.
Modal Logic for
Mixed Strategies
in Games
Language
Given a game G, let
Games
Strategies
Expected utility
Nash equilibrium
Game logics
Syntax and semantics
Proof system
Completeness and
provability
9/19
t ::= aP(π) | t + t
ϕ ::= t ≥ a | ¬ϕ | ϕ ∧ ϕ | [G ]ϕ | [@i ]ϕ | [Ai ]ϕ | [=i ]ϕ
where a ∈ Q, π ∈ Π, i ∈ Ag , and G ⊆ Ag .
e ⊆ Σ - a set of mixed profiles (as functions). Formulas
Fix Σ
e (relations restricted to Σ
e × Σ).
e
evaluated (only) on σ, τ ∈ Σ
Pn
Pn
σ k=1 qk P(π k ) ≥ r iff
k=1 qk σ(πk ) ≥ r
σ [G ]ϕ
iff τ ϕ whenever
for each i ∈ G , σi = τi
σ [@i ]ϕ
iff τ ϕ whenever ui (σ) < ui (τ )
σ [Ai ]ϕ
iff τ ϕ whenever ui (σ) > ui (τ )
σ [=i ]ϕ
iff τ ϕ whenever ui (σ) = ui (τ )
J. Sack & W. van der Hoek. Modal Logic for Mixed
Strategies. Studia Logica, 2014.
Modal Logic for
Mixed Strategies
in Games
Meaning of [G ]
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Syntax and semantics
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Completeness and
provability
[G ]ϕ means that
ϕ is true at any mixed profile where those not in G
potentially switch to different strategies.
[Ag \ {i}]ϕ means that
ϕ is true whenever i potentially switches to a different
strategy.
[∅]ϕ means that
ϕ is true at all profiles.
10/19
Modal Logic for
Mixed Strategies
in Games
Games
Strategies
Expected utility
Nash equilibrium
Abbreviations
Expressing that τ is the mixed profile:
^
def
τ =
P(π) = τ (π)
π∈Π
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Completeness and
provability
11/19
The utility for i is term
def
ui =
X
ui (π)P(π)
π∈Π
The probability i has for playing pure strategy πi is a term:
X
def
Pi (π i ) =
{P(ρ) | ρ ∈ Π, ρi = πi }
Expressing that τi is i’s (mixed) strategy
^
def
τi =
(Pi (π i ) = τi (πi ))
πi ∈Πi
Modal Logic for
Mixed Strategies
in Games
Games
Strategies
Expected utility
Nash equilibrium
Expressing Nash Equilibrium
Definition (Best response)
Given a mixed strategy profile σ, i’s strategy is a best
response if for every formula ϕ, we have
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12/19
σ |= ϕ → [(Ag \ {i})]hvi iϕ.
Given a specific σ we define
bri (σ) ≡ σ → [(Ag \ {i})]hvi iσ.
A Nash equilibrium is a mixed strategy profile, such that
everyone’s strategy is a best response. For each σ, define
^
Nash(σ) ≡
bri (σ).
i∈Ag
So σ is a Nash equilibrium in G if and only if |= Nash(σ)
Modal Logic for
Mixed Strategies
in Games
Axioms
Games
Strategies
Expected utility
Nash equilibrium
Game logics
Syntax and semantics
Proof system
Completeness and
provability
Basic axioms
Global modality axioms
Probability Bounds and Restrictions
Inequality axioms
13/19
Modal Logic for
Mixed Strategies
in Games
Games
Strategies
Expected utility
Nash equilibrium
Game logics
Syntax and semantics
Proof system
Completeness and
provability
Axioms
Basic axioms
Classical Logic Tautologies
[?](ϕ → ψ) → ([?]ϕ → [?]ψ), with ? ∈ {G , @i , Ai , =i }.
[∗]ϕ → ϕ, with ∗ ∈ {G , =i }
(reflexivity)
[i]ϕ → [G ]ϕ (i ∈ G )
[@i ]ϕ → [=i ][@i ]ϕ
(neutrality of =i )
[Ai ]ϕ → [=i ][Ai ]ϕ
±Pi (π i ) ≥ q → [i] ± Pi (π i ) ≥ q ([i] fixes i’s strategy)
±ui ≥ q → [=i ] ± ui ≥ q
([=i ] fixes i’s utility)
ui ≥ q → [@i ]ui > q
(strictness of @i )
ui ≤ q → [Ai ]ui < q
Global modality axioms
Probability Bounds and Restrictions
Inequality axioms
13/19
Modal Logic for
Mixed Strategies
in Games
Axioms
Games
Strategies
Expected utility
Nash equilibrium
Game logics
Syntax and semantics
Proof system
Completeness and
provability
Basic axioms
Global modality axioms
V
[∅](σ → ϕ) → [∅]( i∈G σ i → hG iϕ)
(If ϕ is true at σ, then hG iϕ is true in any τ that agrees
with σ on the strategies of those in G .)
