Non-monotonic Operators in Strategic Games Krzysztof R. Apt CWI - PowerPoint PPT Presentation

Non-monotonic Operators in Strategic Games Krzysztof R. Apt CWI and University of Amsterdam Non-monotonic Operators in Strategic Games – p. 1/2

A Pointer to Maurice’s Work M. Denecker, M. Bruynooghe and V. Marek, Logic Programming Revisited: Logic Programs as Inductive Definitions , ACM Transactions on Computational Logic (TOCL) 2(4), pp. 623 - 654, 2001. Special issue devoted to Robert A. Kowalski. “The thesis is developed that logic programming can be understood as a natural and general logic of inductive definitions. In particular, logic programs with negation represent nonmonotone inductive definitions.” Non-monotonic Operators in Strategic Games – p. 2/2

Strategic Games: 2 Examples Prisoner’s Dilemma C D C 2 , 2 0 , 3 D 3 , 0 1 , 1 A 4 by 3 game X Y Z A 2 , 1 0 , 1 1 , 0 B 0 , 1 2 , 1 1 , 0 C 1 , 1 1 , 0 0 , 0 D 1 , 0 0 , 1 0 , 0 Non-monotonic Operators in Strategic Games – p. 3/2

Strategic Games: Definition Strategic game for n ≥ 2 players: (possibly infinite) set S i of strategies, payoff function p i : S 1 × . . . × S n → R , for each player i . Basic assumptions: players choose their strategies simultaneously, each player is rational: his objective is to maximize his payoff, players have common knowledge of the game and of each others’ rationality. Non-monotonic Operators in Strategic Games – p. 4/2

Dominance by a Pure Strategy X Y A 2 , − 1 , − B 1 , − 0 , − C 2 , − 0 , − A strictly dominates B . A weakly dominates C . Non-monotonic Operators in Strategic Games – p. 5/2

Dominance by a Mixed Strategy X Y A 2 , − 0 , − B 0 , − 2 , − C 0 , − 0 , − D 1 , − 0 , − 1 / 2 A + 1 / 2 B strictly dominates C . 1 / 2 A + 1 / 2 B weakly dominates D . Non-monotonic Operators in Strategic Games – p. 6/2

Iterated Elimination: Example (1) Consider weak dominance. X Y Z A 2 , 1 0 , 1 1 , 0 B 0 , 1 2 , 1 1 , 0 C 1 , 1 1 , 0 0 , 0 D 1 , 0 0 , 1 0 , 0 We first get X Y A 2 , 1 0 , 1 B 0 , 1 2 , 1 C 1 , 1 1 , 0 Non-monotonic Operators in Strategic Games – p. 7/2

Iterated Elimination: Example (2) X Y A 2 , 1 0 , 1 B 0 , 1 2 , 1 C 1 , 1 1 , 0 Next, we get X A 2 , 1 B 0 , 1 C 1 , 1 and finally X A 2 , 1 Non-monotonic Operators in Strategic Games – p. 8/2

4 Operators Given: initial finite strategic game H := ( H 1 , . . ., H n , p 1 , . . ., p n ) . 4 operators on the lattice of all subgames of H : S ( G ) : outcome of eliminating from G all strategies strictly dominated by a pure strategy (once). W ( G ) : . . . weakly dominated by a pure strategy, SM ( G ) : . . . strictly dominated by a mixed strategy, WM ( G ) : . . . weakly dominated by a mixed strategy. Non-monotonic Operators in Strategic Games – p. 9/2

4 Inclusions Note For all G WM ( G ) ⊆ W ( G ) ⊆ S ( G ) , WM ( G ) ⊆ SM ( G ) ⊆ S ( G ) . S W SM WM Non-monotonic Operators in Strategic Games – p. 10/2

Iterated Elimination Do these inclusions extend to the outcomes of iterated elimination? None of these operators is monotonic. Example In X A 1 , 0 B 0 , 0 B is strictly dominated, but not in X B 0 , 0 Non-monotonic Operators in Strategic Games – p. 11/2

Operators T : operator on a finite lattice ( D, ⊆ ) . T 0 = D , T k : k -fold iteration of T , T ω := ∩ k ≥ 0 T k . T is monotonic if G ⊆ G ′ implies T ( G ) ⊆ T ( G ′ ) . Lemma T and U operators on a finite lattice ( D, ⊆ ) . For all G , T ( G ) ⊆ U ( G ) , at least one of T and U is monotonic. Then T ω ⊆ U ω . Non-monotonic Operators in Strategic Games – p. 12/2

Approach Given: two strategy elimination operators Φ and Ψ such that for all G , Φ( G ) ⊆ Ψ( G ) . To prove Φ ω ⊆ Ψ ω we define their ‘global’ versions Φ g and Ψ g and prove instead Φ ω g ⊆ Ψ ω g . Need to show: g = Φ ω and Ψ ω Φ ω g = Ψ ω , for all G , Φ g ( G ) ⊆ Ψ g ( G ) , at least one of Φ g and Ψ g is monotonic. Non-monotonic Operators in Strategic Games – p. 13/2

Global Operators G : a subgame of H . s i , s ′ i : strategies of player i in H . s ′ i ≻ G s i : s ′ i strictly dominates s i over G : Then S ( G ) := G ′ , where G ′ i := { s i ∈ G i | ¬∃ s ′ i ∈ G i s ′ i ≻ G s i } . GS ( G ) := G ′ , where G ′ i := { s i ∈ G i | ¬∃ s ′ i ∈ H i s ′ i ≻ G s i } . Similar definitions for GW , GSM , GWM . Non-monotonic Operators in Strategic Games – p. 14/2

Inclusion 1: SM ω ⊆ S ω Lemma For all G SM ( G ) ⊆ S ( G ) . GS ω = S ω . GSM ω = SM ω . (Brandenburger, Friedenberg and Keisler ’06) For all G GSM ( G ) ⊆ GS ( G ) . GSM and GS are monotonic. Conclusion: GSM ω ⊆ GS ω , so SM ω ⊆ S ω . Non-monotonic Operators in Strategic Games – p. 15/2

Other Inclusions By the same approach the inclusions W ω ⊆ S ω , WM ω ⊆ SM ω hold. Non-monotonic Operators in Strategic Games – p. 16/2

What about WM ω ⊆ W ω ? Consider X Y Z A 2 , 1 0 , 1 1 , 0 B 0 , 1 2 , 1 1 , 0 C 1 , 1 1 , 0 0 , 0 D 1 , 0 0 , 1 0 , 0 Applying WM we get X Y A 2 , 1 0 , 1 B 0 , 1 2 , 1 Another application of WM yields no change. Non-monotonic Operators in Strategic Games – p. 17/2

Conclusion The inclusion WM ω ⊆ W ω does not hold. Non-monotonic Operators in Strategic Games – p. 18/2

Summary S ω S W ω W SM ω SM WM WM Reference K.R. Apt, Relative Strength or Strategy Elimination Procedures , Economics Bulletin, 3(21), pp. 1-9, 2007. Non-monotonic Operators in Strategic Games – p. 19/2

Happy Birthday Maurice Non-monotonic Operators in Strategic Games – p. 20/2

Happy Birthday Maurice Non-monotonic Operators in Strategic Games – p. 21/2