Skip to main content
  • Submit
  • About
    • Editorial Board
    • PNAS Staff
    • FAQ
    • Rights and Permissions
  • Contact
  • Journal Club
  • Subscribe
    • Subscription Rates
    • Subscriptions FAQ
    • Open Access
    • Recommend PNAS to Your Librarian
  • Log in
  • My Cart

Main menu

  • Home
  • Articles
    • Current
    • Latest Articles
    • Special Features
    • Colloquia
    • Collected Articles
    • PNAS Classics
    • Archive
  • Front Matter
  • News
    • For the Press
    • Highlights from Latest Articles
    • PNAS in the News
  • Podcasts
  • Authors
    • Purpose and Scope
    • Editorial and Journal Policies
    • Submission Procedures
    • For Reviewers
    • Author FAQ
  • Submit
  • About
    • Editorial Board
    • PNAS Staff
    • FAQ
    • Rights and Permissions
  • Contact
  • Journal Club
  • Subscribe
    • Subscription Rates
    • Subscriptions FAQ
    • Open Access
    • Recommend PNAS to Your Librarian

User menu

  • Log in
  • My Cart

Search

  • Advanced search
Home
Home

Advanced Search

  • Home
  • Articles
    • Current
    • Latest Articles
    • Special Features
    • Colloquia
    • Collected Articles
    • PNAS Classics
    • Archive
  • Front Matter
  • News
    • For the Press
    • Highlights from Latest Articles
    • PNAS in the News
  • Podcasts
  • Authors
    • Purpose and Scope
    • Editorial and Journal Policies
    • Submission Procedures
    • For Reviewers
    • Author FAQ

New Research In

Physical Sciences

Featured Portals

  • Physics
  • Chemistry
  • Sustainability Science

Articles by Topic

  • Applied Mathematics
  • Applied Physical Sciences
  • Astronomy
  • Computer Sciences
  • Earth, Atmospheric, and Planetary Sciences
  • Engineering
  • Environmental Sciences
  • Mathematics
  • Statistics

Social Sciences

Featured Portals

  • Anthropology
  • Sustainability Science

Articles by Topic

  • Economic Sciences
  • Environmental Sciences
  • Political Sciences
  • Psychological and Cognitive Sciences
  • Social Sciences

Biological Sciences

Featured Portals

  • Sustainability Science

Articles by Topic

  • Agricultural Sciences
  • Anthropology
  • Applied Biological Sciences
  • Biochemistry
  • Biophysics and Computational Biology
  • Cell Biology
  • Developmental Biology
  • Ecology
  • Environmental Sciences
  • Evolution
  • Genetics
  • Immunology and Inflammation
  • Medical Sciences
  • Microbiology
  • Neuroscience
  • Pharmacology
  • Physiology
  • Plant Biology
  • Population Biology
  • Psychological and Cognitive Sciences
  • Sustainability Science
  • Systems Biology

A Social Equilibrium Existence Theorem

Gerard Debreu
PNAS October 1, 1952. 38 (10) 886-893; https://doi.org/10.1073/pnas.38.10.886
Gerard Debreu
Cowles Commission for Research in Economics
  • Find this author on Google Scholar
  • Find this author on PubMed
  • Search for this author on this site
  1. Communicated by J. von Neumann, August 1, 1952

  • Article
  • Authors & Info
  • PDF
Loading

In a wide class of social systems each agent has a range of actions among which he selects one. His choice is not, however, entirely free and the actions of all the other agents determine the subset to which his selection is restricted. Once the action of every agent is given, the outcome of the social activity is known. The preferences of each agent yield his complete ordering of the outcomes and each one of them tries by choosing his action in his restricting subset to bring about the best outcome according to his own preferences. The existence theorem presented here gives general conditions under which there is for such a social system an equilibrium, i.e., a situation where the action of every agent belongs to his restricting subset and no agent has incentive to choose another action.

This theorem has been used by Arrow and Debreu2 to prove the existence of an equilibrium for a classical competitive economic system, it contains the existence of an equilibrium point for an N-person game (see Nash8 and Section 4) and, naturally, as a still more particular case the existence of a solution for a zero-sum two-person game (see von Neumann and Morgenstern, Ref. 11, Section 17.6).

