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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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 Spring '10
 AsuOzdaglar
 Game Theory

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