Salut, fellow game theorists. I have to solve 6 Game Theory problems and fell almost hopeless. Would appreciate any guidance with this one.
A company Zest is actively promoting its services. Everyone who invests in Zest will receive their money back as soon as the next client makes his investment. Besides, 90% of the investment made by the $i+1$ client will be forwarded to the $i$ client. All the residents of the city decide by turn whether they'll invest and if yes, how much. Every resident knows about all the decisions made by other people. If the Company can't fulfill its promises it'll default and it'll be closed and those who didn't receive their money before the default won't ever receive their money back. The company has no own money at all. There are 12 million residents of different wealth in the area.
Questions: How many residents will invest their money in subgame perfect Nash Equilibrium? Does the answer depend on the sequence the investors will come?
The answer for the 2nd question(I think) is No, because even if all the wealthier investors had already invested, a resident still would invest if the following investor invests more than 0, because all his money will be returned +90% of the next investor's input.
How should I solve the 1st question? I probably should start from the end. The 12 000 000th resident won't invest because he's the last one, the 11 999 999th won't invest because he knows that the last resident won't invest but... what do I do next?
This response is only tangential to the original question, but it would be hard to format it as a legible comment.
Supposing some people did invest, the first investor would receive back $a_1 + 0.9\,a_2$ where $a_i$ is the amount invested by the $i$th investor. Now there is only $0.1\,a_2$ in the fund, but the payout for investor $2$ has to be $a_2 + 0.9\,a_3$. In order to prevent the fund from defaulting it is necessary that
$$a_2 + 0.9\,a_3 \leq 0.1\,a_2 + a_3.$$
That is, $a_3 \geq 9a_2.$ After paying investor $2$, the fund has $0.1\,a_3 - 0.9\,a_2$ remaining, so in order to pay investor $3$ without defaulting,
$$a_3 + 0.9\,a_4 \leq 0.1\,a_3 - 0.9\,a_2 + a_4$$
or in other words $a_4 \geq 9(a_2 + a_3) \geq 90a_2.$ Paying investor $3$ leaves $0.1\,a_4 - 0.9(a_2 + a_3)$, and to pay off investor $4$,
$$a_4 + 0.9\,a_5 \leq 0.1\,a_4 - 0.9\,(a_2 + a_3) + a_5$$
so $a_5 \geq 9(a_2 + a_3 + a_4) \geq 900a_2.$ And so forth. The necessary new investment at each step grows exponentially (a factor of ten each time). So if investor $2$ makes the smallest possible investment, let's say one penny, the fund would quickly run out of investors who have sufficient funds to make the next investment (if they were so foolish as to do so).