Tom Wexler

Near-optimal network design with selfish agents

We introduce a simple network design game that models how independent selfish agents can build or maintain a large network. In our game every agent has a specific connectivity requirement, i.e. each agent has a set of terminals and wants to build a network in which his terminals are connected. Possible edges in the network have costs and each agents goal is to pay as little as possible. Determining whether or not a Nash equilibrium exists in this game is NP-complete. However, when the goal of each player is to connect a terminal to a common source, we prove that there is a Nash equilibrium as cheap as the optimal network, and give a polynomial time algorithm to find a (1+ε)-approximate Nash equilibrium that does not cost much more. For the general connection game we prove that there is a 3-approximate Nash equilibrium that is as cheap as the optimal network, and give an algorithm to find a (4.65+ε)-approximate Nash equilibrium that does not cost much more.

The effect of collusion in congestion games

In this paper we initiate the study of how collusion alters the quality of solutions obtained in competitive games. The price of anarchy aims to measure the cost of the lack of coordination by comparing the quality of a Nash equilibrium to that of a centrally designed optimal solution. This notion assumes that players act not only selfishly, but also independently. We propose a framework for modeling groups of colluding players, in which members of a coalition cooperate so as to selfishly maximize their collective welfare. Clearly, such coalitions can improve the social welfare of the participants, but they can also harm the welfare of those outside the coalition. One might hope that the improvement for the coalition participants outweighs the negative effects on the others. This would imply that increased cooperation can only improved the overall solution quality of stable outcomes. However, increases in coordination can actually lead to significant decreases in total social welfare. In light of this, we propose the price of collusion as a measure of the possible negative effect of collusion, specifying the factor by which solution quality can deteriorate in the presence of coalitions. We give examples to show that the price of collusion can be arbitrarily high even in convex games. Our main results show that in the context of load-balancing games, the price of collusion depends upon the disparity in market power among the game participants. We show that in some symmetric nonatomic games (where all users have access to the same set of strategies) increased cooperation always improves the solution quality, and in the discrete analogs of such games, the price of collusion is bounded by two.

Algorithmic Game Theory: Network Formation Games and the Potential Function Method

Abstract Large computer networks such as the Internet are built, operated, and used by a large number of diverse and competitive entities. In light of these competing forces, it is surprising how efficient these networks are. An exciting challenge in the area of algorithmic game theory is to understand the success of these networks in game theoretic terms: what principles of interaction lead selfish participants to form such efficient networks? In this chapter we present a number of network formation games. We focus on simple games that have been analyzed in terms of the efficiency loss that results from selfishness. We also highlight a fundamental technique used in analyzing inefficiency in many games: the potential function method. Introduction The design and operation of many large computer networks, such as the Internet, are carried out by a large number of independent service providers (Autonomous Systems), all of whom seek to selfishly optimize the quality and cost of their own operation. Game theory provides a natural framework for modeling such selfish interests and the networks they generate. These models in turn facilitate a quantitative study of the trade-off between efficiency and stability in network formation. In this chapter, we consider a range of simple network formation games that model distinct ways in which selfish agents might create and evaluate networks.

A network pricing game for selfish traffic

The success of the Internet is remarkable in light of the decentralized manner in which it is designed and operated. Unlike small scale networks, the Internet is built and controlled by a large number of disperate service providers who are not interested in any global optimization. Instead, providers simply seek to maximize their own profit by charging users for access to their service. Users themselves also behave selfishly, optimizing over price and quality of service. Game theory provides a natural framework for the study of such a situation. However, recent work in this area tends to focus on either the service providers or the network users, but not both. This paper introduces a new model for exploring the interaction of these two elements, in which network managers compete for users via prices and the quality of service provided. We study the extent to which competition between service providers hurts the overall social utility of the system.

Triangulation and embedding using small sets of beacons

Concurrent with recent theoretical interest in the problem of metric embedding, a growing body of research in the networking community has studied the distance matrix defined by node-to-node latencies in the Internet, resulting in a number of recent approaches that approximately embed this distance matrix into low-dimensional Euclidean space. Here we give algorithms with provable performance guarantees for beacon-based triangulation and embedding. We show that in addition to multiplicative error in the distances, performance guarantees for beacon-based algorithms typically must include a notion of slack - a certain fraction of all distances may be arbitrarily distorted. For metrics of bounded doubling dimension (which have been proposed as a reasonable abstraction of Internet latencies), we show that triangulation-based reconstruction with a constant number of beacons can achieve multiplicative error 1 + /spl delta/ on a 1 - /spl epsiv/ fraction of distances, for arbitrarily small constants /spl delta/ and /spl epsiv/. For this same class of metrics, we give a beacon-based embedding algorithm that achieves constant distortion on a 1 - /spl epsiv/ fraction of distances; this provides some theoretical justification for the success of the recent global network positioning algorithm of Ng and Zhang, and it forms an interesting contrast with lower bounds showing that it is not possible to embed all distances in a doubling metric with constant distortion. We also give results for other classes of metrics, as well as distributed algorithms that require only a sparse set of distances but do not place too much measurement load on any one node.

