I saw a game theory problem today that I found very interesting, so I tested it out on my WeChat Moments. The problem is quite simple:
Choose any number from 1 to 100, and the one whose final value is closest to two-thirds of the average wins.
The problem lies in how you think about others. If everyone chose 100, the average would be 66.6 = 67. However, to win, no one would choose 100; in that case, everyone would choose 67.
But those who have thought of this step need to think again. Since people have thought of two-thirds of 100, they can still proceed to the next step, two-thirds of 67. So the correct answer becomes 44.
However, some people are still worried. Since I can think of this step, then everyone else can too, so they will keep calculating until they reach 1.
The key to guessing a number lies in considering the rationality of everyone's thinking. It's a pretty fun game; you can try it out on your social media.
The following is a statement circulating online:
Regarding this game theory problem, I have a brilliant and interesting analytical model as follows—
解：
1. The participants' IQs followed a normal distribution. 5% had an IQ ≤ 80, 5% had an IQ ≥ 120, and 90% had an IQ between 80 and 120. The average IQ was 100.
1.1 People with an IQ ≤ 80 tend to guess numbers from one to one hundred based on intuition.
1.2 People with an IQ between 80 and 120 tend to think like this: they choose numbers based on mathematical intuition within the range of likely winning numbers (i.e., 1 to 66).
1.3People with an IQ of 120 or higher cannot be guessed in their thought process, but it is known that their answers are closely scattered around the correct answer.
2. There are some people who provide data without aiming to win, ignoring the victory conditions (but still within the rules); they are…Neuropathy".
2.1 "Mental illness" meets the criteria of a "low-probability event," occurring at a frequency of less than 0.01 or 0.05, meaning it accounts for ≤ 1% or ≤ 5% of the total population.
2.2 The occurrence of "neurotic disorder" is not related to IQ level; it will be randomly distributed in the population.
2.3 There are three types of "neuropathy" —
2.3.1. Those who prank to inflate the average: Select 100 for all.
2.3.2. Those who lower the average through pranks: Select 1 for all.
2.3.3 Those who are not seeking victory and have no purpose: Randomly selected from 0 to 100, with an average value of 50.2.4. The three types of neurosis have an equally high probability of occurrence. If any neurosis is randomly selected from all neurosis cases, there is a 33% chance that it will fall into one of the three types.
In summary, the calculation method is as follows:
Let n be the probability of the occurrence of neurosis, and a be the optimal solution, thenSince the average data selected by those with IQ ≤ 80 is 50, the average data selected by those with IQ between 80 and 120 is 33, the average data selected by those with IQ ≥ 120 is a, and the average data selected by the mentally ill is 1/3 * 100 + 1/3 * 1 + 1/3 * 50 = 50.33, the average data selected by the mentally ill is...
∴ 50*5% (1-n)+ 33*90% (1-n)+a*5% (1-n)+50.33*n=3/2*a ①
When n=5%, substituting into equation ①, we get: 2.375+28.215+2.516=1.5*a-0.0475a ——→1.4525a =33.106
a = 22.7924, choose 23.
When n=1%, substituting into equation ①, we get: 2.475+29.403+0.5033=1.5*a-0.0495a ——→1.4505a=32.3813
a=22.3242, choose 22.
When n=0, substituting into equation ①, we get: 2.5 + 29.7 = 1.5a - 0.05a ——→ a = 22.2068, so choose 22.
It seems that the fewer the number of 2b, the closer the optimal solution is to 22.
So, here's the result: 56 valid numerical messages were received, totaling 2012, with an average of 35.9286. The final two-thirds is 23.95238. It's very close to 22, and I don't know why. Those interested can do some research. Below is the distribution chart and data:
17
10
22
24
100
36
26
23
6
21
19
11
39
50
28
15
76
88
99
17
33
23
66
5
9
32
24
28
2
1
50
45
19
30
15
49
53
14
50
16
66
58
26
68
36
77
22
44
65
66
50
22
24
44
23
30
This siteOriginal articleAll follow "Attribution-NonCommercial-ShareAlike 4.0 License (CC BY-NC-SA 4.0)Please retain the following annotations when sharing or adapting:
Original author:Jake Tao,source:"Game Theory Quiz Results [Guess the number from 1-100, what is two-thirds of the average?]"
Comment list (5 comments)
Furthermore, if you think that choosing 100 is "crazy," then why isn't choosing 99, which also has no chance of winning, crazy? If choosing 1 is "crazy," then why isn't choosing 66 crazy?
P.S. I made a mistake in my previous calculation. If all values are 100, then choosing 67 would be closer to 66.667. So 67 is also possible, and therefore the result would be larger than 22.333.
I don't quite understand why you calculated the three types of "nervousness" separately... I was discussing the people who chose 67-100 separately, so your average for the three types is definitely 50.5...
@anonymous：Wow, that was quite a powerful move! I did a test for fun before. The rating for all three tests together seemed to be 22. This leans more towards psychology; 23 is just a high-probability regression. That's my opinion.
First, assume all integers are randomly generated (1-100). The average is 50.5, and two-thirds of that is 33.667. If everyone pranks and gives the highest value, 100, then two-thirds of the average is 66.667. Therefore, giving [67, 100] will never result in a win.
The problem is to find an integer between 1 and 66 that is closer to two-thirds of the average. Therefore, (1+66)/2*2/3 = 22.333.
Although we've ruled out the possibility of a 67-100 score winning, we can't rule out the possibility that someone might still give a number within that range. In that case, two-thirds of the average of these people would be (67+100)/2*2/3v = 55.667. This would make the actual result slightly larger than the 22.333 obtained above.
Let's assume the result is 23. Then 23 = 22.333*(1-x)+55.667*x, where x=2% is a fairly ideal proportion, so the result is 23.
Awesome