There are k players, with player i having value vi > 0, i = 1, . . . , k. In every period, two of the players play a game.Whoever wins then plays the next game against a randomly chosen one of the other k − 1 players (including the one who has just lost). Suppose that whenever i and j play, i wins with probability vi vi+vj

. Let Xn denote the winner of game n.

(a) Give the transition probabilities of the Markov chain {Xn,n ≥ 1}.

(b) Give the stationarity equations that are uniquely satisfied by the πj .

(c) Give the time reversibility equations.

(d) Find the proportion of all games that are won by j , j = 1, . . . , k.

(e) Find the proportion of games that involve player j as one of the contestants.