A Characterization of Undirected Graphs Admitting Optimal Cost Shares

Abstract

In a seminal paper, Chen, Roughgarden and Valiant [7] studied cost sharing protocols for network design with the objective to implement a low-cost Steiner forest as a Nash equilibrium of an induced cost-sharing game. One of the most intriguing open problems to date is to understand the power of budget-balanced and separable cost sharing protocols in order to induce low-cost Steiner forests. In this work, we focus on undirected networks and analyze topological properties of the underlying graph so that an optimal Steiner forest can be implemented as a Nash equilibrium (by some separable cost sharing protocol) independent of the edge costs. We term a graph efficient if the above stated property holds. As our main result, we give a complete characterization of efficient undirected graphs for two-player network design games: an undirected graph is efficient if and only if it does not contain (at least) one out of few forbidden subgraphs. Our characterization implies that several graph classes are efficient: generalized series-parallel graphs, fan and wheel graphs and graphs with small cycles.

Efficient Black-Box Reductions for Separable Cost Sharing

“…Efficient Black-Box Reductions for Separable Cost Sharing a similar result was shown even for two-player games on undirected graphs [35]. Thus, for general graphs, we cannot expect separable protocols with optimal or close-to-optimal equilibria, or (efficient) black-box reductions.…”

“…As such, the optimal solutions, the Nash equilibria, and their total costs correspond exactly to the ones of the original multi-source game. For the general multi-source case, there are instances with no enforceable optimal solution (even for only two players, see [35]).…”

“…, (s n , t n )) is irredundant, meaning that each edge and each vertex of G is contained in at least one (s i , t i )-path for some player i ∈ N (nodes and edges not used by any player can easily be recognized (and then deleted) by Algorithm Irredundant in [34]; adapted from Algorithm 1 in [18]). For the special case without delays, enforceability was characterized via an LP in [35]. We…”

“…. , P n ) the following cost shares (these cost shares resemble those introduced in [53]; the proof that P is a PNE can be directly adapted from [35]):…”

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.…”