Efron's coins and the Linial arrangement

TL;DR

This paper characterizes which tournaments can be represented as dominance graphs of biased coins, linking them to semiacyclic tournaments, and explores the limitations of such representations with larger tournaments.

Contribution

It establishes a correspondence between dominance graphs of biased coins and semiacyclic tournaments, and demonstrates the existence of tournaments that cannot be represented as such.

Findings

Characterization of dominance graphs as semiacyclic tournaments

Example of a 9-vertex tournament representable with biased coins

Example of an 81-vertex tournament not representable as a dominance graph

Abstract

We characterize the tournaments that are dominance graphs of sets of (unfair) coins in which each coin displays its larger side with greater probability. The class of these tournaments coincides with the class of tournaments whose vertices can be numbered in a way that makes them semiacyclic, as defined by Postnikov and Stanley. We provide an example of a tournament on nine vertices that can not be made semiacyclic, yet it may be represented as a dominance graph of coins, if we also allow coins that display their smaller side with greater probability. We conclude with an example of a tournament with $81$ vertices that is not the dominance graph of any system of coins.