[∅](σ → ϕ) → [∅](ui = ui (σ) → h=i iϕ)
[∅](σ → ϕ) → [∅](ui < ui (σ) → h@i iϕ)
[∅](σ → ϕ) → [∅](ui > ui (σ) → hAi iϕ)
Probability Bounds and Restrictions
Inequality axioms
13/19
Modal Logic for
Mixed Strategies
in Games
Axioms
Games
Strategies
Expected utility
Nash equilibrium
Game logics
Syntax and semantics
Proof system
Completeness and
provability
Basic axioms
Global modality axioms
Probability Bounds and Restrictions
P(π)
P ≥0
π∈Π P(π) = 1
e
¬σ for each σ 6∈ Σ
Inequality axioms
13/19
e ⊆ Σ)
(Σ
Modal Logic for
Mixed Strategies
in Games
Games
Strategies
Expected utility
Nash equilibrium
Game logics
Syntax and semantics
Proof system
Completeness and
provability
13/19
Axioms
Basic axioms
Global modality axioms
Probability Bounds and Restrictions
Inequality axioms
(Permutation)
Pn
Pn
k=1 qjk P(π jk ) ≥ q
k=1 qk P(π k ) ≥ q →
(Adding and deleting zero terms)
t ≥ q ↔ t + 0P(π k+1 ) ≥ q
(Adding
coefficients) P
Pn
n
0
0
q
P(π
k
k) ≥ q ∧
k=1
k=1 qk P(π k ) ≥ q →
Pn
0
0
(q
+
q
)P(π
)
≥
(q
+
q
)
k
k
k=1 k
(Multiplying my a non-negative constant)
t ≥ q ↔ dt ≥ dq where d > 0
(Monotonicity)
(t ≥ q) → (t > q 0 ) where q > q 0
Modal Logic for
Mixed Strategies
in Games
Rules
Games
Strategies
Expected utility
Nash equilibrium
Modus Ponens
A`ϕ→ψ A`ϕ
A`ψ
Game logics
Syntax and semantics
Proof system
Completeness and
provability
Necessitation
`ϕ
` [?]ϕ
(? ∈ {G , =i , @i , Ai })
Monotonicity
`ϕ
A`ϕ
14/19
Modal Logic for
Mixed Strategies
in Games
Pseudo modalities and Existence rule
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Proof system
Completeness and
provability
Pseudo-modalities: Let si ∈ {G , @i , Ai , =i } ∪ L. Define
[(s1 , . . . , sn )]ϕ by
def
[()]ϕ = ϕ
def
[(ψ, s2 , . . . , sn )]ϕ = ψ → [(s2 , . . . , sn )]ϕ
def
[(a, s2 , . . . , sn )]ϕ = [a][(s2 , . . . , sn )]ϕ
Existence rule: For s = (s1 , . . . , sn )
A ` [s](P(π) 6= q) for all q ∈ Q, q 6= p
A ` [s](P(π) = p)
A probability value exists for each pure profile and agent.
15/19
Modal Logic for
Mixed Strategies
in Games
More infinitary rules
Guarded necessitation
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Completeness and
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e such that τi = σi for all i ∈ G
` τ → ϕ for all τ ∈ Σ
` σ → [G ]ϕ
(Compare with theVaxiom
[∅](σ → ϕ) → [∅]( i∈G σ i → hG iϕ))
e such that ui (τ ) = ui (σ)
` τ → ϕ for all τ ∈ Σ
` σ → [=i ]ϕ
e such that ui (τ ) > ui (σ)
` τ → ϕ for all τ ∈ Σ
` σ → [@i ]ϕ
e such that ui (τ ) < ui (σ)
` τ → ϕ for all τ ∈ Σ
` σ → [Ai ]ϕ
16/19
Modal Logic for
Mixed Strategies
in Games
Strong completeness
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Completeness and
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17/19
Theorem (Strong Completeness)
If A |= ϕ then A ` ϕ
Modal Logic for
Mixed Strategies
in Games
Nash equilibrium
Games
Strategies
Expected utility
Nash equilibrium
Recall that
Game logics
Nash(σ) ≡
Syntax and semantics
Proof system
Completeness and
provability
18/19
^
σ → [(Ag \ {i})]hvi iσ.
i∈Ag
Theorem
e contains a Nash equilibrium if and only if for some (or all)
Σ
e
τ ∈ Σ,
e \ {τ }} ` Nash(τ ).
{¬[∅]Nash(σ) | σ ∈ Σ
Modal Logic for
Mixed Strategies
in Games
Remarks
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Strategies
Expected utility
Nash equilibrium
Game logics
Syntax and semantics
Proof system
Completeness and
provability
One game fixes both the language and the model
Soundness and strong completeness is established
Possible further developments might involve
dynamic removal of strategy profiles
non-trivial epistemics (agents may assume certain
strategies of others or might be mistaken about their
own strategy)
extensive game form
imperfect information
19/19