How would you define rationality?

[Mehmet Ismail]

Dear DT Forum Members,

I hope that it’s going to be or it has already become a nice day wherever you may be.

I have finished teaching game theory and cooperation in dynamic games this week. Many students weren't satisfied with the rationality of the equilibrium in the finitely repeated Prisoner’s Dilemma.

To start with, game theory is one of the most diverse modules I’ve taught so far; students come from economics, politics, PPE, political economy, and even some maths, philosophy, and arts backgrounds. However, when students see what we, economists, propose as the ‘rational solution’ in the repeated PD, many are not convinced. And economic theory is even sometimes ridiculed. Common questions I receive include: Why is this rational?, But isn’t game theory about maximizing the utility you get? But it doesn’t make sense to play this, and so on. I cannot blame them, as I had been in a similar situation and struggled about similar questions as a student. Below are some thoughts and even a video which I recorded to draw attention to the issue and my own suggestion to fix it. 

• In the finitely repeated Prisoner’s Dilemma, why is Tit-For-Tat a 'good' strategy in general? In my opinion, it isn't just because the Tit-For-Tat profile’s payoff is greater than the equilibrium payoff. But even if an opponent acts opportunistically -- i.e., makes a unilateral profitable deviation -- the minimal payoff of the Tit-For-Tat player is still greater than the equilibrium payoff. By contrast, unconditional cooperation in the repeated PD could be disastrous if the opponent acts opportunistically by playing defect in every round.

• The issue of rationality in games has a long history, e.g. von Neumann and Morgenstern (1944, p. 32) said this: “But if the superiority of “rational behavior” over any other kind is to be established, then its description must include rules of conduct for all conceivable situations including those where "the others" behaved irrationally, in the sense of the standards which the theory will set for them.”

• Coming back to the repeated PD, the 'irrationals' get much greater payoffs than 'rationals', and even if a rational behaves opportunistically against the irrational, the irrational would still get a greater payoff than the rationals. Under these circumstances, there seem to be some flaws in the definition of the concept of rationality.

• True, unconditional defection is the unique subgame perfect equilibrium, so it is immune to opportunistic behavior (i.e., no unilateral profitable deviation). But as a player, why should I care about the lack of profitable deviation -- especially when I can get greater payoffs by allowing my opponent to behave opportunistically?

• That brings us to my suggestion: the "optimin" reasoning. Roughly speaking, under optimin, players simultaneously maximize minimum utility under the opportunistic behavior of others. Indeed, Tit-For-Tat profile is generally an optimin, because even if a player profitably deviates from the Tit-For-Tat profile, the non-deviator enjoys in general a strictly greater payoff than the unique equilibrium payoff. By contrast, unconditional cooperation is not an optimin because if the opponent behaves opportunistically, the cooperator's payoff is the worst in the repeated Prisoner’s Dilemma.

To draw attention, I have visited four London attractions to give you four reasons to teach optimin reasoning in game theory classes along with equilibrium reasoning. Please see my video below for more information.

https://www.youtube.com/watch?v=kc0tQ3pbmXc

• Perhaps the main reason is the following. Optimin solution arguably gives better predictions than Nash equilibrium in the sense that for every Nash equilibrium there is an optimin point in which every player is weakly better off, and this remains to be true for the optimin players even if some players deviate unilaterally and profitably from the optimin. Moreover, a Nash equilibrium never Pareto dominates an optimin.

• I am aware that optimin is not a 'perfect' solution in part because it allows for opportunistic behavior. However, we might have to admit that every theory has its weaknesses and there is no 'theory of everything' in economics and games. But I believe that this should not preclude us trying to improve the existing theories and find a more reasonable concept of rationality.

All in all, it seems that "rational behavior" -- whatever it may be -- should be about obtaining as high payoffs as possible from a game. However, equilibrium in games cannot capture the benefits from cooperation even when cooperation provides high payoffs under opportunistic behavior as discussed above. 

I have described a new type of rationality concept called optimin based on the idea of maximizing minimum payoffs under opportunistic behavior. To come back to the main question again: Are you satisfied with the definition of rationality in games, and, if not, how would you define it?

Thank you very much for your suggestions and comments.

Mehmet

(human)