Network design is a fundamental problem for which it is important to understand the effects of strategic behavior. Given a collection of self-interested agents who want to form a network connecting certain endpoints, the set of stable solutions—the Nash equilibria—may look quite different from the centrally enforced optimum. We study the quality of the best Nash equilibrium, and refer to the ratio of its cost to the optimum network cost as the price of stability. The best Nash equilibrium solution has a natural meaning of stability in this context—it is the optimal solution that can be proposed from which no user will defect. We consider the price of stability for network design with respect to one of the most widely studied protocols for network cost allocation, in which the cost of each edge is divided equally between users whose connections make use of it; this fair-division scheme can be derived from the Shapley value and has a number of basic economic motivations. We show that the price of stability for network design with respect to this fair cost allocation is $O(\log k)$, 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. This establishes that the fair cost allocation protocol is in fact a useful mechanism for inducing strategic behavior to form near-optimal equilibria. We discuss connections to the class of potential games defined by Monderer and Shapley, and extend our results to cases in which users are seeking to balance network design costs with latencies in the constructed network, with stronger results when the network has only delays and no construction costs. We also present bounds on the convergence time of best-response dynamics, and discuss extensions to a weighted game.

/pdf/the-price-of-stability-for-network-design-with-fair-cost-89te3x19ow.pdf

The Price of Stability for Network Design with Fair Cost Allocation

Network design is a fundamental problem for which it is important to understand the effects of strategic behavior. Given a collection of self-interested agents who want to form a network connecting certain endpoints, the set of stable solutions - the Nash equilibria - may look quite different from the centrally enforced optimum. We study the quality of the best Nash equilibrium, and refer to the ratio of its cost to the optimum network cost as the price of stability. The best Nash equilibrium solution has a natural meaning of stability in this context - it is the optimal solution that can be proposed from which no user will "defect". We consider the price of stability for network design with respect to one of the most widely-studied protocols for network cost allocation, in which the cost of each edge is divided equally between users whose connections make use of it; this fair-division scheme can be derived from the Shapley value, and has a number of basic economic motivations. We show that the price of stability for network design with respect to this fair cost allocation is O(log k), 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. This establishes that the fair cost allocation protocol is in fact a useful mechanism for inducing strategic behavior to form near-optimal equilibria. We discuss connections to the class of potential games defined by Monderer and Shapley, and extend our results to cases in which users are seeking to balance network design costs with latencies in the constructed network, with stronger results when the network has only delays and no construction costs. We also present bounds on the convergence time of best-response dynamics, and discuss extensions to a weighted game.

/pdf/the-price-of-stability-for-network-design-with-fair-cost-4b3bbv50mq.pdf

The price of stability for network design with fair cost allocation

The price of anarchy, defined as the ratio of the worst-case objective function value of a Nash equilibrium of a game and that of an optimal outcome, quantifies the inefficiency of selfish behavior. Remarkably good bounds on this measure are known for a wide range of application domains. However, such bounds are meaningful only if a game's participants successfully reach a Nash equilibrium. This drawback motivates inefficiency bounds that apply more generally to weaker notions of equilibria, such as mixed Nash equilibria and correlated equilibria, and to sequences of outcomes generated by natural experimentation strategies, such as successive best responses and simultaneous regret-minimization.We establish a general and fundamental connection between the price of anarchy and its seemingly more general relatives. First, we identify a “canonical sufficient condition” for an upper bound on the price of anarchy of pure Nash equilibria, which we call a smoothness argument. Second, we prove an “extension theorem”: every bound on the price of anarchy that is derived via a smoothness argument extends automatically, with no quantitative degradation in the bound, to mixed Nash equilibria, correlated equilibria, and the average objective function value of every outcome sequence generated by no-regret learners. Smoothness arguments also have automatic implications for the inefficiency of approximate equilibria, for bicriteria bounds, and, under additional assumptions, for polynomial-length best-response sequences. Third, we prove that in congestion games, smoothness arguments are “complete” in a proof-theoretic sense: despite their automatic generality, they are guaranteed to produce optimal worst-case upper bounds on the price of anarchy.

/pdf/intrinsic-robustness-of-the-price-of-anarchy-2ezupdog91.pdf

Intrinsic Robustness of the Price of Anarchy

Strong price of anarchy

Given a collection of selfish agents who wish to establish links to route traffic among themselves, the set of equilibrium network topologies may appear quite different from the centrally enforced optimum. We study the quality (price of anarchy) of equilibrium networks in a game where links require the consent of both participants and are negotiated bilaterally and compare these networks to those generated by an earlier model due to Fabrikant et al. [6] in which links are formed unilaterally. We provide a characterization of stable and efficient networks in the bilateral network formation game, show that the set of stable networks is richer than those in the unilateral game, and that all stable networks of the unilateral game are also stable in the bilateral game. We also provide an upper and lower bound on the price of anarchy (tight in the size of the network n but not the link cost α) of the bilateral game and show that the worst-case price of anarchy of the bilateral model is worse than for the unilateral model. A careful empirical analysis demonstrates that the average price of anarchy is better in the bilateral connection game than in the unilateral game for small link costs but worse as links become more expensive. In the process, a powerful equivalence between link-based graph stability and two game-theoretic equilibrium notions is also discussed. The equivalence establishes necessary and sufficient conditions for an equilibrium in the bilateral game that helps provide a partial geometric characterization of equilibrium graphs.

/pdf/the-price-of-selfish-behavior-in-bilateral-network-formation-452coukfaj.pdf

The price of selfish behavior in bilateral network formation

We study a network creation game recently proposed by Fabrikant, Luthra, Maneva, Papadimitriou and Shenker. In this game, each player (vertex) can create links (edges) to other players at a cost of α per edge. The goal of every player is to minimize the sum consisting of (a) the cost of the links he has created and (b) the sum of the distances to all other players.Fabrikant et al. conjectured that there exists a constant A such that, for any α > A, all non-transient Nash equilibria graphs are trees. They showed that if a Nash equilibrium is a tree, the price of anarchy is constant. In this paper we disprove the tree conjecture. More precisely, we show that for any positive integer n0, there exists a graph built by n ≥ n0 players which contains cycles and forms a non-transient Nash equilibrium, for any α with 1

/pdf/on-nash-equilibria-for-a-network-creation-game-3cuoumfedc.pdf

On nash equilibria for a network creation game

We study a basic network creation game proposed by Fabrikant et al. [2003]. In this game, each player (vertex) can create links (edges) to other players at a cost of α per edge. The goal of every player is to minimize the sum consisting of (a) the cost of the links he has created and (b) the sum of the distances to all other players.Fabrikant et al. conjectured that there exists a constant A such that, for any α > A, all nontransient Nash equilibria graphs are trees. They showed that if a Nash equilibrium is a tree, the price of anarchy is constant. In this article, we disprove the tree conjecture. More precisely, we show that for any positive integer n0, there exists a graph built by n ≥ n0 players which contains cycles and forms a nontransient Nash equilibrium, for any α with 1

/pdf/on-nash-equilibria-for-a-network-creation-game-43vyjdc3z3.pdf

Stefan Eilts

Papers

On nash equilibria for a network creation game

On Nash Equilibria for a Network Creation Game