# Some impossibilities of ranking in generalized tournaments

## TL;DR

This paper demonstrates fundamental impossibilities in ranking players in generalized tournaments when applying axiomatic principles like self-consistency and order preservation simultaneously.

## Contribution

It proves that the axiomatic properties of self-consistency and order preservation cannot both be satisfied in generalized tournament rankings.

## Key findings

- Self-consistency and order preservation are incompatible axioms.
- The impossibility holds for the universal domain of generalized tournaments.
- Highlights limitations of current ranking axioms in complex competitive settings.

## Abstract

In a generalized tournament, players may have an arbitrary number of matches against each other and the outcome of the games is measured on a cardinal scale with a lower and upper bound. An axiomatic approach is applied to the problem of ranking the competitors. Self-consistency requires assigning the same rank for players with equivalent results, while a player showing an obviously better performance than another should be ranked strictly higher. According to order preservation, if two players have the same pairwise ranking in two tournaments where the same players have played the same number of matches, then their pairwise ranking is not allowed to change in the aggregated tournament. We reveal that these two properties cannot be satisfied simultaneously on this universal domain.