# Can You Guess What Other Readers Are Guessing? – RESULTS

Last week, I asked readers to submit an integer between 0 and 100, and the number they submitted was to represent their best guess at what they thought would be 2/3 of the average of all numbers submitted. (If you missed it, read the question before continuing here.)

Today at 10:15am EST, I collected the results. Here’s what I found:

There were a total of 5,101 reader guesses.

Here’s how many came in each day:

Not all of the 5,101 submissions were legitimate:

44 submissions were blank.

29 had decimals and weren’t integers.

4 submissions were negative numbers. The lowest was -1.0 x 10133. Someone got cute.

48 submissions were higher than 100. The highest was a string of 13,670 nines. Which rounds to 1.0 x 1013,670. This can be simplified to about 101002.067, a number far bigger than a googol. But it’s much, much less than Graham’s Number. It’s falls between 333 and 3333. Thank you for following instructions.

14 submissions were kind of numbers but not really. Things like “2/3 of 30″ and “24,66” and writing out every number from 1 to 100, like “1 2 3 4 5 ……. 99 100.” Nice job.

22 submissions were words, not numbers. Two sounded like an eighth grader saying the dirtiest words he knows, one was a post topic suggestion, and one was this: “Yeah, thanks mum. I’ll just let you know when I’m 5min away” followed by this a few seconds later: “Oops! Sorry! Accidentally typed in the work box! Meant to send that to my mother.”

After eliminating all of those, we were left with 4,940 legitimate submissions.

The people making a legitimate guess had to think hard about what everyone else would do. Or, they had to think hard about what everyone else would think everyone else would do. Or actually, they had to think hard about what everyone else would think everyone else would think everyone else would do.

This kind of head-exploding thought process can go on and on.

One way to approach it is like this—everyone is trying to “undercut” the rest of the group, picking 2/3 of what other people are guessing. But everyone is thinking like that, so the game is actually to undercut everyone else’s attempt to undercut—in other words, you need to pick 2/3 of 2/3 of what everyone else thinks the mean is. But if everyone is thinking that, you’ll lose with that logic. You need to be a further step ahead, picking 2/3 of 2/3 of 2/3 of what everyone else thinks everyone else thinks the mean is. And so on.

If you continue with this logic, you either end up at 0 or 1. You end up at 0 if you just multiply 2/3 by itself a bunch of times. But if you think of it in distinct steps, you realize that if the mean got down to 1, 2/3 of 1 is .666… which rounds to 1, so the mean and 2/3 of the mean are the same and the game stops there, making the right answer 1.

But that logic assumes that almost everyone is A) really smart, B) taking the game dead seriously, and C) is spending a decent amount of time thinking about their answer. So as clever as it is to answer 0 or 1, it’s actually more clever to guess something higher—something that takes into account the whole pool. To do that you have to ask, “Who are the people who read the Wait But Why Dinner Table and how will they treat this exercise?” The question isn’t about what perfect logic says to do if everyone in the game is using perfect logic—it’s about what thousands of Wait But Why readers will end up doing.

And here’s what they did:

Here are those results on a graph, showing the percentage of the whole pool that guessed each number:

Let’s discuss what we see here. There’s the clear high number of 0 and 1 guesses. But what else is happening?

There are some obvious things you’d expect to happen when you have a large number of people submitting a number, regardless of what the question asks readers to do—like the multiples of 10 being represented well:

Also, as expected, there were many more lower answer than higher ones. This is because of the simple logic that the highest the mean could possibly be is 100, and 2/3 of 100 rounds to 67, so all answers 68 or higher have no chance of being the winner. And if you assume that other people are thinking about that, you assume that the mean will likely be well below the highest possible correct answer (67), so 2/3 of the mean will likely be below 50. Again, this logic continues and favors lower answers.

You can also see a common line of thinking take shape in the results. That line starts and says, “If everyone guessed randomly, the mean would be 50 and the correct answer would be 2/3 of 50, which rounds to 33.” Some people stop right there, which is why 33 is the third most common answer. Others say, “33 will be a super common answer because it’s 2/3 of 50, so I’ll pick 2/3 of 33″—which is why 22 is the fourth most common answer. Still others will go a step further and pin the rest of the crowd on the 22 logic, so they’ll answer 15. And so on.

The steps in this line of logic go 50 > 33 > 22 > 15 > 10 > 7 > 5 > 3 > 2 > 1. And you can see that happening here:

Someone choosing 33 tells me that they either didn’t think hard enough about this or they’re not a great game-player or both.

