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Introduction to Game Theory

The Game theory is "a study of mathematical models of conflict and cooperation between intelligent rational decision-makers". Formally, "Game theory is a study of strategic decision making". The alternative word recommended "as a more descriptive name for the discipline" is the interactive decision theory. The Game theory is mainly used in logic, economics, psychology, biology, and political science. The subject at first addressed zero-sum games, like one person's gains exactly equal net losses of the other participant. Though today, game theory applies to a wide variety of class relations and has developed into an umbrella term for the logical side of science, to involve both humans and non-humans, like computers. The classic uses include a sense of balance in many numbers of games, where each person has developed or found a tactic which cannot successfully better his outcomes.

The modern game theory starts with the idea regarding the existence of mixed-strategy equilibria in two-person zero-sum games and it is proved by John von Neumann. John Von Neumann's original proof used the Brouwer's fixed-point theorem on continuous mappings into compact convex sets, which is a standard technique in mathematical economics and game theory. His paper was followed by his book that is Theory of Economic Behavior and Games, with Oskar Morgenstern, that is considered as cooperative games of numerous players. The 2nd edition of this book provided an axiomatic theory of the expected utility that permits mathematical statisticians and economists to treat decision-making under uncertainty.

Representation of games:

Extensive form:

The extensive form of game theory can be used to formalize the games with a time sequence of moves. In an extensive form the Games are played on trees.

The diagram is as shown below; here each vertex (or node) represents a point of choice for a player. The player identified by a number listed by the vertex. The line out of the vertex signifies a possible action for that player. The payoffs are identifying at the bottom of tree. The extensive form can be seen as a multiplayer generalization of a decision tree.

 

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Normal form:

The normal or strategic form of game theory is usually represented by a matrix that shows the strategies, players, and pay-offs. More generally it can be represented by any function which associates a payoff for each player with every possible combination of actions.

Illustration of Game Theory: the Prisoner's Dilemma

The Prisoner's Dilemma, elaborates in figure shown below, and is one of the best-known models in game theory. It elaborates the paradoxical nature of interaction between mutually suspicious participants with opposing interests.

 

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In this hypothetical position, two accomplices to a crime are imprisoned and they forge a pact to not betray one another and not confess to the crime. The strictness of the punishment which each receives is determined not only by her or his behavior, however also by the behavior of his or her accomplice. The two prisoners are separated and they cannot communicate with each other. Each prisoner is informed that there are four possible outcomes:

1.) When one confesses to the crime and turns in the accomplice (defecting from a pact with the accomplice) his sentence will be decreased.

2.) Whenever one confesses while the accomplice does not, then the accomplice cooperates with the pact to not betray each other then the first can strike a deal with the police and will be set free. However the information which he provides will be used to incriminate his accomplice, who will be getting the maximum sentence.

3.) When both prisoners confess to the crime that is, both defect from their pact then each obtains a decreased sentence however no one is set free.

4.) When neither confesses to the crime that is, they collaborate then each prisoner receives the minimum sentence since of the lack of proof. This option might not be as attractive to either individual as the option of striking a deal with the police and being set free at the expense of one's partner. As the prisoners are not capable to communicate with each other, the question of whether to belief the other, not to confess is the vital aspect of this game.

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