标题： 固定比赛操纵：促进竞争性的最佳赛程 显示英文标题

标题： Fixing Match-Fixing: Optimal schedules to promote competitiveness

摘要： 在世界杯小组赛的最后一轮比赛中，那些结果不会影响出线队伍选择的比赛往往缺乏激情。 此外，已经出线的球队可能会考虑其他因素，例如接下来比赛的对手，因此，根据其他小组的结果和下一阶段的赛程安排，赢得比赛可能并不符合其最佳利益。 更严重的是，可能存在一种平局的情况，使得两支球队都能晋级下一轮。 任何两支对抗球队不进行激烈竞争的情况都会对体育运动产生不利影响，而且最重要的是可能导致串通和操纵比赛的机会。 我们在这里开发了一种相对通用的竞争力评估方法，并将其应用于当前的世界杯小组赛形式。 然后，我们提出了对当前形式的修改，以增加小组赛最后一轮比赛的赌注，使比赛更加精彩，同时减少任何串通的机会。 我们使用同样的方法来评估2026年世界杯将引入的“3队小组”形式以及类似当前欧洲足球协会联盟（UEFA）杯的形式。 显示英文摘要

摘要： In the last round of the FIFA World Cup group stage, games for which the outcome does not affect the selection of the qualified teams are played with little enthusiasm. Furthermore, a team that has already qualified may take into account other factors, such as the opponents it will face in the next stage of the competition so that, depending on the results in the other groups and the scheduling of the next stage, winning the game may not be in its best interest. Even more critically, there may be situations in which a simple draw will qualify both teams for the next stage of the competition. Any situation in which the two opposing teams do not play competitively is detrimental to the sport, and, above all, can lead to collusion and match-fixing opportunities. We here develop a relatively general method of evaluating competitiveness and apply it to the current format of the World Cup group stage. We then propose changes to the current format in order to increase the stakes in the last round of games of the group stage, making games more exciting to watch and, at the same time, reducing any collusion opportunities. We appeal to the same method to evaluate a "groups of 3" format which will be introduced in the 2026 World Cup edition as well as a format similar to the one of the current Euro UEFA Cup.