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A contract and balancing mechanism for sharing capacity in a communication network.

Article Excerpt
1. Introduction

In a recent paper of the first three authors (Anderson et al. 2006), a contract and balancing mechanism was proposed as a method for sharing capacity in a communication network. It was shown that, with n [greater than or equal to] 2 players contracting for capacity on a single link over a time period [0, 1], there is a unique Nash equilibrium for the contract quantities:

PROPOSITION 2 (ANDERSON ET AL. 2006). Under price complementarity, and assuming that all players follow a price-taking policy, there is a unique Nash equilibrium for the contract quantities [y.sub.i], i = 1, 2,..., n. At the Nash equilibrium, the time-averaged expected price is equal to the cost per unit of capacity,

[[integral].sub.0.sup.1] E[p(t)]dt = c, (1)

and player i's optimal choice of contract quantity [y.sub.i] satisfies the following equation:

[y.sub.i] = [[[integral].sub.0.sup.1] E[p(t)[D.sub.i](t, p(t))]dt]/[[[integral].sub.0.sup.1] E[p(t)]dt]. (2)

Here, [D.sub.i](t, p(t)) is the price-dependent demand function for player i. Later, a stylized network model was discussed, with a set of links J, and each player associated with a route r which is a subset of J. The following partial generalization of Proposition 2 was stated:

PROPOSITION 3 (ANDERSON ET AL. 2006). If [y.sub.r], r [member of] R, is a Nash equilibrium at which [y.sub.r] > 0, r [member of] R, then [y.sub.r] satisfies the equation

[y.sub.r] = [[[integral].sub.0.sup.1] E[[w.sub.r](t)[D.sub.r](t, [p.sub.r](t))]dt]/[[[integral].sub.0.sup.1] E[[w.sub.r](t)]dt], (3)

where [w.sub.r](t) = [partial derivative][p.sub.r](t)/[partial derivative][y.sub.r]. Further, the time-averaged expected price on link j...

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