lecture8 notes

# Lecture8 notes

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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.

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