There are $r$ players, with player $i$ initially having $n_i$ units, $n_i>0, i=1,2, \cdots, r$. At each stage, two of the players are chosen to play a game, with the winner of the game receiving 1 unit from the loser. Any player whose fortune drops to 0 is eliminated, and this continues until a single player has all $n=\sum_{i=1}^r n_i$ units, with that player designated as the winne r. Note that the mechanism to choose two players at each sta ge is unknown. It can be either deterministic or random. Assu me that the results of successive games are independent and that each game is equally likely to be won by either of its two players.

For any set of players $S \subseteq\{1, \cdots, r\}$, let $X(S)$ denote the $\mathrm{n}$ umber of games involving only members of $S$. Does $E(X(S))$ depend on the player selection mechanism? If you think it doesn't depend, calculate the expectation. If you think it depends, give two mechanisms leading to different expecta tions.