Utilitarian resource assignment

Abstract

This paper studies a resource allocation problem introduced by Koutsoupias and Papadimitriou. The scenario is modelled as a multiple-player game in which each player selects one of a finite number of known resources. The cost to the player is the total weight of all players who choose that resource, multiplied by the "delay" of that resource. Recent papers have studied the Nash equilibria and social optima of this game in terms of the L ∞ cost metric, in which the social cost is taken to be the maximum cost to any player. We study the L 1 variant of this game, in which the social cost is taken to be the sum of the costs to the individual players, rather than the maximum of these costs. We give bounds on the size of the coordination ratio, which is the ratio between the social cost incurred by selfish behavior and the optimal social cost; we also study the algorithmic problem of finding optimal (lowest-cost) assignments and Nash Equilibria. Additionally, we obtain bounds on the ratio between alternative Nash equilibria for some special cases of the problem.

Efficiency analysis of load balancing games with and without activation costs

“…Unfortunately, our results have left a gap: the two parametric bounds are not necessarily asymptotically tight. It is interesting to note that such a gap is very much similar (asymptotically) to the gap left in [4] for the load balancing game as given in the introduction section, though our gap is much smaller.…”

“…In load balancing game, parameter ρ 1 represents the average job length normalized by p min . Note that, as we pointed earlier, in providing the upper bound of 4p max on the PoA, Berenbrink et al [4] assume that all job lengths are at least 1. This assumption has implicitly hidden the possibility of the unboundedness of the PoA when p min → 0, which is indeed the case as we shall show.…”

“…In this study, we consider a utilitarian social objective, which is defined as the average (or 1 metric) of all individual players' costs. Such a social objective is also used in a number of other similar studies (such as [4,15,12] and [21], in which infinitely many jobs are considered) and is a standard assumption in the multi-agent system literature (e.g., [7,18,22]). As shown by Berenbrink et al [4], Nash equilibria behave very differently under 1 metric than ∞ metric.…”

“…Berenbrink et al [4] consider pure strategies in the uniform machine environment and show that, if all job lengths are at least 1, then PoA ≤ 4p max , where p max is the maximum job length, and if additionally all machines are identical, then PoS ≥ √ p max /5. Therefore, it is variability of job lengths, as opposed to machine speeds, that may lead to a big PoS and PoA.…”

Selfish routing with incomplete information

“…For complete information routing games and Social Cost defined as the Expected Maximum Latency, there exist tight bounds of Θ log m log log m for identical links [6,21] and Θ log m log log log m [6] for related links. For complete information routing games and Social Cost defined as the Sum of Individual Costs, Berenbrink et al [3] give a lower bound of n 5 on the Coordination Ratio. Restricting to pure Nash equilibria, they also show an upper bound that solely depends on the user traffics.…”