Prisoner’s Dilemma :: An example of the application of Game Theory

Tucker began with a little story, like this: two burglars, Bob and Al, are captured near the scene of a burglary and are given the “third degree” separately by the police. Each has to choose whether or not to confess and implicate the other. If neither man confesses, then both will serve one year on a charge of carrying a concealed weapon. If each confesses and implicates the other, both will go to prison for 10 years. However, if one burglar confesses and implicates the other, and the other burglar does not confess, the one who has collaborated with the police will go free, while the other burglar will go to prison for 20 years on the maximum charge.

The strategies in this case are: confess or don’t confess. The payoffs (penalties, actually) are the sentences served. We can express all this compactly in a “payoff table” of a kind that has become pretty standard in game theory. Here is the payoff table for the Prisoners’ Dilemma game:

Table 3-1

   

Al

    confess don’t
Bob confess 10,10 0,20
don’t 20,0 1,1

 

The table is read like this: Each prisoner chooses one of the two strategies. In effect, Al chooses a column and Bob chooses a row. The two numbers in each cell tell the outcomes for the two prisoners when the corresponding pair of strategies is chosen. The number to the left of the comma tells the payoff to the person who chooses the rows (Bob) while the number to the right of the column tells the payoff to the person who chooses the columns (Al). Thus (reading down the first column) if they both confess, each gets 10 years, but if Al confesses and Bob does not, Bob gets 20 and Al goes free.

So: how to solve this game? What strategies are “rational” if both men want to minimize the time they spend in jail? Al might reason as follows: “Two things can happen: Bob can confess or Bob can keep quiet. Suppose Bob confesses. Then I get 20 years if I don’t confess, 10 years if I do, so in that case it’s best to confess. On the other hand, if Bob doesn’t confess, and I don’t either, I get a year; but in that case, if I confess I can go free. Either way, it’s best if I confess. Therefore, I’ll confess.”

But Bob can and presumably will reason in the same way — so that they both confess and go to prison for 10 years each. Yet, if they had acted “irrationally,” and kept quiet, they each could have gotten off with one year each.

 

Dominant Strategies

What has happened here is that the two prisoners have fallen into something called a “dominant strategy equilibrium.”

DEFINITION Dominant Strategy: Let an individual player in a game evaluate separately each of the strategy combinations he may face, and, for each combination, choose from his own strategies the one that gives the best payoff. If the same strategy is chosen for each of the different combinations of strategies the player might face, that strategy is called a “dominant strategy” for that player in that game.

DEFINITION Dominant Strategy Equilibrium: If, in a game, each player has a dominant strategy, and each player plays the dominant strategy, then that combination of (dominant) strategies and the corresponding payoffs are said to constitute the dominant strategy equilibrium for that game.

In the Prisoners’ Dilemma game, to confess is a dominant strategy, and when both prisoners confess, that is a dominant strategy equilibrium.

 

Issues With Respect to the Prisoners’ Dilemma

This remarkable result — that individually rational action results in both persons being made worse off in terms of their own self-interested purposes — is what has made the wide impact in modern social science. For there are many interactions in the modern world that seem very much like that, from arms races through road congestion and pollution to the depletion of fisheries and the overexploitation of some subsurface water resources. These are all quite different interactions in detail, but are interactions in which (we suppose) individually rational action leads to inferior results for each person, and the Prisoners’ Dilemma suggests something of what is going on in each of them. That is the source of its power.

Having said that, we must also admit candidly that the Prisoners’ Dilemma is a very simplified and abstract — if you will, “unrealistic” — conception of many of these interactions. A number of critical issues can be raised with the Prisoners’ Dilemma, and each of these issues has been the basis of a large scholarly literature:

     

  • The Prisoners’ Dilemma is a two-person game, but many of the applications of the idea are really many-person interactions.
  • We have assumed that there is no communication between the two prisoners. If they could communicate and commit themselves to coordinated strategies, we would expect a quite different outcome.
  • In the Prisoners’ Dilemma, the two prisoners interact only once. Repetition of the interactions might lead to quite different results.
  • Compelling as the reasoning that leads to the dominant strategy equilibrium may be, it is not the only way this problem might be reasoned out. Perhaps it is not really the most rational answer after all.
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