Because F/N is equal to the average amount of money invested.

The net income of participant 1 is U1 (s1, s-1) = 2F/N-s1. s1 is the amount invested by participant 1.

Assume that the number of people is infinite. It can be seen that the impact of a certain person's strategy (money invested) on the average is negligible, and we will leave it unchanged for now.

At this time, we found from U1 (s1, s-1) = 2F/N-s1 that when s1 = 0, his profit is 2F/N, which is the largest.

In other words, s1=0 is his best strategy.

Because this is a symmetric game.

Therefore, the best strategy for each participant is 0 and not investing a penny.

At this point, we know that all participants will choose not to invest a penny out of their own interests.

So is 0 the best strategy at this time?

We find that when other participants choose strategy 0 and player 1 chooses to invest in s1, F=s1, and U1=2F/N-s1 at this time. Because we assume that N is infinite, 2F/N=0 at this time. So

U1=-s1 means that he will lose as much as he invests. The best strategy at this time is still to choose not to invest a penny.

So we find that all participants invest 0 yuan, and this strategy is the best strategy for each other at this time.

To express it more rigorously, it is because Ui (si, s-i) = 2F/N-si when si = 0, Ui is the largest.

That is, si=0=BR(s-i) and it is true for any participant i.

Therefore, strategy 0 is the Nash equilibrium of the game when the number of participants is infinite.

Note that I only give an explanation of the proof process when the number of participants is infinite, but what about when there are not many participants?

You should think that the strategy si of participant i will have a great impact on the average at this time. In view of this, I will give a strict mathematical proof. The above is for friends who are not good at mathematics to learn.

Ui(si,s-i)=[(F-si)+si]×2/N-si=2(F+si)/n+2si/n-si Because N>2, so 2si/n-si≤0

So when si=0, Ui(si,s-i)=[(F-si)+si]×2/N is the maximum benefit of participant i. Because the game is symmetric, i can belong to any participant.

It can be seen that the Nash equilibrium of this game is that each participant invests 0 yuan.