Energy Structure of Optimal Positional Strategies in Mean Payoff Games

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.