Roger A. McCain, Ph.D.
Professor of Economics Drexel University

The arrows shown leading away from the first circle correspond from top to bottom to these three moves. Thus, if Anna chooses the first move, Barbara will see the two coins shown side by side in the top-circle of the second-column. In that case Barbara has the choice of taking either one or two coins from the second row, leaving either none or one for Anna to choose in the next round, as shown in the top two circles of the third column. Of course, by taking two coins, leaving none for Anna, Barbara will have won the game.

In a similar way, we can see in the diagram how Anna’s other two choices leave Barbara with other alternative moves. Looking to strategy 3, we see that it leaves Barbara with only one possibility; but that one possibility means that Barbara wins. From Anna’s point of view, move 2, in the middle, is the most
interesting. As we see in the middle circle, second column, this leaves Barbara with one coin in each row. Barbara has to take one or the other-those are her only choices. But each one leaves Anna with just one coin to take, leaving Barbara with nothing on her next turn, so Anna wins the game. We can now see that Anna’s best move is to take one coin from the second row, and once she has done that, there is nothing Barbara can do to keep Anna from winning.

Now we know the answers to the questions above. There is a best strategy for the game of Nim. For Anna, the best strategy is, “Take one coin from the second row on the first turn, and then take whichever coin Barbara leaves.” For Barbara, the best strategy is “If Anna leaves coins on only one row, take them. Otherwise, take any coin.” We can also be sure that Anna will win if she plays her best strategy.

John von Neumann was at the Institute for Advanced Study in Princeton. Oskar Morgenstern was at Princeton University. As a result of their collaboration, Princeton was soon buzzing with game theory. Albert Tucker, the chairman of the mathematics department at Princeton, was visiting at Stanford University, and wanted to give a group of psychologists some idea of $=$$=$== what all the buzz was about, without using too much mathematics. The example that he gave them is called the Prisoner’s Dilemma. ${}^3$${}^3$^(3){ }^{3} It is the most studied example in game theory and possibly the most influential half a page written in the twentieth century. You may very well have seen it in some other class. The Prisoner’s Dilemma is presented a little differently from the two previous examples, however.

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 to confess and implicate the other. If neither man confesses, then both will serve 1 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

Payoff Table-A payoff table is a table with the strategies of two or three players along the margins and the payoffs to the players in the cells. Each strategy corresponds to a column or row and so the payoffs in a cell are the payoffs for the strategies that correspond to the column and row.

Table
The Prisoner’s Dilemma

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 comma 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 1 year each.

This remarkable result-that self=interested and seemingly rational action results in both persons being made worse off in terms of their own self-interested purposes-is what has made a wide impact in modern social science. There are many int seem that, from arms races through road conger much like overexploitation of some subsurface and pollution to the depleof fisheries and the face water resources. These are all quite