Further Results on Verification Problems in Extensive-Form Games

Abstract

The computational study of games is receiving increasing attention both in game theory and computer science. The challenge is distinguishing computationally tractable problems (also said easy), admitting polynomial–time algorithms, from the intractable ones (also said hard). In this paper, we focus on extensive form games, as the computational problems defined on such games are largely unexplored. We study the problem (aka verification problem) of certifying that a solution given in input is an equilibrium according to different refinements for extensive form games as the input change. We show that, when the input is a realization plan strategy profi (i.e., strategies for the sequence form representation), deciding whether the input is a Subgame Perfect Equilibrium or is a part of a Sequential Equilibrium is NP-hard even in two-player games (we conjecture the same holds also for Quasi Perfect Equilibrium). This means that there is no polynomial–time algorithm unless P = NP, but it is commonly believed that P ≠ NP. Subsequently, we show that in two-player games, when the input is a behavioral strategy profile there is a polynomial-time algorithm deciding whether the input is a Quasi-Perfect Equilibrium, and a simple variation of the algorithm decides whether the input is part of some Sequential Equilibrium (in games with three or more players, the problem is known to be NP-hard for both Quasi-Perfect Equilibrium and Sequential Equilibrium). Finally, we show that, when the input is an assessment, there is a polynomial-time algorithm deciding whether the input is a Sequential Equilibrium regardless the number of players.

Keywords: Efficient algorithms, extensive–form refinements