You want a number?
The people of the United States of America have the LOWEST average IQ of all OECD countries.
Our average IQ is 98 while Finland’s is 114.
This means that 50% of American citizens have IQs less than 98.
The people of the United States of America have the LOWEST average IQ of all OECD countries.
Our average IQ is 98 while Finland’s is 114.
This means that 50% of American citizens have IQs less than 98.
1
Very cool. I'd love to see more Times features like this. America desperately needs education in statistical and probabilistic thinking.
1
There is the Nash Equilibrium, of course, but there is the "I Trump You" Coefficient: 0.95 of the people who read NYT comments will figure out a way to justify why their answer is correct and "everyone else is a moron".
So, here is my summary:
1. If you guessed >67, then you cannot read or think you are funny.
2. If you guessed 50, you can read, but don't understand what you are reading.
3. If you guessed less than 50, but greater than 1, you understood the question but your reasoning powers or patience are limited.
4. If you guessed 1 or 0, then you over-estimated the intelligence of the average NYT reader is.
Dan: Did I just prove your point?
1. If you guessed >67, then you cannot read or think you are funny.
2. If you guessed 50, you can read, but don't understand what you are reading.
3. If you guessed less than 50, but greater than 1, you understood the question but your reasoning powers or patience are limited.
4. If you guessed 1 or 0, then you over-estimated the intelligence of the average NYT reader is.
Dan: Did I just prove your point?
2
I'm confused. I picked 1 (which as some other comments discuss is a valid Nash equilibrium due to the rounding instructions). The phrase "Sorry, you weren't very close" indicates this is incorrect. The article indicates it is correct. Was the goal to pick the smartest answer (1 or 0) by careful consideration or the true answer (basically a random guess)?
The true answer. It's not a random guess, it's an educated guess. It's based on what you think other people will put. If you think NYTimes readers will totally understand this question you'll put a low number; if you think they'll be confused you'll put a higher one; if you think they think it'll be higher you'll put a higher one; etc.
Unfortunately, the examples I saw didn't make it clear whether rounding or truncation was to be used in calculating 2/3 of the mean - I assumed rounding. I may not have won, but I did pick the best number - 1. This is the only answer that wins if the average is either of two numbers - 1 or 2.
How anyone could pick a number above 67 is beyond me. If everyone picked 100 the winning answer would be 67, so no number larger than 67 can possibly win.
How anyone could pick a number above 67 is beyond me. If everyone picked 100 the winning answer would be 67, so no number larger than 67 can possibly win.
Dear Times,
There are two nash equilibria: all players choosing 1 or all players choosing 0.
If you apply iterated elimination of strictly dominated strategies, such as the article suggests, you will get down to 1 (If everyone said 1, 2/3 of 1 is approximately 0.67, which is closer to 1 than to 0). If everyone were to select 1, you would do so as well (if you're trying to get the closest bet).
The other NE is 0, although you will not get it through IESDS. It is trivial to see that if all choose 0, 0 is also a nash equilibrium.
There are two nash equilibria: all players choosing 1 or all players choosing 0.
If you apply iterated elimination of strictly dominated strategies, such as the article suggests, you will get down to 1 (If everyone said 1, 2/3 of 1 is approximately 0.67, which is closer to 1 than to 0). If everyone were to select 1, you would do so as well (if you're trying to get the closest bet).
The other NE is 0, although you will not get it through IESDS. It is trivial to see that if all choose 0, 0 is also a nash equilibrium.
2
32, the only right answer. What's my prize?
It was immediately clear that the series converges on 0, an intellectual equilibrium that immediately felt comfortable to me. I flirted with 100, but it just felt too phony. I thought the other readers would see right through my feeble attempt of passing myself off as infinitely smarter than I am....
:-)))
:-)))
A lot of people seem to take this as evidence that the Nash answer is wrong or isn't "real world" enough, but ignore the fact that this example is completely contrived. Single stage games almost never occur in the real world - the vast majority of interactions are repeat.
With that in mind, I would suggest the NYT run 10 rounds of this game where people have the option to make a new guess based on the outcome of the previous round. Even with the naive strategy of guessing 2/3 of the prior round's results, you very quickly approach the Nash result.
With that in mind, I would suggest the NYT run 10 rounds of this game where people have the option to make a new guess based on the outcome of the previous round. Even with the naive strategy of guessing 2/3 of the prior round's results, you very quickly approach the Nash result.
3
I went for the Nash Equilibrium. I would have done otherwise if this game is given by, say, MSNBC, or Fox. The distribution of the picks says NYTimes readers are pretty ordinary rational thinker (>3% picked >=66!).
2
I was one of the 5.5% who picked 0, and this puzzle ruined my day. Reading the comments today has ruined this one too at several levels:
1st, it is painful to see so many people disdain the Nash answer and make snide remarks about mathematics/mathematicians (and I am not one). It explains why mathematically gifted children are isolated in school, and called nerds and geeks. NYT readers are quite atypical in educational level and general intelligence, so this disdain is doubly disturbing. It seems like the movie about Nash appeals to us because he had an illness. It's almost as if we need a truly exceptional person to be deficient in some other way. Ditto with Stephen Hawking.
2nd, in this example, the Nash answer was pretty easy to get (even without knowing about the concept of Nash equilibrium). Such widespread innumeracy does not bode well for a technological future. Maybe we shouldn't be so surprised at Donald Trump's fan club.
3rd, the comments suggest that few of us actually indulge in systematic k-step thinking. Rather there is a muddled idea ("reason") of how other people are going to reach a certain answer, and then we give 2/3 of that. A systematic 2-step thinker would almost surely see at once that one has to go all the way.
4th, and worst was realizing that to "win," I had to know that other people would think less logically than me. Maybe this is why swindlers do so well. Lucky for us that Nash, Einstein, Pauling, Salk, Feynman weren't in it for the money!
1st, it is painful to see so many people disdain the Nash answer and make snide remarks about mathematics/mathematicians (and I am not one). It explains why mathematically gifted children are isolated in school, and called nerds and geeks. NYT readers are quite atypical in educational level and general intelligence, so this disdain is doubly disturbing. It seems like the movie about Nash appeals to us because he had an illness. It's almost as if we need a truly exceptional person to be deficient in some other way. Ditto with Stephen Hawking.
2nd, in this example, the Nash answer was pretty easy to get (even without knowing about the concept of Nash equilibrium). Such widespread innumeracy does not bode well for a technological future. Maybe we shouldn't be so surprised at Donald Trump's fan club.
3rd, the comments suggest that few of us actually indulge in systematic k-step thinking. Rather there is a muddled idea ("reason") of how other people are going to reach a certain answer, and then we give 2/3 of that. A systematic 2-step thinker would almost surely see at once that one has to go all the way.
4th, and worst was realizing that to "win," I had to know that other people would think less logically than me. Maybe this is why swindlers do so well. Lucky for us that Nash, Einstein, Pauling, Salk, Feynman weren't in it for the money!
2
I picked 5 and was also in your camp. I figured 0 is the correct answer but not everyone gets it so there are a lot of answers that will be completely wrong so you have average those in too. I just (vastly) overestimated the number of people who would correctly say 0, since it seemed obvious to me.
They could make this an instructive exercise for the people that don't like the Nash result by allowing multiple votes.
They could make this an instructive exercise for the people that don't like the Nash result by allowing multiple votes.
3
I just think you have to approach this game as both a logic puzzle (yes the nash # is zero), but also a psychology game. I.e., how many K-steps will most people take, because not everyone will realize or pick the final equilibrium number of 0 (or 1) as someone else pointed out.
2
I think you completely misunderstood the purpose of the exercise. Your fourth point was the whole point of the puzzle! Yes, if you guessed 0 it means you think almost everyone who answered the puzzle has the mathematical skills to find one of the Nash equilibria. But it also means you think that everyone else thinks that as well.
It's an exercise in human behavior disguised as a math puzzle, not a math puzzle.
It's an exercise in human behavior disguised as a math puzzle, not a math puzzle.
Interesting, though you can never get to 0 , so the article is inaccurate and should be fixed: if you get to 1, then 2/3 is 0.66, which should be rounded again to 1.
When I first loaded the page, the example was 55, whose 2/3 are 36.67, rounded to 37 (which shows that the number should be rounded and not truncated); now that I reloaded the page, I see that they are showing a different example (40, resulting in 27, which proves the same point)
When I first loaded the page, the example was 55, whose 2/3 are 36.67, rounded to 37 (which shows that the number should be rounded and not truncated); now that I reloaded the page, I see that they are showing a different example (40, resulting in 27, which proves the same point)
What do you mean you can't get to 0?? The only reasonable answer to this is to go as low as you can. Since the range that was given was 0-100 (inclusive), then not only 'can' you get to 0, but everybody 'should' get there!
Let's say 3 people were playing this. Two of them followed your logic and guessed 1. Then I guessed 0. Which one of the three of us would be correct? Me!!! I was hoping at least 1/3 of Times readers would understand that, so the correct answer would be 0. Wow was I wrong!
Let's say 3 people were playing this. Two of them followed your logic and guessed 1. Then I guessed 0. Which one of the three of us would be correct? Me!!! I was hoping at least 1/3 of Times readers would understand that, so the correct answer would be 0. Wow was I wrong!
