It is always fascinating for me to discover how irrational we are (obviously I include myself). Multiple situations in everyday life prove it, but there are some cases in which it is certainly surprising. I am going to propose you a very simple problem and you will see what happens to you (I guess). I hasten to tell you that you will solve the problem in some way (it doesn’t matter which one) and I will have a (very easy) way of showing you that it is quite possible that your answer is not the correct one or the most rationally viable one (unless you know the example, in which case all of this doesn’t make sense). Note that I am really daring about the last thing I wrote, but I am extrapolating from my experience with the problem. But what strikes me the most is that even if you know that it will not be okay, if you had to participate in the contest that I am going to propose to you with other people, it is very possible that you would not change anything: you would leave the most irrational answer instead of the other.

Enough introduction. Here I go. Suppose we are in a theater, with thousands of people where a show will take place. In each seat, there is located a paper and a pencil. On stage, on top of a table is a laptop.

Once everyone is settled, the curtain rises and the lead actress says, “Good evening. Before starting the show, I want to do a raffle. The winner will get the computer. What are they supposed to do? We have distributed a paper and a pencil in each seat so that everyone can participate. All I ask is that you write your name and any number between 1 and 100 (natural number, that is, 1, 2, 3, 4,… so on up to 100). But, before each of you chooses the number to write, I will tell you what I am going to do so that you have a better chance of winning the prize.

When my classmates put together all the papers with the numbers that you wrote, I am going to calculate the number that turns out to be the average of those that you wrote down. This is: I am going to add them and then I am going to divide by the number of papers. Once we have calculated it (for everyone to see), I am going to multiply the result by 2/3.

For example, if the average gave 90, then I am going to calculate 2/3 of 90. How is it calculated? You multiply 90 by 2 and then divide by 3, or vice versa, divide 90 first by 3 and then multiply by 2. In any case (and I ask you to verify what I’m saying), that number turns out to be … 60. Another example: if the average between the numbers you chose turns out to be 30, then now 2/3 of 30 is equal to 20 Do you understand me?

Now, who is going to be the winner? The winning person will be the one who has chosen the closest number to the one I found. That is, if in the last case (where 2/3 of the average turned out to be 20) someone wrote just 20, that person will win. If no one wrote 20, I am going to look at those who wrote 21 or 19, and if there is still no winner, I will find out if someone wrote 18 or 22. In the end, it is clear that I will find who (or who ) of you wrote the number closest to 20.

With that said, let’s stop for a moment. If you were in the theater … what number would you choose? Please, before reading further, I ask that you take a little time to think about the problem. If not, what fun would all this be?

I follow. Let’s brainstorm some ideas together What is the maximum number I can get by calculating 2/3 of the average of the numbers I put together? Clearly, the average cannot be greater than 100. It will have to be a number less than or equal to 100. Therefore, 2/3 of the average number will inexorably be a number less than or equal to 66.66. So, if someone is sitting in the theater and intends to win the computer, he deduces that he should choose a number close to 66.66 but ALWAYS LESS. It is clear that 2/3 of the average (whatever this number is, it will NEVER be greater than 66.66). Of course, now another factor appears: “if EVERYONE thought like that, with that logic, the average would be very close to 66.66. Therefore, when I calculate 2/3 of that number, the result will be a number close to 44.44. Again, you should not choose a number close to 66.66, but close to 44.44. But then again, if all / or thought alike (as would be the logical thing), one would assume that everyone is going to write a number that is close to 44.44, so the average will be around that number. That’s fine, but I’m going to calculate after 2/3 of that number, which will be very close to 44.44. When I calculate (as stipulated) 2/3 of the number 44.44… I will obtain a number close to 29.6266…

As you notice, if everyone uses the same logic, you will systematically move forward with this idea, and therefore, in your head at least, you will conclude that you HAVE TO PUT smaller and smaller numbers. This is the only way you will increase your chances of winning the prize. But if everyone uses this idea (which is the RIGHT one), the average will give a smaller number each time, and 2/3 of it will be even smaller, so in the end (as you must already be intuiting), the BEST ALTERNATIVE is to choose the number… ZERO!

Before proceeding, once again, I propose that you reread what I wrote before, until you are convinced that the logical and rational number to choose IS the number ZERO. Here I will stop again: this problem is not my invention. It is not because it even has a name: “Alain Ledoux’s guessing game”, that is, Alain Deloux’s guessing game. The funny thing is that history shows (with the enormous number of cases that can be searched on the internet) that every time someone raises this problem, virtually NO ONE puts zero. It is that the man (or the woman if you prefer), knows that if he puts ZERO he will be more capable and more rational than all those around, but at the same time, one discovers that if wants to win the prize and take the computer, you should choose another number: NOBODY WANTS TO WRITE ZERO! If you put zero you will exclude yourself from the competition. You intuit (and how well you do) that the winner is most likely someone else, EVEN IF YOU ATTEMPT AGAINST THE LOGIC. History indicates that in 1981, the French magazine “Jeux & Stratégie” which is very popular in France, dedicated mainly to mathematical or card games, or even chess or strategy of all kinds, organized a large readership competition [1] which consisted of solving a set of chess, bridge and go problems. Ledoux wrote that out of nearly 15,000 participants, 4,078 tied for first place. Therefore, they had to invent some way to tie the tie. Ledoux came up with the problem I wrote above to try and distinguish a single winner / or.

All the winners of the first part received a letter asking them to write a number chosen not among the first 100 (as in the example I put above) but among the first billion numbers. The winner would be whoever was closer to 2/3 of the average. You … What do you think happened? Unbelievably, the average of the 4,078 readers was 134,822,738.26 (over 134 million). Then 2/3 of that number turned out to be 89,881,825.51. This number is 8.99 percent of the maximum number, which – remarkably – is less than what is normally found in the early rounds of what is known as the Beauty Contest. [2]

In short: if you want to be rational, the number you have to write on the paper is zero. But in this particular case, if you put zero it is very likely that you will not win (unless they are all invulnerable logicians and do not deviate from what rationality indicates). There are many other examples in the literature of the same episode. Two German scientists (Christoph Buhren and Bjorn Frank from the University of Kassel, in Germany), published an article recounting their experiences when they made six thousand chess players participate in the contest! and the result returned to something similar: nobody wrote zero! (despite knowing that that was the correct number) [3]

Humans are unpredictable beings. As I wrote at the beginning, even knowing that what we are going to do is wrong, we do it the same, but curiously (or not), since all the / other / other participants are also human, it is very possible that you (or I) are not be (we are) alone and that there are many people who do the same What is preferable: choose the right thing or win the computer? When one asks: who discovered America? The answer you get is Columbus. However, we all know that the Vikings and perhaps other contingents had arrived before, but still, the only thing we can say is that Columbus was not the first, but that he was the last, since after his arrival, nothing was ever again. the same Is it okay to tell the story like this? To be continue …

[1] The contest was called “Beauty Contest.” An article I wrote here a few months ago appeared in this newspaper: https://www.pagina12.com.ar/360984-como-se-vincula-la-teoria-general-de-keynes-con-riquelme-yp which is Related to this.

[2] https://cheaptalk.org/2009/08/01/grandmasters-play-the-beauty-contest-game/

[3] https://en.chessbase.com/post/6000-che-players-took-part-in-our-beauty-contest

[1] https://www.pagina12.com.ar/360984-como-se-vincula-la-teoria-general-de-keynes-con-riquelme-y-p