# Apply your knowledge: pick half a number

post by Eneasz · 2011-10-04T16:49:59.198Z · score: 6 (8 votes) · LW · GW · Legacy · 25 commentsNPR's Planet Money is running an experiment which could be an interesting way to test your other-people-modeling skills.

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.

http://www.npr.org/blogs/money/2011/10/03/133654225/please-help-us-pick-a-number?sc=fb&cc=fp

The other people guessing are self-selected, I would assume primarily NPR listeners.

## 25 comments

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I'm curious if they've sanitized their inputs on the back end. Interestingly, you *seem* to be able to vote for arbitrarily high numbers or arbitrarily small ones and get the same response as a real vote. I tested, and lets just say that unless they've got more than 100,000 voters, the correct number may end up being *well above* 100 from that vote alone.

This is an interesting experiment. I just hope they've accounted for little bobby tables.

Interestingly, you seem to be able to vote for arbitrarily high numbers or arbitrarily small ones and get the same response as a real vote. I tested, and lets just say that unless they've got more than 100,000 voters, the correct number may end up being well above 100 from that vote alone.

Hope they don't take that seriously, or else the correct number is now essentially 3↑↑↑3, as that's what I voted for.

I doubt that it parsed it correctly, their database probably can't store the number 3↑↑↑3 represents the way it stores other numbers.

Variant: You win a value proportional to the number you picked.

My method and estimate:

- Google "guess half the average number". The first result is Guess 2/3 of the average on Wikipedia.
- Find that "When performed among ordinary people it is usually found that the winner guess is much higher than 0, e.g., 21.6 was the winning value in a large internet-based competition organized by the Danish newspaper Politiken."
- Assume that these participants will follow the same algorithm, and note that 21.6 is slightly under what we'd get if nearly everyone thought "the average of all numbers is 50, so 2/3 that is 33, which will be the average, so I'll pick 2/3 of that which is 22".
- Pick something slightly under 50%
*50%*50 = 12.5. Like 11.

The actual answer was 11.53.

(Full disclosure: actually first I picked 24 = something slightly under 50%*50, because I don't even know why, then later my brain said "hey why'd you do that??" and I picked 11 under a new name Guy "Oops" Srinivasan.)

My mental model of the NPR experimenters suggests that, after running a simulated Keynesian bubble, they are modeling an "anti-bubble", where the participants are expected to pick a very low number, probably 0, which is the number that would let everyone win, were it a cooperative game.

However, my mental model of an average NPR contest participant without any feedback from others is that of a one- or two-leveler (people rarely spend a lot of time thinking about poll answers). The former would expect the average to be 50 and then pick 25, the latter would pick something like 12. Any higher, and you are likely to converge to the fixed point.

The best guess would depend on your exact priors, but, given that there likely to be a fraction of participants who would not recurse all the way, the answer is likely to be non-zero. As any closed-loop model, this one is also subject to positive feedback, so there should be a correction for that, etc.

Of course, if the current average was publicly known, it would quickly drop to zero.

However, my mental model of an average NPR contest participant without any feedback from others is that of a one- or two-leveler (people rarely spend a lot of time thinking about poll answers).

I would be interested in knowing the values of the individual votes, perhaps after the poll is ended. In particular, I'm curious whether anyone picked a number higher than twenty-five - what would you call that, a zero-leveler? I guess someone who picked a number higher than fifty would be a negative-one-leveler.

I'm curious whether anyone picked a number higher than twenty-five - and what would you call that, a zero-leveler?

Actually, a sizable fraction of clever pranksters may get a kick out of thwarting the "rational" choice, since there is no punishment for guessing wrong, and pick 100.

(No, I am not telling you what I picked.)

Every time I've played this in real life there's been someone who's done that.

This thought did occur to me, yes. But I figured the "rational" choice - which is not actually rational, since it's predictably not going to win - of zero was doomed anyway, so chose 1 rather than guarantee a loss with 100 for no purpose.

**Edit:** Oh. There's nothing preventing you from voting multiple times. Hmm... in that case 49 is probably the best bet.

In particular, I'm curious whether anyone picked a number higher than twenty-five - what would you call that, a zero-leveler?

I played this game twice before; once with a high-school group and once with a collegiate. Based on those, I would be totally *not* surprised if a non-trivial number of people picked numbers between 50 and 100. Note that they are not picking 100 for trolling value but rather some 2/3-ish sounding number, like 70. They constitute the "not-quite-grasping-the-game-mechanics" group.

