So here is my question. Suppose we have one person drawn with uniform probability from a population. This person then 'chooses' a proportion (x-1)/(population) of the population in the same period (so at time 1, x/pop have been picked). We are in discrete time with an infinite horizon, the initial people who are chosen is 0.
So, at time t, somebody is drawn. If she has been 'chosen' in the past (either drawn or been chosen by somebody else), I get a payoff of 0, and if she has not been chosen, I get a payoff of y in this period. Now the catch is, if this person who is drawn has already been chosen in the past, she does not choose anybody new this period, and I get 0 (thus the portion of the people who have been chosen next period is the same as this period). If she has not been chosen at anytime in the past, she will choose a new proportion x of the population, and I get a payoff y (the same people can be chosen multiple times, uniform probability). So I want to know what the expected value of y is.
I have been racking my brain for ages on how to express this cleanly. Can anybody provide any guidance for where to even get started? Any help is warmly appreciated.