论文标题
达到最高的ELO评级
Achieving the Highest Possible Elo Rating
论文作者
论文摘要
ELO评级系统测量了游戏或运动中每个竞争对手的大致技能。当竞争对手赢得胜利时，他们的评分会增加并减少。提高评级可能是艰难的工作；必须磨练自己的技能并始终如一地击败竞争。另外，有了足够的钱，您可以操纵游戏的结果以提高评级。本文对ELO评级系统提出了一个自然的问题：说您设法将$ n $人聚集在一起（包括您自己），并获得了足够的钱来钻机$ k $游戏。您可以在$ k $中渐近地获得评级？在这种情况下，您收集的人对游戏的兴趣不是很感兴趣，只有在您付钱的情况下才会玩。本文将$ n = 2 $的问题解决到恒定添加期错误，并为所有其他$ n $提供近距离和下限，包括$ n $任意使用$ k $生长。在$ n = k^{1/3} $上有一个相转换：最高可能的ELO评级从$ n = 2 $增加到$ n = k^{1/3} $，但是（取决于所使用的特定elo系统），对于任何更高的$ n $，却很少会增加。过去的过渡点$ n> k^{1/3} $，最高可能的elo至少为$θ（k^{1/3}）$。相应的上限取决于所使用的特定系统，但对于标准ELO系统，为$θ（k^{1/3} \ log（k）^{1/3}）$。
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 $Θ(k^{1/3}\log(k)^{1/3})$.
