Equilibrium Points in N-Person Games

John F. Nash · Proceedings of the National Academy of Sciences · 1950 · economics · 10.1073/pnas.36.1.48 · record nash1950-equilibrium
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Where this record's text came from. faceted 2026-08 by reading the locally held text (pdftext); every evidence string verified as an exact substring of that text before merging. The extracted text is a raw PNAS page dump: Nash's note is preceded by the reference list of an unrelated MacLane/Whitehead paper and followed by the opening of a Polya note, and OCR has scrambled word order inside two sentences (the limitation quote is a contiguous but internally jumbled span, quoted exactly as it appears). method has no match in the suggested vocabulary - the whole contribution is a two-page existence proof, so 'mathematical-proof' is used. The paper also relates itself to the von Neumann-Morgenstern minimax result, but in the body that work is referred to only as 'the "main theorem"2', with the authors' names appearing solely in the (unquotable) footnote list, so no claim_relation is recorded for it. Replication status is 'untested' because a theorem is not the kind of claim the paper submits to replication and the text says nothing about replication.

The quality of the text these quotes were cut from. corrupt-text-layer: extraction lost inter-word spaces (wordsjammedtogether); 1950 PNAS scan, no clean OA source found. Not enriched. [AMENDED 2026-08-23 - THE COPY WAS NEVER NAMED AND IT IS THE WRONG ONE. The space loss is in the web page we fetched (space ratio 0.034). an internal file is intact at 0.171 and is the copy all 13 live quotes resolve against. 'Not enriched' therefore records a decision taken against the publishing artefact, not against the grounding text.]

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Tags on this paper

Every tag the record carries, each linking to every other paper in the corpus that carries it. A tag appears here whether or not its words occur in the article — the spans below are only the ones a sentence in this paper licenses. Of the 4 concept tags on this paper, none is grounded in a quoted sentence.

Nash equilibrium non-cooperative game mixed strategy fixed-point theorem Because the countering correspondence is convex-valued and has a closed graph, a fixed-point theorem applies to it, and any fixed point of that correspondence is by definition a self-countering n-tuple, i.e. an equilibrium. n-person game equilibrium point pay-off functions mapping of the product space mathematical-proof

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relation to other work	`claim_relations`	how this paper stands to another: agrees, contradicts, extends, qualifies
claim type	`claim_type`	the shape of the claim: existence, mechanism, method, critique, prediction, null
content tag	`content_tags`	a finding, definition, quantity, statistic or variable named in the paper
limitation	`limitations`	what the paper says its evidence cannot support
mechanism	`mechanisms`	how the reported effect is said to come about
method	`method`	how the paper produced its evidence
phenomenon	`phenomenon`	the effect or regularity under study
transfer	`transfers`	a method or concept carried in from another field

The paper

The full text is not reproduced here. This article is not cleared for redistribution, so the page carries the tag layer only. Its evidence sentences are withheld; the grounding status remains with each tag, and the article itself is at https://doi.org/10.1073/pnas.36.1.48. The reading view exists for the papers whose licence permits it.

Every tag on this page

claim_relations 1 · claim_type 1 · content_tags 9 · limitations 1 · mechanisms 1 · method 1 · phenomenon 1 · transfers 1

This record holds no relations between concepts, controls or senses of a term. A layer is empty when the tagger recorded nothing in it — which is not the same as the paper having nothing to record.

facet	tag	field	whose words
claim_type	existence Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`claim_type`	the paper's words
phenomenon	equilibrium points in n-person games Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`phenomenon`	the paper's words
method	mathematical-proof Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: mathematical-proof	`method`	the paper's words
mechanisms	mechanisms Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: Because the countering correspondence is convex-valued and has a closed graph, a fixed-point theorem applies to it, and any fixed point of that correspondence is by definition a self-countering n-tuple, i.e. an equilibrium.	`mechanisms[0]`	the tagger's words
limitations	Equilibrium points are not interchangeable in general: outside the two-person zero-sum case, distinct equilibrium points need not give the players the same expectations. Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`limitations[0]`	the tagger's words
transfers	The existence proof imports Kakutani's fixed-point theorem for convex-valued closed-graph set correspondences and applies it to the game's countering correspondence. Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`transfers[0]`	the tagger's words
claim_relations	builds-on → Kakutani's fixed point theorem Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`claim_relations[0]`	the tagger's words
content_tags	Every finite n-person game has at least one equilibrium point. Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`content_tags.findings[0]`	the tagger's words
content_tags	The set of n-tuples that counter a given n-tuple is convex. Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`content_tags.findings[1]`	the tagger's words
content_tags	Continuity of the pay-off functions implies that the graph of the countering correspondence is closed. Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`content_tags.findings[2]`	the tagger's words
content_tags	Outside the two-person zero-sum case, different equilibrium points need not give the players the same expectations. Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. about this quote: THE QUOTE'S OPENING WAS MOVED BY THE TEXT EXTRACTION: the paper's sentence begins 'In this case any two equilibrium' (printed earlier in this copy, inside the sentence before), and 'this case' is the two-person zero-sum case.	`content_tags.findings[3]`	the tagger's words
content_tags	In the two-person zero-sum case, existence of an equilibrium point is equivalent to von Neumann's main theorem. Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above.	`content_tags.findings[4]`	the tagger's words
content_tags	n-person game Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: n-person game	`content_tags.definitions[0]`	the tagger's words
content_tags	equilibrium point Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: equilibrium point	`content_tags.definitions[1]`	the tagger's words
content_tags	pay-off functions Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: pay-off functions	`content_tags.definitions[2]`	the tagger's words
content_tags	mapping of the product space Quotation withheld — this record is not cleared for redistribution. The sentence is in the paper at the link above. also in: mapping of the product space	`content_tags.definitions[3]`	the tagger's words

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