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
Julia Robinson, 1975

Chapter 8 · Many Agents

Julia Robinson

1919 – 1985

Photograph: George M. Bergman, Julia Robinson (1975, re-scanned), CC BY-SA 4.0, via Wikimedia Commons. Shown toned. File and licence

The problem on the desk

Could George Brown’s repeated best-response procedure actually solve a finite zero-sum game? Robinson proved convergence of the minimax bounds built from accumulated play frequencies. The result does not require the successive pure actions to settle.1

A life in moments

  1. 1919

    St. Louis

    Born on 8 December.2

  2. 1940

    University of California, Berkeley · Berkeley

    Completed her bachelor’s degree.3

  3. 1948

    University of California, Berkeley · Berkeley

    Completed her doctorate with Alfred Tarski.3

  4. 1951

    Published An Iterative Method of Solving a Game.4

  5. 1976

    University of California, Berkeley · Berkeley

    Became a professor.3

  6. 1983–1984

    American Mathematical Society

    Served as society president.5

  7. 1985

    Died on 30 July.5

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

In their own words

What is proved about numbers will be a fact in any universe.

Julia Bowman Robinson, 1919–1985, 1994 (Solomon Feferman)6

A story

The paper’s date matters

The paper was received in July 1950 and published in September 1951. Its opening explicitly credits Brown with the iterative procedure Robinson then analyses.4

Around Robinson

  • built on the work George W. Brown Robinson’s opening attributes the iterative procedure to Brown.4

Advisor in the record Alfred Tarski Mathematics Genealogy Project record

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

The card in the book

Mathematician. She received her doctorate at Berkeley in 1948 under Alfred Tarski, remained associated with Berkeley and became professor there in 1976. Her 1951 paper proved that in every finite zero-sum game the minimax bounds computed from the fictitious-play frequencies converge to the value — the convergence used in this chapter; the actions themselves need not settle — and her work on Hilbert's tenth problem, with Martin Davis and Hilary Putnam, laid the ground for its negative solution by Yuri Matiyasevich in 1970. She was the first woman elected president of the American Mathematical Society (1983–84).

Main works “An iterative method of solving a game” (Annals of Mathematics 54, 1951); “The decision problem for exponential Diophantine equations”, with M. Davis and H. Putnam (1961).

To read further

Sources (6)
  1. An Iterative Method of Solving a Game, Julia Robinson, 1951; Annals of Mathematics 54, pp. 296–301; opening, theorem and proof
  2. Julia Bowman Robinson, 1919–1985, Solomon Feferman, 1994; p. 4
  3. Women at the University of California, Berkeley: Julia Robinson, UC Berkeley Library; Julia Robinson subsection
  4. An Iterative Method of Solving a Game, Julia Robinson, 1951; p. 296
  5. Julia Bowman Robinson, 1919–1985, Solomon Feferman, 1994; p. 23
  6. Julia Bowman Robinson, 1919–1985, Solomon Feferman, 1994; p. 3, epigraph quoting Robinson’s Autobiography, written by Constance Reid

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.