1
But it's not three people, it's 60,000 people (at the moment). If 60,000 people guess 1 then the most correct answer is 1. So which will it converge to? Well, imagine playing the game a thousand times. The first few times I suspect the average guess will get lower and lower but remain higher than 1, so each of those times 1 and not 0 will be the more correct answer (if we play again and you had to choose either 0 or 1, surely you would guess that 1 would be closer?). So it will at least appear to converge to 1. Now think -- at what point do you think that the tens of thousands of readers correctly guessing 1 are going to suddenly jump from 1 to 0?
I think the mathematical answer, as others have said, is that there are two Nash equilibria, 1 and 0, but I think human nature more suggests a convergence towards 1 and not zero.
I think the mathematical answer, as others have said, is that there are two Nash equilibria, 1 and 0, but I think human nature more suggests a convergence towards 1 and not zero.
This is what I loved about the tv show "24," they were always one K ahead of your average guess when it came to plot turns.
Why is the guess of "33" particularly interesting to game theorists and economists?
It's particularly interesting to me how many people guessed '33' because it's a guess that simultaneously shows that those people put some thought into the question, but also really didn't understand it.
If everyone else was guessing random numbers, then the average would be 50 and the correct answer would be 33. It takes a moment to figure that out. But then the people who spent that moment figuring did not bother to ask themselves - is everyone else just going to be guessing a random 0-100 number?
If everyone else was guessing random numbers, then the average would be 50 and the correct answer would be 33. It takes a moment to figure that out. But then the people who spent that moment figuring did not bother to ask themselves - is everyone else just going to be guessing a random 0-100 number?
Hahaha. I went for 33 because I thought many people would deliberately stay in k-1 for the simplicity and beauty of the answer. it's a question of aesthetics, living in an aethestic world, between new york time aethestic readers. Looking at the graph, it seems like it wasn't a bad choice....
22 is the most logical answer. Most people are not going to go beyond two step thinking, so realizing this and not continuing down the line would be three step thinking. Right?
3
I also found that the k+ thinking didn't resonate with me. I picked 22 because I figured I could count on the numbers being randomly distributed between 0 and 66 (since I thought few people would choose above 66), so the average would probably be around 33. 2/3rds of that is 22. I wasn't trying to think steps ahead of others, but trying to find a simple, reasonable way of assuming what the average would be and going off of that.
The explanation seems puzzled that people aren't "smart" enough to think steps and steps ahead of others, but thinking that way doesn't seem to be what the question is asking you to do (at least to me). K+ may make sense for thinking about investments in the stock market, but every human interaction isn't a market interaction and rationality is also about understanding ordinary human actions, which are often based on simple, accessible, and sharable assumptions.
The explanation seems puzzled that people aren't "smart" enough to think steps and steps ahead of others, but thinking that way doesn't seem to be what the question is asking you to do (at least to me). K+ may make sense for thinking about investments in the stock market, but every human interaction isn't a market interaction and rationality is also about understanding ordinary human actions, which are often based on simple, accessible, and sharable assumptions.
3
I think what you just described is a actually a good example of K+ thinking. You decided what you thought everyone else would do, and then you thought one step beyond them (answering 22 instead of 33).
So there are four huge spikes at the moment. 1, 22, 33, 50. I chose 25. Visually it seems I picked right in the middle. I'll just consider my guess the answer a genius would pick who hates math.
I guessed correctly! I started with the first number that popped into my head which was 66. I figured thinking of that number made sense bc the first bold words in the puzzle are "two-thirds" and maybe when I and others see "two-thirds" they think 66. Anyway, I went with that and multiplied by .66. I figured *everyone* would go at least that far (44). But, since these are smart NY Times readers, *most* people would go one step further and enter the answer 29. So I multiplied 29 by .66 = 19. :)
1
...but then you looked at the graph, and saw that your assumption that 'most' people would answer 29 was way off, right? The bar for 29 is a good bit less than 1%!! So, hooray for you, you did guess correctly, but your comment makes it abundantly clear that you did not do so for the correct reason!
1
So true. Just got lucky -- but I might still be encouraged to hit Wall Street.
1
If the question started with the assumption, "Assume that all NYT readers are completely logical, are aware that all NYT readers are completely logical, and cannot help but being completely logical. . . ." then the answer would be 0.
2
As others have pointed out, 1 also works as an equilibrium because of rounding (you can't guess .67).
I got the Nash answer of zero not by thinking I am smart but by thinking that other people are invariably smarter than me. And it appears I was wrong...
I picked 0, but thinking further, isn't 1 a Nash equilibrium point here? From the guidance, it appears we're supposed to round fractions up. So iterating with a 2/3 multiplier, 1 will be an equilibrium point.
4
I think you can read the question two ways.
The average of the total numbers picked OR
The average of each individual number picked by a reader (thus all numbers between 0-100). If so the answer could be:
If all numbers between 0-100 were chosen at least once the average of all numbers chosen is 1+2+3 to 100 is 5050/100 = 50(.5) x 2/4 = 33(.6) = 34
The average of the total numbers picked OR
The average of each individual number picked by a reader (thus all numbers between 0-100). If so the answer could be:
If all numbers between 0-100 were chosen at least once the average of all numbers chosen is 1+2+3 to 100 is 5050/100 = 50(.5) x 2/4 = 33(.6) = 34
1
Seems obvious to me that the equilibrium would be a low number like 1. If we could play the game in multiple trials, that would have been the outcome, since zero is not two-thirds of anything.
I'm shocked at all the high number guesses. I guess there aren't too many Times readers who have actually thought about game theory.
I'm shocked at all the high number guesses. I guess there aren't too many Times readers who have actually thought about game theory.
Wow, I'm at least as smart, or smarter than 86% of NYT's readers! And yet you hardly ever publish my comments!
2
OMG. I don't even understand the question.
1
What's overlooked in this "contest" is the fact that homo sapiens play favorites. Yup. I picked my favorite number.
Numerology is an ancient faith. Kind'a like politics and religion.
Numerology is an ancient faith. Kind'a like politics and religion.
I chose 28, "the average of everyone of everyone else's guess." I've always known I'm average and now I have proof.
1
I like the part where you say "You may play as many times as you like", after you read the answer. Ha ha
3
Th author of this commentary is oversimplifying by effective accusing individual pickers of irrationality for not choosing zero. Choosing zero maybe reaches a Nash equilibrium, but that choice never wins this game. (In previous studies I've read about, hyper-rational MIT student predominantly chose zero but then lost). The rational choice is to make a correct estimate about OTHER players' rationality. Or rather, to make a correct estimate about other player's estimate of other players' rationality. When you judge a person's answer, you are judging how correct they are at guessing the psychology of others, not their ability to calculate a Nash equilibrium.
2
Don't know if I'm smart or just lazy. I googled the number chosen most often between 0 - 100 divided by 2/3. I got 24.
Fascinating
Looking at the chart, it strikes me that the correct way to answer the actual question (which is to guess what thousands of real people will guess) would be lay out a series of bins around a some likely guesses:
-0s and 1s
-33
-50
-teens/twenties
-etc
Then guess what fraction of readers will land in each bin, multiply it out, and come up with your own best guess. Guessing those fractions is essentially deciding how smart you think the other readers are.
Some thoughts looking at the distribution for NYtimes readers:
-I'm shocked by how many guesses were over 66. Those numbers could only be guessed by people who didn't understand the problem, or were trolling, and I wouldn't have expected so many of those here
-The bar at 66 is surprisingly tall as well, since those readers had to assume that everyone else would guess 100
-Looks like 12% of readers guessed 0 or 1 (myself included). In retrospect, that seems like both a smart and a dumb answer. Yes, for a small group of math professors that would clearly be right. But no, a random group of thousands of people are not all going to end up there, so the overall average will have to be higher.
-Doing some counting, it looks like ~40% of people guessed numbers between 2 and 30. Again in retrospect, that's the only reasonable place to be, so I'm fairly impressed. Except.. it's impossible to know if people reasoned their way there, or just guessed a low-ish number. Times - you should have done a follow up survey!
Looking at the chart, it strikes me that the correct way to answer the actual question (which is to guess what thousands of real people will guess) would be lay out a series of bins around a some likely guesses:
-0s and 1s
-33
-50
-teens/twenties
-etc
Then guess what fraction of readers will land in each bin, multiply it out, and come up with your own best guess. Guessing those fractions is essentially deciding how smart you think the other readers are.
Some thoughts looking at the distribution for NYtimes readers:
-I'm shocked by how many guesses were over 66. Those numbers could only be guessed by people who didn't understand the problem, or were trolling, and I wouldn't have expected so many of those here
-The bar at 66 is surprisingly tall as well, since those readers had to assume that everyone else would guess 100
-Looks like 12% of readers guessed 0 or 1 (myself included). In retrospect, that seems like both a smart and a dumb answer. Yes, for a small group of math professors that would clearly be right. But no, a random group of thousands of people are not all going to end up there, so the overall average will have to be higher.
-Doing some counting, it looks like ~40% of people guessed numbers between 2 and 30. Again in retrospect, that's the only reasonable place to be, so I'm fairly impressed. Except.. it's impossible to know if people reasoned their way there, or just guessed a low-ish number. Times - you should have done a follow up survey!
