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Markov Chains for Collaboration

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.