The Inversion Paradox and Ranking Methods in Tournaments
Guillaume Chéze, Etienne Fieux
TL;DR
The paper formalizes the inversion paradox: removing the last-ranked player from a tournament can reverse the order of the remaining players under several ranking methods. It proves an impossibility theorem showing that any natural ranking method that is reducible by Condorcet tournaments and satisfies the long-tournament property must exhibit the paradox for some tournament, and then demonstrates that classical methods such as $Borda$, $Massey$, $Colley$, and $Markov$ can indeed display the paradox. It extends the analysis to cases with few matches using the $\mathcal{Z}_{n,k,l}$ constructions, establishing that for Massey and Colley the paradox occurs iff $k>\ell>0$, and discusses nuanced behavior for Markov. The work emphasizes the inherent instability of ranking in the presence of cycles and near-equal strengths, and discusses practical implications for choosing ranking systems in competitive settings.
Abstract
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.