Game theory

What is game theory?

Game theory describes a method in mathematics in which decision-making behaviour can be studied taking into account several people involved. The assumption is that each player's success depends on both their own actions and the actions of their fellow players.

In its early days, game theory was mainly applied in economics and the social sciences. The starting point of game theory in the 1920s was the description of "homo oeconomicus", an assumption that each player involved maximises his or her benefit. Based on this, considerations about utility-maximising behavioural strategies took place, which were reflected in the further development of game theory.

The number of players considered in a game, as well as the order and number of players, is not limited in game theory.

What are non-cooperative and cooperative game theories?

Game theory is fundamentally divided into the subfields of cooperative and non-cooperative game theory. While in cooperative games all players involved conclude binding contracts, this is not the case in non-cooperative games.

Thus, the Non-cooperative games on the self-interest of each individual player under the assumption of respective utility maximisation.

In cooperative games, individual players can enter into binding contracts and form so-called coalitions in order to generate the greatest collective benefit. In cooperative games, it is also possible that there are side payments among the individual players and thus the greatest possible benefit for each player can be achieved through benefit transformation.

Strategies in play

In game theory, some solution concepts can be found that describe or explain the behaviour of the individual players:

  • Nash equilibrium: The Nash equilibrium (named after its founder John Nash) describes a game strategy in which each individual player behaves optimally and can no longer improve by unilaterally deviating from his chosen strategy.
  • Min-max theoremIn the special case of a two-person zero-sum game, the min-max theorem aims at choosing the strategy in game theory that minimises the opponent's maximum payoff and maximises one's own minimum payoff. Due to the special situation of the zero-sum game, the minimisation of the opponent's utility results in a maximisation of the utility of one's own strategy and vice versa.
  • Shapley valueThe Shapley value is used in cooperative games and describes changes in the utility of individual players when entering into coalitions with other players. The value is also colloquially described as "power". For example, in political situations in parliamentary formations, a large Shapley value can be attributed to a small party if it can achieve the necessary majority.

Examples of games

  • Prisoner's dilemmaIn the best-known example in game theory, it is assumed that two players are accused of a crime. If only one player confesses, he is acquitted under a leniency programme and the other is imprisoned. If neither confesses, both are given a low sentence due to lack of evidence. If both confess, both receive the maximum sentence. Since it is a non-cooperative game where the opponent cannot be bound by contract, both players choose to confess and receive the maximum penalty.
  • Coward's gameThe assumption here is that two players are racing towards each other in a vehicle. If only one player swerves, he is considered a scaredy-cat by his opponent. If no one swerves, both lose their lives. This problem shows three Nash equilibria, as each option of evasion is advantageous over loss of life.
  • Beauty ContestThe beauty contest founded by John Keynes incorporates the behaviour of the other players in the game theory into the choice of strategy, in that the metaphor offers a lottery in which the winner is the player who has chosen the person most often chosen by the other players in a beauty contest.
  • Tragedy of the commons: This problem describes the fact that freely accessible but limited resources are not used efficiently. The individual maximisation of benefits (e.g. through overfishing) leads to the ruin of all.

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