Achieving the Highest Possible Elo Rating

Elo rating systems measure the approximate skill of each competitor in a game or sport. A competitor’s rating increases when they win and decreases when they lose. Increasing one’s rating can be difficult work; one must hone their skills and consistently beat the competition. Alternatively, with enough money you can rig the outcome of games to boost your rating. This paper poses a natural question for Elo rating systems: say you manage to get together n people (including yourself) and acquire enough money to rig k games. How high can you get your rating, asymptotically in k? In this setting, the people you gathered aren't very interested in the game, and will only play if you pay them to. This paper resolves the question for n = 2 up to constant additive error, and provides close upper and lower bounds for all other n, including for n growing arbitrarily with k. There is a phase transition at n = k^{1/3}: there is a huge increase in the highest possible Elo rating from n = 2 to n = k^{1/3}, but (depending on the particular Elo system used) little-to-no increase for any higher n. Past the transition point n > k^{1/3}, the highest possible Elo is at least Θ(k^{1/3}). The corresponding upper bound depends on the particular system used, but for the standard Elo system, is Θ(k^{1/3}log(k)^{1/3}).