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# 0 let si1 bi si and si1 si si0 si si1 we show

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Unformatted text preview: ue NE, it is dominance solvable (and lots of learning and adjustment rules converge to it, e.g., best-response dynamics). 9 Game Theory: Lecture 8 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 inequality follows by the fact that si is not a best 0 response to s−i . Note that si1 ≤ si0 . Iterating this argument, we deﬁne ¯k sik = Bi (s−−1 ), i Sik = {si ∈ Sik −1 | si ≤ sik }. 10 Game Theory: Lecture 8 Supermodular Games Proof Assume s k ≤ s k −1 . Then, by Corollary (Topkis), we have ¯k...
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## This document was uploaded on 03/19/2014 for the course EECS 6.254 at MIT.

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