我们应该如何给运动员和候选人打分：几何评分规则
How should we score athletes and candidates: geometric scoring rules
摘要：
评分规则被广泛用于体育运动中的运动员和选举中的候选人。每个排名中的每个位置都有一定的分数;分数的总和决定了总排名。问题是如何为特定应用程序选择评分规则。首先，我们推导出一个单参数的家庭几何分数，满足两个原则的独立性：一旦一个非常强或弱的候选人被删除，总排名应该保持不变。这个家庭包括博达计数，广义复数（奖牌计数），广义antipultium（阈值规则）作为边缘情况下，我们发现这些规则的额外公理。其次，我们引入了一个单参数的家庭与最佳成绩：运动员应该根据他们的预期的综合素质排名。最后，使用历史数据从冬季两项，高尔夫，田径，我们展示了如何几何和最佳的分数可以简化选择合适的评分规则，显示这些分数非常类似的实际分数由主办方使用，并提供了一个解释的经验现象中观察到的冬季两项和高尔夫锦标赛。我们看到，在运动员成绩分布大致均匀的项目中，几何分数很好地逼近最佳分数。
英文摘要：
Scoring rules are widely used to rank athletes in sports and candidates in elections. Each position in each individual ranking is worth a certain number of points; the total sum of points determines the aggregate ranking. The question is how to choose a scoring rule for a specific application. First, we derive a one-parameter family with geometric scores which satisfies two principles of independence: once an extremely strong or weak candidate is removed, the aggregate ranking ought to remain intact. This family includes Borda count, generalised plurality (medal count), and generalised antiplurality (threshold rule) as edge cases, and we find which additional axioms characterise these rules. Second, we introduce a one-parameter family with optimal scores: the athletes should be ranked according to their expected overall quality. Finally, using historical data from biathlon, golf, and athletics we demonstrate how the geometric and optimal scores can simplify the selection of suitable scoring rules, show that these scores closely resemble the actual scores used by the organisers, and provide an explanation for empirical phenomena observed in biathlon and golf tournaments. We see that geometric scores approximate the optimal scores well in events where the distribution of athletes' performances is roughly uniform.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
