lecture6 notes

Lecture6 notes

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Unformatted text preview: d ui : S → R are bounded (measurable) functions. ˜ We define the distance between the utility function profiles u and u as ˜ max sup |ui (s ) − ui (s )|. ˜ i ∈I s ∈S Consider two strategic form games defined by two profiles of utility functions: ˜ G = �I , (Si ), (ui )�, G = �I , (Si ), (ui )�. ˜ If σ is a mixed strategy Nash equilibrium of G , then σ need not be a mixed ˜ strategy Nash equilibrium of G . ˜ Even if u and u are very close, the equilibria of G and G may be far apart. ˜ For example, assume there is only one player, S1 = [0, 1], u1 (s1 ) = �s1 , and u1 (s1 ) = −�s1 , where � > 0 is a sufficiently small scalar. The ˜ ∗ ˜ unique equilibrium of G is s1 = 1, and the unique equilibrium of G is ∗ = 0, even if the distance b etween u and u is only 2�. ˜ s1 5 Game Theory: Lecture 6 Continuous Games Closeness of Two Games and �-Equilibrium However, if u and u are very close, there is a sense in which the equilibria of ˜ ˜ G are “almost” equilibria of G . Definition (�-equilibrium) Given � ≥ 0, a mixed strategy σ ∈ Σ is called an �-equilibrium if for all i ∈ I and si ∈ Si , ui (si , σ−i ) ≤ ui (σi , σ−i ) + �. Obviously, when � = 0, an �-equilibrium is a Nash equilibrium in the usual sense. 6 Game Theory: Lecture 6 Continuous Games Continuity Property of �-equilibria Proposition (1) Let G be a continuous game. Assume that σk → σ, �k → �, and for each k , σk is an �k -equilibrium of G . Then σ is an �-equilibrium of G . Proof: For all i ∈ I , and all si ∈ Si , we have ui (si , σk i ) ≤ ui (σk ) + �k , − Taking the limit as k → ∞ in the preceding relation, and using the continuity of the utility functions (together with the convergence of probability distributions under weak topology), we obtain, ui (si , σ−i ) ≤ ui (σ ) + �, establishing the result. 7 Game Theory: Lecture 6 Continuous Games Closeness of Two Games We next define formally the closeness of two strategic form games. Definition Let G and G � be two strategic form games with G = ...
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This document was uploaded on 03/19/2014 for the course EECS 6.254 at MIT.

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