Potential Game - Definition

Definition

We will define some notation required for the definition. Let be the number of players, the set of action profiles over the action sets of each player and be the payoff function.

A game is:

an exact potential game if there is a function such that ,

That is: when player switches from action to action, the change in the potential equals the change in the utility of that player.

a weighted potential game if there is a function and a vector such that ,

an ordinal potential game if there is a function such that ,

$u_{i}(a'_{i},a_{-i})-u_{i}(a''_{i},a_{-i})>0 \Leftrightarrow \Phi(a'_{i},a_{-i})-\Phi(a''_{i},a_{-i})>0$

a generalized ordinal potential game if there is a function such that ,

$u_{i}(a'_{i},a_{-i})-u_{i}(a''_{i},a_{-i})>0 \Rightarrow \Phi(a'_{i},a_{-i})-\Phi(a''_{i},a_{-i}) >0$

a best-response potential game if there is a function such that ,

where is the best payoff for player given .

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