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
Abraham Wald

Chapter 1 · Observation, Beliefs, and Decision

Abraham Wald

1902 – 1950

Photograph: Abraham Wald. Photo credited to Konrad Jacobs, copyright MFO, via Wikimedia Commons, CC BY-SA 2.0 DE (https://creativecommons.org/licenses/by-sa/2.0/de/). Indicate any crop or colour treatment. Shown toned. File and licence

The problem on the desk

How can statistical inference be treated as a choice with consequences? Wald brought estimation and testing into a general decision-function framework and developed sequential analysis alongside it. His Columbia work made the procedure itself an object of mathematical study.1

A life in moments

  1. 1902

    Cluj

    Born on 31 October.2

  2. 1934

    Institute for Business Cycle Research · Vienna

    Became a research associate and statistical specialist.2

  3. 1938

    Columbia University · New York

    Arrived as a Carnegie Corporation fellow in the autumn.3

  4. 1943

    Statistical Research Group, Columbia University · New York

    Worked on estimating aircraft vulnerability from damage to survivors.4

  5. 1945

    Columbia University · New York

    Became professor of mathematical statistics.3

  6. 1950

    Died in an aircraft accident on 13 December during a lecture tour.3

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

My interest in statistics and its application in economics dates from this time, when I became the statistical expert of the Institute.

Columbia: Promotions and Memorial Minute for Abraham Wald, 20206

A story

What the surviving aircraft could reveal

The memoranda begin with the number of aircraft sent and the numbers returning with different hit counts. Wald developed a model to infer vulnerability from this incomplete record; the problem was not simply to count visible holes.5

The popular red-dot illustration and a dramatic meeting-room exchange are not established by this consulted passage.

Around Wald

  • co-author Karl Menger Wald reported collaborating with Menger and co-editing the Vienna mathematical colloquium proceedings.2
  • co-author George B. Dantzig Dantzig’s interview documents their joint publication after overlapping statistical results were recognised.7

Recorded as advisor Karl Menger Recorded as students Herman Chernoff, Meyer Girshick, Charles Stein Mathematics Genealogy Project record

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

Two desks

Wald and Stratonovich Choosing well with limited information. The same problem from two desks →

The card in the book

Born in Kolozsvár (Cluj); doctorate in mathematics in Vienna, where he worked on geometry and then, at the Austrian Institute for Business Cycle Research, on economic models, including early existence results for specified models of a competitive economy. He moved to the United States in 1938 and joined Columbia University, where he founded sequential analysis and built statistical decision theory: a statistical procedure is a decision function, judged by its risk, and the minimax decision function is the one that guards against the worst case. He died in an air accident in India in 1950.

Main works “Statistical decision functions which minimize the maximum risk” (Annals of Mathematics 46, 1945); Sequential Analysis (1947); Statistical Decision Functions (1950).

To read further

Sources (7)
  1. Columbia: Promotions and Memorial Minute for Abraham Wald, Abraham Wald; Columbia faculty; archival transcription by Irwin Collier, 2020; Memorial Minute; Annex B and supporting faculty assessment
  2. Columbia: Promotions and Memorial Minute for Abraham Wald, Abraham Wald; Columbia faculty; archival transcription by Irwin Collier, 2020; Annex B
  3. Columbia: Promotions and Memorial Minute for Abraham Wald, Abraham Wald; Columbia faculty; archival transcription by Irwin Collier, 2020; Memorial Minute
  4. A Method of Estimating Plane Vulnerability Based on Damage of Survivors, Abraham Wald, 1943; 1980 CNA reprint, CRC 432; Part I, Introduction, printed p. 1 (PDF p. 10)
  5. A Method of Estimating Plane Vulnerability Based on Damage of Survivors, Abraham Wald, 1943; Part I, Introduction, printed p. 1
  6. Columbia: Promotions and Memorial Minute for Abraham Wald, Abraham Wald; Columbia faculty; archival transcription by Irwin Collier, 2020; Annex B, paragraph beginning “In 1934”
  7. An Interview with George B. Dantzig: The Father of Linear Programming, Donald J. Albers with Constance Reid; interviewee George B. Dantzig, 1986; p. 301

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.