Someone choosing 1 tells me that A) they’re clever enough and spent enough time to figure out the logic for picking 1, but that either B) that they didn’t think hard enough about this to realize that the mean will end up being much higher, or C) they think that almost all Wait But Why readers are similar to them.

Someone choosing 10 or 15 or 22 or another answer in that area tells me that either A) they’re not thinking about this as hard as the people who chose 1 and they didn’t consider how the succession of their logic can continue past where they went, or B) they’re very clever and they thought about all the people who would choose 0 and 1, they thought about all the people who would choose 33, they thought about the inevitable goofballs picking numbers above 50, and they estimated the mean from that information before submitting 2/3 of that estimation as their answer.

And it’s those people who nailed it:

The mean is 22.956, making the winning answer 15. The fact that 15 is one of the numbers you’ll hit in the 50 > 33 > 22 > 15 > 10 > etc. chain is a coincidence, and because the final answer happened to fall on that chain, I suspect we have a higher number of winners than we would if the answer had ended up being 14 or 16.

In any case, congrats to the winners—I saw a number of people comment on last week’s post saying they picked 15.

Oh, and the New York Times? Here were their results:

The two sets of results are very similar, with nearly identical shapes at most points. Two notable differences:

1) Wait But Why readers are less mature than New York Times readers, as evidenced by the higher 69 bar in the WBW graph.

2) Wait But Why readers thought a few steps farther about this, evidenced by the lower mean. Much of this comes from the larger percentage of WBW readers who thought through enough steps to get to 1 or 0. It’s also probably related to the fact that WBW readers think other WBW readers are smart, so those who guessed by trying to size up the crowd of guessers might have gone a bit lower in their guess here than they’d have gone when guessing on the NY Times. I’m one of those people—I guessed 17 when I saw this on the NY Times, and using the same logic, I guessed 13 here.

Finally, I’m leaving the submission form active, because why not, and for anyone who would like to look at the raw data, here’s the spreadsheet that the form results go into.

___________

You can sign up for the Dinner Table email list here to be notified about the new topic each week, and remember to submit future topic suggestions to table@waitbutwhy.com.

• Blrp

OOOOOOOOOHHHH BABY

I was concerned that 15 wasn’t original enough because it was divisible by 5, but I’m glad I didn’t think of that before voting.

The people who voted 0 or 1 are probably thinking “well I’m still smarter than everyone else”. Spoiler alert:

you’re not

• Krzysztof

But if it wasn’t for them, the result would be much higher than 15 and then someone who voted 25 could write:
“The people who voted 20 or less are probably thinking >>well I’m still smarter than everyone else<<. Spoiler alert: you're not"

• Blrp

Well I don’t think correctly predicting the answer to this thing is a good indicator of human intelligence, and it’s not necessarily stupid to think “I’m logical, and I think almost WBW dinner table readers are logical, and I think almost all WBW dinner table readers are like me in that respect” and therefore answering 0 or 1. I’ll say it’s pretty damn naive though.

But what’s stupid is to think “0 or 1 is the most logical so I’m going with that”, then finding out that you’re wrong and still sticking to your guns, thinking you were wrong because you’re logical and everyone else are stupid and it’s their fault. What happened is that you used shitty reasoning and I used better reasoning, and my answer happened to be correct.

• Panda

I do think I’m wrong because I’m logical. But I’m also one that doesn’t think others are stupid for picking other solutions. The game is psychological with no guarantee of an answer. Logical reasoning in deterministic steps can, provided that people use it. That’s all there is to it. I was much more interested in seeing people justification to their solutions rather than the answer itself.

That’s like saying “if it weren’t for the people who put 69 or 100, the average would have been much lower”. It’s true, but it doesn’t make the people who put 69/100 any less immature or stupid. Part of being clever is anticipating the amount of stupid people.

In my opinion, picking 0/1 is stupid if you’re following the game’s stated objective: choose the correct answer. Not what you want the answer to be. Not what the answer is when you’re playing with rational agents. The correct answer.

• DrSuess

I’m feeling pretty good about my original comment of…

“I guessed 22. I assumed a normal distribution of the numbers people had in mind based on lucky numbers, sports jerseys, etc. would end up being 50

2/3 of fifty is 33. Well … expecting other people to think the same as me, I went for 2/3 of that, which is 22.
So, why didn’t I keep going down to 15 (14.6)? How freaking meta are you going to go? It has to stop somewhere! Lolz.

