Tighter Bounds on the Inefficiency Ratio of Stable Equilibria in Load Balancing Games

In this paper we study the inefficiency ratio of stable equilibria in load balancing games introduced by Asadpour and Saberi [3]. We prove tighter lower and upper bounds of 7/6 and 4/3, respectively. This improves over the best known bounds in problem (19/18 and 3/2, respectively). Equivalently, the results apply to the question of how well the optimum for the $L_2$ -norm can approximate the $L_{\infty}$-norm (makespan) in identical machines scheduling

Strong stability of Nash equilibria in load balancing games

We study strong stability of Nash equilibria in the load balancing games of m (m >= 2) identical servers, in which every job chooses one of the m servers and each job wishes to minimize its cost, given by the workload of the server it chooses. A Nash equilibrium (NE) is a strategy profile that is resilient to unilateral deviations. Finding an NE in such a game is simple. However, an NE assignment is not stable against coordinated deviations of several jobs, while a strong Nash equilibrium (SNE) is. We study how well an NE approximates an SNE. Given any job assignment in a load balancing game, the improvement ratio (IR) of a deviation of a job is defined as the ratio between the pre-and post-deviation costs. An NE is said to be a ρ-approximate SNE (ρ >= 1) if there is no coalition of jobs such that each job of the coalition will have an IR more than ρ from coordinated deviations of the coalition. While it is already known that NEs are the same as SNEs in the 2-server load balancing game, we prove that, in the m-server load balancing game for any given m >= 3, any NE is a (5=4)-approximate SNE, which together with the lower bound already established in the literature implies that the approximation bound is tight. This closes the final gap in the literature on the study of approximation of general NEs to SNEs in the load balancing games. To establish our upper bound, we apply with novelty a powerful graph-theoretic tool

Efficiency analysis of load balancing games with and without activation costs

In this paper, we study two models of resource allocation games: the classical load-balancing game and its new variant involving resource activation costs. The resources we consider are identical and the social costs of the games are utilitarian, which are the average of all individual players' costs. Using the social costs we assess the quality of pure Nash equilibria in terms of the price of anarchy (PoA) and the price of stability (PoS). For each game problem, we identify suitable problem parameters and provide a parametric bound on the PoA and the PoS. In the case of the load-balancing game, the parametric bounds we provide are sharp and asymptotically tight

On the Impact of Singleton Strategies in Congestion Games

To what extent does the structure of the players\u27 strategy space influence the efficiency of decentralized solutions in congestion games? In this work, we investigate whether better performance is possible when restricting to load balancing games in which players can only choose among single resources. We consider three different solutions concepts, namely, approximate pure Nash equilibria, approximate one-round walks generated by selfish players aiming at minimizing their personal cost and approximate one-round walks generated by cooperative players aiming at minimizing the marginal increase in the sum of the players\u27 personal costs. The last two concepts can also be interpreted as solutions of simple greedy online algorithms for the related resource selection problem. Under fairly general latency functions on the resources, we show that, for all three types of solutions, better bounds cannot be achieved if players are either weighted or asymmetric. On the positive side, we prove that, under mild assumptions on the latency functions, improvements on the performance of approximate pure Nash equilibria are possible for load balancing games with weighted and symmetric players in the case of identical resources. We also design lower bounds on the performance of one-round walks in load balancing games with unweighted players and identical resources (in this case, solutions generated by selfish and cooperative players coincide)

On Linear Congestion Games with Altruistic Social Context

We study the issues of existence and inefficiency of pure Nash equilibria in linear congestion games with altruistic social context, in the spirit of the model recently proposed by de Keijzer {\em et al.} \cite{DSAB13}. In such a framework, given a real matrix $\Gamma=(\gamma_{ij})$ specifying a particular social context, each player $i$ aims at optimizing a linear combination of the payoffs of all the players in the game, where, for each player $j$, the multiplicative coefficient is given by the value $\gamma_{ij}$. We give a broad characterization of the social contexts for which pure Nash equilibria are always guaranteed to exist and provide tight or almost tight bounds on their prices of anarchy and stability. In some of the considered cases, our achievements either improve or extend results previously known in the literature

Price of anarchy in non-cooperative load balancing games

We investigate the price of anarchy of a load balancing game with K dispatchers. The service rates and holding costs are assumed to depend on the server, and the service discipline is assumed to be processor-sharing at each server. The performance criterion is taken to be the weighted mean number of jobs in the system, or equivalently, the weighted mean sojourn time in the system. Independent of the state of the servers, each dispatcher seeks to determine the routing strategy that optimizes the performance for its own traffic. The interaction of the various dispatchers thus gives rise to a non-cooperative game. For this game, we first show that, for a fixed amount of total incoming traffic, the worst-case Nash equilibrium occurs when each player routes exactly the same amount of traffic, i.e., when the game is symmetric. For this symmetric game, we provide the expression for the loads on the servers at the Nash equilibrium. Using this result, we then show that, for a system with two or more servers, the price of anarchy, which is the worst-case ratio of the global cost of the Nash equilibrium to the global cost of the centralized setting, is lower bounded by K(2√K-1) and upper bounded by √K, independent of the number of servers