I picked 18, with the reasoning that a lot of people would initially think "50" , 2/3 of which is about 33; but then following the instructions they would aim lower and would put 24 (arbitrarily lower I know), so I put 18 because I miscalculated what 2/3 of 24 is. So I guess I'm not so smart after all!!
You have a brilliant future in the hedge fund industry!
:-))
:-))
I enjoy all the finger wagging from the math savvy readers. Now I wonder, how would these people perform on a question regarding fashion or human interaction? Yes, they are just as important. Because, really, what's the point of the Nash Eq, if we have no economy or inadequate human interaction. Cheers to human variety.
2
You missed the whole point of, one, the gulf of difference between being math-savvy and fashion-savvy, and two, the importance of game theory in the economy and human interaction. Though cheers to human variety.
I appreciate the explanation as to why my choice--33--was incorrect, but I think that in my case the explanation is also incorrect. I'm told I am merely a k-1 (i.e., one-step ahead) thinker, which may be true in most instances, but here I'm not sure that it is. My first thoughts followed the logic that the authors later lay out, and I concluded that the answer should be zero, and it was then that I figured that others would figure the same and surmise that it doesn't matter what number they choose, so long as it is 66 or less. With some 44+ K respondents, it seemed reasonable that 2/3 of the average of all those guesses, which I expected (wrongly, it turns out) to be random and therefore evenly distributed, would be a good guess. My problem wasn't, I think, that I'm a k-1 thinker, but that K-infinity thinking got me to an unsustainable end and I looked for something simpler. Oh, well.
2
My problem was less math and more reading comprehension. I thought you were asking us and all readers to pick a number, any number, between 0 to 100. And then, we were supposed to guess the number that represented the best guess of 2/3 of the average of all numbers chosen. NOT all readers were doing the latter, and not the former. So I came up with 32-33, and not 2/3 of that = 22. A comma made all the difference and I think if the sentence had been separated into two sentences instead of one, I would have come to the correct answer.
9
Zero might have been a logical choice but it is actually almost impossible for that to actually be the answer. Since the answer is two-thirds of the average (i.e. mean not median) virtually everyone would have had to pick zero for the answer to actually be zero. For example if one person picked 50, 100 people would have to pick zero in order for the average number to round down to 0.
Betting that >99% of people would select the "right" answer to almost any question is probably not a great bet. Realizing that, people should probably have abandoned 0 even if they followed the logic of 0.
Betting that >99% of people would select the "right" answer to almost any question is probably not a great bet. Realizing that, people should probably have abandoned 0 even if they followed the logic of 0.
3
Of course, but there was no rationale for arriving at another particular number between 1 and 33, not knowing through how many iterations the average reader would go, so going to the logical conclusion was logically more satisfying than a wild guess. Of course, by doing so those of us who entered 0 rested comfortably in the knowledge of giving the mathematically correct, but realistically most likely incorrect, answer. Which in itself is an interesting revelation of how the human mind works.
I picked zero. Most of the rest of you disappoint me - you know who you are. I'm curious what results from the same survey in different newspapers. WSJ, WaPo, etc.
11
How is ZERO = two-thirds of a number?? Oh, I see, it is 2/3's of 0, and you think everyone would guess 0. That's a good strategy.
If a general average is 50, 2/3 = 33. But you might assume that no one should pick a number larger than 67 and that a normal random distribution of the numbers from 0-67 would center on 34, then 2/3 of 34 = 23. Why is this k2 level thinking? How are you "outwitting" anyone? There is a lack of information about how math savvy the population as a whole is, including myself apparently.
2
why would people pick low numbers?
2
Low self esteem, i guess. No wait, I'm always wrong about these things. Never mind what i said.
1
I factored in some human laziness. The thought process was pretty much as described - a general average would be 50, 2/3rd of that would be 33, 2/3rd of that would be 22, and so on. However, I figured most people wouldn't really be bothered to go beyond the first step, and would pick 33. The graph largely supports this. That would make the "right" answer 22, and so I picked a number close to 22 (21).
1
I love this problem because it can be used as a measure of the community's average intelligence AND the average of each members' perception of the community.
As many other commenters have pointed out, the mathematically perfect solution is if everyone inputted zero. You can read their posts for an explanation.
If this were a small group of a dozen professional mathematicians and professional game theorists with money and their reputation on the line, everyone would vote zero and everyone would be correct.
However, those same math professionals would vote a much higher number on this NYTimes article because their expectations of the average NYTimes reader is much lower.
This would be a fun question to pose to various communities. I would argue that the lower average would indicate the community is smarter, has a better reading comprehension, and is less trolling (thanks everyone who voted 99 and 100 to throw off the curve).
To all those who claim that this problem isn't about being smart, you have out done my already low perception of the of the NYTimes community to a level I did not even consider.
As many other commenters have pointed out, the mathematically perfect solution is if everyone inputted zero. You can read their posts for an explanation.
If this were a small group of a dozen professional mathematicians and professional game theorists with money and their reputation on the line, everyone would vote zero and everyone would be correct.
However, those same math professionals would vote a much higher number on this NYTimes article because their expectations of the average NYTimes reader is much lower.
This would be a fun question to pose to various communities. I would argue that the lower average would indicate the community is smarter, has a better reading comprehension, and is less trolling (thanks everyone who voted 99 and 100 to throw off the curve).
To all those who claim that this problem isn't about being smart, you have out done my already low perception of the of the NYTimes community to a level I did not even consider.
4
NY Times readers who made guesses are too smart for their own good, and they think other readers are as smart as they are. I chose 21 (93rd percentile) and at this time 2/3 of the average is 19. I did not take into account the large number of people who would choose 1 or 0, thus bringing the average way down. Of course, those people were way off the actual, because they didn't take into account the large number of people who would guess 4 or higher.
1
You're complaining about the 0s and 1s bringing the average down?! What about people that guessed 99 or 100 (which is literally impossible), or the big chunk of people who guessed 66 (which would only be right if everyone else guessed 100)?? Those guesses drove the average way up!! I mean, congratulations on coming way closer than I did, but sheesh, I was really surprised by the size of the bar at 66.
2
This number puzzle also works for picking Republican presidential candidates:
1. The number of them between 0 and 100, of course, is 17 (candidates).
2. Because all these people are below average, their total average is 0.
3. Two-thirds of 0 = 0.
That leaves absolutely no Republican candidates for me to pick.
(Hmm. Actually, this was true before the puzzle....)
1. The number of them between 0 and 100, of course, is 17 (candidates).
2. Because all these people are below average, their total average is 0.
3. Two-thirds of 0 = 0.
That leaves absolutely no Republican candidates for me to pick.
(Hmm. Actually, this was true before the puzzle....)
13
Not at all! Trump is so full of it, you have to give him a 100 just for that.
That brings their total average to a generously rounded 6.
2/3 of that is 4.
So you can in good conscience now choose between Bush, Kasich, Perry and Fiorina.
But you can eliminate Perry, since he has problems with the number 3.
See, Republican politics isn't so difficult after all....
That brings their total average to a generously rounded 6.
2/3 of that is 4.
So you can in good conscience now choose between Bush, Kasich, Perry and Fiorina.
But you can eliminate Perry, since he has problems with the number 3.
See, Republican politics isn't so difficult after all....
1
JJ, you da Man! Excellent work! However, that said, I'm so sorry, but even that [hilariously deserved] nearly immeasurable score of 3 won't change my decision to subtract ALL Republican presidential candidates from my consideration. P.S.: As for Perry, didn't you mean -3?
1
It didn't occur to me to do iterations. It seemed immediately obvious that the end point (I hadn't heard the term Nash equilibrium) was either 0 or 1, depending on how you round, so the only remaining question was 'how many people WON'T understand that and will guess higher'.
I guessed 5; apparently I underestimated the number of people who wouldn't understand the question. :-)
I guessed 5; apparently I underestimated the number of people who wouldn't understand the question. :-)
6
So apparently we didn't all go with 0 or 1. Sigh.
7
According to the last line of the instructions, you could make your guess and then submit 100 as a (non-counting for you) guess a large number of times to raise the average of all the guesses. Now if everybody did that the average would suddenly rise and the 2/3 number would start rising, potentially close to the 66 threshold.
1
I read the line "only your first guess will count" as meaning that subsequent guesses wouldn't be recorded in the results..
But if you're right, that's rather brilliant. Except, what happens if people do the opposite, guess high and then try to lower the average with a whole bunch of 0s?
But if you're right, that's rather brilliant. Except, what happens if people do the opposite, guess high and then try to lower the average with a whole bunch of 0s?
Smarter people had no interest in this silly proposition and went straight to readers comments in search of clever remarks of which there were few.
15
I got the consolation prize.
3
More proof that I'm not smarter than a 5th grader.
9
i disagree with the complaints that this is a guessing game. This puzzle is like playing the stock market where the trick is in figuring out what others will do.
I hate to break this to you, but until someone comes up with a foolproof algorithm to figure out how other will invest (or ESP), the stock market is in fact a guessing game, as is this exercise. You may make an educated guess and even be correct, but that is not the same as "figuring it out" unless you can do it infallibly.
But there is no way to figure out what others will do. If somebody answers 33 (K1 step) instead of following it down to 1 (or 0) is he stupid or is he assuming everybody else is stupid?
but do you buy stocks?
