On the Complexity of SPEs in Parity Games

TL;DR

This paper investigates the computational complexity of subgame-perfect equilibria in parity games, providing new results that clarify the difficulty of various related decision problems and their tractability under certain parameters.

Contribution

It introduces new complexity classifications for SPE-related problems in parity games, including NP-completeness and fixed-parameter tractability results, advancing theoretical understanding.

Findings

Checking fixed points of the negotiation function is NP-complete.

SPE constrained existence is NP-complete, previously thought easier.

SPE verification with LTL is PSpace-complete.

Abstract

We study the complexity of problems related to subgame-perfect equilibria (SPEs) in infinite duration non zero-sum multiplayer games played on finite graphs with parity objectives. We present new complexity results that close gaps in the literature. Our techniques are based on a recent characterization of SPEs in prefix-independent games that is grounded on the notions of requirements and negotiation, and according to which the plays supported by SPEs are exactly the plays consistent with the requirement that is the least fixed point of the negotiation function. The new results are as follows. First, checking that a given requirement is a fixed point of the negotiation function is an NP-complete problem. Second, we show that the SPE constrained existence problem is NP-complete, this problem was previously known to be ExpTime-easy and NP-hard. Third, the SPE constrained existence problem is fixed-parameter tractable when the number of players and of colors are parameters. Fourth, deciding whether some requirement is the least fixed point of the negotiation function is complete for the second level of the Boolean hierarchy. Finally, the SPE-verification problem -- that is, the problem of deciding whether there exists a play supported by a SPE that satisfies some LTL formula -- is PSpace-complete, this problem was known to be ExpTime-easy and PSpace-hard.