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6 game theory lecture 8 supermodular games

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Unformatted text preview: (t � ) ≥ x(t ). Summary: if f has increasing differences, the set of optimal solutions x (t ) is non-decreasing in the sense that the largest and the smallest selections are non-decreasing. 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 multi-dimensional 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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