Computing Shapley Value in Supermodular Coalitional Games

Coalitional games allow subsets (coalitions) of players to cooperate to receive a collective payoff. This payoff is then distributed “fairly” among the members of that coalition according to some division scheme. Various solution concepts have been proposed as reasonable schemes for generating fair allocations. The Shapley value is one classic solution concept: player i’s share is precisely equal to i’s expected marginal contribution if the players join the coalition one at a time, in a uniformly random order. In this paper, we consider the class of supermodular games (sometimes called convex games), and give a fully polynomial-time randomized approximation scheme (FPRAS) to compute the Shapley value to within a (1 ±e) factor in monotone supermodular games. We show that this result is tight in several senses: no deterministic algorithm can approximate Shapley value as well, no randomized algorithm can do better, and both monotonicity and supermodularity are required for the existence of an efficient (1 ±e)-approximation algorithm. We also argue that, relative to supermodularity, monotonicity is a mild assumption, and we discuss how to transform supermodular games to be monotonic.

A Duopoly Pricing Game for Wireless IP Services

This paper addresses the behavior of the selfish service providers in the form of IP sinks providing high-speed IP access. Service providers compete for mobile users by adjusting the price they charge for their services. Their aim is to maximize the total collected profit. Mobile users are also selfish choosing the service provider offering the best quality of service and price combination. As the service providers come closer to each other, we show the existence of three critical phase transitions in their behavior. Depending on the separation between them, there may exists a unique Nash equilibrium, or a continuum of Nash equilibria, or no Nash equilibrium. We completely characterize the pricing strategies of service providers at Nash equilibria. We also prove that the total social welfare in the presence of selfish providers is close to the maximum social welfare that can reached through non- selfish optimization.

Signed domination numbers of a graph and its complement

Abstract Let G=(V,E) be a simple graph on vertex set V and define a function f:V→{−1,1}. The function f is a signed dominating function if for every vertex x∈V, the closed neighborhood of x contains more vertices with function value 1 than with −1. The signed domination number of G, γs(G), is the minimum weight of a signed dominating function on G. Let G denote the complement of G. In this paper we establish upper and lower bounds on γ s (G)+γ s ( G ) .

The price of civil society

Most work in algorithmic game theory assumes that players ignore costs incurred by their fellow players. In this paper, we consider superimposing a social network over a game, where players are concerned with minimizing not only their own costs, but also the costs of their neighbors in the network. We aim to understand how properties of the underlying game are affected by this alteration to the standard model. The new social game has its own equilibria, and the price of civil society denotes the ratio of the social cost of the worst such equilibrium relative to the worst Nash equilibrium under standard selfish play. We initiate the study of the price of civil society in the context of a simple class of games. Counterintuitively, we show that when players become less selfish (optimizing over both themselves and their friends), the resulting outcomes may be worse than they would have been in the base game. We give tight bounds on this phenomenon in a simple class of load-balancing games, over arbitrary social networks, and present some extensions.

Assignment Games with Conflicts: Robust Price of Anarchy and Convergence Results via Semi-Smoothness

We study assignment games in which jobs select machines, and in which certain pairs of jobs may conflict, which is to say they may incur an additional cost when they are both assigned to the same machine, beyond that associated with the increase in load. Questions regarding such interactions apply beyond allocating jobs to machines: when people in a social network choose to align themselves with a group or party, they typically do so based upon not only the inherent quality of that group, but also who amongst their friends (or enemies) chooses that group as well. We show how semi-smoothness, a recently introduced generalization of smoothness, is necessary to find tight bounds on the robust price of anarchy, and thus on the quality of correlated and Nash equilibria, for several natural job-assignment games with interacting jobs. For most cases, our bounds on the robust price of anarchy are either exactly 2 or approach 2. We also prove new convergence results implied by semi-smoothness for our games. Finally we consider coalitional deviations, and prove results about the existence and quality of strong equilibrium.