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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 multidimensional 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., bestresponse
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 > 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 )
< 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.
 Spring '10
 AsuOzdaglar

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