The Polymatrix replicators form a simple class of o.d.e.’s on prisms defined by
simplexes, which describe the evolution of strategical behaviours within a population
stratified in n>=1 social groups. This class of replicator dynamics contains well known
classes of evolutionary game dynamics, such as the symmetric and asymmetric (or bimatrix)
replicator equations, and some replicator equations for n-person games. In the
1980’s Raymond Redheffer et al. developed a theory on the class of stably dissipative
Lotka-Volterra systems. This theory is based on a reduction algorithm that “infers” the
localization of the system’s attractor in some affine subspace. Waldyr Oliva et al. in 1998
proven that the dynamics on the attractor of such systems is always embeddable in a
Hamiltonian Lotka-Volterra system. We extend these results to Polymatrix replicators.