  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 provide 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 $\Theta(k^{1/3})$. The corresponding upper bound
depends on the particular system used, but for the standard Elo system, is
$\Theta(k^{1/3}\log(k)^{1/3})$.
