Scarf’s lemma
E776131
mathematical lemma
result in combinatorial topology
result in game theory
result in mathematical economics
Scarf’s lemma is a fundamental result in combinatorial topology and game theory that guarantees the existence of approximate solutions to certain systems, underpinning proofs of equilibrium existence in economics and related fields.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Scarf’s lemma canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T9070909 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Scarf’s lemma Context triple: [Herbert Scarf, notableWork, Scarf’s lemma]
-
A.
Sperner's lemma
Sperner's lemma is a fundamental result in combinatorial topology that guarantees the existence of a fully labeled simplex in certain labeled triangulations, and is widely used to prove fixed-point and equilibrium theorems.
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B.
Knaster–Kuratowski–Mazurkiewicz lemma
The Knaster–Kuratowski–Mazurkiewicz lemma is a fundamental result in combinatorial topology that guarantees the existence of a point common to a family of closed sets covering a simplex under certain intersection conditions, and underlies several fixed-point theorems.
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C.
Shapley–Gale theorem
The Shapley–Gale theorem is a foundational result in cooperative game theory that characterizes stable outcomes in assignment and matching problems, underpinning much of modern market design and matching theory.
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D.
Ky Fan’s lemma
Ky Fan’s lemma is a combinatorial topological result that generalizes Tucker’s lemma and provides conditions guaranteeing the existence of certain balanced or fully labeled simplices in labeled triangulations of spheres or simplices.
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E.
Tucker’s lemma
Tucker’s lemma is a combinatorial analog of the Borsuk–Ulam theorem that provides conditions guaranteeing the existence of certain complementary edge labels in triangulated spheres.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Scarf’s lemma Target entity description: Scarf’s lemma is a fundamental result in combinatorial topology and game theory that guarantees the existence of approximate solutions to certain systems, underpinning proofs of equilibrium existence in economics and related fields.
-
A.
Sperner's lemma
Sperner's lemma is a fundamental result in combinatorial topology that guarantees the existence of a fully labeled simplex in certain labeled triangulations, and is widely used to prove fixed-point and equilibrium theorems.
-
B.
Knaster–Kuratowski–Mazurkiewicz lemma
The Knaster–Kuratowski–Mazurkiewicz lemma is a fundamental result in combinatorial topology that guarantees the existence of a point common to a family of closed sets covering a simplex under certain intersection conditions, and underlies several fixed-point theorems.
-
C.
Shapley–Gale theorem
The Shapley–Gale theorem is a foundational result in cooperative game theory that characterizes stable outcomes in assignment and matching problems, underpinning much of modern market design and matching theory.
-
D.
Ky Fan’s lemma
Ky Fan’s lemma is a combinatorial topological result that generalizes Tucker’s lemma and provides conditions guaranteeing the existence of certain balanced or fully labeled simplices in labeled triangulations of spheres or simplices.
-
E.
Tucker’s lemma
Tucker’s lemma is a combinatorial analog of the Borsuk–Ulam theorem that provides conditions guaranteeing the existence of certain complementary edge labels in triangulated spheres.
- F. None of above. chosen
Statements (40)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical lemma
ⓘ
result in combinatorial topology ⓘ result in game theory ⓘ result in mathematical economics ⓘ |
| appliesTo |
balanced games
ⓘ
cooperative games with transferable utility ⓘ systems of linear inequalities ⓘ |
| consequence |
existence of approximate competitive equilibria in exchange economies
ⓘ
nonemptiness of the core for balanced games ⓘ |
| developedBy |
Herbert E. Scarf
ⓘ
linked to:
Herbert Scarf
|
| field |
combinatorial topology
ⓘ
game theory ⓘ general equilibrium theory ⓘ mathematical economics ⓘ |
| guarantees | existence of approximate solutions to certain systems ⓘ |
| hasProperty |
constructive in nature
ⓘ
finite combinatorial formulation ⓘ provides approximate rather than exact solutions ⓘ |
| influenced |
algorithmic game theory
ⓘ
computational general equilibrium analysis ⓘ |
| mathematicalDomain |
combinatorics
ⓘ
economic theory ⓘ optimization theory ⓘ topology ⓘ |
| namedAfter |
Herbert E. Scarf
ⓘ
linked to:
Herbert Scarf
|
| relatedTo |
Brouwer fixed-point theorem
ⓘ
Kakutani fixed-point theorem ⓘ Shapley–Scarf housing market model ⓘ Sperner’s lemma ⓘ
linked to:
Sperner's lemma
core of a cooperative game ⓘ |
| topic |
approximate fixed points
ⓘ
combinatorial representations of equilibria ⓘ |
| underpins | proofs of equilibrium existence in economics ⓘ |
| usedFor |
constructive proofs of equilibrium existence
ⓘ
proving existence of approximate competitive equilibria ⓘ proving existence of core allocations in cooperative games ⓘ proving existence of equilibria ⓘ |
| usedIn |
proofs of core existence theorems
ⓘ
proofs of equilibrium existence in exchange economies ⓘ proofs of equilibrium existence in production economies ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Scarf’s lemma Description of subject: Scarf’s lemma is a fundamental result in combinatorial topology and game theory that guarantees the existence of approximate solutions to certain systems, underpinning proofs of equilibrium existence in economics and related fields.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.