  Consider a system of \(n\) players in which each initially starts on a
different team. At each time step, we select an individual winner and an
individual loser randomly and the loser joins the winner's team. The resulting
Markov chain and stochastic matrix clearly have one absorbing state, in which
all players are on the same team, but the combinatorics along the way are
surprisingly elegant. The expected number of time steps until each team is
eliminated is a ratio of binomial coefficients. When a team is eliminated, the
probabilities that the players are configured in various partitions of \(n\)
into \(t\) teams are given by multinomial coefficients. The expected value of
the time to absorbtion is \((n-1)^2\) steps. The results depend on elementary
combinatorics, linear algebra, and the theory of Markov chains.
