Restoring Pure Equilibria to Weighted Congestion Games

Abstract

Congestion games model several interesting applications, including routing and network formation games, and also possess attractive theoretical properties, including the existence of and convergence of natural dynamics to a pure Nash equilibrium. Weighted variants of congestion games that rely on sharing costs proportional to players' weights do not generally have pure-strategy Nash equilibria. We propose a new way of assigning costs to players with weights in congestion games that recovers the important properties of the unweighted model. This method is derived from the Shapley value, and it always induces a game with a (weighted) potential function. For the special cases of weighted network cost-sharing and weighted routing games with Shapley value-based cost shares, we prove tight bounds on the worst-case inefficiency of equilibria. For weighted network cost-sharing games, we precisely calculate the price of stability for any given player weight vector, while for weighted routing games, we precisely calculate the price of anarchy, as a parameter of the set of allowable cost functions.

Reconciling the Overlay and Underlay Tussle

“…Furthermore, the formulation of our problem considers both the interactions among overlays and the overlays-underlays game. Recently, Kollias and Roughgarden have shown a novel method of finding pure Nash equilibrium for weighted congestion games under Shapley value [15]. We extend their results to show that our model fits under the same general category and thus can serve as the basis of mutual cooperation among the overlays as well as between the overlay and the underlays.…”

“…It is known that weighted congestion games do not have pure Nash equilibrium in general. However, a recent study [15] shows that by constructing the utility function based on Shapley value, pure Nash equilibrium can be restored. We extend this finding to our case and show that with the proper choice of a parameter, the Shapley induced equilibrium is guaranteed to be an operable state as well.…”

“…It is known that weighted congestion games do not generally have pure NE. However, it has been recently shown [15] that instead of directly constructing the player utility function based on a player's perceived payoff/cost, Shapley value [28] can be used as an alternative. The application of Shapley value to our problem context brings two benefits: 1) Shapley cost guarantees the existence of a pure NE in weighed games; 2) Shapley potential allows us to examine the interplay between selfish play and coalitions that are largely intermixed in today's overlay networks.…”

Intrinsic Robustness of the Price of Anarchy

“…Recent applications of the smoothness framework presented here were given by Bhawalkar et al [15] and Kollias and Roughgarden [74] to congestion games with weighted players and arbitrary cost functions; Bhawalkar and Roughgarden [17] to simultaneous second-price auctions (Example 2.7) with bidders with subadditive valuations; Caragiannis et al [25,26] and Lucier and Paes Leme [79] to the extensively-studied "generalized second price" sponsored search auctions; Chen et al [29] to games with altruistic players; Cole et al [40] and Cohen et al [39] to coordination mechanisms [36] in scheduling; Hoeksma and Uetz [70] and Anshelevich et al [7] to classes of scheduling games; and Marden and Roughgarden [81] to subclasses of basic utility games (Example 2.6).…”

Designing Cost-Sharing Methods for Bayesian Games

“…Chen, Roughgarden and Valiant [23] were the first to study the design aspects for this game, identifying the best protocol with respect to the PoA and PoS in various cases, followed by [62] for parallel links, [40,43,54] for weighted congestion games, [29,47,59] for network games, [41] for routing games and [52] for resource allocation. The Bayesian Price of anarchy was first studied in auctions by [26]; see also [58] for routing games, and [61] for the PoS of Shapley protocol in cost-sharing games.…”