Measuring competitive balance in sports leagues that award bonus points, with an application to rugby union
研究了奖励额外积分的体育联赛（如橄榄球联盟）中赛季内竞争平衡的测量方法，推导出“双级联分布”作为最不平衡的积分分布，并应用于六国锦标赛，发现2017年引入额外积分后平衡有所改善，但因果关系尚不明确。
We examine within-season competitive balance measurement in sports leagues where points allocated in each match depend on bonus points awarded as well as the win/draw/loss outcome, as in rugby union. The bilateral nature of matches imposes constraints on the range of competitive balance measures, as teams cannot win matches in which they do not play. Application of normalized competitive balance indices requires knowing the maximum value the index can attain, representing the most unequal distribution of points among teams. However, evaluation of the points distribution consistent with maximum imbalance is non-trivial when the total points allocated are not known in advance, as when bonus points are awarded. We theoretically derive the corresponding perfectly unbalanced distribution of points, characterized as a ‘Double Cascade distribution’, and provide an Excel tool to obtain it for different parameter values. We apply our results to examine variation in within-season competitive balance in the Six Nations Championship in rugby. The degree of competitive balance, as measured by a normalized concentration index, has improved since the award of bonus points in 2017, primarily because the index's maximum value increases with bonus points. However, with limited available data on the bonus-points regime, it is not possible to establish or reject a causal link, due to induced behavioural changes by teams, between the raw concentration index and bonus points. A relatively high level of imbalance still exists compared to other sports leagues, which will be an important consideration in evaluating future developments of the competition.