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Unformatted text preview: n that for all i and all q−i , ui (qi , q−i ) − ui (qi� , q−i ) = Φ∗ (qi , q−i ) − Φ∗ (qi� , q−i ), for all qi , qi� > 0. Φ is an exact potential function for this game. 17 Game Theory: Lecture 8 Potential Games Simple Dynamics in Finite Ordinal Potential Games Deﬁnition
A path in strategy space S is a sequence of strategy vectors (s 0 , s 1 , · · · )
such that every two consecutive strategies diﬀer in one coordinate (i.e., exactly in one player’s strategy). An improvement path is a path (s 0 , s 1 , · · · ) such that, th
uik (s k ) < uik (s k +1 ) where s k and s k +1 diﬀer in the ik coordinate. In
other words, the payoﬀ improves for the player who changes his strategy.
An improvement path can be thought of as generated dynamically by
“myopic players”, who update their strategies according to 1-sided
b etter reply dynamic. 18 Game Theory: Lecture 8 Potential Games Simple Dynamics in Finite Ordinal Potential Games
In every ﬁnite ordinal potential game, every improvement path is ﬁnite.
Proof: Suppose (s 0 , s 1 , ·...
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- Spring '10