On the Fragility of Worst-Case Nash Equilibria in Atomic Congestion Games

Abstract

Aggregate performance in smart mobility systems depends heavily on the emergent behavior of selfish, resource-sharing agents that participate within the systems. As a result, recent work has focused on how a system designer can leverage incentives to influence behavior so that system cost (e.g., traffic congestion) is minimized. These results show that worst-case equilibria can be quite inefficient compared to system-optimal allocations. However, it is unclear to what extent agents are ``satisfied'' with their decisions in these worst-case scenarios. We demonstrate that in any incentivized atomic congestion game, agents' aggregate satisfaction at equilibrium (relative to their actions in an optimal allocation) is correlated with the efficiency of the corresponding system cost, in the sense that if agents are very satisfied with their equilibrium choices, the equilibrium must be relatively efficient. Further, we show that worst-case Nash equilibria are fragile, as every agent is indifferent between their action in a worst-case equilibrium and their action in a system-optimal allocation. In summary, at equilibrium, either agents are highly satisfied with their decisions or their decisions are highly inefficient, but both cannot be true simultaneously. This work adds to recent results for other classes of games which indicate that worst-case equilibrium efficiency guarantees only occur when agents are indifferent about their decisions.