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).
As we note in the introduction, the focus of the works above were on the effect of malice on the equilibrium of the system: how the equilibrium degrades and how the game between the selfish agents and the malicious agent takes place. On the other hand we are more interested in how the presence of malice affects the social optimum itself. Thus our work complements the works of Karakostas and Viglas @cite_7 and Babaioff @cite_1 .
@cite_1 @cite_7
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Over the last years, game theory has provided great insights into the behavior of distributed systems by modeling the players as utility-maximizing agents. In particular, it has been shown that selfishness causes many systems to perform in a globally suboptimal fashion. Such systems are said to have a large Price of Anarchy. In this paper, we extend this active field of research by allowing some players to be malicious or Byzantine rather than selfish. We ask: What is the impact of Byzantine players on the system's efficiency compared to purely selfish environments or compared to the social optimum? In particular, we introduce the Price of Malice which captures this efficiency degradation. As an example, we analyze the Price of Malice of a game which models the containment of the spread of viruses. In this game, each node can choose whether or not to install anti-virus software. Then, a virus starts from a random node and iteratively infects all neighboring nodes which are not inoculated. We establish various results about this game. For instance, we quantify how much the presence of Byzantine players can deteriorate or---in case of highly risk-averse selfish players---improve the social welfare of the distributed system. We study the equilibria of non-atomic congestion games in which there are two types of players: rational players, who seek to minimize their own delay, and malicious players, who seek to maximize the average delay experienced by the rational players. We study the existence of pure and mixed Nash equilibria for these games, and we seek to quantify the impact of the malicious players on the equilibrium. One counter intuitive phenomenon which we demonstrate is the "windfall of malice": paradoxically, when a myopically malicious player gains control of a fraction of the flow, a fraction of the players change from rational to malicious, the new equilibrium may be more favorable for the remaining rational players than the previous equilibrium. We consider a problem at the intersection of distributed computing and game theory, namely: Is it possible to achieve the "windfall of malice" even without the actual presence of malicious players? Our answer to this question is "Yes and No". Our positive result is that for the virus inoculation game, it is possible to achieve the windfall of malice by use of a mediator. Our negative result is that for symmetric congestion games that are known to have a windfall of malice, it is not possible to design a mediator that achieves this windfall. In proving these two results, we develop novel techniques for mediator design that we believe will be helpful for creating non-trivial mediators to improve social welfare in a large class of games. In this paper we initiate the study of how collusion alters the quality of solutions obtained in competitive games. The price of anarchy aims to measure the cost of the lack of coordination by comparing the quality of a Nash equilibrium to that of a centrally designed optimal solution. This notion assumes that players act not only selfishly, but also independently. We propose a framework for modeling groups of colluding players, in which members of a coalition cooperate so as to selfishly maximize their collective welfare. Clearly, such coalitions can improve the social welfare of the participants, but they can also harm the welfare of those outside the coalition. One might hope that the improvement for the coalition participants outweighs the negative effects on the others. This would imply that increased cooperation can only improved the overall solution quality of stable outcomes. However, increases in coordination can actually lead to significant decreases in total social welfare. In light of this, we propose the price of collusion as a measure of the possible negative effect of collusion, specifying the factor by which solution quality can deteriorate in the presence of coalitions. We give examples to show that the price of collusion can be arbitrarily high even in convex games. Our main results show that in the context of load-balancing games, the price of collusion depends upon the disparity in market power among the game participants. We show that in some symmetric nonatomic games (where all users have access to the same set of strategies) increased cooperation always improves the solution quality, and in the discrete analogs of such games, the price of collusion is bounded by two.