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:

    “Mothers often are too easily intimidated by their children’s negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by definition things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.”
    —Elaine Heffner (20th century)

    “Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.”
    —The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on “life” (based on wording in the First Edition, 1935)

    “Scientific method is the way to truth, but it affords, even in
    principle, no unique definition of truth. Any so-called pragmatic
    definition of truth is doomed to failure equally.”
    —Willard Van Orman Quine (b. 1908)