Mathematics

# The Prisoners Dilemma in Game Theory

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When playing against an opponent, how should you strategize? In many fields exists the study of "game theory," which is the study of strategic decision making. Game theory helps reveal which alternatives have greater statistical likelihood of being successful. In economics, for example, game theory is used to plot and predict the courses of actions of firms in a market when they are faced with competition. A firm will have various options based on the possible moves of its competitors. When teaching game theory it is common to begin with a discussion of the "prisoner's dilemma," such as the description provided by Stanford University.

In the prisoner's dilemma two people are arrested by the authorities on suspicion of having committed a serious crime. Both people, now prisoners, are separated and have no knowledge of what the other person is doing or saying. Each prisoner is questioned by the authorities and offered a deal: a minimal sentence of only a few years in prison if he admits to the crime and testifies against his partner. However, if the other prisoner confesses and testifies against the first prisoner, the first prisoner will receive the maximum sentence of 20 years. If both men confess, a middle-ground sentence of seven years in prison can be applied to each. However, if neither man confesses, both will walk free in 48 hours due to lack of evidence.

Without knowing the actions of the other prisoner, what should the first prisoner do? Assuming he does not have much trust in his partner, a bit of statistical analysis would be beneficial. With two prisoners and two options, confess or not confess, a four-quadrant square akin to a Punnett Square can be drawn. The top of the square is labeled "Prisoner 1" and the left side of the square is labeled "Prisoner 2." Prisoner 1 can confess or not confess, and Prisoner 2 can do the same. There is a 25 percent chance of each of the following outcomes: 1 confesses and 2 confesses (top left), 1 does not confess and 2 confesses (top right), 1 confesses and 2 does not confess (bottom left), and 1 does not confess and 2 does not confess (bottom right).

Now look at the best-case scenarios and the worst-case scenarios. If Prisoner 1 talks, his best-case scenario is two years in prison and the worst-case scenario is seven years in prison. Though both involve some prison time, the worst case scenario of not talking lands a sentence of 20 years in prison. Two years or seven years versus zero years or twenty years? Assuming that both prisoners understand their dilemma, it behooves each man to confess.

The concept carries through to economics, where two competing firms will both lower their prices so as not to be undercut and not to be caught flat-footed by the worst-case scenario. When Firm A and Firm B can both either raise or lower their prices, they will both choose to lower prices because the worst-case scenario is far more acceptable than the worst-case scenario of raising one's price and finding that the competitor has done the opposite, winning over most of your clients!

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ARTICLE SOURCES AND CITATIONS
• http://plato.stanford.edu/entries/prisoner-dilemma/