Energy mean-payoff games

TL;DR

This paper investigates energy mean-payoff games on finite graphs, revealing complexity results and strategy requirements, with polynomial-time solutions for one-player cases and co-NP membership for two-player cases.

Contribution

It introduces energy mean-payoff games, analyzes their strategic complexity, and provides algorithms and constructions for optimal strategies, including infinite-memory strategies for the protagonist.

Findings

One-player energy mean-payoff games are solvable in polynomial time.

Two-player energy mean-payoff games are in co-NP.

Optimal strategies for the protagonist may require infinite memory.

Abstract

In this paper, we study one-player and two-player energy mean-payoff games. Energy mean-payoff games are games of infinite duration played on a finite graph with edges labeled by 2-dimensional weight vectors. The objective of the first player (the protagonist) is to satisfy an energy objective on the first dimension and a mean-payoff objective on the second dimension. We show that optimal strategies for the first player may require infinite memory while optimal strategies for the second player (the antagonist) do not require memory. In the one-player case (where only the first player has choices), the problem of deciding who is the winner can be solved in polynomial time while for the two-player case we show co-NP membership and we give effective constructions for the infinite-memory optimal strategies of the protagonist.