Michel Balinski (1933–2019): The Architect of Fair Play
Michel Balinski was a polymathic figure in the world of applied mathematics and operations research. Over a career spanning six decades, he transitioned from solving abstract geometric puzzles to tackling some of the most pressing problems in democratic governance. His work provided the mathematical foundation for how we apportion seats in legislatures and how we might more fairly elect leaders.
1. Biography: A Transatlantic Intellectual Journey
Michel Ludwik Balinski was born on October 6, 1933, in Geneva, Switzerland, into a family of Polish diplomats and intellectuals. His grandfather, Stanisław Wojciechowski, served as the second President of Poland (1922–1926). Following the turmoil of World War II, Balinski moved to the United States, where he would receive a quintessential Ivy League education.
He earned his B.A. from Williams College (1954), an M.S. in Economics from MIT (1956), and his Ph.D. in Mathematics from Princeton University in 1959. At Princeton, he studied under the legendary Albert W. Tucker, a pioneer in game theory and non-linear programming.
Balinski’s academic career was marked by a rare mobility between institutions and continents:
- 1960s–1970s: He held positions at Princeton, the City University of New York (CUNY), and Yale University.
- 1980s–2019: In a significant mid-career shift, he moved to France, becoming a Directeur de recherche at the CNRS and the École Polytechnique in Paris.
This move allowed him to bridge the American school of operations research with the European tradition of social choice theory, a synthesis that would define his later work.
2. Major Contributions: Polytopes and Politics
Balinski’s intellectual output can be divided into two primary phases: polyhedral combinatorics and the mathematics of social choice.
Balinski’s Theorem (1961)
In his early career, Balinski made a landmark discovery in the geometry of convex polytopes. Balinski’s Theorem states that the graph formed by the vertices and edges of a $d$-dimensional convex polytope is $d$-connected. This means that at least $d$ vertices must be removed to disconnect the remaining graph. This result remains a fundamental pillar in the study of linear programming and the structure of algorithms like the Simplex method.
The Balinski-Young Theorem on Apportionment
Collaborating with H. Peyton Young, Balinski revolutionized the study of "apportionment"—the process of dividing legislative seats among states or districts based on population. They proved that no apportionment method can simultaneously satisfy three intuitive "fairness" criteria:
- The Stay-within-Quota Property: A state should never receive more or fewer seats than its population strictly entitles it to (rounded up or down).
- Population Monotonicity: A state should not lose a seat if its population increases while others decrease.
- The Alabama Paradox avoidance: A state should not lose a seat simply because the total number of seats in the legislature increases.
Their work led to the endorsement of the Webster/Sainte-Laguë method as the most mathematically robust way to handle these contradictions.
Majority Judgment
In his final decades, Balinski (alongside Rida Laraki) proposed a revolutionary voting system called Majority Judgment. They argued that traditional voting (where you pick one candidate) is mathematically "noisy" and prone to paradoxes. Instead, they proposed that voters should grade every candidate (e.g., "Excellent," "Good," "Poor"). The winner is the candidate with the highest median grade, a system that effectively eliminates the "spoiler effect" and provides a more nuanced reflection of public will.
3. Notable Publications
Balinski was a prolific author whose books are considered definitive texts in their respective sub-fields:
- "On the Graph of a Convex Polytope" (1961): The seminal paper in Journal of the Society for Industrial and Applied Mathematics that established his reputation in geometry.
- "Integer Programming: Methods, Uses, Computation" (1965): One of the first comprehensive surveys of the field, published in Management Science.
- "Fair Representation: Meeting the Ideal of One Man, One Vote" (1982, with H. Peyton Young): This book is the "bible" of apportionment theory, analyzing the history and mathematics of the US Congress.
- "Majority Judgment: Measuring, Ranking, and Electing" (2010, with Rida Laraki): Published by MIT Press, this work outlines his radical proposal for electoral reform.
4. Awards & Recognition
Balinski’s contributions were recognized by the highest bodies in mathematics and management science:
- The Frederick W. Lanchester Prize (1982): Awarded for Fair Representation, recognizing the best contribution to operations research in the English language.
- The John von Neumann Theory Prize (1997): One of the most prestigious awards in the field, given for his fundamental contributions to optimization and social choice.
- Honorary Doctorate: Awarded by the University of Augsburg (2004).
- INFORMS Fellow: Elected for his life-long impact on the management sciences.
5. Impact & Legacy
Balinski’s legacy is twofold. In pure mathematics, his work on polytopes is taught in every graduate course on combinatorics. In applied policy, he changed how governments view the "fairness" of their structures.
He was the founding editor-in-chief of the journal Mathematical Programming (1971), which became the premier venue for research in optimization. His late-life advocacy for "Majority Judgment" continues to gain traction; the system has been used in various political primaries in France and by professional organizations worldwide to reach more stable consensus than traditional voting allows.
6. Collaborations
Balinski was a deeply collaborative researcher. His most significant partnerships included:
- H. Peyton Young: Together, they solved the centuries-old debate regarding the "correct" way to apportion the US House of Representatives.
- Rida Laraki: His primary collaborator in his later years, with whom he developed the theory of grading-based social choice.
- Albert W. Tucker: His mentor, who integrated him into the "Princeton Circle" of game theorists that included John Nash and Harold Kuhn.
7. Lesser-Known Facts
- A Family of Leaders: His grandfather was the President of Poland, and his father was a high-ranking diplomat. This heritage likely fueled Michel’s lifelong interest in the intersection of mathematics and the "just" administration of a state.
- The "Stable Matching" Refinement: While Gale and Shapley are famous for the "Stable Marriage" algorithm, Balinski provided some of the most elegant proofs and extensions of their work, applying it to real-world problems like college admissions.
- Resistance to "Arrow's Paradox": While many mathematicians were discouraged by Kenneth Arrow’s "Impossibility Theorem" (which suggests no perfect voting system exists), Balinski viewed it as a challenge. He argued that the problem wasn't the impossibility of fairness, but that we were asking the wrong questions by using "ranking" instead of "grading."
Michel Balinski passed away on December 4, 2019, in his home in France. He left behind a world that, thanks to his equations, understands just a little bit more about what it means to be fair.