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:

    “No man, not even a doctor, ever gives any other definition of what a nurse should be than this—”devoted and obedient.” This definition would do just as well for a porter. It might even do for a horse. It would not do for a policeman.”
    —Florence Nightingale (1820–1910)

    “I’m beginning to think that the proper definition of “Man” is “an animal that writes letters.””
    —Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)

    “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)