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 .

Read more about this topic: Potential Game

Famous quotes containing the word definition:

“The very definition of the real becomes: that of which it is possible to give an equivalent reproduction.... The real is not only what can be reproduced, but that which is always already reproduced. The hyperreal.”
—Jean Baudrillard (b. 1929)

“The man who knows governments most completely is he who troubles himself least about a definition which shall give their essence. Enjoying an intimate acquaintance with all their particularities in turn, he would naturally regard an abstract conception in which these were unified as a thing more misleading than enlightening.”
—William James (1842–1910)

“Was man made stupid to see his own stupidity?
Is God by definition indifferent, beyond us all?
Is the eternal truth man’s fighting soul
Wherein the Beast ravens in its own avidity?”
—Richard Eberhart (b. 1904)