definition
staged
Potential Game
For a finite game, a function $P$ on mixed profiles is a potential if every pure unilateral deviation has the same payoff difference and potential difference: $$ g_i(s_i,u_{-i})-g_i(t_i,u_{-i}) = P(s_i,u_{-i})-P(t_i,u_{-i}). $$
MFoGT also gives an evaluation-game version: a $C^1$ function $W$ is a potential for $\Phi$ if, along each player's tangent space, the gradient of $W$ is a positive scalar multiple of the player's evaluation vector.
References
- [MFoGT, Chapter 5, Definitions 5.2.7 and 5.2.8] Laraki, Renault, and Sorin, Mathematical Foundations of Game Theory. Finite and evaluation-game potential functions.