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
John F. Nash Jr., 2005

Chapter 3 · Bargaining, Commitment, and Repeated Cooperation

John F. Nash Jr.

1928 – 2015

Photograph: Videesha, John Nash at Warwick Economics Summit, 2005, via Wikimedia Commons; released into the public domain. Shown toned. File and licence

The problem on the desk

Can a finite game with several players have a profile of choices from which nobody gains by changing alone? Nash framed the answer as a fixed-point problem, extending equilibrium beyond the two-person zero-sum setting.1

A life in moments

  1. 1928

    Bluefield

    Born on 13 June.2

  2. 1948

    Carnegie Institute of Technology · Pittsburgh

    Completed bachelor’s and master’s degrees.3

  3. 1950

    Princeton University · Princeton

    Completed his doctorate on non-cooperative games.3

  4. 1950

    Princeton University · Princeton

    Published his short equilibrium-existence note.1

  5. 1994

    Shared the economics prize with Harsanyi and Selten.3

  6. 2015

    Shared the Abel Prize with Louis Nirenberg.3

  7. 2015

    Monroe Township

    Died on 23 May.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

A self-countering n-tuple is called an equilibrium point.

Equilibrium Points in N-Person Games, 19505

A story

A shorter proof, with a named suggestion

The 1950 note acknowledges David Gale for suggesting Kakutani’s theorem to simplify the proof. Its text begins on page 48 and ends on page 49; the paper is short, but its bibliographic extent is two pages.4

Around Nash

  • suggested the fixed-point route David Gale The equilibrium note explicitly acknowledges Gale’s proof-simplifying suggestion.5
  • shared a prize John C. Harsanyi Shared the 1994 economics prize with Harsanyi and Selten.3
  • shared a prize Reinhard Selten Shared the 1994 economics prize with Selten and Harsanyi.3

Recorded as advisor Albert William Tucker Recorded as students Seth Patinkin 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 who studied at the Carnegie Institute of Technology and received his doctorate from Princeton in 1950 with a thesis on non-cooperative games. He taught at the Massachusetts Institute of Technology and later worked at Princeton. He introduced the equilibrium concept that bears his name and an axiomatic solution of the bargaining problem, and proved fundamental results in differential geometry and partial differential equations. He shared the 1994 Nobel Memorial Prize in Economic Sciences with John Harsanyi and Reinhard Selten, and the 2015 Abel Prize with Louis Nirenberg.

Main works “The bargaining problem” (Econometrica 18, 1950); “Non-cooperative games” (Annals of Mathematics 54, 1951); “Two-person cooperative games” (Econometrica 21, 1953); “The imbedding problem for Riemannian manifolds” (Annals of Mathematics 63, 1956).

To read further

Sources (5)
  1. Equilibrium Points in N-Person Games, John F. Nash Jr., 1950; PNAS 36(1), pp. 48–49; proof and acknowledgement on p. 49
  2. John F. Nash Jr.: Facts, Nobel Prize, 1994; Born; died; prize motivation
  3. Tragic but meaningful life: legendary Princeton mathematician John Nash dies, Morgan Kelly; Princeton University, 2015; 27 May; education, Princeton appointments and prizes
  4. Equilibrium Points in N-Person Games, John F. Nash Jr., 1950; pp. 48–49 and acknowledgement
  5. Equilibrium Points in N-Person Games, John F. Nash Jr., 1950; p. 49

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.