I can’t wait to find out how amazingly wrong I am next week!”

and my reply to one critic …
” … don’t overthink this. Most people here understand that the 2/3 downward spiral eventually leads to 1.
After that, it’s just a case of people…
a) making a guess of what number to start at (50 seems as good as anything);
b) where to hop off the 2/3 train (2 multiplication cycles, ’cause i think people have short attention spans);
c) and maybe bump it up a bit because of goofballs and trolls that pick 100 to screw things up (I didn’t bother).”
It’s just a guess. It’s just a game.”

One step un-meta enough! 🙂

• Blrp

how can a distribution be a number

• Reuben Hopper

Darn, I thought you were going to mention how I commented that I went another 2/3 from 22 and that you’d talk about how frustrated you were that I was actually right. But oh well.

• DrSuess

Hey. No …. it wasn’t you I was talking about! Our methods were the same (high five).
There was a guy who commented after you on my original post that criticized my/ our/ the correct process. I just didn’t mention him by name.

I suspected that I should have gone down the next level to 15.
Well done!

• Taylor Marks

You guys disappointed me. We could have all been winners, but a lot of you are a lot dumber than I thought you were. I picked 1, because I had faith that the readers of this blog were abnormally smart. My faith was misplaced.

• Alison Sanches Krinski

Doesn’t seems like who chose 1 or 0 are smart afterall. They LOST

• Iambic Pentakill

It’s true. In life you must account for the dumbness of others lest ye be dumb as well.

• Don Reba

I would expect 2/3 to fall into the 0 bin.

• Mustafa Ergün

wow. I knew I was gonna fall somewhere near the winning number but I could never say it would be this accurate. Btw either im really luck or there are many predictable people over here.

• Jack Carroll

• Jon Snow

Lol this was fun. Do something like this again sometime will you Tim?

Btw I guessed 9

• Bryson

I would like to point out that the difference between the correct answer and the most picked answer in both the New York Times results and the Wait But Why results is 14 (15-1, 33-19). The difference is that WBW readers underestimated, NYT readers overestimated. I’m surprised that there were so many people entering 22; you can only get to the number through logic (too weird of a number to be a random guess), but it seems like such a weird place to stop. WBW also had a more even distribution, which is interesting.

You also probably should have addressed the fact that Wait But Why had a comments section for this but the New York Times didn’t. I suspect that if there was a comments section for the NYT article, the number of 33 submissions would be far lower. Then again, the more even distribution tells me that people weren’t just copying other submissions.
Edit: There is a comments section in the NYT article. Why is it hidden off in a corner?

Also, it’s weird to think that Wait But Why got almost 10% as many submissions as the New York Times did. For a discussion that was only partially mentioned on the main page of a site that’s barely 2 years old, that’s pretty damn impressive. I’m glad that Wait But Why is attracting such a strong readership, you definitely deserve it.

As for my weird guess/joke/graph thing from last week:
I wasn’t as far off as I thought I would be.

Some Observations:
– There were far more naively rational people than I thought.

– The category I labeled as rational pessimists didn’t exist, but the category I labeled as pessimists contained the correct answer. I find this incredibly amusing.

– The area I meant to take up 0-1 took up 0-5. Doing number scales without any measurements is hard.

• ScribblePouit

I was expecting an update on your graph, and you delivered.

• Garth Peterson

I, for one, was in your “rational pessimists” segment. I figured it would be between 3-10 (I chose 5). There were a lot more odd – as in weird – guesses than I thought there would be. Or the effect of a large number guess is bigger than my mind intuitively thinks as I never did the math.

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• Maverick

Predicted 17. Close 🙂 . i am surprised by the huge number who guessed 0 and 1. Didn’t expect these two categories to be that big !!

• Panda

If you follow the steps as described in the picture above, since you start guessing from highest possible number, every step down you can get a 2/3 of the previous step’s average until you reach down to 1. That is the final step. Taking 2/3 of the step with average of 1 will still give you 1. Keep in mind that the rule stated to submit the 2/3 of all average submission rounded to the nearest integer which removes the possibility of getting down to 0 if you were thinking of it in distinct steps going from high number to low.
0 is a unique case in this game. 2/3 of 0 is still 0, so it is also an answer given that everyone participating knows as much.

• Started at 0.. Got to 22.. And pulled 15 out of my ass but thought that the number really would fall somewhere between 12-17. Great article though!

• now that i stop to think better about this, 15 does look a lot more reasonable than 1. Well

• EZnded

I would like to apologize to everyone who wanted to “win” the game. I’m one of those few people who thought, “whelp, whatever. I’ll just pick a random number just to participate in this game.”

