Mediated Equilibria in Load-Balancing Games

Abstract

Mediators are third parties to whom the players in a game can delegate the task of choosing a strategy; a mediator forms a mediated equilibrium if delegating is a best response for all players. Mediated equilibria have more power to achieve outcomes with high social welfare than Nash or correlated equilibria, but less power than a fully centralized authority. Here we begin the study of the power of mediation by using the mediation analogue of the price of stability—the ratio of the social cost of the best mediated equilibrium $\textsc{bme}$ to that of the socially optimal outcome $\textsc{opt}$. We focus on load-balancing games with social cost measured by weighted average latency. Even in this restricted class of games, $\textsc{bme}$ can range from as good as $\textsc{opt}$ to no better than the best correlated equilibrium. In unweighted games $\textsc{bme}$ achieves $\textsc{opt}$; the weighted case is more subtle. Our main results are (1) that the worst-case ratio $\textsc{bme}/\textsc{opt}$ is at least $(1+\sqrt{2})/2\approx 1.2071$ (and at most 1 + φ ≈ 2.618 [3]) for linear-latency weighted load-balancing games, and that the lower bound is tight when there are two players; and (2) tight bounds on the worst-case $\textsc{bme}/\textsc{opt}$ for general-latency weighted load-balancing games. We also give similarly detailed results for other natural social-cost functions.

All authors were supported in part by NSF grant CCF-0728779. This work was also supported by grants from Carleton College, Oberlin College, and Denison University. Thanks to Bobby Kleinberg for suggesting this line of inquiry to us.