The concept originated in a 1996 paper by Dov Monderer and Lloyd Shapley.
[1] The properties of several types of potential games have since been studied.
In cardinal games, the difference in individual payoffs for each player from individually changing one's strategy, other things equal, has to have the same value as the difference in values for the potential function.
The potential function is a useful tool to analyze equilibrium properties of games, since the incentives of all players are mapped into one function, and the set of pure Nash equilibria can be found by locating the local optima of the potential function.
Thus, through the lens of potential functions, the players become interchangeable (in the sense of one of the definitions above).
Because of this symmetry of the game, decentralized algorithms based on the shared potential function often lead to convergence (in some of sense) to a Nash equilibria.
Using numerical values b1 = 2, b2 = −1, w = 3, this example transforms into a simple battle of the sexes, as shown in Figure 2.
The only stochastically stable equilibrium is (+1, +1), the global maximum of the potential function.
An improvement path (also called Nash dynamics) is a sequence of strategy-vectors, in which each vector is attained from the previous vector by a single player switching his strategy to a strategy that strictly increases his utility.
The opposite is also true: every finite game has the FIP has a generalized-ordinal-potential function.
[4][clarification needed] The terminal state in every finite improvement path is a Nash equilibrium, so FIP implies the existence of a pure-strategy Nash equilibrium.
Moreover, it implies that a Nash equlibrium can be computed by a distributed process, in which each agent only has to improve his own strategy.
A best-response path is a special case of an improvement path, in which each vector is attained from the previous vector by a single player switching his strategy to a best-response strategy.
FBRP is weaker than FIP, and it still implies the existence of a pure-strategy Nash equilibrium.
[5] It means that, for any initial strategy-vector, there exists a finite best-response path starting at that vector.
Weak-acyclicity is not sufficient for existence of a potential function (since some improvement-paths may be cyclic), but it is sufficient for the existence of pure-strategy Nash equilibirum.
It implies that a Nash equilibrium can be computed almost-surely by a stochastic distributed process, in which at each point, a player is chosen at random, and this player chooses a best-strategy at random.