Editing Edgeworth's limit theorem (section)

{{Short description|Mathematical theorem}}
{{refimprove|date=February 2014}}
'''Edgeworth's limit theorem''' is an [[economics|economic]] theorem, named after [[Francis Ysidro Edgeworth]], stating that the [[Core (game theory)|core]] of an economy shrinks to the set of [[Walrasian equilibria]] as the number of [[Agent (economics)|agents]] increases to infinity.

That is, among all possible outcomes which may result from [[free market]] exchange or [[barter]] between groups of people, while the precise location of the final settlement (the ultimate division of goods) between the parties is not uniquely determined, as the number of traders increases, the set of all possible final settlements converges to the set of Walrasian equilibria.

Intuitively, it may be interpreted as stating that as an economy grows larger, agents increasingly behave as if they are price-taking agents, even if they have the power to bargain.

Edgeworth (1881) conjectured the theorem, and provided most of the necessary intuition and went some way towards its proof.<ref>{{Cite book |last=Edgeworth |first=Francis Ysidro |url=https://books.google.com/books?id=CElYAAAAcAAJ&dq=Mathematical+psychics&pg=PA1 |title=Mathematical Psychics: An Essay on the Application of Mathematics to the Moral Sciences |date=1881 |publisher=C. K. Paul |language=en}}</ref> Formal proofs were presented under different assumptions by [[Gérard Debreu|Debreu]] and [[Herbert Scarf|Scarf]] (1963)<ref>{{Cite journal |last1=Debreu |first1=Gerard |last2=Scarf |first2=Herbert |date=1963 |title=A Limit Theorem on the Core of an Economy |url=https://www.jstor.org/stable/2525306 |journal=International Economic Review |volume=4 |issue=3 |pages=235–246 |doi=10.2307/2525306 |jstor=2525306 |issn=0020-6598|url-access=subscription }}</ref> as well as [[Robert Aumann|Aumann]] (1964),<ref>{{Cite journal |last=Aumann |first=Robert J. |date=1964 |title=Markets with a Continuum of Traders |url=https://www.jstor.org/stable/1913732 |journal=Econometrica |volume=32 |issue=1/2 |pages=39–50 |doi=10.2307/1913732 |jstor=1913732 |issn=0012-9682|url-access=subscription }}</ref> both proved under conditions stricter than what Edgeworth conjectured. Debreu and Scarf considered the case of a "replica economy" where there is a finite number of agent types and the agents added to the economy to make it "large" are of the same type and in the same proportion as those already in it. Aumann's result relied on an existence of a [[Continuum (measurement)|continuum]] of agents.