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# I is a wardrop equilibrium if i 1 xi d and i pi li xi

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Unformatted text preview: ion A ﬂow vector x = [xi ]i =1,...,I is a Wardrop equilibrium if ∑I =1 xi ≤ d and i pi + li (xi ) = min{pj + lj (xj )}, j pi + li (xi ) ≤ R , for all i with xi > 0, for all i with xi > 0, with ∑I =1 xi = d if minj {pj + lj (xj )} < R . i 5 Game Theory: Lecture 5 Example Example 1 (Continued) We use the preceding characterization to determine the ﬂow allocation on each link given prices 0 ≤ p1 , p2 ≤ 1: �2 p1 ≥ p2 , 3 (p1 − p2 ), x2 (p1 , p2 ) = 0, otherwise, and x1 (p1 , p2 ) = 1 − x2 (p1 , p2 ). The payoﬀs for the providers are given by: u1 (p1 , p2 ) = p1 × x1 (p1 , p2 ) u2 (p1 , p2 ) = p2 × x2 (p1 , p2 ) We ﬁnd the pure strategy Nash equilibria of this game by characterizing the best response correspondences, Bi (p−i ) for each player. The following analysis assumes that at the Nash equilibria (p1 , p2 ) of the game, the corresponding Wardrop equilibria x satisﬁes x1 > 0, x2 > 0, and x1 + x2 = 1. For the proofs of these statements, see [Acemoglu and Ozdaglar 07]. 6 Game Theory: Lecture 5 Example Example 1 (Continued) In particular, for a given p2 , B1 (p2 ) is the optimal solution set of the following optimization problem maximize 0≤p1 ≤1, 0≤x1 ≤1 subject to p1 x1 3 p1 = p2 + (1 − x1 ) 2 Solving the preceding optimization problem, we ﬁnd that � � 3 p2 B1 (p2 ) = min 1, + . 4 2 Similarly, B2 (p1 ) = p1 2. 7 Game Theory: Lecture 5 Example Example 1 (Continued) Best Response Functions 1 0.8 p2 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 p1 The ﬁgure illustrates the best response correspondences as a function of p1 and p2 . The correspondences intersect at the unique point (p1 , p2 ) = (1, 1 ), which is the unique pure strategy equilibrium. 2 8 Game Theory: Lecture 5 Example Example 2 We next consider a similar example with latency functions given by � 0 if 0 ≤ x ≤ 1/2 l1 (x ) = 0, l2 (x ) = x −1/2 x ≥ 1/2, � for some suﬃciently small � > 0. The following list considers all candidate Nash equilibria (p1 , p2 ) and proﬁtable unilateral deviations for � suﬃciently small, thus establishing the nonexistence of a pure strategy Nash equilibrium: 1 p = p =...
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