Equilibrium (game theory)
In game theory, equilibrium is a state in which players do not deviate from their strategy of their own free will. In a two-person game, equilibria can be identified in normal form (a simplified approach) using a so-called bimatrix . The bimatrix contains external utility values that are modeled by a utility function.
term
The concept of equilibrium originates from classical mechanics . The systems theory has generalized it to what is the state of a system called: A system is in equilibrium when it does not develop forces out of itself, alter the system state, so that a change needs to happen from the outside. In contrast to classical mechanics, the forces involved are self-organized from this point of view .
One possible task of game theory is to identify behavioral recommendations for each participant that will best enable them to pursue their own interests. In game theory language, a list of behavioral recommendations is a balance if the behavioral recommendations are consistent with one another. The first precise formulation of a concept of equilibrium in game theory can be found in a work by John von Neumann for 2-person zero-sum games published in 1928 . The further development of game theory represents the extension of this concept of equilibrium to more general interactive decision problems.
The term equilibrium gained wider popularity in game theory through the work of John Forbes Nash Jr. in the 1950s. Often only the Nash equilibrium is understood here by equilibrium , although other definitions and variants have emerged from it. What they have in common is that under equilibrium in a game the fact that the strategies of players who behave freely and rationally do not change, even if these players do not agree on anything beyond the rules of the game, such as contracts or further agreements would be the case. Equilibrium in the sense of game theory, which tries to find mathematical models for decisions, must be differentiated from other, more concrete concepts of equilibrium, such as market equilibrium . Equilibria in game theory are special forms of the Nash equilibrium, but are often given different names due to other properties.
variants
- Nash equilibrium , a pair of strategies in non-cooperative games
- Strict balance , a pair of strategies in game theory
- Cournot-Nash equilibrium , an equilibrium from the oligopoly theory
- Bertrand equilibrium , an equilibrium in Bertrand competition
- Stackelberg equilibrium , an equilibrium in the Stackelberg duopoly
- Equilibrium in correlated strategies , allows for binding contracts or communication before the decision-making of the players involved
- Equilibrium in evolutionarily stable strategies , see evolutionarily stable strategy
- Equilibrium in mixed strategies , see Mixed Strategy
- Partial game perfect balance , a balance for games in extensive form
- Trembling-hand-perfect balance , a balance involving wrong decisions of the opponent
- Payout- dominant and risk-dominant balance , see also Bimatrix
- Perfect Bayesian balance , a balance for solving dynamic games with incomplete information
- Sequential balance , a balance for dynamic games with incomplete or imperfect information
- Asymptotically stable equilibrium , an equilibrium in game theory and in dynamic systems
- Symmetrical and asymmetrical equilibria
- Quantal Response Balance , a balance related to probit models and logit models
literature
- Christian Rieck: Game Theory: Introduction for Economists and Social Scientists , Springer, Berlin 2013, pp. 155–204. ISBN 978-3322870834
Individual evidence
- ↑ Wolfgang Leininger, Erwin Amann: Introduction to Game Theory , p. 5 ff.