Characterizing the Existence of Potential Functions in Weighted Congestion Games.

Since the pioneering paper of rosenthal a lot of work has been done in order to determine classes of games that admit a potential. First, we study the existence of potential functions for weighted congestion games. Let c c\mathcal{c} be an arbitrary set of locally bounded functions and let g(c) g(c)\mathcal{g}(\mathcal{c}) be the set of weighted congestion games with cost functions in c c\mathcal{c}. We show that every weighted congestion game g?g(c) g?g(c)g\in\mathcal{g}(\mathcal{c}) admits an exact potential if and only if c c\mathcal{c} contains only affine functions. We also give a similar characterization for w-potentials with the difference that here c c\mathcal{c} consists either of affine functions or of certain exponential functions. We finally extend our characterizations to weighted congestion games with facility-dependent demands and elastic demands, respectively.