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Unformatted text preview: , this yields f (max(x , x � ), t � ) − f (x � , t � ) ≥ 0. Thus max(x , x � ) maximizes f (·, t � ), i.e, max(x , x � ) belongs to x (t � ). Since x � is the greatest element of the set x (t � ), we conclude that max(x , x � ) ≤ x � , thus x ≤ x � . Since x is an arbitrary element of x (t ), this implies x (t ) ≤ x (t � ). A similar ¯ ¯ argument applies to the smallest maximizers. 28 Game Theory: Lecture 7 Supermodular Games Supermodular Games Deﬁnition The strategic game �I , (Si ), (ui )� is a supermodular game if for all i ∈ I : Si is a compact subset of R [or more generally Si is a complete lattice in Rmi ]; ui is upper semicontinuous in si , continuous in s−i . ui has increasing diﬀerences in (si , s−i ) [or more generally ui is supermodular in (si , s−i ), which is an extension of the property of increasing diﬀerences to games with multi-dimensional strategy spaces]. 29 Game Theory: Lecture 7 Supermodular Games Supermodular Games Applying Topkis’ theorem implies that each player’s “best response correspondence is increasing in the actions of other players”. Corollary Assume �I , (Si ), (ui )� is a supermodular game. Let Bi (s−i ) = arg max ui (si , s−i ). s i ∈ Si Then: 1 2 ¯ Bi (s−i ) has a greatest 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. 30 Game Theory: Lecture 7 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 unique NE, it is dominance solvable (and lots of learning and adjustment rules converge to it, e.g., best-response dynamics). 31 Game Theory: Lecture 7 Supermodular Games Proof We iterate the best response mapping. Let S 0 = S , and let 0 s 0 = (s1 , . . . , sI0 ) be the largest element of S . ¯0 Let si1 = Bi (s−i ) and Si1 = {si ∈ Si0 | si ≤ si1 }. We show that any si &gt; si1 , i.e, any si ∈ Si1 , is strictly dominated by si1 . For / all s−i ∈ S−i , we have ui (si , s−i ) − ui (si1 , s−i ) 0 0 ≤ ui (si , s−i ) − ui (si1 , s−i ) &lt; 0, where the ﬁrst inequality follows by the increasing diﬀerences of ui (si , s−i ) in (si , s−i ), and the strict inequ...
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## This document was uploaded on 03/19/2014 for the course EECS 6.254 at MIT.

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