Potential game

An order or a potential order potential function is in game theory a special function on the set of strategy combinations of a game. This feature makes the strategy combination are arranged according to their payout to the players. A strategy combination if and only here has a higher value if it leads to every player to a higher payout. By strict binds order potential function of the payoff functions, we obtain the special cases of the weighted potential and the exact potential. The latter is also referred to simply as potential or potential function.

However, most games do not have a potential order. From Dov Monderer therefore 1988 and 1996, the following classes were introduced by Play:

• Game with potential order
• Game with a weighted potential
• Playing with ( exact ) potential

A potential function was first used in games in 1973 by Robert W. Rosenthal, to show that congestion games have a Nash equilibrium in pure strategies.

Definition

For all three definitions was a game in normal form. Next be an arbitrary but fixed strategy profile and the profile to be created by the transfer of a player 's strategy.

Order potential

An order potential function is a function that applies for that

Weighted potential

A weighted potential function is a function in which there is a number for each player, so it always holds true that

In this case it is called a weighted potential game. The weights form a vector. Knowing these numbers, it is called a potential and speaks of a game with potential.

Accurate potential

An (exact ) potential function is a function for which it holds that

The exact potential function is thus a special case of a weighted function of potential, in which all weights. It is considered that each load game has an exact potential function, conversely, every finite game, which has an exact potential function, is isomorphic to a Load game.

Properties

Every finite game with potential order has a Nash equilibrium in pure strategies.

Two potential functions and a game differ only by a constant:

This means that for two combinations strategy and is

Swell

• Game theory