Achieving the Highest Possible Elo Rating

TL;DR

This paper investigates how much a player can artificially inflate their Elo rating by rigging games with a limited budget, revealing a phase transition at a specific number of players and providing bounds for various scenarios.

Contribution

It provides the first asymptotic bounds on maximum Elo ratings achievable through game rigging with a given budget, including a phase transition phenomenon.

Findings

For two players, the maximum rating is determined up to a constant error.

A phase transition occurs at n=k^{1/3}, drastically increasing the maximum rating.

For n > k^{1/3}, the maximum Elo rating scales at least as Θ(k^{1/3}).

Abstract

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 $\Theta(k^{1/3}\log(k)^{1/3})$.