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

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Dominance by a Pure Strategy

X Y A 2, − 1, − B 1, − 0, − C 2, − 0, − A strictly dominates B. A weakly dominates C.

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Inherent Dominance

L R T 2, − 1, − M 1, − 2, − B 1, − 3, − 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 T 2, − 1, − B 1, − 1, − B is not inherently weakly dominated but is weakly

dominated.

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Dominance by a Mixed Strategy

X Y A 2, − 0, − B 0, − 2, − C 0, − 0, − D 1, − 0, −

1 2A + 1 2B strictly dominates C. 1 2A + 1 2B weakly dominates D.

Order Independence – p. 5/2

Never Best Responses

X Y A 4, − 2, − B 2, − 4, − C 3, − 3, − D 0, − 0, − C is a never best response to a pure strategy. D is a never best response to a mixed strategy.

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Order Independence

We consider finite strategic games. A strategy elimination procedure is order independent if all roads lead to Rome.

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Order Independence

We consider finite strategic games. A strategy elimination procedure is order independent if all roads lead to Rome.

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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 T 1, 1 1, 1 B 1, 1 0, 0

can be reduced to

L R T 1, 1 1, 1

and to

L T 1, 1 B 1, 1

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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.

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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.

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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 OI [Börgers ’90]: equal to SM 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]

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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.

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What’s new?

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Good News

For finite games you don’t need to read [1], [2] and [3]!

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Good news for you

For finite games you don’t need to read [1], [2] and [3]! It all follows from one generic result.

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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.

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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} ∀si ∈ Gi \ G′

i si is D-dominated in G.

We call a dominance relation D hereditary if

G →D G′, si from G′, si is D-dominated in G,

implies

si is D-dominated in G′.

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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.

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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

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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.

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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.

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THANK YOU

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