Technically, it's inaccurate to say that there is ONE Nash equilibrium. There are two equilibria: 1 and 0. 1 is an equilibrium solution because the problem has rounding. 2/3rds of 1 is 2/3, which rounds to 1. Then, if everyone's rational, it becomes a coordination problem. Each of us would have no reason to change his or her strategy if we all converged on either 1 or 0, but we all have the most reason to choose among the two the number we expect everyone else to choose.
If we treat it as a Bayesian game, though, then yeah, 0 has a higher expected payoff, since only 1/3 of players would need to pick it for it to be the solution.
If we treat it as a Bayesian game, though, then yeah, 0 has a higher expected payoff, since only 1/3 of players would need to pick it for it to be the solution.
11
But wouldn't 0 be the only Nash equilibrium? If enough people pick 0, the average will become <0.5, rounded to 0, times 2/3 = 0
2
But the definition of a Nash Equilibrium (NE), given here and elsewhere, doesn't say anything about the assessment of how likely other players, rational or not, are to play a certain strategy. It's worth here pointing out that not all NEs are created equal, and not all are worth playing.
The definition of a Bayesian Nash Equilibrium (BNE) is any strategy played such that each player, were all to choose it, would have the highest possible expected payoff. If we make certain assumptions about what can go into each player's expectations (no collusion, no signal favoring 1, full rationality), then 0 is the only BNE.
The definition of a Bayesian Nash Equilibrium (BNE) is any strategy played such that each player, were all to choose it, would have the highest possible expected payoff. If we make certain assumptions about what can go into each player's expectations (no collusion, no signal favoring 1, full rationality), then 0 is the only BNE.
1
Glad to see someone make this point -- I tried in reply to someone else, but the Times didn't publish my post. Given absolute numbers (or using the logic of your second paragraph) 0 is probably the "better" answer, but under the rules of the game where rounding is required 1 also has the same advantages -- even if everyone picks it, it's still 2/3ds of itself!
So, over time, the average 2/3rd non-zero response approaches 1. The later you respond, more likely it is to have the answer 1. and 2/3rd of 1 rounds off to 1 as well. However, if people pick zero, it's a whole another story.
4
I read the NYT for news. Stick to that and forego the games.
3
Who on earth could understand these instructions?
29
Well, JMF (nice initials by the way) - that explains why some people guessed 66 or higher!!
2
"For [me], k is 2." I'm not sure that assessment is correct. My process was to estimate how many steps ahead the respondents (including myself) were likely to think, then to calculate an answer based on that estimate.
After my embarrassing defeat re the last question in yesterday's test, I'm not playing your stupid games any more.
10
95th percentile for me, with a secret sauce formula. Yaaay!
Dammit! Dammit, Mark! I only got better than 93rd %-ile, and thought I was pretty smart. So I phoned my wife and told her the good news. She retorted, instantly, "Oh yeah? Then how come you didn't go to Special Class (a programme that ran for gifted students in the public-school system in the city I grew up in)?"
the fact that there were significant numbers of people who guessed above 66, which is *impossible*, reveals america's very sad math and logic deficits. :(
16
Or, trolling, as others have pointed out.
No, they just knew a lot more than the rest of us did.
Or just that they didn't read the instructions
The average of 19 roughly equals 0.66 raised to the 4th. So does that mean the average respondent has K=4?
1
No, for a variety of reasons. We don't know, for instance, how those who selected 19 arrived at that choice. Beyond that, though, even if all selectors had arrived at their choices by guessing the number of iterations each player had followed the mean number of iterations followed will not necessarily concur with the mean value of the number guessed. The limiting case, zero, corresponding approximately to +infinity iterations, illustrates why this is so.
http://www.nytimes.com/interactive/2015/08/13/upshot/are-you-smarter-tha...
potentiality #1
it's safe to say no one would intentionally guess over 66 or 67 unless they were intentionally trying to lose.
so the average of any kind of guess under 66 would be 33. two thirds of 33 is 22
potentiality #2
everyone would just go for 66 without thinking about anything.
two thirds of 66 is 44. or if many people think along these lines, a better option might be two thirds of 44.
potentiality #3
everyone would just turn off their brains and go for two thirds of 50.
now new york times readers probably aren't stupid enough for #3. #1 requires the most intelligence. however there might be some, so perhaps these should be accounted for when calculating the final possibility
the other thing is that the longer people spend thinking about it and trying to outsmart each other by trying to work out how deep they will go, then the lower the number will go
so based on maybe 20 minutes of thinking about it and maybe longer of unconscious processing, i would bet somewhere between 15 and 35
" your entry is within five of the mark. though not the winning entry, it's still closer than about 82 percent of other readers. "
the other thing is it could just be a proof of how well you fit the average mentality of a new york times reader - smart enough to go deep but prioritize enough to not go insanely deep, and to know that others will be the same
potentiality #1
it's safe to say no one would intentionally guess over 66 or 67 unless they were intentionally trying to lose.
so the average of any kind of guess under 66 would be 33. two thirds of 33 is 22
potentiality #2
everyone would just go for 66 without thinking about anything.
two thirds of 66 is 44. or if many people think along these lines, a better option might be two thirds of 44.
potentiality #3
everyone would just turn off their brains and go for two thirds of 50.
now new york times readers probably aren't stupid enough for #3. #1 requires the most intelligence. however there might be some, so perhaps these should be accounted for when calculating the final possibility
the other thing is that the longer people spend thinking about it and trying to outsmart each other by trying to work out how deep they will go, then the lower the number will go
so based on maybe 20 minutes of thinking about it and maybe longer of unconscious processing, i would bet somewhere between 15 and 35
" your entry is within five of the mark. though not the winning entry, it's still closer than about 82 percent of other readers. "
the other thing is it could just be a proof of how well you fit the average mentality of a new york times reader - smart enough to go deep but prioritize enough to not go insanely deep, and to know that others will be the same
1
Hey! I'm stupid enough for option 3, at least when I'm giving it only 1.3 seconds of thought, based on the fact that there was nothing riding on the test and nothing to lose. In real tests like the LSAT and GRE I'm used to getting a 99% right rating though.
4
I wonder if those who picked numbers greater than 66 were trolling or just innumerate...
21
Or maybe they thought this game was so stupid that only one other person would play. They also estimated that the other player was so anchored to how much smarter they than every other reader that they used 100 as their predicted value. Knowing this, the "troll" reader correctly submitted 66!
6
"So far, the average of all numbers has been 28, which means two-thirds of the average is 19 and your entry of 14 is within five of the mark. Though not the winning entry, it's still closer than about 80 percent of other readers."
Oh, happy guessing! What are the odds. I just gambled. My lucky number is seven, so three times seven - easy. 2/3s is 14. Not bad for ten seconds, I guess. ;-)
Oh, happy guessing! What are the odds. I just gambled. My lucky number is seven, so three times seven - easy. 2/3s is 14. Not bad for ten seconds, I guess. ;-)
1
This is how, race to the bottom begins - everyone guessing how low others would go.
5
The difficult matter here was guessing (1) what proportion of people could do the arithmetic properly and (2) how many steps out they would carry the iteration. Is there some sort of general knowledge about what proportion of folks carry thinking forward 1, 2, 3 or more steps in cases like this? Such what be fairly useful in planning, for example, dynamic pricing algorithms.
2
Who would anyone recognize the need to iterate, but stop after a small (and totally arbitrary) number of iterations?
1
You'll be amused to hear that's exactly what I did. I presumed that the general assumption would be that everyone wouldn't iterate to zero and just guessed at how many times I thought most people would attempt.
1
As a small time real estate investor living in Boston, I actually give a fair amount of thought to that kind of behavior as it applies to land development. Construction is going gangbusters in some waterfront areas that are only a few feet above current sea level, in spite of dire warnings about the Northeast great exposure to sea level rise. While there is a lot of uncertainty as to how much and how fast the water line is going to rise, mainstream scientists tell us that it is extremely likely to happen with force within the coming decades. This fact seems completely lost on the vast majority of real estate people, city planners, and the public at large, here, and in much of the rest of the word. I am waiting to see that moment when one influential investor will decide to withdraw from low lying coastal areas. The repercussions will be interesting to watch…
5
So I thought that this little test was great, I had fun thinking about it. At the end I decided to just go with the example average that was 45 but I didn't realize that it changes every time. I Think one of the reasons that I liked these game so much is because when growing up I liked to put myself in someone else's mind.
I think that if people tried to think like other people and by that understand their emotions and their problems the world would be a better place. If we applied the same exercise these test made us do to our everyday interaction with people we will be able to stop many conflicts by understanding the person instead of judging them. So in summary I think that putting ourselves in other people's shoes is a good thing we should do more often.
I think that if people tried to think like other people and by that understand their emotions and their problems the world would be a better place. If we applied the same exercise these test made us do to our everyday interaction with people we will be able to stop many conflicts by understanding the person instead of judging them. So in summary I think that putting ourselves in other people's shoes is a good thing we should do more often.
1
Of course we are all above average. Isn't everybody?
5
Laugh out loud (or cry) at the statistics trolls that guessed 0 or 100.
