How Should We Score Athletes and Candidates: Geometric Scoring Rules

Scoring in Multi-Event Tournaments How much is a first place worth? Is the athlete who came first once and third twice better than the athlete who came second three times? Different sports value these positions differently: (60, 54, 48) in IBU biathlon, (25, 18, 15) in F1 racing, (8, 7, 6) in Diamond League athletics. Are these choices based on anything? Is there even a rational way to choose a scoring vector of arbitrary length to rank any number of athletes? In “How should we score athletes and candidates: Geometric scoring rules,” authors A.Y. Kondratev, E. Ianovski, and A.S. Nesterov investigate two approaches to how the problem of choosing a scoring vector for a tournament can be reduced to the choice of a single parameter: an axiomatic approach based on eliminating spoilers and an optimization approach based on maximizing the expected quality of the winner. Intriguingly, the vectors generated by the second approach are uncannily similar to those used in real sporting events. Scoring rules are widely used to rank athletes in sports and candidates in elections. Each position in each individual ranking is worth a certain number of points; the total sum of points determines the aggregate ranking. The question is how to choose a scoring rule for a specific application. First, we derive a one-parameter family with geometric scores that satisfies two principles of independence: once an extremely strong or weak candidate is removed, the aggregate ranking ought to remain intact. This family includes Borda count, generalized plurality (medal count), and generalized antiplurality (threshold rule) as edge cases, and we find which additional axioms characterize these rules. Second, we introduce a one-parameter family with optimal scores: the athletes should be ranked according to their expected overall quality. Finally, using historical data from biathlon, golf, and athletics, we demonstrate how the geometric and optimal scores can simplify the selection of suitable scoring rules, show that these scores closely resemble the actual scores used by the organizers, and provide an explanation for empirical phenomena observed in biathlon and golf tournaments. We see that geometric scores approximate the optimal scores well in events in which the distribution of athletes’ performances is roughly uniform. Funding: This work was supported by HSE University (Basic Research Program, Priority 2030 Program). Supplemental Material: The online appendix is available at https://d