## Game Theory: An Example Game

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Introduction
The Terminology
Basic Concepts of Game Theory
Types of Games
Applications of Game Theory
A Simple Example

The basics of game theory can be demonstrated by a non-zero-sum game called the "Prisoner's Dilemma", which is demonstrates payoffs, cooperation, and level of information.

The two players in the game can choose between two moves, either "cooperate" or "defect". The idea is that each player gains when both cooperate, but if only one of them cooperates, the other one, who defects, will gain more. If both defect, both lose (or gain very little) but not as much as the "cheated" cooperator whose cooperation is not returned. The whole game situation and its different outcomes can be summarized by table 1, where hypothetical "points" are given as an example of how the differences in result might be quantified.

The name comes from a hypothetical situation: two criminals are arrested for committing a crime in unison, but the police do not have enough proof to convict either. Thus, the police separate the two and offer a deal: if one testifies to convict the other, he would get a reduced sentence or go free. Here the prisoners do not have information about the other's "move" as they would in chess. The payoff if they both say nothing (and thus cooperate with each other) is good, since neither can be convicted without further proof. If one of then betrays but the other remains silent, then he benefits because he goes free while the other is imprisoned because there is sufficient proof to convict the silent one. If they both betray the each other, they both get reduced sentence, which can be described as a null result. The obvious dilemma is the choice between two options, where a good decision cannot be made without information.

This decisions in terms of the outcomes of the decisions of the prisoner may be assigned arbitrary point values and described with a table:

 Action of Prisoner A Cooperate with B Betray B Action of Prisoner B Cooperate with A Equivalent Gain [+ 5] Loss [- 10] Betray A Win [+ 10] No Gain [0]

We might consider this a zero-sum game in that there is no mutual cooperation: either each gets 0 when both betray, or when one of them cooperates, the defector gets +10, and the cooperator -10, in total 0. On the other hand, if both cooperate the sum becomes positive because each gets 5 to total 10. However, this gain is offset by the equal losses/gains (10) in for betrayal, leading to a temptation to defect, or a tendency to attempt to seize the winning move. This assumption is not generally valid. For example, it is easy to imagine that two wolves together would be able to kill an animal that is more than twice as large as the largest one each of them might have killed on his own.

Such is not often the case, since the gains usually outweigh the ambivalent results. There are also time effects involved (since many tries may lead to trust/distrust), thus making it suitable for short term studies only. Also, we are not assuming rationality. Normally, one would tend to shy away from any possibly losing moves. Thus, it would be rational to always betray, which would eliminate the 10 point loss and the 5 point gain from the game entirely. If player A always betrayed, if B cooperated, A will gain 10. If B betrays, A will gain nothing. However, with 2 rational players, both would always betray, and neither will ever gain. However, if one or both were irrational from time to time, they would both gain.

1. Dauben, Joseph W "Game Theory." Microsoft Encarta. Microsoft Corp, 1999
2. McCain, Roger A. "Game Theory: An Introductory Sketch." Hypertext. 1999 (August 1999)