Saddle Point
Back to the cover

Part I · Foundations

  1. 0 Mathematical Foundations
  2. 1 Observation, Beliefs, and Decision

Part II · Engines

  1. 2 Competing for Allocation: The Blotto Game
  2. 3 Bargaining, Commitment, and Repeated Cooperation
  3. 4 Signals, Disclosure, and Persuasion
  4. 5 Different Models of One Game
  5. 6 Leadership and Rules: Inspection
  6. 7 Decisions in Time
  7. 8 Many Agents
  8. 9 Networks, Dependence, and Resilience
  9. 10 Aggregating Information

Part III · Systems

  1. 11 Architecture of Decision Support
  2. 12 Simulation and Wargaming with AI Agents

Part IV · Limits

  1. 13 Where the Mathematics Stops
Émile Borel, 1918

Chapter 2 · Competing for Allocation: The Blotto Game

Émile Borel

1871 – 1956

Photograph: Auguste Léon, Émile Borel, 6 January 1918. Musée départemental Albert-Kahn / Wikimedia Commons. Public domain in France. Shown toned. File and licence

The problem on the desk

How should a player choose among methods of play when chance and an opponent’s choices both matter? Borel approached symmetric games through probabilities of winning and calculated small cases of randomised play. The general minimax theorem remained a separate achievement.R1

A life in moments

  1. 1871

    Saint-Affrique

    Born on 7 January.2

  2. 1889

    École normale supérieure · Paris

    Entered the École normale supérieure.2

  3. 1920

    Faculty of Sciences · Paris

    Took the chair of probability and mathematical physics.2

  4. 1921–1927

    Académie des sciences · Paris

    Published notes investigating games and randomised methods of play.3

  5. 1928

    Institut Henri Poincaré · Paris

    Served as the first director when the institute opened on 17 November.4

  6. 1956

    Paris

    Died on 3 February.2

Every line rests on a dated source (numbered below) unless it carries a mark: R for an inference, S for a story as told.

A story

An institute rather than a single grant

Asked by George Birkhoff how the International Education Board could support mathematics, Borel proposed an institute. The Institut Henri Poincaré opened in 1928 with probability among its central subjects.4

Around Borel

  • priority question John von Neumann Fréchet’s 1953 advocacy of Borel’s priority prompted a published reply from von Neumann.5

Advisor in the record Gaston Darboux Students in the record Henri Lebesgue, Paul Montel, Georges Valiron Mathematics Genealogy Project record

The genealogy lists only what the database records; a missing name is not a missing teacher.

Two desks

Borel and Neumann Randomisation in adversarial games. The same problem from two desks →

The card in the book

French mathematician, trained at the École Normale Supérieure and later professor at the Faculty of Sciences in Paris, where he held the chair of probability and mathematical physics; he initiated the Institut Henri Poincaré and became its first director in 1928. He contributed to measure theory and probability, studied divergent series and their summation, and proved that almost all real numbers are normal. His notes of 1921–1927 on games of skill developed mixed strategies and asked whether a minimax solution exists in general; von Neumann proved the theorem for finite two-person zero-sum games in 1928.

Main works Leçons sur la théorie des fonctions (1898); “Les probabilités dénombrables et leurs applications arithmétiques” (1909); “La théorie du jeu et les équations intégrales à noyau symétrique” (Comptes Rendus 173, 1921).

To read further

Sources (5)
  1. John von Neumann’s Conception of the Minimax Theorem: A Journey Through Different Mathematical Contexts, Tinne Hoff Kjeldsen, 2001; pp. 62–64
  2. Fonds Émile Borel: repères biographiques, Florence Greffe; Académie des sciences, 2020; Repères biographiques, pp. 2–3
  3. John von Neumann’s Conception of the Minimax Theorem: A Journey Through Different Mathematical Contexts, Tinne Hoff Kjeldsen, 2001; sections 6.1–6.2
  4. History, Institut Henri Poincaré; Origins: inauguration, first director and Birkhoff’s approach
  5. John von Neumann’s Conception of the Minimax Theorem: A Journey Through Different Mathematical Contexts, Tinne Hoff Kjeldsen, 2001; section 6.2

The facts on this page were gathered for the site from the sources listed, each checked against its locator; the book's card is reproduced unchanged.