Choice Aggregation Paradoxes when Ties are Possible: Novel Computational and Empirical Analyses of Weak Cycles

Abstract The extant literature on choice aggregation paradoxes focuses on strict instances given their empirical availability, despite the prevalence of ties in many settings. In this paper we consider the incidence of weak transitivity violations (cycles) in rank sum aggregation in a unique empirical setting, the IAAF Cross Country World Championships . In this setting, rank sum scoring is used to aggregate 4 team members per group into a group score. Unlike other prominent empirical cases of rank sum scoring, the even number of constituents per group means pairwise group ties, and therefore weak cycles, are possible and may be probable. We compute the empirical likelihood of 3-group weak cycles from pairwise rank sum scoring across 83 Championships. We also compute combinatorial likelihoods of said violations in this setting using a novel, large sample space enumerative algorithm for statistical comparison. Like the concept of a p-value , cycle incidence sheds light on the robustness of a given statistical result. We find regular occurrences of both weak and strict cycles, and that weak cycles occur more regularly than strict cycles in theory and our data. Both weak and strict cycles are under-represented in our empirical setting relative to the corresponding theoretical incidence—in which all rank-sum outcome sequences are equally likely—due to correlated within-group ability levels. While the degree of ambiguity for a weak cycle is less relative to a strict cycle, weak cycles are more prevalent in the data and may therefore introduce a greater aggregate level of ambiguity.