The Inversion Paradox and Ranking Methods in Tournaments

This article deals with ranking methods. We study the situation where a tournament between $n$ players $P_1$, $P_2$, \ldots $P_n$ gives the ranking $P_1 \succ P_2 \succ \cdots \succ P_n$, but, if the results of $P_n$ are no longer taken into account (for example $P_n$ is suspended for doping), then the ranking becomes $P_{n-1} \succ P_{n-2} \succ \cdots \succ P_2 \succ P_1$. If such a situation arises, we call it an inversion paradox. In this article, we give a sufficient condition for the inversion paradox to occur. More precisely, we give an impossibility theorem. We prove that if a ranking method satisfies three reasonable properties (the ranking must be natural, reducible by Condorcet tournaments and satisfies the long tournament property) then we cannot avoid the inversion paradox, i.e., there are tournaments where the inversion paradox occurs. We then show that this paradox can occur when we use classical methods, e.g., Borda, Massey, Colley and Markov methods.