The strategy improvement algorithm for mean payoff games and parity games is a local improvement algorithm, just like the simplex algorithm for linear programs. Their similarity has turned out very useful: many lower bounds on running time for the simplex algorithm have been created, by first constructing lower bounds for strategy improvement in a game, and then transforming these into a linear program. However, earlier connections between these algorithms required constructing an intermediate Markov decision process, which is not always possible. We prove a formal, direct connection between the two algorithms, showing that many variants of strategy improvement for parity and mean payoff games are truly an instance of the simplex algorithm, under mild nondegeneracy assumptions. The linear programming formulation of these games leads to what we call hypercubelike polyhedra. These polyhedra allow us to show combinatorial properties of various games on graphs. In particular, we show a connection between strategy sets of games and lopsided sets.

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