Term Paper on "Game Theory Is a Separate and Interdisciplinary"

Term Paper 6 pages (1953 words) Sources: 1+

[EXCERPT] . . . .

Game theory is a separate and interdisciplinary approach to the study of human behavior. The areas of study most involved in game theory are mathematics, social and behavioral sciences such as economics. It can be used specifically for conflict resolution. Game theory was founded by mathematician John von Neumann. His first important book was The Theory of Games and Economic Behavior, written in collaboration with mathematical economist, Oskar Morgenstern. Game Theory, Analysis of Conflict by Myerson offers an overview of the topic and its development over the past couple of decades.

A game refers to a strategic situation that involves at least two rational and intelligent individuals called players. The fundamental result of decision theory, which forms the foundation of game theory as well, is that each player's goal is to maximize the expected value of his or her own payoff. These payoffs are measured on some utility scale, which is merely a numeric depiction of each outcome that can be gained through the player's actions. Individuals have preferences that give them the opportunity to rank the outcomes with respect to one other. For each pair of outcomes, a player can say whether he or she likes one better than the other or whether he or she is indifferent about the two.

The logical roots for game theory are in Bayesian decision theory. In fact, game theory can be seen as an extension of the decision theory (Myerson, 1991, p.5). In general, a decision theory is an interdisciplinary area of study for practitioners in mathematics, statistics, economics, philosophy, management and psychology. It is concerned with how people make decisions and how optimal decisions can be reached. Bayesian theory is named after Thomas Bayes, who proved a special case of what is called Bayes' theorem. This is a mathematical formula used for calculating conditional probabilities. It figures prominently in subjectivist or Bayesian statistics and inductive logic. Subjectivists, who maintain that rational belief is governed by the laws of probability, stress conditional probabilities in theories of evidence and empirical learning. Bayes' Theorem is central to these enterprises since it simplifies the calculation of conditional probabilities as well as significant features of subjectivist position (Wikipedia).

Bayesian games are a combination of game theory and probability theory that allow taking incomplete information into account. In Bayesian games, each player can have some private information that impacts the overall game play, but is not known by the others. However, others are assumed to have beliefs about the private information. These beliefs are represented using probability distributions and are updated using the Bayes' rule whenever new information is available. Optimality in this setting demands that players act optimally according to their beliefs and their private information.

In the mathematical theory of games as described by von Neumann and Morgenstern (1944), the development is recognized to proceed in two major steps: 1) The presentation of an all-inclusive formal characterization of a general n-person game, 2)the introduction of the concept of pure strategy that makes possible a radical simplification of this scheme, replacing an arbitrary game by a suitable prototype game. They called these two descriptions the extensive and normalized forms of a game. The extensive form is best suited for game theory.

This form specifies: the players of a game; for every player every opportunity they have to move; what each player can do at each of their moves; what each player knows for every move; and the payoffs received by every player for every possible combination of moves. The game is represented by a game tree. Each node represents every stage of the game as it is played. The game above has two players: 1 and 2. The initial node belongs to player 1, indicating that that player moves first. Play according to the tree is as follows: player 1 chooses between U. And D; player 2 observes player 1's choice and then chooses between U' and D' . The payoffs are as specified in the tree. There are four outcomes represented by the four terminal nodes of the tree: (U, U'), (U, D'), (D, U') and (U, U'). The payoffs associated with each outcome respectively are as follows (0,0), (2,1), (1,2) and (3,1).If player 1 plays D, player 2 will play U' to maximize his payoff, and so player 1 will only receive 1. However, if player 1 plays U, player 2 maximizes his payoff by playing D' and player 1 receives 2. Player 1 prefers 2 to 1 and so will play U. And player 2 will play D' . This is the subgame perfect equilibrium.

As noted above, a strategic game is a much simpler form. There are three elements: Who is on the list of players? What is the set of strategies available to each player? What are the payoffs associated with any strategy? The normal or strategic form game is a matrix that shows the players, strategies, and payoffs Here there are two players, one chooses the row and theother chooses the column. Each player has two strategies, which is specified by the number of rows and the number of columns. The payoffs are provided in the interior. The first number is the payoff received by the row player. The second is the payoff for the column player. It is presumed that each player acts simultaneously or, at least, without knowing the actions of the other. If players have some information about the choices of other players, the game is usually presented in extensive form.

The Nash equilibrium, named after John Nash, is an optimal collective strategy in a game with two or more players, where no player has anything to gain by changing only his or her own strategy. If each player has chosen a strategy and no player can benefit by changing his or her strategy while the other players keep theirs unchanged, then the current set of strategy choices and the corresponding payoffs constitute a Nash equilibrium (Myerson, 1991, p. 95).

There are a wide variety of games, like the Prisoners' Dilemma, where the Nash equilibria yield very low player payoffs. If possible, the players would want to change the game to have better outcomes. They can communicate and coordinate their moves, perhaps signing contractual agreements. (Wikipedia)

The classical prisoner's dilemma has one Nash equilibrium: when both players defect. However, "both defect" is inferior to "both cooperate," in the sense that the total jail time served by the two prisoners is greater if both defect. The strategy "both cooperate" is unstable, as a player could do better by defecting while their opponent still cooperates. Thus, "both cooperate" is not an equilibrium.

Prisoner A Stays Silent

Prisoner A Betrays

Prisoner B. Stays Silent

Both serve six months

Prisoner B. serves ten years; Prisoner A goes free

Prisoner B. Betrays

Prisoner A serves ten years; Prisoner B. goes free

Both serve two years

Person A / Person B

Confess

Don't Confess

Confess

5,5

2,10

Don't Confess

10,2

3,3

If both persons confess to the crime (not knowing what the other will do) they both get 5 years. If only one person confesses ("we did it"), he gets a lighter sentence for cooperation, and the partner gets a longer sentence with a conviction based on solid evidence. If neither confess, it is more difficult for the state to present the case and expected sentences [pr (conviciton)x (length)] will be lighter. Extensions in this case would be the nature of agreements, contracts, or collusion between the two players. The game theorist would attempt to define how such agreements would be sustainable and to what degree contracts could be enforced. Enforcement might retaliation if one player defected from the agreement or made possible through repetition of the game.

If people are going to have a long-term relationship with someone, they may behave differently than a one-time occurrence. In a repeated game, there is an infinite sequence of rounds or points in time when players may obtain information and choose moves. A player must always consider the effect that his or her current move might have on the future. This may make players more cooperative or belligerent. For example, repeating the prisoners' dilemma game often enough may give rise to many equilibria where both prisoners never confess.

Nash also proposed that cooperation between players can be studied by using the same basic concept of Nash equilibrium that underlies all areas of noncooperative game theory. He argued that cooperative actions result from bargaining among the cooperating players, and in this process each player should be expected to behave in some bargaining strategy and be able to predict outcome.

The Nash bargaining game is a two-player noncooperative game where two players attempt to divide a good like a cake between them. Each player requests an amount of the cake. If their requests are compatible, each player receives the amount requested; if not, each player receives nothing. The simplest form of the Nash bargaining game assumes the utility function for each player to be a linear function… READ MORE

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