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Unformatted text preview: (t � ) ≥ x(t ). Summary: if f has increasing diﬀerences, the set of optimal solutions
x (t ) is nondecreasing in the sense that the largest and the smallest
selections are nondecreasing.
6 Game Theory: Lecture 8 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]. 7 Game Theory: Lecture 8 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...
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 Spring '10
 AsuOzdaglar

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