New results on the verification of Nash refinements for extensive-form games

The computational study of strategic interaction situations has recently deserved a lot of attention in multi--agent systems. A number of results on strategic--form games and zero--sum extensive--form games are known in the literature, while general--sum extensive--form games are not studied in depth. We focus on the problem to decide whether or not a solution is a refinement of the Nash equilibrium (NE) for extensive--form games. Refinements are needed because the NE concept is not satisfactory for this game class. While verifying whether a solution is an NE is in P, verifying whether it is a NE refinement may be not (all the results known so far show NP--hardness). In this paper, we provide the first positive result, showing that verifying a sequential equilibrium with any number of agents and a quasi perfect equilibrium with two agents are in P. We show also that when the input is expressed in (non--perturbed) sequence form even the problem to verify a subgame perfect equilibrium is NP--complete and that sequence form, if applicable, must be rethought to verify (and therefore to compute) an extensive--form perfect equilibrium.