Saturday, February 19, 2022

What is the game theory?

Game theory is the study of mathematical models of strategic interactions among people that always aims to perform optimal actions based on given premises and information. It has applications in all fields of social science, as well as in logic, systems science and computer science.

In the 21st century, game theory applies to a wide range of behavioral relations, and is now an umbrella term for the science of logical decision making in humans, animals, and computers.

Modern game theory began with the idea of mixed-strategy equilibria in two-person zero-sum game and its proof by John von Neumann. Von Neumann’s original proof used the Brouwer fixed-point theorem on continuous mappings into compact convex sets, which became a standard method in game theory and mathematical economics.

Game theory was developed extensively in the 1950s by many scholars. It was explicitly applied to evolution in the 1970s, although similar developments go back at least as far as the 1930s. Game theory has been widely recognized as an important tool in many fields.

Originally, it addressed two-person zero-sum games, in which each participant’s gains or losses are exactly balanced by those of other participants. But as the time flew more and more game types have been discovered.

Some of these game types are cooperative / non-cooperative, symmetric / asymmetric, Simultaneous / sequential and many more. Not all game types will be mentioned.

A game is cooperative if the players are able to form binding commitments externally enforced (e.g. through contract law). A game is non-cooperative if players cannot form alliances or if all agreements need to be self-enforcing (e.g. through credible threats).

Cooperative games are often analyzed through the framework of cooperative game theory, which focuses on predicting which coalitions will form, the joint actions that groups take, and the resulting collective payoffs.

A symmetric game is a game where the payoffs for playing a particular strategy depend only on the other strategies employed, not on who is playing them. That is, if the identities of the players can be changed without changing the payoff to the strategies, then a game is symmetric. Many of the commonly studied 2×2 games are symmetric.

The standard representations of chicken, the prisoner’s dilemma, and the stag hunt are all symmetric games.

The most commonly studied asymmetric games are games where there are not identical strategy sets for both players. For instance, the ultimatum game and similarly the dictator game have different strategies for each player.

Simultaneous games are games where both players move simultaneously, or instead the later players are unaware of the earlier players’ actions.

Sequential games (or dynamic games) are games where later players have some knowledge about earlier actions.

This need not be perfect information about every action of earlier players; it might be very little knowledge. For instance, a player may know that an earlier player did not perform one particular action, while they do not know which of the other available actions the first player actually performed.

The Cournot competition model involves players choosing quantity of a homogenous product to produce independently and simultaneously, where marginal cost can be different for each firm and the firm’s payoff is profit.

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