One of the many lessons that one learns in prison is, that things are what they are and will be what they will be.
~ Oscar Wilde
The prisoner’s dilemma (PD) is a famous, even notorious, example of a “non-cooperative game.” It needs no re-introduction. However, if you are on the fence about reading on, see here for a description. Merrill Flood described the game in a RAND Corporation memorandum in 1952, and William Poundstone (1992) reports in his excellent book that it was a certain Professor Albert Tucker who came up with the prisoner scenario to introduce his students to the logic of non-cooperative game theory. Let’s be clear: There is nothing special about prisoners in the PD; prisoners entered the picture – and hence our collective imagination – as a pedagogical device thought up by a creative professor.
Now Khadjavi & Lange (KL, 2013) come along and do something interesting, and arguably overdue. They study the PD with a sample of convicts (women incarcerated in Germany with sentences ranging from a few days to life). There is a critical difference, though. The original PD does not involve convicts, but detainees. The point of the dilemma set up by the prosecutor is to get at least one of them to confess to a major offense. If both cooperate with each other and keep quiet, they get away with a short sentence on a minor count. KL, however, treat their convicts to the same dilemma to which they subject their university subject controls. Well, sort of. Whereas the students receive money, the convicts receive coffee or cigarettes. The story could end right there because economists, of all academics, would clamor that the currency should be the same for participants in all conditions. If differences are observed between students and convicts (there were just a few), who is to say that the difference in currency is not responsible? Conversely, for those data that came out the same for the two groups, who is to say that the difference in currency does not hide true substantive differences? Choosing not to gnash my teeth over this, I move on.
In the canonical, simultaneous, one-shot, anonymous game, KL find that 37% (rounded) of the students and 56% of the cons cooperated. This looks like a hefty difference, but the low-powered study only yielded marginal significance. In a sequential game, where the first mover must choose between cooperation and defection, knowing that the second mover might betray her trust, 63% of the students and 46% of the cons cooperated. Again, a marginal difference. KL proceed to declare that the students, but not the cons, showed significantly more cooperation in the sequential game than in the simultaneous game. Quibble alert: With four chi-square tests conducted on a set of four proportions, there are no grounds to infer differences between two groups from the juxtaposition of one significant result in one group with a nonsignificant result in another. Nonetheless, KL cheerfully declare that “inmates are therefore able to better solve their classical dilemma than students” (p. 164; italics in the original). As second movers who have learned that first movers cooperated, students (62%) and cons (60%) also show the same rate of cooperation. KL suggest that this finding supports their prediction that second movers reciprocate first movers’ cooperation at a rate that is higher than the rate of cooperation in the simultaneous game. Perhaps, but their data suggest that this is true only for students and not cons.
There is little to be learned about prisoners here; there were no specific hypotheses about how prisoners might differ from students in their strategic thinking. KL write that they are concerned with bounded self-interest, and from this, one might derive the naïve expectation that cons will act more selfishly than students. KL never say so explicitly, which leaves one wondering why they went to the trouble of recruiting inmates for research. Is it only for the news value that prisoners can play in their own dilemma?
The interest in bounded self-interest opens the door to a moral interpretation of the findings. KL adapt Fehr & Schmitt’s (1999) utility theory, which assumes that at least some people are motivated to act in such a way as to reduce or eliminate self-other differences in payoffs. Do the findings say anything about the idea that inequality-aversion contributes to cooperation? No!
KL derive what they call “equilibria based on Fehr and Schmitt (1999)" (p. 170). Let’s take a look at their derivations for the simultaneous game. Things do not get better for the sequential game; so I will not discuss it. Working with the payoffs of $9 for unilateral defection, $7 for mutual cooperation, $3 for mutual defection, and $1 for unilateral cooperation, KL show that a player will cooperate if α ≤ (3μ + 4βμ – 4)/(4(1-μ)), where α denotes the weight given to inequality disadvantageous to the self (being suckered), β denotes the weight given to inequality advantageous to the self (suckering), and μ denotes the subjective probability that the other person will cooperate. There are thus 3 free parameters; each one of them can be derived if one pretends to know the other two. But one does not. So what is the point? The use of Fehr & Schmitt’s utility function degenerates into an exercise of post-hoc fitting and not making any predictions. Correction: There isn’t even any post-hoc fitting. If there were, we might learn whether students or cons are more moral in the sense of being more inequality-averse (in one of the two senses of the term: guilt or envy). Lastly, the exercise in applied math does not say anything about an equilibrium in the game-theoretical sense. It only shows for which α, β, or μ (considering one at a time) the expected value of cooperation is equal to the expected value of defection. A game-theoretic equilibrium could be found if α and β were set and μ ignored. Then, we could derive the probability with which a rational player would cooperate (Binmore, 2007). As it is, there is no use for utility in this paper.
As Luke would say: "What we've got here is a failure to communicate."
Binmore, K. (2007). Game theory: A very short introduction. New York, NY: Oxford University Press.
Fehr, E., & Schmitt, K. M. (1999). Theory of fairness, competition, and cooperation. The Quarterly Journal of Economics, 114, 817- 868.
Flood, M. M. (1952). Some experimental games. Research memorandum RM-789. RAND Corporation, Santa Monica, CA.
Khadjavi, M., & Lange, A. (2013). Prisoners and their dilemma. Journal of Economic Behavior & Organization, 92, 163-175.
Poundstone, W. (1992). Prisoner's Dilemma. New York: Doubleday.