Pseudopolynomial iterative algorithm to solve total-payoff games and min-cost reachability games

Abstract

Quantitative games are two-player zero-sum games played on directed weighted graphs. Total-payoff games-that can be seen as a refinement of the well-studied mean-payoff games-are the variant where the payoff of a play is computed as the sum of the weights. Our aim is to describe the first pseudo-polynomial time algorithm for total-payoff games in the presence of arbitrary weights. It consists of a non-trivial application of the value iteration paradigm. Indeed, it requires to study, as a milestone, a refinement of these games, called min-cost reachability games, where we add a reachability objective to one of the players. For these games, we give an efficient value iteration algorithm to compute the values and optimal strategies (when they exist), that runs in pseudo-polynomial time. We also propose heuristics to speed up the computations.

On Relevant Equilibria in Reachability Games

“…gain) that Player M in (resp. Player M ax) can ensure to obtain from v. Moreover, as quantitative coalitional games are determined these values always exist and can be computed in polynomial time [7,8,13].…”

Optimal Reachability in Divergent Weighted Timed Games

“…, and refer to it as the value of v in G. Finite weighted games are known to be determined [15]. If the game is clear from the context, we may drop the index G from all previous notations.…”

“…In the presence of negative weights, a pseudo-polynomial-time (i.e. polynomial with respect to the game where weights are stored in unary) solution has been given in [15], based on a fixed point computation with value iteration techniques. Moreover, the value problem with threshold −∞ is shown to be in NP∩coNP, and as hard as solving mean-payoff games.…”

“…In this work, we introduce a generalisation of the strictly non-Zeno cost hypothesis in the presence of negative weights, that Deciding the divergence Untimed pseudo-poly. [15] PTIME-complete NL-complete (unary), PTIME(binary) Timed Undecidable [12] 2-EXPTIME, EXPTIME-hard PSPACE-complete we call divergence. We show the decidability of the class of divergent weighted timed games, with a 2-EXPTIME complexity (and an EXPTIME-hardness lower bound).…”

“…The property of divergence is also interesting in the absence of time. Indeed, weighted games with reachability objectives have been recently explored as a refinement of mean-payoff games [14,15]. A pseudo-polynomial time (i.e.…”