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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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