The NPR blog Planet Money is asking you to guess a number:
This is a guessing game. To play, pick a number between 0 and 100. The goal is to pick the number that’s closest to half the average of all guesses.
So, for example, if the average of all guesses were 80, the winning number would be 40.
The game will close at 11:59 p.m. Eastern time on Monday, October 10. We’ll announce the winner — and explain why we’re doing this — on Tuesday, October 11.
This is a famous game that has been used in numerous experiments investigating whether real people are as rational as game theory and economic theory assumes they are. Powerful logic suggests that you should guess the number zero:
- For sure the average will be no greater than 100 so half the average will be no greater than 50.
- Anybody who is smart enough to figure this out will guess something no greater than 50 so the average will be no greater than 50 and half the average will be no greater than 25.
- Anybody who is smart enough to figure this out will guess something no greater than 25, etc.
Of course time after time in experiments the actual guesses are very far from zero, demonstrating that people are in fact less rational than economic theory assumes.
Planet Money, however is an intelligent blog and when they analyze the results of their experiment, they won’t jump to that conclusion. They will be insightful enough to see past the straw man.
It all starts at point 2. It is true that people who are smart enough to figure out point 1 will guess something no greater than 50, but almost all of those people are also smart enough to know that there is a sizeable proportion of people who are not that smart. And thus these smart people, if they are rational, will not deduce in point 2 that the average will be no greater than 50. The induction will not take them past point 2.
In fact, some of the smartest and most rational people in the world, professional chess players, guess numbers around 23 when they play these experiments. (To be precise, the chess players were playing a version of the Beauty Contest were you are supposed to guess 2/3 of the average. Their guesses would be somewhat lower in the Planet Money version, see below.) And that is because if someone is indeed as rational as game theory and economic theory assumes she is, and also she is smart enough to know that
- Not everybody is that rational,
- Most of the rational people know that not everybody is that rational,
- Most of the rational people know that most (but not all) of the rational people know that not everybody is that rational
etc., then she will never choose anything close to zero. Indeed, according to my calculations, the ultra-rational guess in the Planet Money Beauty Contest is about 16. Here is how I came up with that number.
I think that
- About 2/5 of the Planet Money readers will be confused by the rules of the game and guess 50.
- Another 3/10 will be smart enough to know that the rational thing to do is to guess something less than 50, and reasoning as in the straw-man argument they will guess 25.
- The remaining 3/10 of the population are the really smart ones.
12 comments
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October 5, 2011 at 1:54 am
James Lee
I bet only 0.5% of listeners will actually go to the website and participate. Of that 0.5% of listeners, they’ll skew towards the “smart” or “really smart”. Piling on top of your analysis, I’ll guess:
1. 1/5 pick 50
2. 2/5 pick 25
3. 2/5 pick x
Yielding x = 12.5. I’ll go down to 12.
October 5, 2011 at 8:40 am
Scott
You seem to assume that everyone will try to play rationally, or at least play to win. What about trolls who vote 100 to skew the average? If there are many participants, then the odds of winning the (likely modest) prize are small, and the private benefits of messing with the game are greater.
October 5, 2011 at 8:47 am
Peter
We should also look to the past. Some years ago in an article for the Financial Times on behavioral finance, Professor Richard Thaler asked the same question and got an interesting dispersion of answers.
This is simply a numerical version of Keynes’ beauty contest and any ‘rationale’ estimate is just that. A guess as to what an uncertain outcome will be.
I, for one, do not have any prior knowledge as to what people will guess, rationally or otherwise, and whether participants have come across this particular game before (which will influence their choices based on the learning effect).
It certainly won’t show rationality or irrationality at work.
October 5, 2011 at 9:35 am
Joshua Gans
I did a similar exercise and got 12.5. Different priors on unsophisticated behaviour I guess.
October 5, 2011 at 11:40 am
Lones Smith
Jeff, knowing how awesome smart is your logic here, and also knowing that you have market power to bid likewise, I have just bid 15.
October 5, 2011 at 11:55 am
Planet Money is Running a Beauty Contest Experiment « Cheap Talk | 冰力/专注于摄影,图像美化
[…] here to see the original: Planet Money is Running a Beauty Contest Experiment « Cheap Talk 相关日志2011 年 10 月 5 日 — Planet Money is Running a Beauty Contest Experiment « Cheap […]
October 5, 2011 at 12:02 pm
dan
this has already been done in the field (with stronger incentives)
http://www.jstor.org/stable/3083273
October 5, 2011 at 12:38 pm
David Miller
In the classroom I play this with the 2/3 rule. There is usually a big spike at 2/3, containing 10-15% of the class. But then there is a trough, rather than a spike, at 4/9. Instead the second peak (much broader) is around 1/3, which is 2/3 of 1/2. So the ones who are confused about the rules (we tend to call them “level zeroes”) aren’t just guessing randomly; they’re confused about the rules in a particular way. But the “level ones” don’t reason about them that way; the level ones seem to ascribe random behavior to the level zeroes, even though empirically the random background noise is only about 5% of the participants. The winning guess is usually somewhere near “level two,” at 2/9.
So if I plug in 5% background noise (averaging out to 50), 15% guessing 50 by misunderstanding the rules, 40% level ones guessing 25, 5% hyper-rationals guessing zero, and 35% smarty-pants correct guessers, I get a smarty-pants guess of 12.12. I figure the NPR audience is pretty similar to UCSD undergrads when it comes to this game, so I would go with 13 because I think it will be less common than 12 even if 12 is the best answer in a vacuum.
October 5, 2011 at 1:20 pm
Sean
I think 100% of Planet Money readers will read this post, so I’m guessing 8.
Of course, that will only work if none of them read my comment!
October 6, 2011 at 10:52 am
Anonymous
We played this game (the 2/3 version) in the first class of microeconomics at CEMFI, I guessed 18,5 and win. I applied a somewhat different analysis than Jeff.
You look around and see how many people write their answers in the first 5 seconds, those will misunderstand the rules and play 50. Then you wait a little bit more, and count people that just smile, those are the level 0s, and will play 25. This way of reasoning will take you about 1 minute or so, until you can do the math, and play your bid (in my case 18,7).
There are of course some other people that dont get it at all, or think it is not really an strategy game but pure luck, you cant count on them…
Of course there is also some people that either have already play the game or they are really “smart” and play 0.
Summarizing, Jeff strategy is a good approximation (3rd order?) of what the rational “guesser” would do, assuming that he knows the real distribution of peoples rationalization level
December 1, 2011 at 2:14 pm
Here’s The Winner In Our Pick-A-Number Game | My Blog
[…] at Cheap Talk, the economist Jeff Ely has a very nice discussion of this sort of experiment. He does a little […]
January 18, 2015 at 8:19 pm
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