  In this work we offer an $O(|V|^2 |E|\, W)$ pseudo-polynomial time
deterministic algorithm for solving the Value Problem and Optimal Strategy
Synthesis in Mean Payoff Games. This improves by a factor $\log(|V|\, W)$ the
best previously known pseudo-polynomial time upper bound due to Brim,~\etal The
improvement hinges on a suitable characterization of values, and a description
of optimal positional strategies, in terms of reweighted Energy Games and Small
Energy-Progress Measures.
