The Will Rogers Phenomenon and Sequential Migration

Summary

The Will Rogers phenomenon describes when elements are moved or migrate from one set to another and the averages in both sets increase. We provide a mathematical condition for when the phenomenon occurs and when sequential migration results in the phenomenon occurring at each step for any permutation of the migrating elements. We use an example to explore other questions, including the likelihood of the phenomenon occurring and its relationship to Simpson’s paradox.

Acknowledgment

Thank you to two referees for their suggestions to improve the exposition of this paper, especially for the suggestion of the lemma as a way to simplify the proof of Proposition 2.

Additional information

Notes on contributors

Michael A. Jones

Michael A. Jones ([email protected], MR ID 640157) is the Managing Editor and an Associate Editor at the American Mathematical Society’s Mathematical Reviews in Ann Arbor. He is a past editor of Math Magazine and has taught the Mathematics of Decisions, Elections and Games in the Michigan Math and Science Scholars program at the University of Michigan since 2009. He is interested in surprising or paradoxical behavior in probability, fair division and social choice.

Allison Mocny

Allison Mocny ([email protected], MR ID 1584385) is a recent graduate from Loyola Academy in Wilmette, Illinois and a freshman at Case Western Reserve University in Cleveland, Ohio, where she is studying statistics. She was a student in Michael Jones’ Michigan Math and Science Scholars class in 2022.

Jennifer Wilson

Jennifer Wilson ([email protected], MR ID 848485) is Associate Professor of Mathematics at Eugene Lang College, The New School, where she teaches and conducts research in game theory, social choice theory, and fair division. She is particularly interested in the connections between continuous and discrete problems, which offers complementary ways to think about paradoxes such as the Will Rogers phenomenon.