6
I was at first tempted to put zero but then figured there would always be somebody that put a non-zero answer. I assumed (incorrectly) that would only be a few percent, which would bring the average above non-zero. Only 5% choosing something random between 0 and 66 would give an answer of 1. Of course if other people choose 1 then the answer is also 1 so it seemed logical. So much for that line of thinking. Who the heck put 100?
4
Why would anyone assume that most people would pick a random number? It seems logical that the majority of people would put at least some effort into playing a game.
So I looked at it as 4 categories:
1) group 1: Random (small) averaged to 50
2) group 2: People putting effort in but assuming that most others picked randomly. (large) picked 33.3
3) group 3: People putting effort in and assuming that most others also put effort in. (significant) 22
4) group 4: People putting effort in and guessing that most others low balled it. (small) 14
For my final guess overall I assumed avg close to 33.3 but pulled down a bit to account for groups 3 and 4. I should have ended up with closer to 22 but my intuition told me to pull a bit lower and I lucked out and got it right.
What I did not account for is an unusually large percentage that understood this as a math problem and (mistakenly) applied a little known abstract perfect world concept that assumes the answer will end at 0. That percentage is audience based and tied to the higher than avg level of education in Times readers. My guess is that if you did this test on a random population you would end up with something higher and closer to 22. That mistaken percentage also illustrates the gap between understanding theory and its application in the real world. It was probably amplified by those that were drawn to 0 as the most unintuitive answer.
So I looked at it as 4 categories:
1) group 1: Random (small) averaged to 50
2) group 2: People putting effort in but assuming that most others picked randomly. (large) picked 33.3
3) group 3: People putting effort in and assuming that most others also put effort in. (significant) 22
4) group 4: People putting effort in and guessing that most others low balled it. (small) 14
For my final guess overall I assumed avg close to 33.3 but pulled down a bit to account for groups 3 and 4. I should have ended up with closer to 22 but my intuition told me to pull a bit lower and I lucked out and got it right.
What I did not account for is an unusually large percentage that understood this as a math problem and (mistakenly) applied a little known abstract perfect world concept that assumes the answer will end at 0. That percentage is audience based and tied to the higher than avg level of education in Times readers. My guess is that if you did this test on a random population you would end up with something higher and closer to 22. That mistaken percentage also illustrates the gap between understanding theory and its application in the real world. It was probably amplified by those that were drawn to 0 as the most unintuitive answer.
6
The only possible answer where everyone can guess what 2/3 of everyone else guesses is 0 and that's no fun. The results for the first few people would bounce around wildly as occasional truly random guesses changed the range of the numbers included thus far. For the people who used any logic at all that would push them toward lower numbers because you want to be below what most other people put in. How low you go is probably a function of how long you sit there and munch on the problem realizing that all those other people out there are already thinking their number should be lower (but not giving in to that silly 0 answer).
1
I assumed that most people are very suggestible, not especially original (on average), and would be likely to pick some varient of the examples given by the authors, so I calculated my answer based on those assumptions. My answer was 1 below the winning number, better than 95% of the guesses.
2
While I certainly understand that if one gets stuck in an infinite loop, eventually the answer is going to be zero. Hopefully, there are readers who are smart enough to not do the same thing over and over again, expecting a different result. Therefore, whatever YOU picked, and then multiplied by 2/3, is the correct answer, assuming you can do that correctly. Those that chose the mathematically correct answer are certifiably insane.
1
curiously i just checked now and two thirds below the answer i chose would actually have been the same distance away from the answer at the time
Why insane? Because we had more faith in our fellow NYT readers than was justified? That can be equally said about every Obama voter.
OK, I get the author's point. And if I were in the market for housing or buying stocks I certainly would want expert advice, attempt to understand the possible consequences and know what my finances could afrord. But on a test with no consequences and that should eat up no more than a few minutes of my time, not so much. Yup, I picked 33. Time to get dressed and go to work.
2
My answer was based exactly on your behavior! I figured most respondents wouldn't put in the time or thought to go beyond 33. So my answer was 21 (I picked a random number close to 22).
Looks like I was right on the mark.
Looks like I was right on the mark.
On the other hand, I aced the last (?) test that the NYT presented. The one concerning a 3rd grader's reading comprehension, that many NYT's readers didn't ace. Guess that just shows where my head is at and what my educational background is (not in economics or psychology).
Putting aside those who picked a number larger than 34 for whatever reason, I see that most people are 1, 2 or infinite steppers. There obviously is a large group of people who think they are smart.
I begin this comment by stating that I selected 42 as an answer.
But, it seems to me, that the headline come-on encouraging NYT readers to measure intelligence by meeting postulates given in the Nash equilibrium test asks an incorrect question, "are you smarter than other New York Times readers?, whose participants are not pre selected but a number of random readers who choose to participate.
If the outcome of the test is really to measure who is more or less smart, the actual question would be something like "Are You Smarter Than Other New York Times' Readers in Regard to X (or X-Y, or Y)?" which, of course, cannot be rationally measured by a NYT quiz.
I do not dispute the Nash test, but I do dispute the notion that anyone who guesses an average number (even using Nash's postulates) and reduce that number by a third without regard to any other information has done nothing more than guesses well.
If we are talking about negotiations, however, or stock or asset purchases, or markets' behavior, then Nash's theory has many applications. I like to think that in the negotiations I have participated in, that, while I may have out foxed others, I achieved something far more tangible and important than guessing an average: a positive result for the party on whose behalf I've negotiated.
As I think more about it, I insulted myself by taking the test based on thinking I might be so much smarter than others. That's a little narcissistic. Too much Donald Trump in that thought.
But, it seems to me, that the headline come-on encouraging NYT readers to measure intelligence by meeting postulates given in the Nash equilibrium test asks an incorrect question, "are you smarter than other New York Times readers?, whose participants are not pre selected but a number of random readers who choose to participate.
If the outcome of the test is really to measure who is more or less smart, the actual question would be something like "Are You Smarter Than Other New York Times' Readers in Regard to X (or X-Y, or Y)?" which, of course, cannot be rationally measured by a NYT quiz.
I do not dispute the Nash test, but I do dispute the notion that anyone who guesses an average number (even using Nash's postulates) and reduce that number by a third without regard to any other information has done nothing more than guesses well.
If we are talking about negotiations, however, or stock or asset purchases, or markets' behavior, then Nash's theory has many applications. I like to think that in the negotiations I have participated in, that, while I may have out foxed others, I achieved something far more tangible and important than guessing an average: a positive result for the party on whose behalf I've negotiated.
As I think more about it, I insulted myself by taking the test based on thinking I might be so much smarter than others. That's a little narcissistic. Too much Donald Trump in that thought.
1
Freddy is faster than you. NYT's response read: "Congratulations! You could not have done any better than you did. You correctly guessed two-thirds of the average of 18,491 New York times readers .."
1
So what are your stock tips?
Ah but since there are over 1,000,000 NYT readers online, you are only correctly guessing 2/3 of the average of what 1.8491% of the readers guessed, which is not statistically significant as all polls have a margin of error of at least a few percentage points.
I'm going to stick with the formula that has been my fall back position since the day I left college. You're going to have to pay me to ever do math again.
4
I really underestimated readers, too. I also thought that the sample numbers might bias other players (but not smart me, of course). I "lost" big time.
4
The sample numbers change every time you open the page.
Three men traveling together rent a shared room in a hotel for the night -
They pay $30 for the room - each man giving the desk clerk $10...
Later that evening, the clerk realizes he overcharged the men by $5 -
He gives the bellboy five one-dollar bills - and instructs the bellboy to return the overcharged funds to the three gentlemen -
The Bellboy goes upstairs knocks on their door - and the three men all come to the door - where the bellboy explains the error and hands them the five one-dollar bills for their refund...
The men are thankful for - and hand two of the dollar bills back to the bellboy as a tip for his troubles - while each
man takes one of the three remaining dollar bills for himself --
So --
Having originally paid $10 for the room - the three men have now paid $9 each after having received a one-dollar refund --
$9 x 3 = $27 - Plus the two dollars they returned to the bellboy = $29 --
!!!
Where is the missing thirtieth dollar...??
They pay $30 for the room - each man giving the desk clerk $10...
Later that evening, the clerk realizes he overcharged the men by $5 -
He gives the bellboy five one-dollar bills - and instructs the bellboy to return the overcharged funds to the three gentlemen -
The Bellboy goes upstairs knocks on their door - and the three men all come to the door - where the bellboy explains the error and hands them the five one-dollar bills for their refund...
The men are thankful for - and hand two of the dollar bills back to the bellboy as a tip for his troubles - while each
man takes one of the three remaining dollar bills for himself --
So --
Having originally paid $10 for the room - the three men have now paid $9 each after having received a one-dollar refund --
$9 x 3 = $27 - Plus the two dollars they returned to the bellboy = $29 --
!!!
Where is the missing thirtieth dollar...??
They paid out $9 each, $25 for the room and $2 to the bellboy = $27. The other $3 are in their wallets.
3
Um. There's only $30 in play. After all the hubbub, the hotel has $25, the men $3, and the bellboy $2. That's $30. Someone's math skills have gone awry. The refund ($5) was divided among 4, unequally, not 3 as your $9x3 expression indicates.
