On the Complexity of Nash Equilibrium Reoptimization

We provide, to the best of our knowledge, the first study about reoptimization complexity of game-theoretical solutions. In a reoptimization problem, we are given an instance, its optimal solution, and a local modification, and we are asked to find the exact or an approximate solution to the modified instance. Reoptimization is crucial whenever an instance needs to be solved repeatedly and, at each repetition, its parameters may slightly change. In this paper, we focus on Nash equilibrium, being the central game-theoretical solution. We study the reoptimization of Nash equilibria satisfying some properties (i.e., maximizing/minimizing the social welfare, the utility of a player or the support size) for some different local modifications of the game (i.e., modification of a payoff or addition/removal of an action), showing that such problems are NP-hard. Furthermore, we assess the approximation complexity of the aforementioned problems, showing that it matches the complexity of the original (non-reoptimization) problems. Finally, we show that, when finding a Nash equilibrium is thought as an optimization problem, reoptimization is useful for finding approximate solutions. Specifically, it allows one to find epsilon-Nash equilibria with smaller epsilon than that of the solutions returned by the best known approximation algorithms.