Order Independence Krzysztof R. Apt CWI and University of Amsterdam - PowerPoint PPT Presentation

Order Independence Krzysztof R. Apt CWI and University of Amsterdam Order Independence – p. 1/2

Preliminaries Order Independence – p. 2/2

Dominance by a Pure Strategy X Y 2 , − 1 , − A 1 , − 0 , − B 2 , − 0 , − C A strictly dominates B . A weakly dominates C . Order Independence – p. 3/2

Inherent Dominance L R 2 , − 1 , − T 1 , − 2 , − M 1 , − 3 , − B M is not strictly dominated. M is weakly dominated by T given { L } , and weakly dominated by B given { R } (or { L, R } ). M is inherently dominated. Example L R 2 , − 1 , − T 1 , − 1 , − B B is not inherently weakly dominated but is weakly dominated. Order Independence – p. 4/2

Dominance by a Mixed Strategy X Y 2 , − 0 , − A 0 , − 2 , − B 0 , − 0 , − C D 1 , − 0 , − 1 2 A + 1 2 B strictly dominates C . 1 2 A + 1 2 B weakly dominates D . Order Independence – p. 5/2

Never Best Responses X Y 4 , − 2 , − A 2 , − 4 , − B 3 , − 3 , − C D 0 , − 0 , − C is a never best response to a pure strategy. D is a never best response to a mixed strategy. Order Independence – p. 6/2

Order Independence We consider finite strategic games. A strategy elimination procedure is order independent if all roads lead to Rome. Order Independence – p. 7/2

Order Independence We consider finite strategic games. A strategy elimination procedure is order independent if all roads lead to Rome. Order Independence – p. 8/2

It is well known that . . . Strict dominance by a pure strategy is order independent ([Gilboa, Kalai and Zemel, ’90]). Strict dominance by a mixed strategy is order independent ([Osborne and Rubinstein ’94]). Weak dominance is not order independent: L R 1 , 1 1 , 1 T 1 , 1 0 , 0 B can be reduced to L L R and to 1 , 1 T 1 , 1 1 , 1 T 1 , 1 B Order Independence – p. 9/2

Our Previous Work [1] K. R. Apt. Uniform proofs of order independence for various strategy elimination procedures. The B.E. Journal of Theoretical Economics, 4(1) , 2004. Article 5, 48 pages. Dealt with various dominance notions on finite games. [2] K. R. Apt. Order independence and rationalizability. In Proceedings TARK ’05 , pages 22–38, 2005. Dealt with iterated elimination of never best responses in finite and infinite games. [3] K. R. Apt. The many faces of rationalizability. The B.E. Journal of Theoretical Economics, 7(1) , 2007. Article 18, 39 pages. Dealt with strict dominance and iterated elimination of never best responses in infinite games. Order Independence – p. 10/2

Results of [1]: Glossary – S : strict dominance, – W : weak dominance, – NW : nice weak dominance ([Marx and Swinkels ’97]), – PE : payoff equivalence, – RM : the ‘mixed strategy’ version of the dominance relation R , – inh-R : the ‘inherent’ version of the (mixed) dominance relation R ([Börgers ’90]), – OI: order independence, – ∼ -OI: order independence up to strategy renaming. Order Independence – p. 11/2

Summary of Results of [1] Notion Property Result originally due to S OI [Gilboa, Kalai and Zemel, ’90] [Stegeman ’90] inh-W OI [Börgers ’90] inh-NW OI SM OI [Osborne and Rubinstein ’94] inh-WM [Börgers ’90]: equal to SM OI inh-NWM OI PE ∼ -OI S ∪ PE ∼ -OI NW ∪ PE ∼ -OI [Marx and Swinkels ’97] PEM ∼ -OI SM ∪ PEM ∼ -OI NWM ∪ PEM ∼ -OI [Marx and Swinkels ’97] Order Independence – p. 12/2

Summary of Results of [2] and [3] – global strict dominance by a pure strategy ([Milgrom and Roberts ’90, Ritzberger ’02, Chen, Long and Luo, ’07]), – global strict dominance by a mixed strategy ([Brandenburger, Friedenberg and Keisler ’07, Apt ’07]), – never best response to a pure/mixed/correlated strategy, – global never best response to a pure/mixed/correlated strategy ([Bernheim, ’84]). Theorem All these dominance relations are order independent on finite games. Order Independence – p. 13/2

What’s new? Order Independence – p. 14/2

Good News For finite games you don’t need to read [1], [2] and [3]! Order Independence – p. 15/2

Good news for you For finite games you don’t need to read [1], [2] and [3]! It all follows from one generic result. Order Independence – p. 16/2

Good news for you For finite games you don’t need to read [1], [2] and [3]! It all follows from one generic result. Theorem Assume the game is finite. If the dominance relation is hereditary, then it is order independent. Order Independence – p. 17/2

The Result Theorem Assume the game is finite. If the dominance relation is hereditary, then it is order independent. Given two games G and G ′ , G → D G ′ when G � = G ′ , G ′ ⊆ G and ∀ i ∈ { 1 , . . . , n } ∀ s i ∈ G i \ G ′ i s i is D -dominated in G. We call a dominance relation D hereditary if G → D G ′ , s i from G ′ , s i is D -dominated in G , implies s i is D -dominated in G ′ . Order Independence – p. 18/2

Proof Idea (1) Treat the dominance notion as a reduction relation on the set of restrictions of an initial game. Then order independence = unique normal form property. Use Newman’s Lemma. Order Independence – p. 19/2

Weak Confluence A a set, → a binary (reduction) relation on A . → ∗ : the transitive reflexive closure of → . b is a → -normal form of a if a → ∗ b , no c exists such that b → c . → is weakly confluent if ∀ a, b, c ∈ A a ւ ց b c implies that for some d ∈ A b c ց∗ ∗ ւ d Order Independence – p. 20/2

Unique Normal Form Property If each a ∈ A has a unique normal form, then ( A, → ) satisfies the unique normal form property. Newman’s Lemma (’42) Consider ( A, → ) such that no infinite → sequences exist, → is weakly confluent. Then → satisfies the unique normal form property. Order Independence – p. 21/2

Proof Idea (2) Lemma Assume the game is finite. If the dominance relation is hereditary, then it is weakly confluent. All the relevant dominace relations are hereditary. This yields direct order independence proofs for (global) strict dominance by a pure/mixed strategy, iterated elimination of (global) never best responses to a pure/mixed/correlated strategy, iterated elimination of inherent dominance by a pure/mixed strategy. Order Independence – p. 22/2

THANK YOU Order Independence – p. 23/2