My goal wasn’t to “win”. Winning here doesn’t really amount to anything, nor does it prove that I’m smarter than anyone else, cause I’m not.

I’m just curious, that’s all.

• Jeffrey Quave

curiosity killed the cat

• Not Schrödinger’s cat!

• Brian Wilson

Or did it? =)

• Blrp

The task included taking into account all the people who picked random numbers, so I think you’re fine.

Also, it doesn’t sound like you’re actually sorry.

• RJ
• Richard Kenneth Niescior

I was one of the bastards who through in a 69 for the hell of it 🙂

• betty

Threw

• Richard Kenneth Niescior

The internet nearly collapsed due to an automated correction error, your contribution has allowed thousands of twats to understand the message I attempted to articulate for they are incapable of using their brains to interpret ‘through’ as a homophone for ‘threw’.

• Jim Schwarzbach

I picked 19 because I had seen the NYT article and that was the winning number there, so . . .

• Iambic Pentakill

I’d bet that you would find that the people that picked 0 or 1 tend toward optimism more than the average person.

Also, I wonder what xkcd readers would have chosen.

• Lukas

Now, the really interesting experiment would be Round 2, where people entering the numbers know the results of Round 1 and the logic behind it.

• Bryson

Even better: instead of disabling comments, encourage them. I’d love to see how people would go about trying to win with knowledge of what other people are thinking / want other people to think they’re thinking.

• Don Reba

0 is also the “how can we all win?” answer. If everyone picked it, everyone would have won.

• Stijn

I was thinking: if everyone follows the logic iteration to the end, we would all pick 1 and we would all win – everybody happy!

• Jeffrey Quave

but everyone gets greedy and wants more than one

Alright. Who put “yogurt”?

• Ben Yacobi

i actually picked 23 – which makes me very average. I figured quite a few people would misunderstand the instructions but I was much too pessimistic. Out of interest, I wonder what would the result have been had the submission with the 13,670 9s been included?

• Jeffrey Quave

that’s like making every answer 100+ in this sample size

• Ezo

I picked 14, so close 🙁

• Suraj Rao

Picked 15 ?
Estimated that many people would pick some higher numbers, so didn’t go into single digits. Kudos to my intuition though ☺

• picked 8. guess i was kinda low, but i knew that the answer wouldn’t be 1. didn’t think that many ppl would choose numbers this big. or that there’s a slight spike of ppl choosing numbers around 67.

• wobster109

Were all 4940 submissions from unique readers?

• Another Batshit Citizen

And can you do an English translation of this post next week for those of us that are still picking up our brains off the floor from trying to understand it? 🙂

• Conker The Squirrel

Tim, thank you for this awesome entertaining post.
I still don’t get why we had to use integers and not floating point (especially, when we multiply by 2/3) 😛
Like Taylor Marks said: You guys disappointed me….”

Anyway, i have now a feature request:
I want to be with my “naively rational people” and ignore the rest,
please install a filter, which only shows comments from users who chose the same (or almost the same) number 😛

(btw, thanks for the labels, Bryson 😉 )

• Jeffrey Russell

• Tim Urban

Using the spreadsheet data, a reader sent me an interesting analysis of how the winning answer changed with time. Thought I’d share:

• Jeff

ARE YOU KIDDING!?I PICKED 10! SO DAMN CLOSE!

• Jeff

I thought people would be more logical…..

• girly freak

It was not that I did not think logical. I just thought the others don’t think logical. 😀 So I picked far too high.

And I really really wonder, why so many people chose 0 or 1 because they did not have any chance to win this. I did not think, that would happen because nobody can really think that everybody thinks logical. If they assume 80% being logical, the other 20% destroy their chances picking 1 or 0. In my opinion those people are as logical as the people picking 100. 😛

• anonymous user

I picked 17… so… close… AUGHHHH!

• Abby Braunsdorf

Me too!

• qwerty

Aww I chose 2/3 of 50 – 33 :p

• spelling error:

“It’s falls between 3^3^3 a 3^3^3^3.”

should be: “It falls between 3^3^3 and 3^3^3^3.”

• Patrick McNeil

Yes! I chose 15!

• Bug

I chose 60 and still understand none of this. I despise numbers. 🙁

• John Lovén

This is by far the most fascinating social experiment I’ve ever come across, much because of the simplicity of the question and the complexity of the answer. What I find most interesting is that the result doesn’t reflect how smart the people in the poll are; it reflects how smart the people in the poll think other people in the poll are.

• Jay

backwards induction babyyyyy.

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