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Unformatted text preview: ¯k
sik +1 = Bi (s−i ) ≤ Bi (s−−1 ) = sik . i
This shows that the sequence {sik } is a decreasing sequence, which is
bounded from below, and hence it has a limit, which we denote by s¯i . Only
¯
the strategies si ≤ si are undominated. Similarly, we can start with
0
s 0 = (s1 , . . . , sI0 ) the smallest element in S and identify s.
To complete the proof, we show that s and s are NE. By construction, for all
¯
i and si ∈ Si , we have
k
k
ui (sik +1 , s−i ) ≥ ui (si , s−i ). Taking the limit as k → ∞ in the preceding relation and using the upper
semicontinuity of ui in si and continuity of ui in s−i , we obtain ¯¯
¯
ui (si , s−i ) ≥ ui (si , s−i ), showing the desired claim.
11 Game Theory: Lecture 8 Potential Games Potential Games A strategic form game is a p otential game [ordinal potential game,
exact potential game] if there exists a function Φ : S → R such that
Φ (si , s−i ) gives information about ui (si , s−i ) for each i ∈ I .
If so, Φ is referred to as the p otential function.
The potential function has a natural analogy to “energy” in...
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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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