Aleksei Y. Kondratev (National Research University Higher School of Economics, Saint Petersburg, Russia): How should we score athletes and candidates: geometric scoring rules

We study how to rank candidates based on individual rankings via positional scoring rules. Each position in each individual ranking is worth a certain number of points; the total sum of points determines the aggregate ranking. Our selection principle is consistency: once one of the candidates is removed, we want the aggregate ranking to remain intact. This principle is crucial whenever the set of the candidates might change and the remaining ranking guides our actions: whom should we interview if our first choice got a better offer? Who gets the cup once the previous winner is convicted of doping? Which movie should a group watch if everyone already saw the recommender system’s first choice? Will adding a spoiler candidate rig the election? Unfortunately, no scoring rule is completely consistent, but there are weaker notions of consistency we can use. There are scoring rules which are consistent if we add or remove a unanimous winner — such as an athlete with suspiciously strong results. Likewise, consistent for removing or adding a unanimous loser — such as a spoiler candidate in an election. While extremely permissive individually, together these two criteria pin down a one-parameter family with the geometric sequence of scores. These geometric scoring rules include Borda count, generalised plurality (medal count), and generalised antiplurality (threshold rule) as edge cases, and we provide elegant new axiomatisations of these rules. Finally, we demonstrate how the one-parameter formulation can simplify the selection of suitable scoring rules for particular scenarios.

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