Cyclical games with prohibitions

The authors consider a certain combinatorial game on a digraph for two cases of the price function. For one case the game in question extends the cyclical game studied in Ehrenfeucht and Mycielski and Gurvitch, Karzanov and Khachiyan which, in its turn, is a generalization of the well-known problem of finding a minimum mean cycle in an edge-weighted digraph. The authors prove the existence of optimal uniform stationary strategies for both cases and give algorithms to find such strategies.