AI 中文总结
该研究探讨体育联赛排名的积分规则，提出积分成为充分统计量的条件，分析多联赛排名差异并验证统一积分规则更贴合估计技能。
AI 中文摘要
体育联赛为每场比赛结果分配固定积分点，并根据累计积分对运动员或球队进行排名。由于积分点的设定未考虑概率模型，我们研究使积分成为数据的合适摘要的模型。我们发现，当且仅当积分规则为固定和（即每场比赛在参与者之间分配固定数量的积分）且结果遵循相邻类别模型（其斜率参数为积分点本身）时，积分是充分统计量。国家冰球联盟和足球联赛不满足固定和条件，足球联赛胜场奖励3分，而非充分性要求的2分。我们还确定，当两名运动员与其他对手相遇的频率相同，且每个运动员的主场和客场比赛次数相同时，积分排名与估计技能排名完全一致；场地平衡的循环赛对所有配对都满足该条件，此时Elo排名会产生相同顺序。在来自足球、冰球和排球的9个联赛中，两种排名顺序仅在赛程条件不满足时存在差异，且除联赛已采用统一规则外，为每种结果等级分配1分的统一规则比各联赛自身规则更符合估计技能。
英文摘要
Sport leagues assign a fixed score-point to each match outcome and rank the players (or teams) by their accumulated scores. Since the score-points are mandated without a probabilistic model in mind, we investigate models that make the score the right summary of the data. We find the score to be a sufficient statistic if and only if the points rule is constant-sum, so that every match distributes a fixed number of points between its participants, and the outcomes follow the adjacent categories model whose slope parameters are the score-points themselves. The constant-sum condition fails in the National Hockey League and in football, which awards three points for a win instead of the two required by sufficiency. We also establish that when two players meet every other opponent equally often, and each plays as often at home as away, they are ordered identically by the score and by the estimated skills; a venue-balanced round-robin satisfies the condition for all pairs. The Elo ranking then produces the same order. In nine leagues from football, ice hockey and volleyball, the two orders differ only where the schedule condition fails, and the uniform rule awarding one point per outcome level agrees with the estimated skills more closely than each league's own rule, unless that rule is already the uniform one.
