Achieving the Highest Possible Elo Rating

TL;DR

This work studies the maximal attainable Elo rating when $n$ players, starting from equal ratings, participate in $k$ games with a fixed pot function $\sigma$. It reveals a phase transition at $n\asymp k^{1/3}$: with few players the growth is limited, but as $n$ grows beyond the transition, the highest possible rating scales like a fractional power of $k$ modulated by the left-tail of $\sigma$. The key method combines a lower-bound strategy family that achieves $\Theta(\min(n, k^{1/3}))$ growth and a sophisticated upper-bound via a path-length and potential function analysis using $f(x)=\int_{0}^{x} 1/\sigma(-\tau)\,d\tau$. The results explicitly depend on the tail behavior of $\sigma$ (e.g., logistic, Gaussian, uniform) through $f^{-1}$, yielding tight or near-tight bounds in various regimes, and they connect to the mass-movement view of the classic maximum overhang problem. The findings illuminate how ability to manipulate outcomes (via rigging) interacts with the structure of Elo updates, and they suggest several open questions about tightening gaps and exploring related variants.

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

Theorems & Definitions (24)

• Theorem 1.1
• proof
• Remark
• Theorem 2.1
• proof
• Theorem 3.1
• proof
• Remark
• Theorem 3.2
• proof