Potential (game theory)

An order of potential or order potential function is in game theory , a special function on the set of strategy combinations of a game . This feature arranges the strategy combination according to its payout to the players. A strategy combination has a higher value if it leads to a higher payout for each player. By tying the ordering potential function more strictly to the disbursement functions, the special cases of the weighted potential and the exact potential are obtained . The latter is also simply referred to as potential or potential function.

Most games, however, have no potential for order. By Dov Monderer why 1988 and 1996, the following classes were introduced games:

• Playing with the potential for order
• Game with weighted potential
• Playing with (exact) potential

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

definition

With all three definitions a game is in normal form . Furthermore, let there be any fixed strategy profile and the profile that arises when a player changes his strategy from to . ${\displaystyle \Gamma =(N,\Sigma ,u)}$ ${\displaystyle \sigma \in \Sigma }$ ${\displaystyle \sigma ':=(\sigma ^{-i},\sigma _{i}')}$ ${\displaystyle i\in N}$ ${\displaystyle \sigma _{i}}$ ${\displaystyle \sigma _{i}'}$

Order potential

An order potential function is a function for which it holds that ${\displaystyle P}$ ${\displaystyle P:\Sigma \rightarrow \mathbb {R} }$

${\displaystyle u_{i}(\sigma ')-u_{i}(\sigma )>0\quad \Leftrightarrow \quad P(\sigma ')-P(\sigma )>0}$

Weighted potential

A weighted potential function is a function in which there is a number for each player , so that it always holds that ${\displaystyle P}$ ${\displaystyle P:\Sigma \rightarrow \mathbb {R} }$ ${\displaystyle i\in N}$ ${\displaystyle w_{i}>0}$

${\displaystyle u_{i}(\sigma ')-u_{i}(\sigma )=w_{i}\cdot (P(\sigma ')-P(\sigma ))}$

In this case it is called a weighted potential game. The weights form a vector . If one knows these numbers, one calls a -potential and speaks of a game with -potential. ${\displaystyle \Gamma }$ ${\displaystyle w_{1},w_{2},\ldots ,w_{n}}$ ${\displaystyle w}$ ${\displaystyle P}$ ${\displaystyle w}$ ${\displaystyle w}$

Exact potential

An (exact) potential function is a function for which it holds that ${\displaystyle P}$ ${\displaystyle P:\Sigma \rightarrow \mathbb {R} }$

${\displaystyle u_{i}(\sigma ')-u_{i}(\sigma )=P(\sigma ')-P(\sigma )}$

The exact potential function is therefore a special case of a weighted potential function in which all are weights . It is true that every load game has an exact potential function; conversely, every finite game that has an exact potential function is isomorphic to a load game. ${\displaystyle w_{i}=1}$

properties

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

Two potential functions and a game differ only in one constant: ${\displaystyle P_{1}}$ ${\displaystyle P_{2}}$

${\displaystyle P_{1}(\sigma )=P_{2}(\sigma )+c}$

This means that for two strategy combinations and applies ${\displaystyle \sigma ^{*}}$ ${\displaystyle \sigma ^{**}}$

${\displaystyle P_{1}(\sigma ^{*})-P_{1}(\sigma ^{**})=P_{2}(\sigma ^{*})-P_{2}(\sigma ^{**})}$

swell

1. ^ ^{A b} Dov Monderer, Lloyd S. Shapley: Potential Games.  ( Page no longer available , search in web archives )  Info: The link was automatically marked as defective. Please check the link according to the instructions and then remove this notice. (PDF; 200 kB) Games and Economic Behavior 14, 1996, pp. 124-143  
2. ^ Robert W. Rosenthal: A Class of Games Possessing Pure-Strategy Nash Equilibria. In: International Journal of Game Theory. No. 2, 1973, pp. 65-67