Given the ranking of competitors, how should the prize endowment be allocated? This paper introduces and axiomatically studies the prize allocation problem. We focus on consistent prize allocation rules satisfying elementary solidarity and fairness principles. In particular, we derive several families of rules satisfying anonymity, order preservation, and endowment monotonicity, which all fall between the equal division rule and the winner-takes-all rule. Our results may help organizers to select the most suitable prize allocation rule for rank-order competitions. This paper was accepted by Manel Baucells, behavioral economics and decision analysis. Funding: A. Y. Kondratev is supported by HSE University [Basic Research Program and “Priority 2030” Program].

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.

The paper is devoted to game-theoretic methods for community detection in networks. The traditional methods for detecting community structure are based on selecting dense subgraphs inside the network. Here we propose to use the methods of cooperative game theory that highlight not only the link density but also the mechanisms of cluster formation. Specifically, we suggest two approaches from cooperative game theory: the first approach is based on the Myerson value, whereas the second approach is based on hedonic games. Both approaches allow to detect clusters with various resolutions. However, the tuning of the resolution parameter in the hedonic games approach is particularly intuitive. Furthermore, the modularity-based approach and its generalizations as well as ratio cut and normalized cut methods can be viewed as particular cases of the hedonic games. Finally, for approaches based on potential hedonic games we suggest a very efficient computational scheme using Gibbs sampling.