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In this paper, we provide an effective characterization of all the subgame-perfect equilibria in infinite duration games played on finite graphs with mean-payoff objectives. To this end, we introduce the notion of requirement and the notion of negotiation function. We establish that the set of plays that are supported by SPEs are exactly those that are consistent with the least fixed point of the negotiation function. Finally, we show that the negotiation function is piecewise linear and can be analyzed using the linear algebraic tool box.

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.

In this paper, we provide an effective characterization of all the subgameperfect equilibria in infinite duration games played on finite graphs with mean-payoff objectives. To this end, we introduce the notion of requirement, and the notion of negotiation function. We establish that the plays that are supported by SPEs are exactly those that are consistent with a fixed point of the negotiation function. Finally, we use that characterization to prove that the SPE threshold problem, who status was left open in the literature, is decidable.This paper is an enhanced version of [BRvdB21]. All the proofs are now included in the main body of the paper, and some of them have been rewrote to be more readable, especially those of Theorems 7.3 and 8.3. The proof of Theorem 4.4 has been modified in order to relax its hypotheses, via a change in the definition of steady negotiation (Definition 3.5). We have also taken advantage of the new available space to add examples in the main body of this paper, especially Example 5.2.