Solution concept

In game theory, a solution concept can be described as criteria that explain the behavior of the agents. The problem here is that, normatively, very simple assumptions about human behavior have to be made. The results of experimental economic research often deviate considerably from the predictions of the commonly accepted solution concepts.

Dominance

Dominance is the strictest criterion. A distinction is made between strong and weak dominance. For the criterion of stoachastic dominance , see Ordinary Stochastic Order .

A course of action for players is very dominant when all alternatives and all possible counter-responses applies: The option brings players a greater benefit than the alternative , d. H. . ${\ displaystyle s_ {i}}$ ${\ displaystyle i}$ ${\ displaystyle s_ {i} '}$ ${\ displaystyle s ^ {*}}$ ${\ displaystyle s_ {i}}$ ${\ displaystyle i}$ ${\ displaystyle s_ {i} '}$ ${\ displaystyle u_ {i} (s_ {i}, s ^ {*})> u_ {i} (s_ {i} ', s ^ {*})}$
A course of action for players is weak dominant when all alternatives and all possible counter-responses applies: The option brings to players one at least as great benefit as the alternative , d. H. , and the strict inequality holds for at least one answer . ${\ displaystyle s_ {i}}$ ${\ displaystyle i}$ ${\ displaystyle s_ {i} '}$ ${\ displaystyle s ^ {*}}$ ${\ displaystyle s_ {i}}$ ${\ displaystyle i}$ ${\ displaystyle s_ {i} '}$ ${\ displaystyle u_ {i} (s_ {i}, s ^ {*}) \ geq u_ {i} (s_ {i} ', s ^ {*})}$ ${\ displaystyle s ^ {*}}$ ${\ displaystyle u_ {i} (s_ {i}, s ^ {*})> u_ {i} (s_ {i} ', s ^ {*})}$

Several weakly dominant strategies can exist in a game, while a strongly dominant strategy, if it exists, is always unique.

Under the usual assumptions in game theory, it follows that rational players only interested in their own good would play a dominant solution.

In a quasi-linear environment, the Vickrey-Clarke-Groves mechanisms implement efficient solutions in weakly dominant strategies.

Nash equilibrium

The Nash equilibrium is named after one of the 1994 Nobel Prize winners , John Nash , who established this criterion. A Nash equilibrium is a combination of strategies in which each player's strategy is optimal with respect to the strategies of their opponents. As a rule, so-called mixed strategies are also taken into account, in which several pure strategies are played with a positive probability. If a game can be solved by the iterative elimination of strictly dominated strategies , the dominant solution is also a Nash equilibrium.

This solution concept is powerful because it can be shown that for a large and important class of games, including all games with a finite number of players and strategies, there is at least one Nash equilibrium in mixed strategies. The problem is that this concept only offers a clear solution in exceptional cases; it usually allows several strategy combinations as solutions, sometimes all of them.

Refinements of the Nash equilibrium

If the Nash equilibrium permits several solutions, refinements come into play. These are: Trembling-hand-perfect balance , protects against suboptimal opposing behavior - this concept was introduced into the debate by Reinhard Selten (also Nobel Prize winner 1994) - strict Nash equilibrium , which demands that a balance is strictly better than its immediate one Surroundings; Risk dominance ; Pareto efficiency versus all other Nash equilibria, evolutionary stability .

Bayesian Nash equilibrium

In a Bayesian game , player preferences are participants' private information. To calculate the optimal strategy, the players therefore make assumptions such that the unknown preferences of the other players can be represented as random variables with a known probability distribution. The size to be strategically optimized is then the expected benefit of an option for action. A Bayesian Nash equilibrium is a Nash equilibrium with respect to Bayesian game.

Dynamic games

The subgame-perfect Nash equilibrium exists especially for the extensive form . For games that are both dynamic and Bayesian, sequential equilibrium and perfect Bayesian equilibrium exist .

Equilibrium in Correlated Strategies

The equilibrium in correlated strategies is a solution concept developed by the mathematician Robert Aumann , which enables the strategies to be harmonized . In contrast to the Nash equilibrium, which allows neither binding contracts nor communication before the decision-making of the players involved and thus the choice of strategy of one player remains unaffected by the choice of strategy of the other player, the equilibrium in correlated strategies enables the strategies to be correlated with one another.

Maximin / Minimax solution

With the Maximin solution , two-person zero-sum games could already be solved satisfactorily before the Nash criterion was established, since the Max-Min solution in this class is a Nash equilibrium. But this solution is also sometimes considered for non-zero-sum games, although in this case it does not guarantee optimality, since it is sometimes less risky than the Nash equilibrium.

Solutions for cooperative games

Solution concepts have been developed for cooperative game theory . Among other things, imputation amount , nucleolus , Nash negotiation solution , Kalai-Smorodinski solution , the Shapley value or the mean-voter solution .

Web links

Gambit - a comprehensive game theory software under the GPL

Individual evidence

↑ Holler, Manfred / Illing, Gerhard: Introduction to game theory. 6th, revised edition, Springer Verlag, Berlin and Heidelberg, 2006. S. 87ff.

[1] Holler, Manfred / Illing, Gerhard: Introduction to game theory. 6th, revised edition, Springer Verlag, Berlin and Heidelberg, 2006. S. 87ff.