  We study the class of potential games that are also graphical games with
respect to a given graph $G$ of connections between the players. We show that,
up to strategic equivalence, this class of games can be identified with the set
of Markov random fields on $G$.
  From this characterization, and from the Hammersley-Clifford theorem, it
follows that the potentials of such games can be decomposed to local
potentials. We use this decomposition to strongly bound the number of strategy
changes of a single player along a better response path. This result extends to
generalized graphical potential games, which are played on infinite graphs.
