Exact complexity of computing simulation equilibria in mixed-strategy simulation games

Determine the exact worst-case computational complexity of finding a simulation equilibrium (i.e., a Nash equilibrium in which player one uses the mixed-strategy simulation action with non-zero probability) in the mixed-strategy simulation game G^m defined in Definition 1, for general two-player normal-form base games G and arbitrary positive simulation cost c.

Background

The paper introduces mixed-strategy simulation games, where player one can pay a fixed cost to simulate player two’s mixed strategy and best-respond, breaking ties in player two’s favor. Although these games are formally infinite, the authors prove a reduction to a finite subgame and provide an upper bound on the difficulty of solving such games. They also show that deciding whether enabling mixed-strategy simulation introduces Pareto-improving equilibria is NP-hard.

Despite these results, the precise computational complexity of computing a simulation equilibrium remains unresolved. The authors note that while any Nash equilibrium of the base game persists in the simulation game, the existence and computation of equilibria in which simulation is used is more subtle, and their exact complexity is left open.