In the traditional setting of routing games, the standard assumption is that selfish agents are unconcerned with the performance of their competitors in the network. We propose an extension to this setting by modeling agents to consider a combination of their own performance as well as that of their rivals. Per agent, we parameterize this trade-off, thereby allowing agents to be partially selfish and partially malicious. We consider two types of routing games based on the structure of the agents' performance objectives, namely bottleneck routing games and additive routing games. For bottleneck routing games, the performance of an agent is determined by its worst-case link performance, and for additive routing games, performance is determined by the sum of its link performances. For the bottleneck routing scenario we establish the existence of a Nash equilibrium and show that the Price of Stability is equal to 1. We also prove that the Price of Anarchy is unbounded. For additive routing games, we focus on the fundamental load balancing game of routing over parallel links. For an interesting class of agents, we prove the existence of a Nash equilibrium. Specifically, we establish that a special case of the Wardrop equilibrium is likewise a Nash equilibrium. Moreover, when the system consists of two agents, this Nash equilibrium is unique, and for the general case of N agents, we present an example of its non-uniqueness.
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