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

The Quality of Equilibria for Set Packing Games

We introduce set packing games as an abstraction of situations in which $n$ selfish players select subsets of a finite set of indivisible items, and analyze the quality of several equilibria for this class of games. Assuming that players are able to approximately play equilibrium strategies, we show that the total quality of the resulting equilibrium solutions is only moderately suboptimal. Our results are tight bounds on the price of anarchy for three equilibrium concepts, namely Nash equilibria, subgame perfect equilibria, and an equilibrium concept that we refer to as $k$-collusion Nash equilibrium

The sequential price of anarchy for affine congestion games with few players

This paper determines the sequential price of anarchy for Rosenthal congestion games with affine cost functions and few players. We show that for two players, the sequential price of anarchy equals 1.5, and for three players it equals approximately 2.13. While the case with two players is analyzed analytically, the tight bound for three players is based on the explicit computation of a worst-case instance using linear programming. The basis for both results are combinatorial arguments to show that finite worst-case instances exist

The Inefficiency of Nash and Subgame Perfect Equilibria for Network Routing

This paper provides new bounds on the quality of equilibria in finite congestion games with affine cost functions, specifically for atomic network routing games. It is well known that the price of anarchy equals exactly 5/2 in general. For symmetric network routing games, it is at most (5n−2)/(2n+ 1), where n is the number of players. The paper answers to two open questions for congestion games. First, we show that the price of anarchy bound (5n−2)/(2n+ 1) is tight for symmetric network routing games, thereby answering a decade-old open question. Secondly, we ask if sequential play and subgame perfection allows to evade worst-case Nash equilibria, and thereby reduces the price of anarchy. This is motivated by positive results for congestion games with a small number of players, as well as recent results for other resource allocation problems. Our main result is the perhaps surprising proof that subgame perfect equilibria of sequential symmetric network routing games with linear cost functions can have an unbounded price of anarchy. We complete the picture by analyzing the case with two players: We show that the sequential price of anarchy equals 7/5, and that computing the outcome of a subgame perfect equilibrium is NP-hard

Sequential Scheduling on Identical Machines

We study a sequential version of the well-known KP-model: Each of n agents has a job that needs to be processed on any of m machines. Agents sequentially select a machine for processing their jobs. The goal of each agent is to minimize the finish time of his machine. We study the corresponding sequential price of anarchy for m identical machines under arbitrary and LPT orders, and suggest insights into the case of two unrelated machines. Keywords: sequential price of anarchy, machine scheduling, congestion games, load balancing, subgame-perfect equilibrium, makespan minimization.