Energy Structure of Optimal Positional Strategies in Mean Payoff Games

TL;DR

This paper explores the structure of optimal positional strategies in mean payoff games, revealing a unique decomposition into extremal energy measures, and provides an efficient recursive enumeration method for these structures.

Contribution

It introduces the concept of an energy-lattice of extremal-SEPMs and a recursive procedure to enumerate all elements, advancing the understanding of strategy structures in MPGs.

Findings

The energy-lattice of OPSs is characterized by least-SEPMs of basic subgames.

A pseudo-polynomial recursive enumeration procedure is proposed.

The complexity of enumeration is polynomial in game size and energy measures.

Abstract

This note studies structural aspects concerning Optimal Positional Strategies (OPSs) in Mean Payoff Games (MPGs), it is a contribution to understanding the relationship between OPSs in MPGs and Small Energy-Progress Measures (SEPMs) in reweighted Energy Games (EGs). Firstly, it is observed that the space of all OPSs, $\texttt{opt}_{\Gamma}\Sigma^M_0$, admits a unique complete decomposition in terms of so-called extremal-SEPM{s} in reweighted EG{s}; this points out what we called the "Energy-Lattice $\mathcal{X}^*_{\Gamma}$ of $\texttt{opt}_{\Gamma}\Sigma^M_0$". Secondly, it is offered a pseudo-polynomial total-time recursive procedure for enumerating (w/o repetitions) all the elements of $\mathcal{X}^*_{\Gamma}$, and for computing the corresponding partitioning of $\texttt{opt}_{\Gamma}\Sigma^M_0$. It is observed that the corresponding recursion tree defines an additional lattice $\mathcal{B}^*_{\Gamma}$, whose elements are certain subgames $\Gamma'\subseteq \Gamma$ that we call basic subgames. The extremal-SEPMs of a given \MPG $\Gamma$ coincide with the least-SEPMs of the basic subgames of $\Gamma$; so, $\mathcal{X}^*_{\Gamma}$ is the energy-lattice comprising all and only the least-SEPMs of the \emph{basic} subgames of $\Gamma$. The complexity of the proposed enumeration for both $\mathcal{B}^*_{\Gamma}$ and $\mathcal{X}^*_{\Gamma}$ is $O(|V|^3|E|W |\mathcal{B}^*_{\Gamma}|)$ total time and $O(|V||E|)+\Theta\big(|E| \mathcal{B}^*_{\Gamma}|\big)$ working space. Finally, it is constructed an \MPG $\Gamma$ for which $|\mathcal{B}^*_{\Gamma}| > |\mathcal{X}^*_\Gamma|$, this proves that $\mathcal{B}^*_{\Gamma}$ and $\mathcal{X}^*_\Gamma$ are not isomorphic.