We study restricted improvement cycles (ri-cycles) in finite positional
-person games with perfect information modeled by directed graphs (digraphs) that may contain cycles. We obtain criteria of restricted improvement acyclicity (ri-acyclicity) in two cases: for
= 2 and for acyclic digraphs. 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.