Solution concept

As a solution concept in game theory can describe criteria that explain the behavior of the agents. The problem here is that, normatively, very simple assumptions must be made about human behavior. The results of experimental economics often deviate significantly from the predictions of the commonly accepted solution concepts.

Dominance

Domination is the sharpest criterion. We distinguish between strong and weak dominance.

• A treatment option for players is strongly dominant if for all alternatives and all possible counter- responses: The option brings greater benefits for players than the alternative, ie.
• A treatment option for players is weakly dominant if for all alternatives and all possible counter- responses: The option takes for a player at least as great benefit as the alternative of, And for at least one answer applies strict inequality.

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

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

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

Nash equilibrium

The Nash equilibrium is named after a Nobel Prize winner of 1994, John Nash, who has 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 the opponents. As a rule, also known as mixed strategies are considered where several pure strategies are played with positive probability. If a game is solvable by dominance, the dominant solution is both a Nash equilibrium.

Mighty is this solution concept, since it can be shown that for a large and important class of games, including for all games with finite number of players and strategies, at least one Nash equilibrium exists in mixed strategies. The problem is that this concept only in exceptional cases provides a unique solution, usually it allows for multiple strategy combinations as solutions, sometimes all.

Refinements of the Nash equilibrium

Does the Nash equilibrium solutions to more, so come refinements to the train. These are: Trembling -hand perfect equilibrium, protects from suboptimal to enemy behavior - this concept was introduced by Reinhard Selten ( also Nobel Prize winner 1994) in the debate -, strictness, which requires that an equilibrium is strictly better than its immediate surroundings; Risk dominance; Pareto efficiency over all other Nash equilibria, evolutionary stability.

Bayesian Nash equilibrium

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

Dynamic Games

Especially for the extensive form, there is a subgame perfect Nash equilibrium. For games that are both dynamic as well bayesianisch exist, the sequential equilibrium and perfect Bayesian equilibrium.

Balance in correlated strategies

The balance in correlated strategies is a process developed by the mathematician Robert Aumann solution concept, by which a harmonization of policies is possible. In contrast to the Nash equilibrium, which allows neither binding contracts or communication before the decision making of the players involved and thus the strategy choice of one of the strategy choice of the other player remains unaffected, the balance in correlated strategies allows a correlation of strategies among themselves.

Maximin-/Minimax-Lösung

With the maximin solution could be two-person zero -sum games already satisfactorily solved before established the Nash criterion, since in this class, the Max -Min solution is a Nash equilibrium. But even for non- zero-sum games sometimes comes this solution into account, although it ensures in this case, no optimality, since it sometimes is less risky than the Nash equilibrium.

Solutions for cooperative games

For the cooperative game theory has developed its own solutions. Among other Imputationsmenge, nucleolus, Nash bargaining solution, Kalai - Smorodinski solution, the Shapley value or the mean- voter solution.