I want to expand on what I wrote previously in “A Simple But Challenging Game: Part II, this time focusing on Rosenthal’s Centipede Game. To remind you of the rules, in that game there are two players. The players, named Mutt and Jeff, start out with $2 each, and they alternate rounds. On the first round, Mutt can defect by stealing $2 from Jeff, and the game is over. Otherwise, Mutt cooperates by not stealing, and Nature gives Mutt $1. Then Jeff can defect and steal $2 from Mutt, and the game is over, or he can cooperate and Nature gives Jeff $1. This continues until one or the other defects, or each player has $100.

As I previously wrote, in this game, the Nash equilibrium is that Mutt should immediately defect on his first turn. This result is obtained by induction. When both players have $99, it is clearly in Mutt’s interest to steal from Jeff, so that the he will end with $101, and Jeff will end with $97. But that means that when Jeff has $98 and Mutt has $99, Jeff knows what Mutt will do if he cooperates, and can see that he should steal from Mutt, so that he will end with $100 and Mutt will end with $97. But of course that means that when both players have $98, Mutt can see that he should steal from Jeff, and so on, until one reaches the conclusion that Mutt should start the game by stealing from Jeff.

Of course, this Nash equilibrium behavior doesn’t really seem very wise (not to mention ethical), and experiments show that humans will not follow it. Instead they usually will cooperate until the end or near the end of the game, and thus obtain much more money than would “Nashists” who rigorously follow the conclusions of theoretical game theory.

Game theorists often like to characterize the behavior of Nashists as “rational,” which means that they need to explain the “irrational” behavior of humans in the Rosenthal Centipede Game. See for example, this economics web-page, which gives the following “possible explanations of ‘irrational’ behavior”:

There are two types of explanation to account for the divergence. The first assumes that the subject pool contains a certain proportion of altruists who place a positive weight in their utililty function on the payoff of their opponent. Also to the extent that selfish players believe that there is some probability that other players are altruists, they have an incentive to mimic altruistic behaviour by passing.

The second explanation considers the possibility of action errors. Errors in action, or ‘noisy’ play, may result from subjects experimenting with different strategies. Or simply from subjects pressing the wrong key.

Let’s step back for a second and consider what “rational” behavior should mean. A standard definition from economics is that a rational agent will act so as to maximize his expected utility. Let’s accept this definition of “rational.”

The first thing we should note is that “utility” is not usually the same as “pay-off” in a game. As noted in the first explanation above, many people get utility from helping other people get a pay-off. But there are many other differences between pay-offs and utility. You might lose utility from performing actions that seem unethical or unjust, and gain utility from performing actions that seem virtuous or just. You might want to minimize the risk in your pay-off as well as maximize the expected pay-off. You might value pay-offs in a non-linear way, so that the difference between $101 and $100 is very small in terms of utility.

Of course, this difference between pay-off and utility is very annoying theoretically. We’d really like the pay-offs to strictly represent utilities, but unfortunately for experiments, it is only possible to hand out dollars, not some abstract “utils.”

But suppose that the pay-offs in the Rosenthal Centipede Game really did represent utils. Would the game theory result really be “rational” even in that case? Would the only remaining explanation of cooperating behavior be that the players just don’t understand the situation and are making an error?

No. Remember that to be “rational,” an agent should maximize his expected utility. But he can only do that conditioned on some belief about the nature of the person he is playing with. That belief should take the form of a probability distribution for the possible strategies of his opponent. A Nashist rigidly reasons by backward induction that his opponent must always defect at the first opportunity. He even believes this if he plays second, and his opponent cooperates on the first turn! But is this the most accurate belief possible, or the one that will serve to maximize utility? Probably not.

A much more accurate belief could be based on the understanding that even people who understand the backward induction argument can reason beyond it and see that many of their opponents are nevertheless likely to cooperate for a long time, and therefore it pays to cooperate. If you believe that your opponent is likely to cooperate, it is completely “rational” to cooperate. And if this belief that other players are likely to cooperate is backed by solid evidence such as the fact that they started the game by cooperating, then the behavior of the Nashist, based on inaccurate beliefs that cannot be updated, is in fact quite “irrational,” because it does not maximize his utility.

Sophisticated game theorists do in fact understand these points very well, but they muddy the waters by unnecessarily overloading the term “rational” with a second meaning beyond the definition above; they in essence say that “rational” beliefs are those of the Nashist. For example, take a look at this 1995 paper about the centipede game by Nobel Laureate Robert Aumann. Aumann proves that “Common Knowledge of Rationality” (by which he which he essentially means the certain knowledge that all players must always behave as Nashists) will imply backward induction. He specifically adds the following disclaimer at the end of his paper:

We have shown that common knowledge of rationality (CKR) implies backward induction. Does that mean that in perfect information games, only the inductive choices are appropriate or wise? Would we always recommend the inductive choice?

Certainly not. CKR is an ideal (this is not a value judgement; “ideal” is meant as in “ideal gas”) condition that is rarely met in practice; when it is not met, the inductive choice may be not only unreasonable and unwise, but quite simply irrational. In Rosenthal’s (1982) centipede games, for example, even minute departures from CKR may make it incumbent on rational players to “stay in” until quite late in the game (Aumann, 1992); the resulting outcome is very far from that of backward induction. What we have shown is that if there is CKR, then one gets the backward induction outcome; we do not claim that CKR obtains or “should” obtain, and we make no recommendations.

This is all well and good, but why use the horribly misleading name “Common Knowledge of Rationality” for something that would be more properly called “Universal Insistence on Idiocy?”

I hope it is obvious by now why I am skeptical of explanations of various types of human behavior that are based on assuming that all humans are always Nashists, and even more skeptical of recommendations about how we should behave that are based on those same assumptions.

[Acknowledgement: I thank my son Adam for discussions about these issues.]

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Tags: economics, Game Theory, Nash equilibrium, rationality, super-rationality, utility