By going back in time and reasoning that "originally" each of the three men paid $10, but now paid $9, is distorting the arithmetic. Just before the tip was provided, they each had paid $8.33 for the room ($10-$1.66) to the hotel -- the hotel, then, had $25, the men $5, and the bellboy $0.
By going back in time and reasoning that "originally" each of the three men paid $10, but now paid $9, is distorting the arithmetic. Just before the tip was provided, they each had paid $8.33 for the room ($10-$1.66) to the hotel -- the hotel, then, had $25, the men $5, and the bellboy $0.
4
I took it from the bellboy, in a clever bit of pickpocketing.
Or you could just look up 2/3 challenges on Wiki and, using this information, pick a number around 19.
1
Maybe I'm missing something... but zero *is* an integer. So not clear why some have said the Nash equilibrium should be one.
Plus the series converges to zero, not one. Thus - zero.
I haven't followed economic theory in a looong time, but this is turning out to be fascinating and I'm now about to follow up on k-step thinking. (That was one of the reasons I didn't end up as an economist: it didn't seem to be very related to the real world.)
Hmmm...
Plus the series converges to zero, not one. Thus - zero.
I haven't followed economic theory in a looong time, but this is turning out to be fascinating and I'm now about to follow up on k-step thinking. (That was one of the reasons I didn't end up as an economist: it didn't seem to be very related to the real world.)
Hmmm...
2
The reason for 1 is that 2/3 of 1 rounds to 1. So, with repetition the answer never gets below 1.
No, don't agree; the answer converges to zero, not 2/3.
Believe me, I actually did this to 1,000 rounds (because I'm like that). At that point the answer is 5.4 E-175.
Believe me, I actually did this to 1,000 rounds (because I'm like that). At that point the answer is 5.4 E-175.
Generally speaking, NY Times readers make no bones about how erudite they consider themselves. They often do little to keep up appearances with humility, as I've seen in reader comments over the years.
Apparently, the newspaper couldn't resist appealing to their vanity with this thinly disguised click bait piece.
Apparently, the newspaper couldn't resist appealing to their vanity with this thinly disguised click bait piece.
12
that's a good point. i'm guessing you guessed far away from the number. also it's not a test of intelligence but how well you fit the average mindset a new york times reader.
I am completely non-mathematical, so I didn't play the game. But reading the comments, I have started to understand the concept of why the answer would approach 1 (or 0) over repeated attempts.
So while I can't count myself in with the smart guys, at least I learned something. It made me wonder about economics models, so I am hoping the Times will publish the results (as a time trend) and give us information on how this is useful.
So while I can't count myself in with the smart guys, at least I learned something. It made me wonder about economics models, so I am hoping the Times will publish the results (as a time trend) and give us information on how this is useful.
They distorted the test by adding "Stop and think for a second. What is everyone else going to do?," which is likely to add one K step to most people's answer.
1
I am 75 yo and this article is completely bewildering to me. I have read and reread the instructions and can't understand why I'm asked to pick a fraction of a number rather than just guessing the number.
And why when the article is redisplayed do the numbers in the "For example ..." change?
Would someone please explain what I am missing?
And why when the article is redisplayed do the numbers in the "For example ..." change?
Would someone please explain what I am missing?
The point is that by being asked to guess 2/3 of the average of the other guesses, you supposed to put in a number that differs (is 2/3) from the average answer.
Hi John, I'll try. A simple game might be pick a number from 0 to 100. After everyone picks a number, all the numbers that were picked are averaged and the person who picked a number that turned out to be the closest to the average number wins. But this game says to pick a number from 0 to 100. After everyone picks a number, all the numbers are averaged. But then that average number is multiplied by 2/3. The winner in this game is the person who picked a number that turned out to be closest to that average multiplied by 2/3. So the first thing you might realize is if everyone picked the biggest number, 100, the average would be 100. So 2/3 of the average, 100, is 66. So the winning guess has to be 66 or less. Not sure if this helps you. Hope it does!
Ok let me try:
It is partly a math problem, but it's mostly a behavioral problem. So your confusion is actually a good thing, because you recognize that it's not just a straightforward math problem.
If you were asked to guess the average number, it would probably be something close to 50, right? Because that's the middle of the spectrum from 0 and 100, right?
Instead, you're being asked something more sneaky: 2/3 of the average. 2/3 of 50 is 33. So instead of picking 50, everyone should pick 33, right? Wrong! Because if *everyone* picks 33, then the correct answer would be 2/3 of 33, which is 22! So everyone should be picking 22, right? Wrong! Because if everyone picks 22, then 2/3 of 22 is 14! If you continue along this line of thought you'll get that everyone should be picking 1 (or 0). So why isn't 1 the right answer? Well, because a lot of people decided that other people won't pick 1 -- they decided the average reader won't go so many step ahead in their logic. So you see, the question isn't just about picking a number -- it's about guessing how far ahead other people reading this article will think, and perhaps how confused they'll become in the process.
The reason the numbers in the "For example" change is probably because they don't want to bias the results by leading people to think in a certain direction. If they always gave the example of 33, there might be a spike at 22, which is something they're trying to avoid.
It is partly a math problem, but it's mostly a behavioral problem. So your confusion is actually a good thing, because you recognize that it's not just a straightforward math problem.
If you were asked to guess the average number, it would probably be something close to 50, right? Because that's the middle of the spectrum from 0 and 100, right?
Instead, you're being asked something more sneaky: 2/3 of the average. 2/3 of 50 is 33. So instead of picking 50, everyone should pick 33, right? Wrong! Because if *everyone* picks 33, then the correct answer would be 2/3 of 33, which is 22! So everyone should be picking 22, right? Wrong! Because if everyone picks 22, then 2/3 of 22 is 14! If you continue along this line of thought you'll get that everyone should be picking 1 (or 0). So why isn't 1 the right answer? Well, because a lot of people decided that other people won't pick 1 -- they decided the average reader won't go so many step ahead in their logic. So you see, the question isn't just about picking a number -- it's about guessing how far ahead other people reading this article will think, and perhaps how confused they'll become in the process.
The reason the numbers in the "For example" change is probably because they don't want to bias the results by leading people to think in a certain direction. If they always gave the example of 33, there might be a spike at 22, which is something they're trying to avoid.
I would like to see the graph plotted over time - that is the answer verses number of respondents.
If New York Times Online readers were hyper-rational, they would get back to work.
29
I felt fine with 22 as a guess. It's "only 2" steps ahead according to the study, but for me it was a question of how smart you think the rest of the readers are at solving the riddle as well as doing the math. The writer didn't bold the "chosen in the contest" part so I assumed that would throw many people off. If he had bolded the "chosen in the contest" I would have guessed lower. Like 12 or so. I was also trying to factor in people missing what the question was asking which I thought would be high. I figured there'd be a lot of 50s and 33s and much fewer 22s or lower.
This proves the value of not thinking. I did not and ended up being beating 93% of the readers. Use the force Luke!
2
I over-estimated the intelligence of NYT readers and went with a K-5 guess. In retrospect, based on some of the comments to other stories, I should have known not many would go that deep.
1
again it's not a test of intelligence, it's a test of intelligence combined with how much people are likely to care. so it's a test of how much you fit the average mentality of a nytimes reader !
Really NYT? This is the best "The Upshot" can come up with, especially on the heels of the Tuesday's "New York State Test Questions" for 3rh graders about keeping secrets? This little blurb is on a par with "My phone is smarter than your honor student" bumper sticker. If you want to really test the smarts of New York Times readers, give us something with substance and real intelligence. Other than that, have a really nice day!
1
This explains everything! Finally!
since this number changes depending on sample size, shouldn't that be included?
1
I don't think that the answer here has much to do with Nash equilibrium.
The way I remember game theory is that the number of states is finite, and that a finite number of participants have a clear payout function. In this case, not only the sample is floating, but the winning conditions can change over time as more people play (and, presumably, read the comments section).
However, the stock analogy is spot on, particularly in terms of autocorrelation - i.e. if NYT puts an article on this test with the correct answer, and if real money is on the line, the crowd will rush towards the answer, so whoever is betting on 2/3rds of it with the right timing will win.
The way I remember game theory is that the number of states is finite, and that a finite number of participants have a clear payout function. In this case, not only the sample is floating, but the winning conditions can change over time as more people play (and, presumably, read the comments section).
However, the stock analogy is spot on, particularly in terms of autocorrelation - i.e. if NYT puts an article on this test with the correct answer, and if real money is on the line, the crowd will rush towards the answer, so whoever is betting on 2/3rds of it with the right timing will win.
Well, I began by guessing 50.5 as the average (because I forgot 0). And I rounded 50.5 to 51.
The, instead of dividing by 3 and multiplying by 2 to get 2/3.
I had a drink, got sloppy and just divided by 3.
And that was 17
So I did pretty good.
Now, where'd I put my drink?
The, instead of dividing by 3 and multiplying by 2 to get 2/3.
I had a drink, got sloppy and just divided by 3.
And that was 17
So I did pretty good.
Now, where'd I put my drink?
3
Weird little test. On a whim I picked 33, as 2/3 of 50, the natural average of random numbers between 1 and 100. Thanks to that graph I see when I picked it, this was the most popular number choice by far, probably people using the same reasoning as me.
