We introduce the concept of a sink equilibrium. A sink equilibrium is a strongly connected component with no outgoing arcs in the strategy profile graph associated with a game. The strategy profile graph has a vertex set induced by the set of pure strategy profiles; its arc set corresponds to transitions between strategy profiles that occur with nonzero probability. (Here our focus will just be on the special case in which the strategy profile graph is actually a best response graph; that is, its arc set corresponds exactly to best response moves that result from myopic or greedy behaviour.) We argue that there is a natural convergence process to sink equilibria in games where agents use pure strategies. This leads to an alternative measure of the social cost of a lack of coordination, the price of sinking, which

In many applications, wireless ad-hoc networks are formed by devices belonging to independent users. Therefore, a challenging problem is how to provide incentives to stimulate cooperation. In this paper, we study ad-hoc games—the routing and packet forwarding games in wireless ad-hoc networks. Unlike previous work which focuses either on routing or on forwarding, this paper investigates both routing and forwarding. We first uncover an impossibility result—there does not exist a protocol such that following the protocol to always forward others’ traffic is a dominant action. Then we define a novel solution concept called cooperation-optimal protocols. We present Corsac, a cooperation-optimal protocol consisting of a routing protocol and a forwarding protocol. The routing protocol of Corsac integrates VCG with a novel cryptographic technique to address the challenge in wireless ad-hoc networks

We study the impact of combinatorial structure in congestion games on the complexity of computing pure Nash equilibria and the convergence time of best response sequences. In particular, we investigate which properties of the strategy spaces of individual players ensure a polynomial convergence time. We show that if the strategy space of each player consists of the bases of a matroid over the set of resources, then the lengths of all best response sequences are polynomially bounded in the number of players and resources. We also prove that this result is tight, that is, the matroid property is a necessary and sufficient condition on the players ’ strategy spaces for guaranteeing polynomial time convergence to a Nash equilibrium. In addition, we present an approach that enables us to devise hardness proofs for various kinds of combinatorial games, including first results about the hardness of market sharing games and congestion games for overlay network design. Our approach also yields a short proof for the PLS-completeness of network congestion games. In particular, we show that network congestion games are PLS-complete for directed and undirected networks even in case of linear latency functions.

A separable assignment problem (SAP) is defined by a set of bins and a set of items to pack in each bin; a value, fij, for assigning item j to bin i; and a separate packing constraint for each bin – i.e. for bin i, a family Ii of subsets of items that fit in bin i. The goal is to pack items into bins to maximize the aggregate value. This class of problems includes the maximum generalized assignment problem (GAP) 1) and a distributed caching problem (DCP) described in this paper. Given a β-approximation algorithm for finding the highest value packing of a single bin, we give 1. A polynomial-time LP-rounding based ((1 − 1 e)β)approximation algorithm. 2. A simple polynomial-time local search ( β approximation algorithm, for any ɛ> 0. β+1 − ɛ)-Therefore, for all examples of SAP that admit an approximation scheme for the single-bin problem, we obtain an LPbased algorithm with (1 − 1 e − ɛ)-approximation and a local search algorithm with ( 1 2 −ɛ)-approximation guarantee. Furthermore, for cases in which the subproblem admits a fully polynomial approximation scheme (such as for GAP), the LP-based algorithm analysis can be strengthened to give a guarantee of 1 − 1 e. The best previously known approximation algorithm for GAP is a 1 2-approximation by Shmoys and Tardos; and Chekuri and Khanna. Our LP algorithm is based on rounding a new linear programming relaxation, with a provably better integrality gap. To complement these results, we show that SAP and DCP cannot be approximated within a factor better than 1 − 1 e unless NP ⊆ DTIME(n O(log log n)), even if there exists a polynomial-time exact algorithm for the single-bin problem.

We analyze replication of resources by server nodes that act selfishly, using a game-theoretic approach. We refer to this as the selfish caching problem. In our model, nodes incur either cost for replicating resources or cost for access to a remote replica. We show the existence of pure strategy Nash equilibria and investigate the price of anarchy, which is the relative cost of the lack of coordination. The price of anarchy can be high due to undersupply problems, but with certain network topologies it has better bounds. With a payment scheme the game can always implement the social optimum in the best case by giving servers incentive to replicate.

We propose weakening the assumption made when studying the price of anarchy: Rather than assume that self-interested players will play according to a Nash equilibrium (which may even be computationally hard to find), we assume only that selfish players play so as to minimize their own regret. Regret minimization can be done via simple, efficient algorithms even in many settings where the number of action choices for each player is exponential in the natural parameters of the problem. We prove that despite our weakened assumptions, in several broad classes of games, this “price of total anarchy ” matches the Nash price of anarchy, even though play may never converge to Nash equilibrium. In contrast to the price of anarchy and the recently introduced price of sinking [15], which require all players to behave in a prescribed manner, we show that the price of total anarchy is in many cases resilient to the presence of Byzantine players, about whom we make no assumptions. Finally, because the price of total anarchy is an upper bound on the price of anarchy even in mixed strategies, for some games our results yield as corollaries previously unknown bounds on the price of anarchy in mixed strategies. 1

We study the speed of convergence to approximately optimal states in two classes of potential games. We provide bounds in terms of the number of rounds, where a round consists of a sequence of movements, with each player appearing at least once in each round. We model the sequential interaction between players by a best-response walk in the state graph, where every transition in the walk corresponds to a best response of a player. Our goal is to bound the social value of the states at the end of such walks. In this paper, we focus on two classes of potential games: selfish routing games, and cut games (or party affiliation games [7]).

Additionally, our analysis of player-specific matroid congestion games yields a polynomial time algorithm for computing pure equilibria. We also address questions related to the convergence time of such games. For player-specific matroid congestion games, in which the best response dynamics may cycle, we show that from every state there exists a short sequences of better responses to an equilibrium. For weighted matroid congestion games, we present a superpolynomial lower bound on the convergence time of the best response dynamics showing that players do not even converge in pseudopolynomial time.

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. 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

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