How did you compute the average number of games until convergence for the TrueSkill ranking system?

One way to think about the TrueSkill ranking system is that it attempts to identify the correct ordering of n players in terms of 50 skill levels. If each ordering is equally likely, a computer would need log2(50) many bits of information to uniquely encode the skill level of a player. Now, assume that 2 players play a Head-to-Head game.

Disregarding draws, the game outcome can provide 1 bit of information (which of the two players was the winner). Since each of these games requires 2 players, the system needs 2*log2(50) many Head-to-Head games per player. Note that the particular Head-to-Head games have to be chosen such that they, in fact, do carry one bit of information.

Interestingly, every match-made game where the game outcome is not predictable ahead of time ensures that the game is informative! In general, with k teams of m players in each team, one game outcome provides log2(k!) bits but it needs k*m players per game so in the most general case, the system needs k*m*log2(n)/ ...

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