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2 p p 0 let x b e the corresponding ow allocation if

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Unformatted text preview: 0: A small increase in the price of provider 1 will generate 1 2 positive profits, thus provider 1 has an incentive to deviate. 2 p = p > 0: Let x b e the corresponding flow allocation. If x = 1, 1 2 1 then provider 2 has an incentive to decrease its price. If x1 < 1, then provider 1 has an incentive to decrease its price. 3 0 ≤ p < p : Player 1 has an incentive to increase its price since its 1 2 flow allocation remains the same. 4 0 ≤ p < p : For � sufficiently small, the profit function of player 2, 2 1 given p1 , is strictly increasing as a function of p2 , showing that provider 2 has an incentive to increase its price. 9 Game Theory: Lecture 5 Existence Results Existence Results We start by analyzing existence of a Nash equilibrium in finite (strategic form) games, i.e., games with finite strategy sets. Theorem (Nash) Every finite game has a mixed strategy Nash equilibrium. Implication: matching pennies game necessarily has a mixed strategy equilibrium. Why is this important? Without knowing the existence of an equilibrium, it is difficult (perhaps meaningless) to try to understand its properties. Armed with this theorem, we also know that every finite game has an equilibrium, and thus we can simply try to locate the equilibria. 10 Game Theory: Lecture 5 Existence Results Approach Recall that a mixed strategy profile σ∗ is a NE if ui (σi∗ , σ∗ i ) ≥ ui (σi , σ∗ i ), − − for all σi ∈ Σi . ∗ In other words, σ∗ is a NE if and only if σi∗ ∈ B−i (σ∗ i ) for all i , − ∗ ( σ ∗ ) is the b est response of player i , given that the other where B−i −i players’ strategies are σ∗ i . − We define the correspondence B : Σ � Σ such that for all σ ∈ Σ, we have (1) B (σ) = [Bi (σ−i )]i ∈I The existence of a Nash equilibrium is then equivalent to the existence of a mixed strategy σ such that σ ∈ B (σ): i.e., existence of a fixed point of the mapping B . We will establish existence of a Nash equilibrium in finite games using a fixed point theorem. 11 Game Theory: Lecture 5 Existence Results Definitions A set in a Euclidean space is compact if and only if it is bounded and closed. A set S is convex...
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This document was uploaded on 03/19/2014 for the course EECS 6.254 at MIT.

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