But also from the graph I see that a lot of jokers picked 0 and 1, which could not possibly have been the right answer. 0 is 2/3 of 0, so everyone would have had to choose 0 for that to be right. And 1 is 2/3 of 1.5, so for that to be the average nearly everyone would have to pick numbers between 0 and 4 or so, and mostly 2 or 3.
Also some silly folks picked high numbers, and that wouldn't have been right no matter what; if the average was 100 (everyone choosing 100 without fail) then the right answer would have been about 67.
So the wiser choices were to go below 50 by a good bit, I didn't go far enough, but this relies not only on figuring out what other people would be likely to choose, but whether they'd be right in their assumptions. Knowing what I know now, if I could do it all over again, I wouldn't have taken the test, thus sparing myself the agony of the feet or something.
But also from the graph I see that a lot of jokers picked 0 and 1, which could not possibly have been the right answer. 0 is 2/3 of 0, so everyone would have had to choose 0 for that to be right. And 1 is 2/3 of 1.5, so for that to be the average nearly everyone would have to pick numbers between 0 and 4 or so, and mostly 2 or 3.
Also some silly folks picked high numbers, and that wouldn't have been right no matter what; if the average was 100 (everyone choosing 100 without fail) then the right answer would have been about 67.
So the wiser choices were to go below 50 by a good bit, I didn't go far enough, but this relies not only on figuring out what other people would be likely to choose, but whether they'd be right in their assumptions. Knowing what I know now, if I could do it all over again, I wouldn't have taken the test, thus sparing myself the agony of the feet or something.
2
0 is not a jokers choice, it is the mathematically correct choice. Many people picked the mathematically incorrect choices of not 0, therefore, the actual answer was closer to 19. If everyone had logically followed the problem to its conclusion we all would have chosen 0 and that would be the mathematically, and the actually, correct answer.
24
The 0 answer is only mathematically correct in a world in which every participant can make the assumption that every other participant completely understands the logical conclusion of this type of problem and will likewise make that assumption.
We do not live in that world.
We do not live in that world.
36
Thanks guys, and Meredith you're right except that SteveS is more right; my own answer was based on the assumption that a lot of people (or most) simply wouldn't understand the test, and it's a reasonable assumption considering things like Kansas and the popularity of Trump.
6
This abstract test only exposes our degree of trust in the intelligence of others. The more cynical we are, the higher the average. Truly random guesses are just noise. And there is a difference between the number of steps one considers and the number of steps one actually applies to his or her final guess, so the test doesn't measure one's own intelligence but one's bias about other's intelligence. If we each thought that everyone else was rational, the average would be 0. It looks Times readers don't think very highly of each other. But random guesses could be obscuring the truth. We'll never know for sure.
20
I picked one. I went with the idea that enough people would want to make it simple for themselves to calculate two-thirds of a number and so would initially think to figure two-thirds of the number three, which would give 2. So then two-thirds of that is rounded to the number one. I did think about the possibility of having to continuously break down your guesses by more and more two-thirds in order to stay in step with the thinking ahead, but that hurts too much, so one it is.
I failed to realize the average would be 50, but that's only assuming everyone gave random guesses without the two thirds hitch. The fact that people did indeed us 50 as a base answer is more forced through a sort of logical begging of the question rather than an average of random guesses of a number between 0 and 100. Interesting
I failed to realize the average would be 50, but that's only assuming everyone gave random guesses without the two thirds hitch. The fact that people did indeed us 50 as a base answer is more forced through a sort of logical begging of the question rather than an average of random guesses of a number between 0 and 100. Interesting
2
I knew I wasn't going to test smarter before I tested.
Then I picked 50.
So my K =2
Booyah all you K=1 losers!
Then I picked 50.
So my K =2
Booyah all you K=1 losers!
This is not testing "smartness."
32
or never bet : )
I think it was Feynman who said that in a bet one is either a cheater or a fool.
I think it was Feynman who said that in a bet one is either a cheater or a fool.
Having picked 1 as our first guess, we hope NYTimes readers keep on picking and picking in k-step fashion :)
5
Well, I looked at the examples and guessed that people would go for the examples. So I picked the middle example. 20. Which was apparently close!
Rather than going through Vizzini's convoluted reasoning, though, I tend to go on intuition and instinct. Obviously, I'm *never* going to play the stock market.
Rather than going through Vizzini's convoluted reasoning, though, I tend to go on intuition and instinct. Obviously, I'm *never* going to play the stock market.
5
The examples change each time you look at the article.
Considering the restriction to integers, I get a Nash equilibrium of 1. I see a peak in the answers at 1.
I severely underestimated the other players.
I severely underestimated the other players.
13
I think you are right. It depends on the algorithm: whether you iteratively pick a new answer at each step k (answer is 1), or whether you take k -> infinity directly (in which case the answer is still 0). The fact that the Nash equilibrium is dependent on this means the answer, over the integers, must be ill-posed.
2
me too -- I thought 1 was the nash equilibrium. If people's further guesses were counted, the average would converge on 1, not zero.
4
That is why I picked 1 also. I think we "overestimated" the other players.
2
This average is calculated based on reader responses, not on plausible guesses.
So the average includes people who have made guesses that range from impossible to extremely unlikely-- and that affects the range and average ( how much is beyond my fathoming.)
For instance nearly 5% answered zero as the 2/3 of the average guess-- that is not a possible answer, but people answered it nevertheless. Beyond 66 or 67 are implausible answers because that implies that the guesser believed that everyone would answer 100 and then took 2/3 of that -- unlikely, but answers greater than 67 are not possible, for it implies taking 2/3 of a number larger than 100. To give an answer of 100, the guesser is stating that the group of Times readers all started with the number 150 .
So this average is based on a group of people who have varying degrees of ability with math and logic. I'm not the greatest, but at least I can exclude the impossible answers.
So the average includes people who have made guesses that range from impossible to extremely unlikely-- and that affects the range and average ( how much is beyond my fathoming.)
For instance nearly 5% answered zero as the 2/3 of the average guess-- that is not a possible answer, but people answered it nevertheless. Beyond 66 or 67 are implausible answers because that implies that the guesser believed that everyone would answer 100 and then took 2/3 of that -- unlikely, but answers greater than 67 are not possible, for it implies taking 2/3 of a number larger than 100. To give an answer of 100, the guesser is stating that the group of Times readers all started with the number 150 .
So this average is based on a group of people who have varying degrees of ability with math and logic. I'm not the greatest, but at least I can exclude the impossible answers.
4
Sorry, Mom, but your logic is faulty. Although 0 is not a plausible answer (because we have to assume some irrationality in people's choices) it is not an impossible answer--if everyone chose 0 (which would be smart) then the answer would be 0. You are right, though, that answers greater than 67 are impossible (and thus very bad choices).
2
0 is 2/3 of 0. But otherwise, agreed.
1
Zero is possible, and it's what I guessed. It is the Nash equilibrium. I didn't think that people would be smart enough to start there, but I thought over time the answers would get better and move towards zero - but I didn't think through that there isn't really a mechanism for that - not sure if subsequent answers are counted in the average - probably that's what they meant by they don't "count" but I didn't get that.
1
This is actually a rather simple exercise. I missed by one number (96%), but aside from that, it's easy to understand the thinking patterns of most people when it comes to numbers. 95% of the population is going to choose a number between 0 and 50. Don't ask me the phycology behind that, but just from general experience this is what people will choose. Numbers they are more familiar with. Most will make a gut guess. As for those that chose above 50 through 100, I noticed a spoke in the around 70 range, which if I made an assumption (could be wrong) these are guesses from people of that age group who choose more on the merits of their birthdate (age) than the logic in understanding the exercise. When looking at it like this, if I guess that most people will chose between 0-50, I'm basically looking at two quadrants of numbers 16 an 32, with the two-thirds average being somewhere in the range of 25. The winning number will likely fluctuate between 15-20 throughout the time this test remains open. I'll admit, once I got to that point, I made a guess. No math was done. For those that get the exact number, based on their $400k ivy league math degree, well, I got pretty close with my $3500 community college education..
9
Sigh..psychology not phycology. Thanks spell check! That $3500 education didn't cover enough spelling and grammar.
1
Agreed, though, while you may have meant it ironically, the spike at 66/67 is clearly real. People who have no idea how to go about solving the problem picked 2/3 of 100, because there are 100 possible choices. I know that doesn't make "logical sense," but having tutored math, let me say, it makes intuitive sense to plenty of people.
1
got an ivy league degree albeit in chemistry, pretty certain this is a psychology test more than a math test. bravo for your good skills.
This isn't a test of intelligence. It's more about having notion of what others guess. Tangentially, it's about perhaps having a sensitivity to the patterns of behavior of a population that can be associated with intelligence. Still, this is not what it was being "sold" as on the website front page. I don't find being able to guess the number other's guess so I'm not playing. That's more about picking out data to be conformist, average. I'm not a lemming who jumps over the cliff along with everyone else. That is the sort of intelligence that interests me. So I'm not playing.
6
This is so true. It is utterly ridiculous that the test is described as measuring intelligence.
.66*66/2.78 almost got it. I forgot about the knuckleheads who thought that 100 could be 2/3 of any guess.
5
Got it Woo-hoo! Now I just have to learn where to put commas and some basic grammer and I'll convince myself I'm smart.
4
...and spelling.
3
Add spelling to your little list there.
