To put this in a simple math framework, here is the experiment.
There are six players $\{1,2,3,4,5,6\}$ and player $n$ has $M(n)$ dollars to play with. A die is rolled, and if it comes up $n$, then player $n$ is the winner for that roll. Each of the other players hands over an amount of $M(n)$ to this winner. If a player $k$ has less than or equal that amount, $M(k) \le M(n)$, they lose and leave the game.
The game is played until only one player is left - the TAKE ALL WINNER.
What is the expected value for each of the six players when the game is over?
Since each play is fair, it is obvious that $E(n) = M(n)$. I know that the calculation for one 'roll of the die' would work out and give this result. But how to you mathematically describe this process. We have no doubt that that the game will conclude with a single winner, and we can find a math model to calculate the expected number of rolls before the game ends.
So, do you have to employ some 'heavy' theory, perhaps modeling this with a stochastic processes, to formally calculate expected values here. Or is there a simpler way of viewing the problem?