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# The Price of Stability for Network Design with Fair Cost Allocation

SIAM J. Comput., no. 4 (2008): 1602-1623

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Abstract

Network design is a fundamental problem for which it is important to understand the effects of strategic behavior. Given a collection of selfinterested agents who want to form a network connecting certain endpoints, the set of stable solutions 驴 the Nash equilibria 驴 may look quite different from the centrally enforced optimum. We study t...More

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Introduction

- The system behavior arises from the actions of a large number of independent agents, each motivated by self-interest and optimizing an individual objective function.
- The authors show that the price of stability for network design with respect to this fair cost allocation is O, where k is the number of users, and that a good Nash equilibrium can be achieved via best-response dynamics in which users iteratively defect from a starting solution.

Highlights

- In many network settings, the system behavior arises from the actions of a large number of independent agents, each motivated by self-interest and optimizing an individual objective function
- We show that the price of stability for network design with respect to this fair cost allocation is O, where k is the number of users, and that a good Nash equilibrium can be achieved via best-response dynamics in which users iteratively defect from a starting solution
- Recent theoretical work has framed this type of question in the following general form: how much worse is the solution quality at a Nash equilibrium1, relative to the quality at a centrally enforced optimum? Questions of this genre have received considerable attention in recent years, for problems including routing [24, 25, 4], load balancing [5, 6, 16, 23], and facility location [26]
- Since the set of possible solutions is finite, it follows that any sequence of improving moves leads to a Nash equilibrium
- We obtain stronger bounds in the case where users experience only delays, not construction costs; this includes a result that relates the cost at the best Nash equilibrium to that of an optimum with twice as many players, and a result that improves the potential-based bound on the price of stability for the single-source delay-only case
- We show that the Nash equilibrium is not affected by the change, and the optimum can only improve

Results

- The H(k) bound on the price of stability extends directly to the case in which users are selecting arbitrary subsets of a ground set, rather than paths in a graph; it extends to the case in which the cost of each edge is a non-decreasing concave function of the number of users on it.
- The authors obtain stronger bounds in the case where users experience only delays, not construction costs; this includes a result that relates the cost at the best Nash equilibrium to that of an optimum with twice as many players, and a result that improves the potential-based bound on the price of stability for the single-source delay-only case.
- Theorem 2.2 Take a fair connection game with each edge having a nondecreasing concave cost function ce(x), where x is the number of players using edge e.
- If the utility function of each player depends on a concave cost and delay that is independent of the number of users on the edge, the authors get that the price of stability is at most H(k) as the authors have shown at the end of the previous section.
- Theorem 3.4 If in a single source fair connection game all costs are delays, and all delays are from a set D satisfying the above condition, the price of stability is at most α(D).
- Consider the Nash equilibrium obtained via a minimum cost flow computation as in the proof of Theorem 3.3, let xe be the number of paths using edge e, and D(v) be the length of the shortest path from s to v in the residual graph.
- Theorem 4.1 In the two player fair connection game, best response dynamics starting from any configuration converges to a Nash equilibrium in polynomial time.

Conclusion

- Theorem 5.1 In a weighted game where each edge e is in the strategy spaces of at most two players, there exists a potential function for this game, and a Nash equilibrium exists.
- Theorem 5.3 For any weighted game in which all players have the same source s and sink t, best response dynamics converges to a Nash equilibrium, and Nash equilibria exist.

Related work

- Network design games under a different model were considered by a subset of the authors in [1]; there, the setting was much more “unregulated” in that users could offer to pay for an arbitrary fraction of any edge in the network. This model resulted in instances where no pure Nash equilibrium existed; and in many cases in [1] when pure Nash equilibria did exist, certain users were able to act as “free riders,” paying very little or nothing at all. The present model, on the other hand, ensures that there is always a pure Nash equilibrium within a logarithmic factor of optimal, in which users pay a fair portion of the resources they use. Network creation games of a fairly different flavor — in which users correspond to nodes, and can build subsets of the edges incident to them — have been considered in [2, 7, 10]. The model in this paper associates users instead with connection requests, and allows them to contribute to the cost of any edge that helps form a path that they need.

The bulk of the work on cost-sharing (see e.g. [9, 11] and the references there) tends to assume a fixed underlying set of edges. Jain and Vazirani [12] and Kent and Skorin-Kapov [15] consider costsharing for a single source network design game. Cost-sharing games assume that there is a central authority that designs and maintains the network, and decides appropriate cost-shares for each agent, depending on the graph and all other agents, via a complex algorithm. The agents’ only role is to report their utility for being included in the network.

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