The prisoners’ dilemma and the mini-max strategy by Charles Warner The Prisoners’ Dilemma

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Charles Warner
The Prisoners’ Dilemma
The Prisoners’ Dilemma is perhaps the best-known strategic game, as suggested by Dixit and Nalebuff,i and it illustrates how cooperation is often the best strategy. The Prisoners’ Dilemma also is useful for demonstrating how to use two very useful decision tools, a decision tree and a payoff matrix, and how to employ a mini-max strategy.

Suppose that in Russia during the Stalin era, a conductor of an orchestra was traveling by train and was reading the score of the music he was to conduct at his next engagement. Two KGB policemen watched him reading and, thinking that the musical notations were some secret code, arrested him as a spy. The conductor protested that it was only Tchaikovsky’s Violin Concerto, but with no success. On the second day of his imprisonment, an interrogator walked up to the conductor and said confidently, “You had better tell us everything you know. We have caught your friend Tchaikovsky, and he is already talking.”

The KGB had, in fact, picked up a man whose only offense was that he was named Tchaikovsky, and they were subjecting him to the same kind of intense interrogation. If the two innocents withstand this treatment and confess nothing, they will both get off with a relatively mild three-year sentence (the standard punishment at that time for doing nothing). On the other hand, if the conductor makes a false confession and implicates Tchaikovsky while Tchaikovsky holds out, the conductor gets a reduced sentence of one year and Tchaikovsky gets the maximum sentence of 25 years for being recalcitrant. Of course, the tables will be turned if the conductor stands firm and Tchaikovsky gives a false confession and implicates the conductor (25 years for the conductor, one year for Tchaikovsky). If both give false confessions and implicate the other person, then both get a reduced sentence of 10 years. If neither one of them confesses nor implicates the other, they each get three years. These options are clearly laid out for the two prisoners, who, of course, are never allowed to talk to each other.

The conductor reasons as follows: He knows Tchaikovsky is either (a) confessing and implicating him or (b) holding out. If Tchaikovsky confesses and implicates him, the conductor gets 25 years by holding out, but only 10 years by confessing and implicating the other person, so it is to his advantage to confess. If Tchaikovsky is holding out, the conductor gets three years if he holds out and only one year if he confesses and implicates Tchaikovsky, so it is to his advantage in this scenario to confess and implicate Tchaikovsky. Thus, confession is clearly the conductor's best strategy.

Tchaikovsky is no dummy; he’s sitting in his cell doing the same mental calculations. He comes to the same conclusion. The result is, of course, that both men confess and implicate the other and are sent to Siberia for 10 years (the KGB have played this game many times and know they will get something on both men, regardless if it is true or not, and be able to fill their quota of prisoners).

When the two men meet in the Gulag Archipelago, they compare stories and realize that they are both innocent and that they have been duped. If they had both held out and said nothing, they each would have gotten only three years instead of the 10 they wound up with. However, the temptation to get sent away for only one year by confessing and implicating the other was so overwhelmingly tempting at the time that they could not resist, and thus were in for 10 years.

The decisions the prisoners had to contemplate is best illustrated by using two decision tools, a decision tree (Figure 1 below) and a payoff matrix (Figure 2) below. The term “rat” or “not rat” is used to indicate whether or not one prisoner will implicate the other by making a false charge against him. The numbers in the decision tree and the payoff matrix are arranged with the conductor’s consequences of ratting or not ratting first, then Boris’s consequences of ratting or not ratting.

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