1
The number of people who participate should figure into the guess, as well as knowing the average k value for the population at large from the two previous runs. But FT reader average k value may differ from NY Times readers. When I entered my guess, I am inherently guessing the average of other people's k values (I guessed it was 1 without knowing that's what I was doing), a mostly losing proposition.
1
Interesting than the correct answer is one of the least common guesses. This seems at odds with the 'Delphi' effect where large groups tend to pick the right answer, even for topics they are ignorant about.
2
I think the Delphi effect relies on a bit of communication between the parties, so it's not quite the same this situation.
I'm watching to see if my original number will eventually be a winner. Unfortunately, I think I need to get the word out on another platform along with a hint at the method so that we can get "better" (i.e. more like me) guessers on here.
I'm watching to see if my original number will eventually be a winner. Unfortunately, I think I need to get the word out on another platform along with a hint at the method so that we can get "better" (i.e. more like me) guessers on here.
1
My understanding of the Delphi effect is that the average converges to the correct answer. It's likely that such an effect would be masked by the fact that the correct answer here is itself a function of people's average answer. If the average answer were 19, that would no longer be correct.
1
In this case the Delphi effect can't work, because if the large group picks x as the (average) best answer, then the best answer is 2/3 x.
My guess, and I suspect the guesses of many, was a function of my estimate of the intelligence of fellow participants. I didn't think many would get passed the second level. I should have assumed that some would and guessed a little lower.
Turns out I underestimated the intelligence of my fellow participants, not by very much, but underestimate I did. For this I am happy. Improves my opinion of the NYT readers who participated.
I would love to see the results of this based on a population of Republicans, Democrats, Independents, Fox News watchers, former :( Daily Show and Colbert Report watchers, and Trump supporters.
Turns out I underestimated the intelligence of my fellow participants, not by very much, but underestimate I did. For this I am happy. Improves my opinion of the NYT readers who participated.
I would love to see the results of this based on a population of Republicans, Democrats, Independents, Fox News watchers, former :( Daily Show and Colbert Report watchers, and Trump supporters.
7
It's funny, Steve, I overestimated the intelligence of my fellow participants and came in below the correct answer. I knew everyone wouldn't figure out the Nash equilibrium, but I figured quite a few would, so I guessed in the high single digits. As a result, I was somewhat disappointed by fellow NYT readers.
I totally agree that it would be interesting to see results of the test across media outlets. As it stands currently, it looks like FT readers are a bit smarter (at least with regard to numbers) that those at the NYT.
I totally agree that it would be interesting to see results of the test across media outlets. As it stands currently, it looks like FT readers are a bit smarter (at least with regard to numbers) that those at the NYT.
But a random guess that is correct regardless of its randomness can hardly be said to be a guess by a person “fatigued, clueless, overwhelmed, uncooperative, or simply more willing to make a random guess in the first period of a game and learn from subsequent experience than to think hard before learning.” Why? Because a "guess" may not be a guess, but may be thinking so intuitive to the individual that there is no "thought" per se other than "but of course this is the answer."
Camerer, Ho, and Chong could not possibly control for that, much less include it in their "calculations".
Camerer, Ho, and Chong could not possibly control for that, much less include it in their "calculations".
1
It's too early to think, I haven't had my third cup of coffee yet!
4
I don't have to play. I already know that I'm smarter.
21
I find it interesting that the winning answer was higher in the NY Times then the Financial Times. While people might jump to say this proves the Financial Times audience "better"; I suspect that it's an example on how context affects our expectations of rationality.
9
No.
3
You fell victim to one of the classic blunders - The most famous of which is "never get involved in a land war in Asia" - but only slightly less well-known is this: "Never go in against a Sicilian when death is on the line"!
35
Inconceivable!
Yeah, but the character who said that dropped dead moments later.
Two quick comments:
1) I'm not sure the game is well posed unless payout is specified. My best guess is that the game is intended to be binary payout: success for picking the correct number, and nothing for picking the wrong number. But it would be equally reasonable to specify a payout that increases decreasing distance to the correct number, in which case, the game actually changes (especially if one includes expected behavior).
2) I'm not sure this answer is right over the integers. Over the rationals, it clearly is, but (2/3)^10 * 50 = 0.86, for example. Given that we are playing over the integers, there has to be some rounding or truncation at the final step, and arguably 1 is closer to 2/3 than 0. Of course, in the limit of k -> infinity steps, people shouldn't truncate until the end, in which case 0 may still be correct, but again, I'm not sure the question is well-posed.
1) I'm not sure the game is well posed unless payout is specified. My best guess is that the game is intended to be binary payout: success for picking the correct number, and nothing for picking the wrong number. But it would be equally reasonable to specify a payout that increases decreasing distance to the correct number, in which case, the game actually changes (especially if one includes expected behavior).
2) I'm not sure this answer is right over the integers. Over the rationals, it clearly is, but (2/3)^10 * 50 = 0.86, for example. Given that we are playing over the integers, there has to be some rounding or truncation at the final step, and arguably 1 is closer to 2/3 than 0. Of course, in the limit of k -> infinity steps, people shouldn't truncate until the end, in which case 0 may still be correct, but again, I'm not sure the question is well-posed.
11
If I pick the right lottery number, does that make me smarter than other lottery entrants?
3
This is a guessing game. It has nothing to do with being smart, or smarter than anyone else. I expect a little more from the Times than this type of click bait.
50
This game provides a simplified example of theories that have great predictive value in price equilibrium. If it were a guessing game then the stock market as we know it wouldn't exist.
10
You missed the point, and it is a very big point.
For some participants it is a very simple guessing game.
For other participants it is a guess about the intelligence in terms of K level of their fellow participants.
My guess was based on the average Times reader having a K of 1.
I should have guessed a little higher average K level.
But many participants, perhaps knowing the type of problem, incorrectly guessed a K level that was much too high for their fellow participants. That is also an error even though they understood the theory.
Part of the point is developing a reasonably good understanding of the intelligence of fellow participants, in this, politics, the economy (stock and housing markets) , and life.
It has everything to do with being smart.
For some participants it is a very simple guessing game.
For other participants it is a guess about the intelligence in terms of K level of their fellow participants.
My guess was based on the average Times reader having a K of 1.
I should have guessed a little higher average K level.
But many participants, perhaps knowing the type of problem, incorrectly guessed a K level that was much too high for their fellow participants. That is also an error even though they understood the theory.
Part of the point is developing a reasonably good understanding of the intelligence of fellow participants, in this, politics, the economy (stock and housing markets) , and life.
It has everything to do with being smart.
22
It's not just a guessing game. The trick to figure out how MOST people will think, then think one step ahead of that. Right of the bat, everyone should realize that the highest possible average is 100 - i.e. if everyone chose 100 as their guess. Two-thirds of a hundred (rounded) is 67, so 67 is the highest possible "right" answer. However, we know that it is very unlikely that everyone will pick 100.
If we ASSUME that most people will realize that picking any number greater than 67 doesn't make any sense, it follows that the true average will be much less than 67. Here is where a lot of people's reasoning will diverge. I figured everyone's guess will be scattershot between 0-67. If the distribution of those guesses is random, the average of all those numbers will be 34. However, I figured the distribution will be skewed towards 0 because a lot of people will try to outsmart themselves. From that, I guessed that the true average is 30, two-thirds of which is 20. Turns out I was only off by one.
If we ASSUME that most people will realize that picking any number greater than 67 doesn't make any sense, it follows that the true average will be much less than 67. Here is where a lot of people's reasoning will diverge. I figured everyone's guess will be scattershot between 0-67. If the distribution of those guesses is random, the average of all those numbers will be 34. However, I figured the distribution will be skewed towards 0 because a lot of people will try to outsmart themselves. From that, I guessed that the true average is 30, two-thirds of which is 20. Turns out I was only off by one.
1
Since my guess is lower than 2/3's of the average, then 2/3's of the average of others' guesses should logically continue to decrease. Wouldn't my guess eventually be precise? Does that mean I'll be smarter than most later in the day? And if that is so, once the correct guess goes even lower than my guess, will I become "stupider"?
4
If I believed that others are hyper-rational, I would end up with a 0. But it is arguable that this is correct. If I account for the fact that in a real world, there are some who won't be hyper-rational (but neither are not rational), and others who make mistakes, then it is actually not rational to choose 0. Rationality should take into account the fact that there are some would not choose the result predicted by a complete application and reach nash equilibrum. Ironically, it is actually more rational to include those who select the less than rational figure somewhere between 33 and below, because it is more rational, as the skewed chart shows. When information includes the fact that there are others who won't completely (but possibly) apply rationality, that is more precise.
23
Larry Summers applied this thinking to market equilibrium with the catchy "There are idiots, look around!" argument. The paper he opened with this line followed up the quote with much more complex math of course.
And apparently, the even less rational, weaker tail all the way to 66 ... at least. A great model would also incorporate the "I just want the answer, so I'll select 100" folks. This is basically equivalent to the stock pickers who choose stocks because the ticker is their initials.
1
But before I made my guess, I did realize that the correct answer was 50*(2/3)^n where n = a very very high number.
Once I wrote that expression and saw that it goes to 0 as n goes to infinity (as opposed to converging to some other positive number), I made a guess about how far all the other guessers would go