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# lecture5 notes - 6.254 Game Theory with Engineering...

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6.254 : Game Theory with Engineering Applications Lecture 5: Existence of a Nash Equilibrium Asu Ozdaglar MIT February 18, 2010 1

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Game Theory: Lecture 5 Introduction Outline Pricing-Congestion Game Example Existence of a Mixed Strategy Nash Equilibrium in Finite Games Existence in Games with Infinite Strategy Spaces Reading: Fudenberg and Tirole, Chapter 1. 2
Game Theory: Lecture 5 Example Introduction In this lecture, we study the question of existence of a Nash equilibrium in both games with finite and infinite pure strategy spaces. We start with an example, pricing-congestion game , where players have infinitely many pure strategies. We consider two instances of this game, one of which has a unique pure Nash equilibrium, and the other does not have any pure Nash equilibria. 3

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Game Theory: Lecture 5 Example Pricing-Congestion Game Consider a price competition model studied in [Acemoglu and Ozdaglar 07]. 1 unit of traffic Reservation utility R Consider a parallel link network with I links. Assume that d units of ﬂow is to be routed through this network. We assume that this ﬂow is the aggregate ﬂow of many infinitesimal users. Let l i ( x i ) denote the latency function of link i , which represents the delay or congestion costs as a function of the total ﬂow x i on link i . Assume that the links are owned by independent providers. Provider i sets a price p i per unit of ﬂow on link i . The effective cost of using link i is p i + l i ( x i ) . Users have a reservation utility equal to R , i.e., if p i + l i ( x i ) > R , then no traﬃc will be routed on link i . 4
Game Theory: Lecture 5 Example Example 1 We consider an example with two links and latency functions l 1 ( x 1 ) = 0 and l 2 ( x 2 ) = 3 2 x 2 . For simplicity, we assume that R = 1 and d = 1. Given the prices ( p 1 , p 2 ) , we assume that the ﬂow is allocated according to Wardrop equilibrium , i.e., the ﬂows are routed along minimum effective cost paths and the effective cost cannot exceed the reservation utility. Definition A ﬂow vector x = [ x i ] i = 1, ... , I is a Wardrop equilibrium if I i = 1 x i d and p i + l i ( x i ) = min { p j + l j ( x j ) } , for all i with x i > 0, j p i + l i ( x i ) R , for all i with x i > 0, with I = d if min j { p j + l j ( x j ) } < R. i = 1 x i 5

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Game Theory: Lecture 5 Example Example 1 (Continued) We use the preceding characterization to determine the ﬂow allocation on each link given prices 0 p 1 , p 2 1: x 2 ( p 1 , p 2 ) = 2 3 ( p 1 p 2 ) , 0, p 1 p 2 , otherwise , and x 1 ( p 1 , p 2 ) = 1 x 2 ( p 1 , p 2 ) . The payoffs for the providers are given by: u 1 ( p 1 , p 2 ) = p 1 × x 1 ( p 1 , p 2 ) u 2 ( p 1 , p 2 ) = p 2 × x 2 ( p 1 , p 2 ) We find the pure strategy Nash equilibria of this game by characterizing the best response correspondences, B i ( p i ) for each player. The following analysis assumes that at the Nash equilibria ( p 1 , p 2 ) of the game, the corresponding Wardrop equilibria x satisfies x 1 > 0, x 2 > 0, and x 1 + x 2 = 1. For the proofs of these statements, see [Acemoglu and Ozdaglar 07].
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