We did this on Less Wrong a couple years ago; Warrigal called it The Aumann's agreement theorem game (guess 2/3 of the average). The average guess was about 13.2, making the winning number (2/3 the average in that game) about 8.8.

I wonder if repeated trials would drive the average down or up?

Playing against sufficiently rational agents the correct guess is zero. But assuming that not every other player is completely rational, applies sufficient recursion etc, you have to take into account their irrationality. If some non negligible proportion answer anything other than zero, and at least one participant guesses a very small non-zero number, every player choosing zero is guaranteed to miss. I don't know how much I trust the NPR readership to take recursion to the limit.

I asked a friend just now. She (immediately) answered 5.

Sounds like the Less Wrong readership may overestimate the rationality of the average participant.

If some non negligible proportion answer anything other than zero, the rational players choosing zero will get it wrong.

Will they, though? The object is to choose the number *closest to half* the average. If ten people pick zero, five pick twelve and five pick twenty-five, the average will be 9.25, half of which is 4.625, which is closer to zero than twelve.

(I mean, the people choosing zero *will* get it wrong, but it's because at least one person - me - thought that far ahead and picked 1. fiddlemath picked 2, presumably due to a different estimate of the ratio of LWers/NPRers in the poll.)

I have a feeling the half-average "should" be in the interval between 0 and 1.

I will be very surprised if the half-average is zero.

I'm guessing there is both literature and mathematical models for this interesting problem, but I'm not going to go ahead and Google it. Not knowing is way more fun.

My thought process when I saw this:

Okay, if most people on there are NPR listeners, they'll think: "The average will be around 50, so I'll pick 25." The smarter NPR commenters and all the LW commenters will iterate it one more time, and pick 11. If about 2/3 will do the first, and the other 1/3 the second, the average will be [grabs calculator] about 20, so I'll pick 10. Or better yet 7 or 8, because I'm hardly going to be the first to think this. Oh darn, comments are closed.

ETA (rot13'd because it contains the answer): V infgyl haqrerfgvzngrq gur cbchynevgl bs Cynarg Zbarl naq gurersber bs gur cebcbegvba bs ACE yvfgraref, jvgu gur erfhyg gung V thrffrq n yvggyr ybjre guna gur jvaavat inyhr bs ryrira naq bar unys.

This is a cool game.

For those who are interested: Here are the results.

**[deleted]**· 2011-10-05T19:15:34.058Z · score: 1 (1 votes) · LW · GW

My estimate:

- Lets say 40% of the people are going to do a naive estimate of 25 based on half of the average of random guesses between 1-100
- Lets say 20% recurse one level and guess 12.5
- Lets say 15% think they're clever and guess super low like 0/.5/1
- Lets say 15% of people are going to be anchored to 40 and guess 20.
- And lets say the remaining 10% just guess more or less randomly, the average being 50

This results in an 'average' guess of ~20.5, half of which is 10.25. I actually guessed 10.5 as I used slightly different estimates.

Of course, had I known this was posted on LW, I'd have used higher probabilities of low guesses.

I hope Less Wrong doesn't end up accounting for a significant proportion of the entries. I suspect the average member is likely to apply a greater level of recursion.

**[deleted]**· 2011-10-04T17:05:17.919Z · score: 3 (5 votes) · LW · GW

I took the unique fixed point: 0.

Everyone is picking this, right?

No. I expect the actual average to be greater than 1. If we did the same thing here, I'd probably pick a value very near 1; there, I picked a value very near 2. After further thought, I now suspect that choice is a bit low.

Certainly, it's true that 0 is the unique fixed point, and it's easy to see this if you think about if for a minute or two. On the other hand, how often do you think it will happen that someone will simply *misread* the question, and answer something else?

I briefly misread the problem as "pick the guess closest to the average guess;" which is actually a sort of interesting question in itself. (I suspect the answer would be lower than 50, as smaller numbers are actually more mentally available than large numbers.) It was only when I was double-checking the text that I realized I was thinking about the wrong sort of thing. I'm fairly certain that I'm better at catching my own mistakes than most, so I expect this and similar mistakes to actually happen.

If similar misreadings happen even once per 25 respondents, then 1 will probably be closer to the average than 0.

It's not only a schelling point, it's the only correct response if everyone picks it - half of the average of zero is zero.

(This is why I chose 1.)

Ok i've read all the comments and have no idea what everyone is talking about and how to rationally go about making a guess. Could someone post a link or something that might explain?