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Unformatted text preview: and least element, denoted by Bi (s−i ) and Bi (s−i ).
¯
¯
If s � ≥ s−i , then Bi (s � ) ≥
Bi (s−i ) and Bi (s � ) ≥ Bi (s−i ).
−i −i −i ¯
Applying Tarski’s ﬁxed point theorem to B establishes the existence of a
pure Nash equilibrium for any supermodular game. We next pursue a diﬀerent approach which provides more insight into the structure of Nash equilibria. 8 Game Theory: Lecture 8 Supermodular Games Supermodular Games
Theorem (Milgrom and Roberts)
Let �I , (Si ), (ui )� be a supermodular game. Then the set of strategies that
survive iterated strict dominance in pure strategies has greatest and least elements
s and s, coinciding with the greatest and the least pure strategy Nash Equilibria. ¯
Corollary
Supermodular games have the following properties:
1 Pure strategy NE exist. 2 The largest and smallest strategies are compatible with iterated strict
dominance (ISD), rationalizability, correlated equilibrium, and Nash
equilibrium are the same. 3 If a supermodular game has a uniq...
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This document was uploaded on 03/19/2014 for the course EECS 6.254 at MIT.
 Spring '10
 AsuOzdaglar

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