On acyclicity of games with cycles

We study restricted improvement cycles (ri-cycles) in finite positional nn-person games with perfect information modeled by directed graphs (di-graphs) that may contain directed cycles (di-cycles). We assume that all these di-cycles form one outcome cc, for example, a draw. We obtain criteria of restricted improvement acyclicity (ri-acyclicity) in two cases: for n=2n=2 and for acyclic di-graphs. We provide several examples that outline the limits of these criteria and show that, essentially, there are no other ri-acyclic cases.We also discuss connections between ri-acyclicity and some open problems related to Nash-solvability.