The Effect of Malice on the Social Optimum in Linear Load Balancing Games

TL;DR

This paper analyzes how malicious players influence the social optimum in linear load balancing games, showing that malicious strategies are equivalent to selfish behavior and introducing a measure called Cost of Malice.

Contribution

It demonstrates the existence of pure strategy Nash equilibrium in the presence of malicious players and bounds the impact of malice with a new ratio called Cost of Malice.

Findings

Malicious players cannot cause more harm than selfish agents.

Pure strategy Nash equilibrium exists in the game.

Cost of Malice is bounded by (1+f/2).

Abstract

In this note we consider the following problem to study the effect of malicious players on the social optimum in load balancing games: Consider two players SOC and MAL controlling (1-f) and f fraction of the flow in a load balancing game. SOC tries to minimize the total cost faced by her players while MAL tries to maximize the same. If the latencies are linear, we show that this 2-player zero-sum game has a pure strategy Nash equilibrium. Moreover, we show that one of the optimal strategies for MAL is to play selfishly: let the f fraction of the flow be sent as when the flow was controlled by infinitesimal players playing selfishly and reaching a Nash equilibrium. This shows that a malicious player cannot cause more harm in this game than a set of selfish agents. We also introduce the notion of Cost of Malice - the ratio of the cost faced by SOC at equilibrium to (1-f)OPT, where OPT is the social optimum minimizing the cost of all the players. In linear load balancing games we bound the cost of malice by (1+f/2).