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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., bestresponse
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.
 Spring '10
 AsuOzdaglar

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