While I correctly analysed the situation fully before calling it, I expected most of NYT's readers to use zero-step thinking. It turns out that they do one better, so I guess I'm the typical NYT reader, in a sense.
General comment: As different things get "strongly" coupled to each other, one needs to take more and more steps into account to get a reasonable answer.
The question I have is this: Is there any way to figure out how many steps the other people in the situation will be using?
Most people are 1-2 step thinkers. [Citation Needed] But I'll explain why I think this:
Many people will assume nobody will think about the problem (1 step thinkers). A few people will assume someone will think about the problem who thought about the people who won't think about the problem (2 step thinkers).
I think most 3+ step thinkers will take the problem to it's most logical inclusion (making them n-step thinkers). Why stop arbitrarily at 3 or 4? They've recognized that people who exist who will think other people exist who thought about the problem. At that point they should be able to recognize that this pattern will continue to n. While I am sure there are 3->(n-1) thinkers, they'll be more rare than n-step thinkers, I'd reason.
There are a number of 0 step thinkers who read the question and answered 33 though.
Now I have a question: Is the person who answers 0 smarter than the people who happened to guess the right answer at the proper point of time, even if at the current time (and possibly indefinitely) they got the problem wrong? aka "I only got the answer wrong because other people didn't think hard enough."
I was a 0-step thinker assuming that the majority of people doing this would pick random numbers bringing the average to 50. I expected that on a popular website that will likely get shared on Facebook, random would be the most likely outcome.
I think I either underestimated how many people would share this on Facebook, or perhaps, the audience that will go to a New York Times page is different than the audience that will go to "Only geniuses will be able to get this..." type quizzes. Of course, it could be that HN skewed the results in favor of a true mathematical + predictive model.
This is similar to what I have found in poker. The tougher the game, the more steps that the opponents take.
What cards do I have (1S), what does my opponent have (2S), what does my opponent think that I have (3S), and what do I think that my opponent thinks that I have (4S). I have found that beyond 4S, the analysis really loses its value even in the toughest games.
Paradoxically, I have found these tools are a detriment to easier games because the game is dominated by 1S and 2S thinkers and a 4S analysis often results in the wrong conclusion.
I figured most people will think people are picking 42 (meaning of life and all) so they would pick 28. So to get ahead of them I picked 19.
http://i.imgur.com/Ke1m0jp.png
Well, you were right, but not for the right reason. The distribution has strong peaks at 33 and 22, which are one and two iterations of 2/3 multipliers away from 50. There's no peak whatsoever at 42 or 28. You just happened to catch the right spot between the biggest peaks, not your own peak.
(And the peak at 66 is rather disconcerting; there's no rational reason for that choice other than innumeracy or misreading the question.)
> Many people believe that it’s better to sell a house in the spring, for example, because more buyers are looking then. But what really matters is the ratio between buyers and sellers. You’d rather sell your house when your town has 20 buyers and only 2 sellers than when it has 100 buyers and 500 sellers.
That's a great observation, and I wonder if anyone has done a serious study on the optimal season to sell a house.
I'm sure that plenty of people have examined statistical data to make conclusions of this sort. The interesting question would be whether anyone seriously studied the effect that the buyer:seller ratio has on prices.
I also used the same round-up logic w/ Nash equilibrium and picked one. Clearly over-estimating how much thought the readers have put into answering the puzzle.
The NYT glibly informed me my choice was 'not even close'.
I was too optimistic. 16 when the answer was 19 (43,884 answers so far). This is more a test how much faith you have in the general intelligence in our society. People who played it to the end game nash equilibrium must think that they are surrounded by geniuses.
Wonder how much anchoring effect there is (sample numbers are given in the original question). I note the sample numbers are generated by javascript -- are they testing the impact of those numbers?
Also, I don't see why N-step arguments are invalid; everyone who went for the Nash Equilibrium is wrong (the average is not 3/2 * 0); you just need to predict how many step-thinkers people are :P
(I guessed 6, which is approx 2/3 of 2/3 of the winning answer... so presumably I'm two steps too deep :P)
21 comments
[ 3.0 ms ] story [ 64.1 ms ] threadSo the answer to this headline, finally, is "yes." Eat it, Betteridge.
General comment: As different things get "strongly" coupled to each other, one needs to take more and more steps into account to get a reasonable answer.
The question I have is this: Is there any way to figure out how many steps the other people in the situation will be using?
Many people will assume nobody will think about the problem (1 step thinkers). A few people will assume someone will think about the problem who thought about the people who won't think about the problem (2 step thinkers).
I think most 3+ step thinkers will take the problem to it's most logical inclusion (making them n-step thinkers). Why stop arbitrarily at 3 or 4? They've recognized that people who exist who will think other people exist who thought about the problem. At that point they should be able to recognize that this pattern will continue to n. While I am sure there are 3->(n-1) thinkers, they'll be more rare than n-step thinkers, I'd reason.
There are a number of 0 step thinkers who read the question and answered 33 though.
Now I have a question: Is the person who answers 0 smarter than the people who happened to guess the right answer at the proper point of time, even if at the current time (and possibly indefinitely) they got the problem wrong? aka "I only got the answer wrong because other people didn't think hard enough."
I think I either underestimated how many people would share this on Facebook, or perhaps, the audience that will go to a New York Times page is different than the audience that will go to "Only geniuses will be able to get this..." type quizzes. Of course, it could be that HN skewed the results in favor of a true mathematical + predictive model.
What cards do I have (1S), what does my opponent have (2S), what does my opponent think that I have (3S), and what do I think that my opponent thinks that I have (4S). I have found that beyond 4S, the analysis really loses its value even in the toughest games.
Paradoxically, I have found these tools are a detriment to easier games because the game is dominated by 1S and 2S thinkers and a 4S analysis often results in the wrong conclusion.
(And the peak at 66 is rather disconcerting; there's no rational reason for that choice other than innumeracy or misreading the question.)
That's a great observation, and I wonder if anyone has done a serious study on the optimal season to sell a house.
The NYT glibly informed me my choice was 'not even close'.
Also, I don't see why N-step arguments are invalid; everyone who went for the Nash Equilibrium is wrong (the average is not 3/2 * 0); you just need to predict how many step-thinkers people are :P
(I guessed 6, which is approx 2/3 of 2/3 of the winning answer... so presumably I'm two steps too deep :P)