*Dov Monderer*

We introduce and analyze *q*-potential games and *q*-congestion games, where *q* is a positive integer. A 1-potential (congestion) game is a potential (congestion) game. We show that a game is a *q*-potential game if and only if it is (up to an isomorphism) a *q*-congestion game. As a corollary, we derive the result that every game in strategic form is a *q*-congestion game for some *q*. It is further shown that every *q*-congestion game is isomorphic to a *q*-network game, where the network environment is defined by a directed graph with one origin and one destination. Finally we discuss our main agenda: The issue of representing *q*-congestion games with non-negative cost functions by congestion models with non-negative and monotonic facility cost functions. We provide some initial results in this regard.