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Network Design with Weighted Players

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Abstract

We consider a model of game-theoretic network design initially studied by Anshelevich et al. (Proceedings of the 45th Annual Symposium on Foundations of Computer Science (FOCS), pp. 295–304, 2004), where selfish players select paths in a network to minimize their cost, which is prescribed by Shapley cost shares. If all players are identical, the cost share incurred by a player for an edge in its path is the fixed cost of the edge divided by the number of players using it. In this special case, Anshelevich et al. (Proceedings of the 45th Annual Symposium on Foundations of Computer Science (FOCS), pp. 295–304, 2004) proved that pure-strategy Nash equilibria always exist and that the price of stability—the ratio between the cost of the best Nash equilibrium and that of an optimal solution—is Θ(log k), where k is the number of players. Little was known about the existence of equilibria or the price of stability in the general weighted version of the game. Here, each player i has a weight w i ≥1, and its cost share of an edge in its path equals w i times the edge cost, divided by the total weight of the players using the edge.

This paper presents the first general results on weighted Shapley network design games. First, we give a simple example with no pure-strategy Nash equilibrium. This motivates considering the price of stability with respect to α-approximate Nash equilibria—outcomes from which no player can decrease its cost by more than an α multiplicative factor. Our first positive result is that O(log w max )-approximate Nash equilibria exist in all weighted Shapley network design games, where w max  is the maximum player weight. More generally, we establish the following trade-off between the two objectives of good stability and low cost: for every α=Ω(log w max ), the price of stability with respect to O(α)-approximate Nash equilibria is O((log W)/α), where W is the sum of the players’ weights. In particular, there is always an O(log W)-approximate Nash equilibrium with cost within a constant factor of optimal.

Finally, we show that this trade-off curve is nearly optimal: we construct a family of networks without o(log w max / log log w max )-approximate Nash equilibria, and show that for all α=Ω(log w max /log log w max ), achieving a price of stability of O(log W/α) requires relaxing equilibrium constraints by an Ω(α) factor.

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Authors and Affiliations

  1. Department of Computer Science, Stanford University, 393 Terman Engineering Building, Stanford, CA, 94305, USA

    Ho-Lin Chen

  2. Department of Computer Science, Stanford University, 462 Gates Building, Stanford, CA, 94305, USA

    Tim Roughgarden

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  1. Ho-Lin Chen
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  2. Tim Roughgarden
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Correspondence to Tim Roughgarden.

Additional information

Research of H.-L. Chen supported in part by NSF Award 0323766.

Research of T. Roughgarden supported in part by ONR grant N00014-04-1-0725, DARPA grant W911NF-04-9-0001, and an NSF CAREER Award.

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Chen, HL., Roughgarden, T. Network Design with Weighted Players. Theory Comput Syst 45, 302–324 (2009). https://doi.org/10.1007/s00224-008-9128-8

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  • DOI: https://doi.org/10.1007/s00224-008-9128-8

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