The "No Justice in the Universe" phenomenon: why honesty of effort may not be rewarded in tournaments

TL;DR

This paper investigates the paradox where honest and symmetric tournaments can still be unfair, favoring worse players, and constructs explicit examples demonstrating this phenomenon for any number of players.

Contribution

It proves the existence of honest, symmetric yet unfair tournaments and provides explicit constructions for any number of players, extending previous results.

Findings

Explicit examples of unfair tournaments for any number of players.

For three players, the set of win-probability vectors forms a convex polygon.

Partial results and conjectures for larger numbers of players.

Abstract

In 2000 Allen Schwenk, using a well-known mathematical model of matchplay tournaments in which the probability of one player beating another in a single match is fixed for each pair of players, showed that the classical single-elimination, seeded format can be "unfair" in the sense that situations can arise where an indisputibly better (and thus higher seeded) player may have a smaller probability of winning the tournament than a worse one. This in turn implies that, if the players are able to influence their seeding in some preliminary competition, situations can arise where it is in a player's interest to behave "dishonestly", by deliberately trying to lose a match. This motivated us to ask whether it is possible for a tournament to be both honest, meaning that it is impossible for a situation to arise where a rational player throws a match, and "symmetric" - meaning basically that the rules treat everyone the same - yet unfair, in the sense that an objectively better player has a smaller probability of winning than a worse one. After rigorously defining our terms, our main result is that such tournaments exist and we construct explicit examples for any number n >= 3 of players. For n=3, we show (Theorem 3.6) that the collection of win-probability vectors for such tournaments form a 5-vertex convex polygon in R^3, minus some boundary points. We conjecture a similar result for any n >= 4 and prove some partial results towards it.