Characterizing competition ranks within a comprehensive family of position operators

There is only one way to assign positions to objects arranged in linear orders: following the sequence of natural numbers (1, 2, 3, 4, …). However, in weak orders, where ties arise, there are different possibilities to assign positions to tied objects. In this paper, we focus mainly on three relevant cases: the standard, modified, and fractional ranks. They are differentiated by the spaces that appear after, before, or on either side of the position values corresponding to the objects that are in a tie. For instance, if two objects are tied and are located immediately below the top object, these ranks assign the positions (1, 2, 2, 4, …), (1, 3, 3, 4, …), and (1, 2.5, 2.5, 4, …), respectively. Collectively, and because of the common properties shown here, we call them “competition ranks”. In this paper, we characterize a parameterized family of position operators which includes the competition ranks. We also provide specific axiomatizations of each of them, taking into account the spaces in the sequence of assigned position numbers. It is shown why the dense rank (1, 2, 2, 3, …), another position operator where gaps do not appear, is an essentially different approach. Furthermore, interesting duality relationships are revealed between the competition ranks and between the properties introduced to characterize them, which allow us to understand their internal logic and connections. Different examples, mainly from sports, bibliometrics, etc., illustrate the introduced concepts.