Winning by Losing: Incentive Incompatibility in Multiple Qualifiers

Abstract

In sport tournaments, the rules are presumably structured in a way that any team cannot be better off (e.g., to advance to the next round of competition) by losing instead of winning a game. However, the organizers may overlook a drawback in the competition rules which allows for the possibility of perverse incentives. The existing national rules of awarding places for the UEFA Champions League and the UEFA Europa League, which are based on the results of the national championship, a round-robin tournament, and the national cup, a knock-out tournament, might produce a situation where a team will be strictly better off by losing a game. Competition rules of the European qualification tournament to the World Cup 2014 and qualification tournament to Euro 2016 suffer from the same problem. We consider arbitrary qualification systems (r, k) consisting of r round-robin and k knock-out tournaments, k,r ≥ 0, k r ≥ 1, and show that incentive incompatibility arises in two cases: (i) if r ≥ 2 or (ii) if r = 1, k ≥ 1 and the allocation of vacant slots, which appear when the intersection of a pair of sets of winners in all tournaments is non-empty, favors knock-out tournaments.