Potential Games

A game with finite players is a potential game if there exists a potential function $\Phi :S\to \mathbb{R}$ such that $\Phi (s_i,s_{-i})$ provides information about $u_i(s_i,s_{-i})$ for all $i$. $\Phi$ is an ordinal potential function if

$$u_i(s_i,s_{-i})-u_i(s_i^{\prime },s_{-i})>0⟺\Phi (s_i,s_{-i})-\Phi (s_i^{\prime },s_{-i})>0,\forall s_i,s_i^{\prime },s_{-i},i.$$

$\Phi$ is an exact potential function if

$$u_i(s_i,s_{-i})-u_i(s_i^{\prime },s_{-i})=\Phi (s_i,s_{-i})-\Phi (s_i^{\prime },s_{-i}),\forall s_i,s_i^{\prime },s_{-i},i.$$

For games with an infinite strategy space, we further require $\Phi$ to be continuous.

The global maxima of a potential function are PSNEs.

• A Cournot Competition with homogeneous costs is an ordinal potential game, and is an exact potential game if further the demand is linear.
• Routing Games and Cost Sharing Games are exact potential games.