Chris and Pat get 15 years
After Chris and Pat are arrested, neither knows whether the other will confess or really will stay silent as promised. What Chris knows is that if Pat is true to his word and doesn’t talk, Chris can get off with time served by betraying Pat. If instead Chris stays faithful to her promise and keeps silent too, she can expect to get five years. Remember, game theory reasoning takes a dim view of human nature. Each of the crooks looks out for numero uno. Chris cares about Chris; Pat looks out only for Pat. So if Pat is a good, loyal buddy—that is, a sucker—Chris can take advantage of the chance she’s been given to enter a plea. Chris would walk and Pat would go to prison for life.
Of course, Pat works out this logic too, so maybe instead of staying silent, Pat decides to talk. Even then, Chris is better off confessing than she would be by keeping her mouth shut. If Pat confesses and Chris stays silent, Pat gets off easy—that’s neither here nor there as far as Chris is concerned—and Chris goes away for a long time, which is everything to her. If Chris talks too, her sentence is lighter than if she stayed silent while Pat confessed. Sure, Chris (and Pat) gets fifteen years, but Chris is young, and fifteen years, with a chance for parole, certainly beats life in prison with no chance for parole. In fact, whatever Chris thinks Pat will do, Chris’s best bet is to confess.
This produces the dilemma. If both crooks kept quiet they would each get a fairly light sentence and be better off than if both confessed (five years each versus fifteen). The problem is that neither one benefits from taking a chance, knowing that it’s always in the other guy’s interest to talk. As a consequence, Chris’s and Pat’s promises to each other notwithstanding, they can’t really commit to remaining silent when the police interrogate them separately.
IT’S ALL ABOUT THE DOG THAT DIDN’T BARK
The prisoner’s dilemma illustrates an application of John Nash’s greatest contribution to game theory. He developed a way to solve games. All subsequent, widely used solutions to games are offshoots of what he did. Nash defined a game’s equilibrium as the planned choice of actions—the strategy—of each player, requiring that the plan of action is designed so that no player has any incentive to take an action not included in the strategy. For instance, people won’t cooperate or coordinate with each other unless it is in their individual interest. No one in the game-theory world willingly takes a personal hit just to help someone else out. That means we all need to think about what others would do if we changed our plan of action. We need to sort out the “what ifs” that confront us.
Historians spend most of their time thinking about what happened in the world. They want to explain events by looking at the chain of things that they can observe in the historical record. Game theorists think about what did not happen and see the anticipated consequences of what didn’t happen as an important part of the cause of what did happen. The central characteristic of any game’s solution is that each and every player expects to be worse off by choosing differently from the way they did. They’ve pondered the counterfactual—what would my world look like if I did this or I did that?—and did whatever they believed would lead to the best result for them personally.
Remember the very beginning of this book, when we pondered why Leopold was such a good king in Belgium and such a monster in the Congo? This is part of the answer. The real Leopold would have loved to do whatever he wanted in Belgium, but he couldn’t. It was not in his interest to act like an absolute monarch when he wasn’t one. Doing some counterfactual reasoning, he surely could see that if he tried to act like an absolute ruler in Belgium, the people probably would put someone else on the throne or get rid of the monarchy altogether, and that would be worse for him than being a constitutional monarch. Seeing that prospect, he did good works at home, kept his job, and freed himself to pursue his deepest interests elsewhere. Not facing such limitations in the Congo, there he did whatever he wanted.
This counterfactual thinking becomes especially clear if we look at a problem or game as a sequence of moves. In the prisoner’s dilemma table I showed what happens when the two players choose without knowing what the other will do. Another way to see how games are played is to draw a tree that shows the order in which players make their moves. Who gets to move first matters a lot in many situations, but it does not matter in the prisoner’s dilemma because each player’s best choice of action is the same—confess—whatever the other crook does. Let’s have a look at a prospective corporate acquisition I worked on (with the details masked to maintain confidentiality). In this game, anticipating what the other player will do is crucial to getting a good outcome.
The buyer, a Paris-based bank, wanted to acquire a German bank. The buyer was prepared to pay a big premium for the German firm but was insistent on moving all of the German executives to the corporate headquarters in Paris. As we analyzed the prospect of the acquisition, it became apparent that the price paid was not the decisive element for the Heidelberg-based bank. Sure, everyone wanted the best price they could get, but the Germans loved living in Heidelberg and were not willing to move to Paris just for money. Paris was not for them. Had the French bankers pushed ahead with the offer they had in mind, the deal would have been rejected, as can be seen in the game tree below. But because their attention was drawn to the importance the Germans attached to where they lived, the offer was changed from big money to a more modest amount—fine enough for the French—but with assurances that the German executives could remain in Heidelberg for at least five years, which wasn’t ideal for the French, but necessary for their ends to be realized.
FIG. 3.1. Pay Less to Buy a Bank
The very thick, dark lines in the figure show what the plans of action were for the French buyer and the German seller. There is a plan of action for every contingency in this game. One aspect of the plan of action on the part of the executives in Heidelberg was to say nein to a big-money offer that required them to move to Paris. This never happened, exactly because the French bankers asked the right “what if” question. They asked, What happens if we make a big offer that is tied to a move to Paris, and what happens if we make a more modest money offer that allows the German bank’s management to stay in Heidelberg? Big money in Paris, as we see with the thick, dark lines, gets nein and less money in Heidelberg encourages the seller to say jawohl. Rather than not make the deal at all, the French chose the second-best outcome from their point of view. They made the deal that allowed the German management to stay put for five years. The French wisely put themselves in their German counterparts’ shoes and acted accordingly.
By thinking about the strategic interplay between themselves and the German executives, the French figured out how to make a deal they wanted. They concentrated on the all-important question, “What will the Germans do if we insist they move to Paris?” No one actually moved to Paris. Historians don’t usually ask questions about things that did not happen, so they would probably overlook the consequences of an offer that insisted the German management relocate to France. They might even wonder why the Germans sold so cheaply. In the end, the Germans stayed in Heidelberg.
Why should we care about their moving to Paris when in fact they didn’t? The reason they stayed in Heidelberg while agreeing to the merger is precisely because of what would have happened had the French insisted on moving them to France: no deal would have been struck, and so there would have been no acquisition for anyone to study.