Epistemic Analysis of Strategic Games with Arbitrary Strategy Sets - PowerPoint PPT Presentation

Epistemic Analysis of Strategic Games with Arbitrary Strategy Sets Krzysztof R. Apt CWI and University of Amsterdam Epistemic Analysis of Strategic Games with Arbitrary Strategy Sets – p.1/14

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. Epistemic Analysis of Strategic Games with Arbitrary Strategy Sets – p.2/14

Main Result Assume an arbitrary strategic game. RAT ( φ ) : each player i uses property φ i to select his strategy (‘each player i is φ i -rational’). Suppose each φ i 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 φ i -optimal. The latter requires transfinite iterations . Epistemic Analysis of Strategic Games with Arbitrary Strategy Sets – p.3/14

Preliminaries: Operators ( D, ⊆ ) : a complete lattice with the largest element ⊤ , T : an operator on ( D, ⊆ ) , i.e., T : D → D . T is monotonic if ∀ G 1 , G 2 ( G 1 ⊆ G 2 ⇒ T ( G 1 ) ⊆ T ( G 2 )) . 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 . Epistemic Analysis of Strategic Games with Arbitrary Strategy Sets – p.4/14

Strategic games Fix a strategic game H := ( T 1 , . . ., T n , p 1 , . . ., p n ) . So T i is the set of strategies of player i and p i : T 1 × . . . × T n → R his payoff. A restriction of H is a sequence ( S 1 , . . ., S n ) such that S i ⊆ T i for all i . Restrictions of H ordered by the componentwise set inclusion ( S 1 , . . ., S n ) ⊆ ( S ′ 1 , . . ., S ′ n ) iff S i ⊆ S ′ i for all i form a complete lattice with H the largest element . Epistemic Analysis of Strategic Games with Arbitrary Strategy Sets – p.5/14

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. Epistemic Analysis of Strategic Games with Arbitrary Strategy Sets – p.6/14

Rationality, ctd Let us generalize. Given player i in H := ( T 1 , . . ., T n , p 1 , . . ., p n ) we formalize his notion of rationality as a property φ i ( s i , G, G ′ ) where s i ∈ T i and G , G ′ are restrictions of H . Intuition : φ ( s i , G, G ′ ) holds if s i is an optimal strategy for player i in G in the context of G ′ , assuming he uses φ to select optimal strategies. Epistemic Analysis of Strategic Games with Arbitrary Strategy Sets – p.7/14

Examples sd ( s i , G, G ′ ) iff s i is not strictly dominated on G by any strategy from the restriction G ′ := ( S ′ 1 , . . ., S ′ n ) of H , (assuming H is finite) msd ( s i , G, G ′ ) iff s i is not strictly dominated on G by any of its mixed strategy from the restriction G ′ := ( S ′ 1 , . . ., S ′ n ) of H , br ( s i , G, G ′ ) iff s i is a best response in the restriction G ′ := ( S ′ 1 , . . ., S ′ n ) of H to some belief µ i held in G . Two natural possibilities: G ′ = H or G ′ = G . We abbreviate: φ ( s i , G, H ) to φ g ( s i , G ) , φ ( s i , G, G ) to φ l ( s i , G ) . Epistemic Analysis of Strategic Games with Arbitrary Strategy Sets – p.8/14

Iterated Elimination of Strategies Each φ := ( φ 1 , . . ., φ n ) determines an operator T φ : T φ ( G ) := ( S ′ 1 , . . ., S ′ n ) , where G := ( S 1 , . . ., S n ) and S ′ i := { s i ∈ S i | φ i ( s i , G ) } . φ i ( · , · ) is monotonic if ∀ G 1 , G 2 ∀ s i ∈ T i ( G 1 ⊆ G 2 ∧ φ ( s i , G 1 ) ⇒ φ ( s i , G 2 )) . If each φ i is monotonic, then T φ is monotonic and by Tarski’s Theorem T ∞ φ is the largest fixpoint of T φ . Epistemic Analysis of Strategic Games with Arbitrary Strategy Sets – p.9/14

Possibility Correspondences Given H = ( T 1 , . . ., T n , p 1 , . . ., p n ) . We assume a space Ω of states . In each state ω ∈ Ω each player i chooses strategy s i ( ω ) ∈ T i . Example : Ω = T 1 × . . . × T n with s i ( ω ) := s i , 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). Epistemic Analysis of Strategic Games with Arbitrary Strategy Sets – p.10/14

Common knowledge and common belief Let PE := { ω ∈ Ω | ∀ i ∈ [1 ..n ] P i ( ω ) ⊆ E } . Event : a subset of Ω . Event F is evident if P ⊆ PF . Suppose each P i 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 P i 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. Epistemic Analysis of Strategic Games with Arbitrary Strategy Sets – p.11/14

Games and Knowledge/Belief Each event E determines a restriction G E of H G E := ( S 1 , . . ., S n ) , where S j := { s j ( ω ′ ) | ω ′ ∈ E } . Suppose player i uses φ i ( · , · ) to select his optimal strategies. Player i is φ i - rational in ω if φ i ( s i ( ω ) , G P i ( ω ) ) holds. Intuition : In ω player i only knows/believes that the state of the world is in P i ( ω ) . So G P i ( ω ) is the game he knows/believes in. If φ i ( s i ( ω ) , G P i ( ω ) ) and player i uses φ i ( · , · ) to select his optimal strategy, then in ω he indeed acts ’rationally’. Epistemic Analysis of Strategic Games with Arbitrary Strategy Sets – p.12/14

Common Knowlege/Belief of Rationality Given possibility correspondences P 1 , . . ., P n : RAT ( φ ) := { ω ∈ Ω | each player i is φ i -rational in ω }, CK ( φ ) := { ω ∈ Ω | for some knowledge correspondences P 1 , . . ., P n ω ∈ K ∗ RAT ( φ ) }, CB ( φ ) := { ω ∈ Ω | for some belief correspondences P 1 , . . ., P n ω ∈ RAT ( φ ) and ω ∈ B ∗ RAT ( φ ) }. Epistemic Analysis of Strategic Games with Arbitrary Strategy Sets – p.13/14

Main Results Suppose each property φ i is monotonic. Then G CK ( φ ) = G CB ( 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 G CK ( br g ) = G CB ( br g ) ⊆ T ∞ φ holds. In general transfinite iterations of T φ are necessary, i.e. φ � = T ω T ∞ φ . Epistemic Analysis of Strategic Games with Arbitrary Strategy Sets – p.14/14