  This study strengthens the links between Mean Payoff Games (\MPG{s}) and
Energy Games (EG{s}). Firstly, we offer a faster $O(|V|^2|E|W)$
pseudo-polynomial time and $\Theta(|V|+|E|)$ space deterministic algorithm for
solving the Value Problem and Optimal Strategy Synthesis in \MPG{s}. This
improves the best previously known estimates on the pseudo-polynomial time
complexity to: \[ O(|E|\log |V|) + \Theta\Big(\sum_{v\in
V}\texttt{deg}_{\Gamma}(v)\cdot\ell_{\Gamma}(v)\Big) = O(|V|^2|E|W), \] where
$\ell_{\Gamma}(v)$ counts the number of times that a certain energy-lifting
operator $\delta(\cdot, v)$ is applied to any $v\in V$, along a certain
sequence of Value-Iterations on reweighted \EG{s}; and
$\texttt{deg}_{\Gamma}(v)$ is the degree of $v$. This improves significantly
over a previously known pseudo-polynomial time estimate, i.e.
$\Theta\big(|V|^2|E|W + \sum_{v\in
V}\texttt{deg}_{\Gamma}(v)\cdot\ell_{\Gamma}(v)\big)$ \citep{CR15, CR16}, as
the pseudo-polynomiality is now confined to depend solely on $\ell_\Gamma$.
Secondly, we further explore on the relationship between Optimal Positional
Strategies (OPSs) in \MPG{s} and Small Energy-Progress Measures (SEPMs) in
reweighted \EG{s}. It is observed that the space of all OPSs,
$\texttt{opt}_{\Gamma}\Sigma^M_0$, admits a unique complete decomposition in
terms of extremal-SEPM{s} in reweighted EG{s}. This points out what we called
the "Energy-Lattice $\mathcal{X}^*_{\Gamma}$ associated to
$\texttt{opt}_{\Gamma}\Sigma^M_0$". Finally, 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 $\texttt{opt}_{\Gamma}\Sigma^M_0$.

Data Structures and Algorithms