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 running 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 golf tournaments. We see that geometric scores approximate the optimal scores well in events where the distribution of athletes' performances is roughly uniform.
翻译:分级规则被广泛用于体育运动员和选举候选人的排名。 每个人排名中的每一职位都值得一定数量的分数; 点数总和决定总排名。 问题是如何为特定应用选择一个评分规则。 首先, 我们产生一个单数家族, 分数符合两个独立原则: 一旦一个非常强或弱的候选者被撤走, 总计排名应该保持完整。 这个家族包括波尔达计分、 通用的多元( medal 计数) 和通用的反多元性( 超值规则) 等边缘案例, 我们发现这些规则是另外的。 其次, 我们引入一个一等数家族, 最优分数家族: 运动员应该按其预期的整体质量排序 。 最后, 我们使用比德特伦、 高尔夫的历史数据, 并运行我们演示几何和最佳评分可以简化合适的评分规则的选择, 表明这些分与组织者使用的实际得分非常接近, 并且为高尔夫锦标赛中观察到的经验性现象提供了解释。 我们发现, 几度得分近于运动员成绩最优分, 。