In Section 1 the topological concepts to be used are defined. In Section 2 an abstract definition of equilibrium is presented with a proof of the theorem. In Section 3 saddle points are presented as particular cases of equilibrium points and in connection with the closely related MinMax operator. Section 4 concludes with a short historical survey of results about saddle points, fixed points for multi-valued transformations and equilibrium points.

Only subsets of finite Euclidean spaces will be considered here.

1. Topological Concepts

Two sets in R …

View Full Text
PreviousNext
Back to top
Article Alerts
Email Article

Thank you for your interest in spreading the word on PNAS.

NOTE: We only request your email address so that the person you are recommending the page to knows that you wanted them to see it, and that it is not junk mail. We do not capture any email address.

Enter multiple addresses on separate lines or separate them with commas.
A Social Equilibrium Existence Theorem
(Your Name) has sent you a message from PNAS
(Your Name) thought you would like to see the PNAS web site.
Citation Tools
A Social Equilibrium Existence Theorem
Gerard Debreu
Proceedings of the National Academy of Sciences Oct 1952, 38 (10) 886-893; DOI: 10.1073/pnas.38.10.886

Citation Manager Formats

  • BibTeX
  • Bookends
  • EasyBib
  • EndNote (tagged)
  • EndNote 8 (xml)
  • Medlars
  • Mendeley
  • Papers
  • RefWorks Tagged
  • Ref Manager
  • RIS
  • Zotero
Request Permissions
Share
A Social Equilibrium Existence Theorem
Gerard Debreu
Proceedings of the National Academy of Sciences Oct 1952, 38 (10) 886-893; DOI: 10.1073/pnas.38.10.886
del.icio.us logo Digg logo Reddit logo Twitter logo CiteULike logo Facebook logo Google logo Mendeley logo
  • Tweet Widget
  • Facebook Like
  • Mendeley logo Mendeley

More Articles of This Classification

  • Mathematical picture language program
  • What Chern–Simons theory assigns to a point
  • History in the diagrams of Euclid's Elements
Show more

Related Content

  • Debreu's social equilibrium existence theorem
  • PubMed
  • Google Scholar

Cited by...

  • Debreus social equilibrium existence theorem
  • Speaking Axiomatically: Citation Patterns to Early Articles in General Equilibrium Theory
  • Google Scholar

Similar Articles

You May Also be Interested in

Recent flooding events highlight why flood-risk governance in the United States needs a major overhaul. They also suggest why the necessary refocus on shared responsibility will not be easy.
Opinion: How to achieve better flood-risk governance in the United States
Recent flooding events highlight why flood-risk governance in the United States needs a major overhaul. They also suggest why the necessary refocus on shared responsibility will not be easy.
Image courtesy of Shutterstock/michelmond.
Bridget Scanlon discusses the use of global hydrologic models for studying changes in water storage worldwide.
Global hydrologic models and water storage
Bridget Scanlon discusses the use of global hydrologic models for studying changes in water storage worldwide.
Listen
Past PodcastsSubscribe
PNAS Profile of Dorothy L. Cheney and Robert M. Seyfarth.
PNAS Profile
PNAS Profile of Dorothy L. Cheney and Robert M. Seyfarth.
Researchers estimate the risk of infectious disease transmission on board transcontinental airline flights.
Infectious disease transmission on airplanes
Researchers estimate the risk of infectious disease transmission on board transcontinental airline flights.
Image courtesy of Pixabay/PublicDomainPictures.
Researchers report early evidence of Maya animal management.
Early Maya animal rearing and trade
Researchers report early evidence of Maya animal management.
Proceedings of the National Academy of Sciences: 115 (16)
Current Issue

Submit

Sign up for Article Alerts

Jump to section

  • Article
    • 1. Topological Concepts
    • 2. Equilibrium Points
    • 3. Saddle Points and MinMax Operator
    • 4. Historical Note
    • Acknowledgments
    • Footnotes
    • References
  • Authors & Info
  • PDF
Site Logo
Powered by HighWire
  • Submit Manuscript
  • Twitter
  • Facebook
  • RSS Feeds
  • Email Alerts

Articles

  • Current Issue
  • Latest Articles
  • Archive

PNAS Portals

  • Classics
  • Front Matter
  • Teaching Resources
  • Anthropology
  • Chemistry
  • Physics
  • Sustainability Science

Information for

  • Authors
  • Reviewers
  • Press

Feedback    Privacy/Legal

Copyright © 2018 National Academy of Sciences.