Repeated Prisoners Dilemma with a Last Period

Repeated Prisoners’ Dilemma with a Last Period

What would happen if you played the prisoners’ dilemma game in Figure 39 100 times with the same opponent? If you played the game only once, you should always be mean. If your opponent plays mean, you score 1 if you’re mean and 0 if you’re nice. If your opponent is nice, you score 3 if you are mean and 2 if you are nice. Thus, regardless of what your opponent does, you are always better off being mean if the game is played just once. What if you are going to play it 100 times? Would it be reasonable for you to use the following logic?

Figure 39

If we are both mean throughout the entire game, we will both get a payoff of only 1 each period. If, however, we are both nice all the time, our payoff per round will be 2. If I start out playing mean, my opponent will be mean to me, and we will be stuck getting a payoff of only 1. I would rather start being nice and hope that he too is nice. It’s true that if he is nice, I could take advantage of him and be mean for one round. After this one round he would not keep being nice, however, and then I would be stuck with getting only 1 each period because he would probably always be mean after that. As a result I should play nice at least until he is mean to me.

Unfortunately, last-period problems would prevent parties known to be rational from being nice to each other even in round one. Consider what strategy you should employ in period 100, the last period. Playing mean always gives you a higher payoff in the round you played it than being nice would. The only possible reason you might play nice in a given round is so your opponent will play nice in the next round. (Recall that in simultaneous move games such as this, your opponent can’t see how you’re going to move when you move; thus, your choice in any given round can’t influence your opponent’s move in that same round.) In the last period, however, there are obviously no other rounds to care about. Thus, you should always play mean in round 100 and so should your opponent. Knowing this, what should you do in round 99? Playing mean will always give you a higher payoff in period 99. The only reason you might not want to be mean in round 99 is so that your opponent will be nice to you in round 100. We have already concluded, however, that your opponent should be mean to you in round 100 no matter what. Thus, everyone should be mean in round 99. Of course, this means that in round 98 both of you should also be mean because both players will always be mean in rounds 99 and 100. You can continue to apply this logic backward to establish that you should be mean in round 1! Consequently, even if this prisoners’ dilemma game is played 100, 1,000, or a billion times, rational players should be mean in every round.

Irrationality and the Prisoners’ Dilemma

Game theory teaches that when two people play a finitely repeated prisoners’ dilemma game, they should always be mean. Economics, however, is supposed to be a science, and in science you test your theories. Unfortunately for game theory, when real people play a finitely repeated prisoners’ dilemma game they often are nice to each other, especially in the early rounds.^[10] Why the divergence between theory and reality?

Obviously, reality is wrong and should adjust itself to theory. Of course, it’s also possible that game theorists are in error in assuming that people are always rational.

Imagine that you are playing a repeated prisoners’ dilemma game 100 times with someone you know is not entirely rational. Many people, outside of game theory land, are nicer than they should be, but don’t like to be taken advantage of. Let’s say that you think your opponent will start out playing nice. You suspect that if you start being mean to him, however, he will be mean to you. How should you now play? You should probably play nice until the last period. In the last period, of course, you should always betray your fellow player.

This betrayal in the final round is why two parties who are known to be rational could never be nice to each other in a finitely repeated prisoners’ dilemma game. Since a rational opponent would always betray you in round 100, you should betray him in round 99. Similarly, since you will betray him in round 99, he should be mean to you in round 98, which, of course, means that . . . . If some doubt exists as to your opponent’s rationality, however, you might want to play nice in round one. This doesn’t mean that your opponent would benefit from being irrational, but rather that he would benefit from being seen as irrational.

Interestingly, even if you both are rational, it’s possible to get the nice, nice outcome until the last rounds. If both players are rational but neither is 100 percent sure that the other is rational, then both players might, rationally, play nice until the last few rounds.

Repeated Prisoners’ Dilemma with No Last Period

If a prisoners’ dilemma game has no last period, then the nice, nice outcome is achievable. Since the final period always brings betrayal, a final period makes it impossible for players ever to be nice. Many games in real life, however, have no final period. If you play a prisoners’ dilemma game forever, you might reasonably adopt a strategy of always being nice unless your opponent is mean to you. If both players adopt this strategy, they will achieve a good outcome every round. The nice, nice outcome is reachable even if the prisoners’ dilemma game doesn’t go on forever. All that’s needed is no definite ending date. For example, imagine two people playing a prisoners’ dilemma game and then flipping a coin to determine if they should play again. If they don’t stop playing until their coin comes up heads, then the game has no known last period where betrayal must manifest.

In a repeated prisoners’ dilemma game with no last period, your ideal outcome would be for you always to be mean and your opponent always to be nice. This outcome is almost certainly not achievable, however. What might be obtainable is for both players always to be nice. Remember that the only reason any rational person should ever be nice in prisoners’ dilemma is to induce his fellow opponent to be nice in the next round. Consequently, to induce your opponent to play nice, he must think that you will be mean if he turns mean. In game theory land people will be nice only if it’s in their interest to be nice. Unfortunately, just because a repeated prisoners’ dilemma game goes on forever doesn’t mean that the players will always be nice to each other.

It’s easiest to betray those who trust you, though once you have betrayed someone they are unlikely ever to trust you again. Treachery is justified, however, if it brings you a large enough one-time benefit. Let’s compare the games in Figures 40 and 41. Both are prisoners’ dilemma games. In these two games the players get 5 each if they are both mean and 10 each if they are both nice. Obviously, the players would rather be in a situation where they are always nice to each other than always mean. The difference between the games manifests if one player betrays the other. If you are mean, and he is nice, you do far better in the game in Figure 41 than in the one in Figure 40. Thus, there is a much greater incentive to betray your opponent in the game in Figure 41.

Figure 40

Figure 41

Smoking gives pleasure today and causes health problems in the future. Smoking can be a rational decision for someone who places far greater value on the present than the future. Like smoking, betraying someone in a repeated prisoners’ dilemma game helps you today but harms you in future periods. Thus, the less someone cares about the future the more likely he is to betray you in a repeated prisoners’ dilemma game. For example, a supplier who might go bankrupt or a lawyer considering retirement would value the present more than the future because he won’t be playing in future periods. You should consequently place greater trust in someone who has a long time horizon.

People’s actions often betray how much they care about the future relative to today. For example, you should have limited trust in smokers because they obviously care far more about the present than the future. Conversely, someone who exercises is willing to make sacrifices today for future benefits and thus is less likely to betray you for a short-term gain.

If you are convinced that another player will soon cheat you, don’t necessarily try to change his mind. It might well be in his self-interest to be mean to you. If you suspect that your fellow player is someday going to betray you, your optimal response is probably to betray him first.

^[10]Gibbons (1992), 224–225.