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 definition of good prose is proper words in their proper places; of good verse, the most proper words in their proper places. The propriety is in either case relative. The words in prose ought to express the intended meaning, and no more; if they attract attention to themselves, it is, in general, a fault.”
    —Samuel Taylor Coleridge (1772–1834)

    “The physicians say, they are not materialists; but they are:MSpirit is matter reduced to an extreme thinness: O so thin!—But the definition of spiritual should be, that which is its own evidence. What notions do they attach to love! what to religion! One would not willingly pronounce these words in their hearing, and give them the occasion to profane them.”
    —Ralph Waldo Emerson (1803–1882)

    “It is very hard to give a just definition of love. The most we can say of it is this: that in the soul, it is a desire to rule; in the spirit, it is a sympathy; and in the body, it is but a hidden and subtle desire to possess—after many mysteries—what one loves.”
    —François, Duc De La Rochefoucauld (1613–1680)