Oct 162012

A few years ago Richard Thaler asked readers of the Financial Times to participate in this game:

Choose an integer number between 0 and 100. You win a prize if your number is equal or closest to 2/3 of the average number chosen by all other participants.

What number would you choose?

If you think that the other participants will choose a random number within the range, the average will be 50. Hence you choose 33.

But hang on. Just as you chose 33, so presumably will other participants, at least on average, based on your same line of reasoning. So if the average number chosen by all participants is 33, then the smart thing to do is to choose 22.

But hang on. Do you really think you are smarter than the others? Just as you figured out that 22 is the smart choice, so will others, at least on average. So the super smart thing to do is to choose 15.

But hang on…

You know where we are heading. We are heading towards 0 (you get there after 12 iterations). Zero is the only rational choice to make if you don’t think you are smarter than the other participants.

But hang on. Somehow that can’t be right. You get the strong feeling that if you choose 0 you are not going to win the prize. This is because, although you don’t think you are smarter than most, it is reasonable to assume that at least some of the players are not as rational. For example, if 10% of players are totally naive and choose a random number – 50 on average – then the overall average will be 5 and the right answer will be 3. However, if the rest of the players share your thoughts and assumptions, they will also choose 3, thereby increasing the average to 8 and the right answer to 5. Then you answer 5, but so will the rest, thus increasing the right answer to 6. The process converges to 8. So 8 is the right answer if 90% of players are as smart as you are and 10% are totally naive. If 20% are naive, the process converges to 14; with 30% it converges to 18, and so on. But then it may also be the case that the less rational players are not totally naive (Level 0 rationality) but, for example, exhibit Level 1 rationality, where the average answer is 33. In this case, with 10% Level 1 players the process converges to 5; with 20% to 9; with 30% to 12, and so on. Of course, there are plenty more combinations, with varying proportions of players at Level 0, Level 1, Level 2 and so on.

Thaler’s game is known as a Beauty Contest, in reference to Keynes’ famous passage in Chapter 12 of the General Theory:

Professional investment may be likened to those newspaper competitions in which the competitors have to pick out the six prettiest faces from a hundred photographs, the prize being awarded to the competitor whose choice most nearly corresponds to the average preferences of the competitors as a whole; so that each competitor has to pick, not those faces which he himself finds prettiest, but those which he thinks likeliest to catch the fancy of the other competitors, all of whom are looking at the problem from the same point of view. It is not a case of choosing those which, to the best of one’s judgment, are really the prettiest, not even those which average opinion genuinely thinks the prettiest. We have reached the third degree where we devote our intelligences to anticipating what average opinion expects the average opinion to be. And there are some, I believe, who practise the fourth, fifth and higher degrees.

Beauty Contests have no right solutions, only experimental answers. In Thaler’s experiment, with 1,476 participants, the winning number was 13. This is roughly consistent with all players exhibiting Level 3 rationality, but also with 80% being fully rational and 20% totally naive, or 70% fully rational and 30% Level 1, as well as with many other more complex combinations. Analogous results have been obtained in many similar experiments.

Interestingly, once the answers are revealed, the winner is rewarded and the game is fully elucidated to all participants, playing it a second time will typically produce a winning number which is closer, but not equal to zero. It takes several repetitions for the game to finally converge to the zero solution.

Changing coordinates, the gap between the winning number and zero resembles the gap between the market price of a stock and its intrinsic value. An Efficient Market theorist playing the Beauty Contest game would choose 0 and lose (and convince himself he has won). The higher the winning number, the larger is the percentage of less rational players in the game. Likewise, the larger the percentage of less rational investors in the stock market, the wider is the gap between the stock price and its intrinsic value. And the larger is the gain of the investor who has paid the former and gets the latter.

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