I will introduce multi-matrix games which include Replicator equations and Bimatrix games . These are ordinary differential equations defined on cartesian products of simplexes. They govern a kind of "replicator" time evolution for groups of players which interact with themselves and other groups. If the loss of one player is the gain of the opponent, called zero-sum games, the system has an integral of motion. I will discuss the question:

Are zero-sum multi-matrix games Hamiltonian?