ÉtudeScience

Mathematics · 1950

Equilibrium Points in N-Person Games

John F. Nash

Princeton University

Proceedings of the National Academy of Sciences, vol. 36, pp. 48–49 ·

Cited by 8,000+Open access
View original paper

Opens the version of record via DOI

Nash's one-page note proved that every finite game with any number of players has at least one equilibrium — a set of strategies where no player can do better by unilaterally changing. The 'Nash equilibrium' became the central solution concept of game theory.

Foundational to game theory; won the 1994 Nobel Memorial Prize in Economics.

  • Proved existence of an equilibrium point for any finite n-player game.
  • Generalised von Neumann's two-player result to many players and non-zero-sum games.
  • Defined the Nash equilibrium, where no player benefits from a unilateral deviation.

A concise mathematical proof applying the Kakutani (Brouwer) fixed-point theorem to the players' best-response correspondence to guarantee an equilibrium exists.

Keywords

Computer Science & AI

A Note on Two Problems in Connexion with Graphs

Dijkstra · 1959 · Numerische Mathematik

Dijkstra's three-page note introduced two classic graph algorithms, including the shortest-path algorithm that now bears his name. It finds the minimum-cost route from a source to all other nodes in a weighted graph.

Cited by 26,000+

Computer Science & AI

A Mathematical Theory of Communication

Shannon · 1948 · The Bell System Technical Journal

Shannon founded information theory. He defined information quantitatively in terms of entropy, introduced the 'bit', and proved fundamental limits on how much data a noisy channel can carry — and that error-free communication is possible up to that limit.

Cited by 73,000+

Computer Science & AI

On Computable Numbers, with an Application to the Entscheidungsproblem

Turing · 1936 · Proceedings of the London Mathematical Society

Turing introduced an abstract machine — now called the Turing machine — to make 'computation' precise. He used it to prove that some problems are undecidable, settling Hilbert's Entscheidungsproblem in the negative and founding the theory of computation. (Read in 1936; published in the 1937 volume.)

Cited by 9,000+

Étude Science indexes and summarises this work; it is not the publisher. The summary above is written by Étude. For the definitive text, figures, and data, please consult the original publication via the link above. Nash (1950) hold the rights to the original work.