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18 game theory lecture 8 potential games simple

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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 Proposition In every finite ordinal potential game, every improvement path is finite. Proof: Suppose (s 0 , s 1 , ·...
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