Beauty Contested

Rationality breaks down in social prediction games.

Posted Jan 27, 2020

J. Krueger
Eat 2/3 of what your friend is eating!
Source: J. Krueger

I look at things for the art sake and the beauty sake and for the deal sake. – D. J. Trump, showing linguistic flexibility, New York Magazine, 11 July 1988, p. 24.

Richard Thaler has gathered and annotated many a phenomenon that (neo)classical economics cannot explain (Thaler, 2015; reviewed by Krueger & Kutzner, 2017). Psychology has to come to the rescue.

Does this mean that the anomalous is the province of psychology or that its province is simply that which is anomalous by the lights of conventional economics? Most psychological theories are descriptive rather than normative; if a psychological theory can explain weird economic behavior, that behavior is not anomalous psychologically speaking. Economic behavior typically refers to the modal behavior expected or seen in a crowd. A few oddballs behave in unique ways, but their behavior is not anomalous given a good psychological theory allowing individual differences; odd behavior is only socially, or statistically, anomalous.

Perhaps the best-known anomaly, predating Thaler’s own discoveries, is the beauty contest, first described in Keynes’s General Theory. Keynes (1936) observed that in a modern stock market (and he wrote a century ago), value is no longer directly observable. What we see instead is other people’s perceptions of other people’s perceptions (etc.) of value. This infinite regress prohibits that all, or even most, can be ahead of all, or even most, others. Some may be able to look deeper into the rabbit hole than others, but they can be overtaken by those others if those get wise to it. In the long run, all die of exhaustion.

Keynes’s beauty contest, intended to model the stock market by metaphor, morphed into a beauty contest game, which goes like this: all of you (e.g., students in a class) pick a number between 1 and 100. There is a prize for the person who picks the number identical to or closest to 2/3 of the average picked number [a wicked modification of this game sets the prize money to the winning number]. What’s a poor boy to do? When the game is played, the result is usually a right-skewed distribution with a mode of or close to zero and an average of about 20. A few wags offer the number 100, and for reasons that need not be elaborated, these are anomalous by both economic and psychologic lights. In short, the modal person is being economically rational, but, by definition, the 2/3-times-the-average guesser wins.

Economics invokes game theory to explain why the rational guess should be zero. Any positive number a player might consider should immediately be revised to 2/3 of its value, and this process should be repeated indefinitely. The number thus drifts toward zero without ever reaching it, as long an infinite number of decimal places is allowed. Thaler and others take this to mean that 0 is the Nash equilibrium. When all players settle on 0, no one will want to switch to a positive number and leave the cash prize to others. Then again, if all propose 0, no one gets anything, which suggests that a positive guesser is not stupid but altruistic.

Why should a player guess 2/3 the value of others’ guesses? The exact fraction is irrelevant for the point of the thought experiment as long as it is smaller than 1 and greater than 0. Regardless of the size of the fraction, the Nash equilibrium of 0 remains. Psychologically, though, it matters. As players will likely anchor their guesses on the fraction, larger fractions will result in right-shifted distributions (Krueger, 2019). Consider what would happen if the fraction were greater than 1. Imagine you were told you would win the jackpot if you provided a number twice as large as the average number proposed by a group of players who are given the same task as you have. To put it more bluntly, imagine you and your bestie as players, and that the one writing down the larger number wins. Absurd, right? No one in their right mind would play such a game, and no economist – however psychologically naïve – would propose such a game to make a point, or to metaphorically illuminate economic behavior in the wild.

Why are we so sure that a game with a fraction > 1 is absurd, whereas a game with a fraction < 1 is not? This is itself a psychological puzzle. Is it sufficient to note that there is no limit if the fraction > 1? Why would the existence of a floor of 0 for a game with the fraction < 1 turn the game into a credible psychological experiment? And if the game as stated, with a fraction of 2/3, is indeed absurd, how can it model economic behavior? Behavioral economists like Thaler will say that the observed data show that most people do provide positive numbers as estimates, and thereby keep the game going. If we accept the beauty contest game as a model of economic behavior, the implication for economic theory is devastating. The game does not merely suggest that economic theory fails to predict economic behavior, but that if people did behave in the way considered rational by economic theory, there would be no economy, that is, there would be no trade or exchange. A theory denying the existence of the phenomenon it purports to model is a bad theory.

This conclusion does not need to horrify us as much as it may first seem. There may certainly be something wrong with economic theory, but perhaps there is also something wrong with the game. Let us ask again, why are players instructed to guess a number that is 2/3 of the average of all the guessed numbers? And why is this fractionality crucial for a model of the stock market? Do investors win in the market if they value stocks less than the average person does? Is it not these investors who are more likely to buy stocks and thereby drive the market, perhaps into a bubble?

Why should the game not simply instruct players to estimate the average estimate? Suppose we revisit Galton’s (1907) bovine adventure. If 100 herdsmen estimate an oxen’s weight, the average of their estimates will be closer to the beast’s true weight than individual estimates will be on average (Krueger & Chen, 2014). Might there be a further accuracy gain if the herdsmen estimate the average estimate? This is possible, but the gain is unlikely to be large. The distribution of the estimates will shrink drastically, but there is no obvious reason why the bias, that is, the difference between the average estimate and the true value, should shrink. If this is so, why play the beauty contest game at all? Perhaps we should leave such games to TV personalities and politicians.

When did Thomas Mann receive the Nobel Prize?

My friend W. and I were wondering when Thomas Mann received the Nobel in literature. Remembering that he published his epic saga Buddenbrooks around the turn of the century, we collectively homed in on the 1920s. W., being a loss-averse Swabian and me being a tight-fisted Westphalian, we did not want to bet any money, but our competitive nerve was being stimulated. Who would make the better prediction?

Our conversation revealed that W. was considering an earlier date than I was. We agreed that he whose guess was closer to the googled date would be the winner. After a quick tactical reflection, I moved to let him go first. Was this a good move? I think it was. Whatever date W. chose – as long as it was lower than my best private guess – I would take his guess and add one year. Thus, I would increase my chances of winning.

To see that this must be so, consider two bettors whose best private estimates are modeled as the true date plus random error. If bettor 1 makes his private estimate public – without attempting to outflank bettor 2 – bettor 2 takes bettor 1’s estimate and adds one year if his, bettor 2’s, private estimate is greater than bettor 1’s; otherwise he subtracts a year. If the mean and the variance of the true-value-plus-error distribution are known, or reasonably assumed, the bettor 2’s probability of winning can be estimated.

This logic worked for me. W. offered the estimate of 1922 and I countered with 1923. The actual year of Mann’s Nobel was 1929. W. could have done better if he had beat me to offering to go second. Alternatively, he could have upped his estimate to shrink my range. The trouble here is that he would not know by how much.

References

Galton, F. (1907). Vox populi. Nature, 75, 450- 451.

Keynes, J. M. (1936). The general theory of employment, interest and money. London: Macmillan.

Krueger, J. I. (2019; September 19). Anchor What? Psychology Today Online. https://www.psychologytoday.com/us/blog/one-among-many/201909/anchor-what

Krueger, J. I., & Chen, L. J. (2014). The first cut is the deepest: Effects of social projection and dialectical bootstrapping on judgmental accuracy. Social Cognition, 32, 315-335.

Krueger, J. I., & Kutzner, F. (2017). Homo Anomalus: Richard Thaler’s Kuhnian adventure. Review of ‘Misbehaving: The making of behavioral economics’ by Richard Thaler. American Journal of Psychology, 130, 385-389.

Thaler, R. H. (2015). Misbehaving: The making of behavioral economics. Princeton, NJ: Princeton University Press.