One of the more famous and intriguing results of game theory is that cooperation can be sustained in a repeated prisoners dilemma as long as nobody knows when the last game will be played. To set out the basic issue consider the following game between Bob and Francesca.
If they both cooperate they get a nice payoff of 10 each. If they both defect they get 0 each. Clearly mutual cooperation is better than mutual defection. But, look at individual incentives. If Francesca cooperates then Bob does best to defect and get 15 rather than 10. If Francesca defects then Bob does best to defect and get 0 rather than -5. Bob has a dominant strategy to choose defect. So does Francesca. We are likely to end up with mutual defection.
But what if Bob and Francesca are going to play the game repeatedly with each other? Intuitively there is now an incentive to cooperate in one play of the game in order to encourage cooperation in subsequent plays of the game. To formalize that logic suppose that whenever Bob and Francesca interact then with probability p they will interact again tomorrow. Also suppose that both Bob and Francesca employ a grim trigger strategy - I will cooperate unless you defect and if you defect I will defect forever after. Can this sustain cooperation?
If Bob and Francesca cooperate in every play of the game the expected payoff of Bob is 10 for as long as they keep on playing, which gives
If Bob defects now his expected payoff is 15 because he gets a one time benefit and then has to settle for mutual defection from then on. It follows that cooperating makes sense if 10/(1 - p) > 15 or p > 0.33. It should be emphasized that this story relies on Bob and Francesca both using a grim trigger strategy and both expecting the other to use it. Even so, cooperation can, in principle, be sustained if p is high enough. By contrast, if people know when the end is likely to come (p is small) then there is no hope of sustaining cooperation.
What of the evidence? A paper recently published by Pedro Dal Bo and Guillaume Frechette in the Journal of Economic Literature surveys the evidence. They fit a model to a meta-data set of over 150,000 choices from relevant studies. The Figure below summarizes some of the findings that come out of that model. In interpreting this figure we need to understand that in most experiments subjects play the repeated game several times against different opponents. So, Bob plays with Francesca for, say, 10 rounds (determined randomly according to p). This is supergame 1. He then plays with Claire for, say, 5 rounds (again randomly determined according to p). This is supergame 2. And so on.
The figure below shows the fitted probability of a subject cooperating in round 1 of supergame 1 and of round 1 of supergame 15. Look first at supergame 1. Here we can see that around 50-60% of subjects cooperate - which is quite high - and the probability of cooperating does not depend much on p. This is inconsistent with the theory because we would not expect such high levels of cooperation when p is low. What about in supergame 15? Here we see a much higher dependence on p. This is starting to look more consistent with the theory because we see low levels of cooperation when p is low.
So, can cooperation be sustained in a repeated prisoners dilemma? The relatively high levels of cooperation seen in the above figure may give some optimism. But it is important to appreciate that cooperation is only going to be sustained if both people cooperate. If there is a 50% chance a random individual will cooperate then there is only a 25% chance they will start with mutual cooperation. This does not look so good. And it turns out that 'always defect' consistently shows up as the most popular strategy employed when playing the prisoners dilemma. The chances of sustained cooperation among two strangers seem, therefore, somewhat remote.
All hope, though, is not lost because life is not only about interaction between strangers. Once we add in reputation, choosing who your friends are, and so on, there are various mechanisms that may be able to sustain cooperation. And, even putting that aside, there are still strong arguments to try and cooperate with strangers. As David Kreps, Paul Milgrom, John Roberts and Robert Wilson pointed out in a well-cited paper it is not always in your interest to defect in the first round of a prisoners dilemma. Basically, if the other person wants to be cooperative then by defecting you miss out long term. Better to cooperate and give the other person a chance.
If they both cooperate they get a nice payoff of 10 each. If they both defect they get 0 each. Clearly mutual cooperation is better than mutual defection. But, look at individual incentives. If Francesca cooperates then Bob does best to defect and get 15 rather than 10. If Francesca defects then Bob does best to defect and get 0 rather than -5. Bob has a dominant strategy to choose defect. So does Francesca. We are likely to end up with mutual defection.
But what if Bob and Francesca are going to play the game repeatedly with each other? Intuitively there is now an incentive to cooperate in one play of the game in order to encourage cooperation in subsequent plays of the game. To formalize that logic suppose that whenever Bob and Francesca interact then with probability p they will interact again tomorrow. Also suppose that both Bob and Francesca employ a grim trigger strategy - I will cooperate unless you defect and if you defect I will defect forever after. Can this sustain cooperation?
If Bob and Francesca cooperate in every play of the game the expected payoff of Bob is 10 for as long as they keep on playing, which gives
What of the evidence? A paper recently published by Pedro Dal Bo and Guillaume Frechette in the Journal of Economic Literature surveys the evidence. They fit a model to a meta-data set of over 150,000 choices from relevant studies. The Figure below summarizes some of the findings that come out of that model. In interpreting this figure we need to understand that in most experiments subjects play the repeated game several times against different opponents. So, Bob plays with Francesca for, say, 10 rounds (determined randomly according to p). This is supergame 1. He then plays with Claire for, say, 5 rounds (again randomly determined according to p). This is supergame 2. And so on.
The figure below shows the fitted probability of a subject cooperating in round 1 of supergame 1 and of round 1 of supergame 15. Look first at supergame 1. Here we can see that around 50-60% of subjects cooperate - which is quite high - and the probability of cooperating does not depend much on p. This is inconsistent with the theory because we would not expect such high levels of cooperation when p is low. What about in supergame 15? Here we see a much higher dependence on p. This is starting to look more consistent with the theory because we see low levels of cooperation when p is low.
So, can cooperation be sustained in a repeated prisoners dilemma? The relatively high levels of cooperation seen in the above figure may give some optimism. But it is important to appreciate that cooperation is only going to be sustained if both people cooperate. If there is a 50% chance a random individual will cooperate then there is only a 25% chance they will start with mutual cooperation. This does not look so good. And it turns out that 'always defect' consistently shows up as the most popular strategy employed when playing the prisoners dilemma. The chances of sustained cooperation among two strangers seem, therefore, somewhat remote.
All hope, though, is not lost because life is not only about interaction between strangers. Once we add in reputation, choosing who your friends are, and so on, there are various mechanisms that may be able to sustain cooperation. And, even putting that aside, there are still strong arguments to try and cooperate with strangers. As David Kreps, Paul Milgrom, John Roberts and Robert Wilson pointed out in a well-cited paper it is not always in your interest to defect in the first round of a prisoners dilemma. Basically, if the other person wants to be cooperative then by defecting you miss out long term. Better to cooperate and give the other person a chance.
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