Guess 2/3 average,% username%
In 2005, the Danish newspaper Politiken offered its readers to play the following game: anyone could send a real number from 0 to 100 to the editors. He whose number would be closest to 2/3 of the average of the numbers sent won 5000 DKK ( at that time about $ 800).
This game is known in the theory of games called "guess 2/3 of the average." It demonstrates the difference between absolutely rational behavior and real actions of players.
Let us imagine that all participants in the game act completely rationally and, last but not least, know that the others also act rationally and do not agree with each other. What is the best number in this situation?
Obviously, it makes no sense to call numbers greater than 66. (6), since arithmetic average can not be more than 100. But, if all players argue in this way, then all numbers will not be greater than 66. (6), then the arithmetic average will not exceed this number, and therefore call more than 2/3 * 66. (6) = 44. (4) again there is no point. Repeating this argument infinitely many times, we come to the conclusion that the only correct move will be the number 0. Thus, if all the players reason rationally, then they all have to choose the number 0.
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However, in real life the situation is different. Even if a player is rational, he knows that many of his opponents are not rational, which means he will have to take into account that their numbers will be greater than 0. We can assume that the majority will send more or less random numbers, then the average will be 50, two thirds from 50 approximately 33. If we go further and assume that quite a few people would guess up to the number 33, then we can choose two thirds of 33, i.e. 22. Further iterations will give ~ 15, ~ 10, etc., but it seems unlikely that a sufficiently large number of players will figure that far.
Let's return to the beginning of the article. What is the number won in Denmark? Below you can see a histogram of the game in which 19,196 people took part.
The first thing that catches the eye is the expected peaks at points 22 and 33. The winning number turned out to be slightly less than 22, most likely as a result of the fact that most of the participants understood the meaninglessness of the choice of numbers greater than 66. (6). It is curious that there were those who sent 67 and more, including 100. Interestingly, they did it without seeking to win or simply did not understand the futility of such a move? It is also interesting, were those who sent 0 guided by absolutely rational reasoning, or simply chose a round number?
Another interesting point: if the problem condition limits the choice to only integers, then there are two rational winning strategies: 0 and 1. The fact is that because of the discreteness of the integers, the multiplication by 2/3 will not be repeated infinitely many times. . When we get to 1, the next iteration will yield 2/3, but, rounding to integers, we again get 1.
I propose to play the game on Habré. Send me a habrakpote real numbers from 0 to 100. Attention: do not write numbers in the comments, because An important part of the game is the players' ignorance of the numbers of others. I will announce the results when there are enough votes or a long period of time has passed. I have a theory about what the winning number will be, but I will hold it for now :)
UPD: unregistered on the habr, send the numbers to the box twothirds.habr@gmail.com
UPD2: there were significantly more participants than I expected (at the moment - about 350), so the processing of the results was delayed. They will be a separate post tomorrow (today), at about the same time of day as the original post.
UPD3: results are processed, the winner is determined. Do not send numbers anymore :) The results are here: habrahabr.ru/blogs/wisdom_of_the_crowds/62789
This game is known in the theory of games called "guess 2/3 of the average." It demonstrates the difference between absolutely rational behavior and real actions of players.
Let us imagine that all participants in the game act completely rationally and, last but not least, know that the others also act rationally and do not agree with each other. What is the best number in this situation?
Obviously, it makes no sense to call numbers greater than 66. (6), since arithmetic average can not be more than 100. But, if all players argue in this way, then all numbers will not be greater than 66. (6), then the arithmetic average will not exceed this number, and therefore call more than 2/3 * 66. (6) = 44. (4) again there is no point. Repeating this argument infinitely many times, we come to the conclusion that the only correct move will be the number 0. Thus, if all the players reason rationally, then they all have to choose the number 0.
')
However, in real life the situation is different. Even if a player is rational, he knows that many of his opponents are not rational, which means he will have to take into account that their numbers will be greater than 0. We can assume that the majority will send more or less random numbers, then the average will be 50, two thirds from 50 approximately 33. If we go further and assume that quite a few people would guess up to the number 33, then we can choose two thirds of 33, i.e. 22. Further iterations will give ~ 15, ~ 10, etc., but it seems unlikely that a sufficiently large number of players will figure that far.
Let's return to the beginning of the article. What is the number won in Denmark? Below you can see a histogram of the game in which 19,196 people took part.
The first thing that catches the eye is the expected peaks at points 22 and 33. The winning number turned out to be slightly less than 22, most likely as a result of the fact that most of the participants understood the meaninglessness of the choice of numbers greater than 66. (6). It is curious that there were those who sent 67 and more, including 100. Interestingly, they did it without seeking to win or simply did not understand the futility of such a move? It is also interesting, were those who sent 0 guided by absolutely rational reasoning, or simply chose a round number?
Another interesting point: if the problem condition limits the choice to only integers, then there are two rational winning strategies: 0 and 1. The fact is that because of the discreteness of the integers, the multiplication by 2/3 will not be repeated infinitely many times. . When we get to 1, the next iteration will yield 2/3, but, rounding to integers, we again get 1.
I propose to play the game on Habré. Send me a habrakpote real numbers from 0 to 100. Attention: do not write numbers in the comments, because An important part of the game is the players' ignorance of the numbers of others. I will announce the results when there are enough votes or a long period of time has passed. I have a theory about what the winning number will be, but I will hold it for now :)
UPD: unregistered on the habr, send the numbers to the box twothirds.habr@gmail.com
UPD2: there were significantly more participants than I expected (at the moment - about 350), so the processing of the results was delayed. They will be a separate post tomorrow (today), at about the same time of day as the original post.
UPD3: results are processed, the winner is determined. Do not send numbers anymore :) The results are here: habrahabr.ru/blogs/wisdom_of_the_crowds/62789
Source: https://habr.com/ru/post/62696/
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