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Epistemic Analysis of Strategic Games with Arbitrary Strategy Sets

Krzysztof R. Apt CWI and University of Amsterdam

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Executive Summary

We provide an epistemic analysis of arbitrary strategic games based on the possibility correspondences. It calls for the use of transfinite iterations of the corresponding operators. This approach is based on Tarski’s Fixpoint Theorem. It applies both to the notions of rationalizability and the iterated elimination of strictly dominated strategies.

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Main Result

Assume an arbitrary strategic game. RAT(φ): each player i uses property φi to select his strategy (‘each player i is φi-rational’). Suppose each φi is monotonic. Then the following sets of strategy profiles coincide: those that the players choose in the states in which RAT(φ) is common knowledge, those that the players choose in the states in which RAT(φ) is true and is common belief, those that remain after the iterated elimination of the strategies that are not φi-optimal. The latter requires transfinite iterations.

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Preliminaries: Operators

(D, ⊆ ): a complete lattice with the largest element ⊤, T: an operator on (D, ⊆ ), i.e., T : D → D. T is monotonic if ∀G1, G2 (G1 ⊆ G2 ⇒ T(G1) ⊆ T(G2)). G is a fixpoint of T if G = T(G).

Transfinite iterations of T on D:

T 0 := ⊤, T α+1 := T(T α),

for limit ordinal β, T β :=

α<β T α,

T ∞ :=

α∈Ord T α.

Tarski’s Theorem For a monotonic operator T on (D, ⊆ ),

T ∞ is the largest fixpoint of T.

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Strategic games

Fix a strategic game H := (T1, . . ., Tn, p1, . . ., pn). So Ti is the set of strategies of player i and

pi : T1 × . . . × Tn → R

his payoff. A restriction of H is a sequence (S1, . . ., Sn) such that

Si ⊆ Ti for all i.

Restrictions of H ordered by the componentwise set inclusion

(S1, . . ., Sn) ⊆ (S′

1, . . ., S′ n) iff Si ⊆ S′ i for all i

form a complete lattice with H the largest element.

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Rationality

Basic assumption: each player is rational. What does it mean? Some natural possibilities: he does not choose a strategy strictly dominated by another pure/mixed strategy, he chooses only best replies to the (beliefs about the) strategies of the opponents.

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Rationality, ctd

Let us generalize. Given player i in H := (T1, . . ., Tn, p1, . . ., pn) we formalize his notion of rationality as a property

φi(si, G, G′)

where si ∈ Ti and G, G′ are restrictions of H. Intuition: φ(si, G, G′) holds if si is an optimal strategy for player i in G in the context of G′, assuming he uses

φ to select optimal strategies.

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Examples

sd(si, G, G′) iff si is not strictly dominated on G by any strategy from the restriction G′ := (S′

1, . . ., S′ n) of H,

(assuming H is finite) msd(si, G, G′) iff si is not strictly dominated on G by any of its mixed strategy from the restriction G′ := (S′

1, . . ., S′ n) of H,

br(si, G, G′) iff si is a best response in the restriction

G′ := (S′

1, . . ., S′ n) of H to some belief µi held in G.

Two natural possibilities: G′ = H or G′ = G. We abbreviate:

φ(si, G, H) to φ g(si, G), φ(si, G, G) to φ l(si, G).

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Iterated Elimination of Strategies

Each φ := (φ1, . . ., φn) determines an operator Tφ:

Tφ(G) := (S′

1, . . ., S′ n),

where G := (S1, . . ., Sn) and

S′

i := {si ∈ Si | φi(si, G)}.

φi(·, ·) is monotonic if ∀G1, G2 ∀si ∈ Ti (G1 ⊆ G2 ∧ φ(si, G1) ⇒ φ(si, G2)).

If each φi is monotonic, then Tφ is monotonic and by Tarski’s Theorem T ∞

φ is the largest fixpoint of Tφ.

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Possibility Correspondences

Given H = (T1, . . ., Tn, p1, . . ., pn). We assume a space Ω of states. In each state ω ∈ Ω each player i chooses strategy si(ω) ∈ Ti. Example: Ω = T1 × . . . × Tn with si(ω) := si, where

ω := s.

A possibility correspondence: a mapping from Ω to

P(Ω).

(i) for all ω, P(ω) = ∅, (ii) for all ω and ω′, ω′ ∈ P(ω) implies P(ω′) = P(ω), (iii) for all ω, ω ∈ P(ω). (i),(ii): belief correspondence (a frame for KD45), (i),(ii),(iii): knowledge correspondence (a frame for S5).

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Common knowledge and common belief

Let PE := {ω ∈ Ω | ∀i ∈ [1..n] Pi(ω) ⊆ E}. Event: a subset of Ω. Event F is evident if P ⊆ PF. Suppose each Pi is a knowledge correspondence. Event E is a common knowledge in ω ∈ Ω (ω ∈ K∗E) if for some evident event F

ω ∈ F ⊆ PE.

Suppose each Pi is a belief correspondence. Event E is a common belief in ω ∈ Ω (ω ∈ B∗E) if for some evident event F

ω ∈ F ⊆ PE.

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Games and Knowledge/Belief

Each event E determines a restriction GE of H

GE := (S1, . . ., Sn),

where

Sj := {sj(ω′) | ω′ ∈ E}.

Suppose player i uses φi(·, ·) to select his optimal strategies. Player i is φi-rational in ω if φi(si(ω), GPi(ω)) holds. Intuition: In ω player i only knows/believes that the state of the world is in Pi(ω). So GPi(ω) is the game he knows/believes in. If φi(si(ω), GPi(ω)) and player i uses φi(·, ·) to select his

• ptimal strategy, then in ω he indeed acts ’rationally’.

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Common Knowlege/Belief of Rationality

Given possibility correspondences P1, . . ., Pn: RAT(φ) := {ω ∈ Ω | each player i is φi-rational in ω},

CK(φ) := {ω ∈ Ω | for some knowledge

correspondences P1, . . ., Pn

ω ∈ K∗RAT(φ)}, CB(φ) := {ω ∈ Ω | for some belief

correspondences P1, . . ., Pn

ω ∈ RAT(φ) and ω ∈ B∗RAT(φ)}.

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Main Results

Suppose each property φi is monotonic. Then

GCK(φ) = GCB(φ) = T ∞

φ .

Properties sd g, msd g and br g are monotonic. sd l and msd l are not monotonic. For them only the inclusion

GCK(brg) = GCB(brg) ⊆ T ∞

φ

holds. In general transfinite iterations of Tφ are necessary, i.e.

T ∞

φ = T ω φ .

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