## “A Continuous Dilemma,” D. Friedman & R. Oprea (2011)

I feel pretty confident that the two lab experiment papers I will write about today will represent the only such posts on that field here for quite a while. Both results are interesting, but as an outsider to experimental econ, I’m quite surprised that these represent the “state of the art”, and at some level both must since both are forthcoming in the AER.

In the present paper, Friedman and Oprea run three versions of the prisoner’s dilemma: a one-shot game, a one-minute continuous time game where players must “wait” 7.5 seconds to react to an opponent’s change of strategy, and a one-minute continuous time game with no limit on reaction speed aside from human reaction time. We’ve known since Nash that finitely-repeated prisoner’s dilemmas can only support defect every period as an equilibrium (by a simple backward induction unraveling argument), but that infinitely-repeated prisoner’s dilemmas can support any payoffs from the cooperate payoff to the defect payoff in equilibrium (by the Fudenberg-Maskin folk theorem). Two results from the 1980s save us a bit here. First, as the underrated Radner has pointed out, if you can react quickly to an opponent’s deviation, then you can only lose a tiny bit by cooperating and hoping your opponent cooperates also. That is, with a very high number of periods, cooperate until almost the very end is an “almost” dominant equilibria. If your opponent defects, you defect almost immediately afterward and thereafter both players play the “unique” equilibrium defect-defect. If your opponent does not defect, you both continue to cooperate until the very end. Regardless of your opponent’s strategy, “cooperate until opponent defects the first time” gains only a tiny bit less than the maximal payoff from using defect every period. Second, Simon and Stinchcombe (1989) show that in continuous time games, induction cannot be used and something like the folk theorem applies.

Friedman and Oprea test this in a lab. Basically none of their subjects cooperate in the one-shot game, and cooperation steadily increases as the minimum wait to react drops from 30 seconds to nearly continuous. In the example where the only restriction on reaction time is human response time, cooperation occurs 80-90% of the time, essentially encompassing the entire game in every example except for the last few seconds. A modification of Radner’s insight shows that this type of cutoff strategy is an epsilon-equilibrium, and that expected cooperation given the limits on reaction time are reasonable. The authors do not fully solve for the (epsilon)-equilibria of their game – I have no idea how they got away with this, but I would love to know what they said to the referees! In any case, the intuition for why cutoff strategies are nearly dominant equilibria seems reasonable, although it should be noted that this intuition is essentially Radner’s intuition and not anything novel to the present paper.

So what’s the takeaway? For a theoretically-minded reader, I think the experimental results here are simply more justification for taking care in interpreting Nash predictions for actions in lengthy, finitely-repeated games. Even for modeling purposes, it might be reasonable to see more work on epsilon-equilibria in, say, oligopoly behavior; cartel pricing is much easier to support when prices and quantities are very quickly reported if we look at that type of equilibria. I still find it a bit strange that the authors do not, as far as I can tell, attempt to distinguish between different types of theoretical explanation for high rates of cooperation in repeated games. Is there infection from beliefs a la the Kreps’ et al Gang of Four paper? (This does not appear to be the case to me, since I believe Gang of Four can sustain cooperation all the way to the horizon.) Would bounded rationality matter? (Both players’ complete action profile over time is available throughout the game in the present paper.) There are many other explanations that could be tested here. (Indeed, Bigoni et al have a new paper following up the present results with some discussion of infinite versus finite horizon continuous time games.)

http://faculty.arts.ubc.ca/roprea/prisonerEX.pdf (Dec 2010 working paper. Final version forthcoming in the AER. If you’re coming from a theory background, there are many norms in experimental econ that will strike you as strange – writing about an experiment with 36 American undergraduates who self select into lab studies as if it representative of human behavior, for example – but I’m afraid that battle has already been lost. Best just to read experimental work for what it is; some interesting insights for theory lie inside even despite these peccadillos.)