There are $N$ people and the amount of money hold by the $i_{th}$ person is a prior given parameter denoted by $x_{1,i}\in\mathbb{R}$, where $i=1,\cdots,N$. Here we assume that the wealth can be expressed by real numbers and can be divided in arbitrary way such that any real number of money makes sense. For instance we admit $\cfrac{1}{7}$ dollar.
Now the personal wealth would vary in a stochastic way in the sense that in the $n$-th round, the $i$-th person will be chosen with probability $$\frac{e^{-x_{n,i}}}{\sum\limits_{j=1}^{N}e^{-x_{n,j}}}$$ where $x_{n,i}\in\mathbb{R}$ which can be negative stands for the wealth of the $i$-th person before the $n$-th round. Then the chosen person will spread his/her money to the other $N-1$ people equally. After that we would give him $d\in \mathbb{N}$ amount of money and then take $d/N$ from each of the $N$ persons. And finally the total money hold by them remains the same after each round.
Notice that the probability that each of them is choosen will be related to $x_{n,i}$. So can we get the expectation of $E(x_{n,i})$ for each $i$? Here the variable $x_{n,i}$ is viewed as a random variable over $n$. What kind of mathematics can be applied to this problem? Probability theory, Random processes Markov chains or something else?