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

翻译：Elo评级系统衡量游戏或运动中每位参赛者的近似技能水平。参赛者的评级在获胜时上升，在落败时下降。提升自身评级是一项艰巨的任务——必须磨练技能并持续击败对手。然而，若拥有足够资金，你也能通过操纵比赛结果来提升评级。本文针对Elo评级系统提出一个自然问题：假设你召集了$n$人（包括你自己）并获取足够资金操纵$k$场比赛，你的评级最高能达到多少（渐近于$k$）？在此设定下，你所召集的人对游戏兴趣寥寥，只有在获得报酬时才愿意参赛。本文在$n=2$情形下解决了该问题（误差为常数加法项），并为所有其他$n$值（包括随$k$任意增长的情形）提供了紧密的上下界。在$n=k^{1/3}$处存在相变：从$n=2$到$n=k^{1/3}$，最高可能Elo评级出现显著跃升，但（取决于具体Elo系统）对于更大$n$值，评级提升幅度极小甚至为零。当超过相变点$n>k^{1/3}$时，最高可能Elo至少为$\Theta(k^{1/3})$。对应的上界取决于具体系统，但在标准Elo系统中，该值为$\Theta(k^{1/3}\log(k)^{1/3})$。