The scenario goes like this: You're in a group of 92 people. Each person picks a number from 0 to 100. Then the arithmetic mean is calculated from everyone's answers, and it is multiplied by 2/3. Then, whoever's number is closest to this number, that is 2/3 of everyone's average, wins the game. What number do you choose?
I'm really struggling with this problem. Please help! Thanks!
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It's explained pretty well here:
As a speed summary:
Suppose everyone picks randomly the first time. Then the average of the numbers will be around $50$ and thus you should pick $50 \cdot \frac{2}{3} \approx 33$. Assuming everyone is logical everyone is thinking of this, so they will all pick $33$. But then the winning number would be $33 \cdot \frac{2}{3} = 22$. Assuming everyone is logical everyone is thinking of this, so they will all pick $22$. The process repeats, with everyone picking smaller and smaller numbers until everyone picks $0$. Thus $0$ is the answer.
The point is: this is a question of psychology, more than of mathematics. Let's start from the assumption that people choose randomly. Assuming a Gaussian distribution, you would get something around 50 for the average, hence the "correct answer" would be$\frac{2}{3}50\cong 30$ (I'm approximating to simplify). But assuming that everybody has thought about this, you would get your prize with $\frac{2}{3}30=20$. So on like that: you get that if every player has reasoned like that, everybody will play a $0$ and everybody (or nobody) will win. Of course, it is impossible to foresee what strategy the other people will choose, and to play $0$ is quite risky. As it says in the link posted by Hway-Ray Tung, the "best choice" would be, empirically, around 19, so just choose some number around 19 and make a sacrifice to the goddess Fortune ;)