Eckmann's impossibility theorem

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Eckmann's impossibility theorem (1954) is a mathematical theorem that implies the following conclusion about social choice theory: it is impossible to combine multiple choices into one in a continuous manner unless the space of choices is trivial in a certain sense (no cycles). In this context, the theorem is a topological analog of the famous Arrow's impossibility theorem (1950).

The theorem was independently rediscovered by Chichilnisky and her collaborators in the 1980s.

Averaging numbers and even locations is a familiar task. For example, the arithmetic mean of $m$ values is computed by this standard formula:

${\text{ the mean of }}x_{1},\ ...\ ,x_{m}={\tfrac {1}{m}}(x_{1}+...+x_{m})\,.$

This formula might not work in a more complex setting. If these are just elements of a group, the group may have no division. If these are points in a subset of a vector space, the set may be non-convex. Here is an axiomatic approach.

Definition. For a set $X$ and integer $m>1$, a function $f:X^{m}\to X$ is called a mean of degree $m$ on $X$ if it is

1. symmetric: $f(s(u))=f(u)$, for any $u\in X^{m}$, and any permutation $s$; and
2. diagonal: $f(x,x,\ ...\ ,x)=x$, for any $x\in X$.

Theorem. There is a mean of degree $m$ on an abelian group $G$ that is also a homomorphism if and only if $G$ allows division by $m$. In that case, the mean is unique and is given by the standard formula above.

An application of the homology functor to the above theorem produces the desired result.

Theorem (Eckmann, 1954). Suppose $X$ is a polyhedron with finitely generated homology groups. If there is a mean of degree $m$ for each $m$ on $X$ that is also a continuous function, then $X$ is contractible.

In the context of social choice theory, a mean is a choice function: $f:X^{m}\to X;$ it combines the choices within $X$ of the $m$ participants, $x_{1},...,x_{m}$, into a single one, $f(x_{1},...,x_{m})$.

In this spirit, the conditions of the theorem are renamed as follows:

• continuity = stability
• symmetry = anonymity
• diagonality = unanimity

The meaning of the theorem is then as follows. It is obvious that there can be no "democratic" procedure that can help the driver and the passenger to decide to go left or right: here $X$ consists of two points. But also there is not such procedure for a group of hikers to decide on the direction to take in the forest: $X$ is the circle. Or for a consortium of companies to decide where on orbit to place their jointly owned satellite: $X$ is the sphere. These three spaces of choices have topological features (cycles or "holes") of three different kinds.

Chichilnisky (1980) proved a version of this theorem in the case when the space of choices $X$ is the space of orderings of alternatives (such as candidates). Baryshnikov (1993) proved that Chichilnisky's theorem implies Arrow's Impossibility Theorem.

References and further reading

• B. Eckmann, Raume mit Mittelbildungen, Comment. Math. Helv. 28 (1954), 329-340 (the original source)
• B. Eckmann, T. Ganea and P. J. Hilton, Generalized means, in: Studies in Mathematical Analysis and Related Topics, Stanford University Press (1962), 82-92
• B. Eckmann, Social choice and topology: A case of pure and applied algebra, Expo. Math. 22 (2004), 385-393 (a review)
• Chichilnisky, Graciela (August 1980). "Social choice and the topology of spaces of preferences" (PDF). Advances in Mathematics. 37 (2): 165–176. doi:10.1016/0001-8708(80)90032-8. ISSN 0001-8708. (the second original source)
• Chichilnisky, Graciela; Heal, Geoffrey (October 1983). "Necessary and sufficient conditions for a resolution of the social choice paradox" (PDF). Journal of Economic Theory. 31 (1): 68–87. doi:10.1016/0022-0531(83)90021-2. ISSN 0022-0531.
• Chichilnisky, Graciela (1998). Topology and Markets. American Mathematical Society. ISBN 9780821871300. Search this book on (a review)
• Y. M. Baryshnikov, Unifying Impossibility Theorems: A Topological Approach, Advances in Applied Mathematics 14. (1993), 404-415 (a review)
• Y. M. Baryshnikov. Topological and Discrete Social Choice: in search of a theory, Social Choice and Welfare 14. (1997), 199-209 (a review)
• Lauwers, L., Topological social choice. Mathematical Social Sciences, (2000), 40(1), 1-39. (collection of papers)
• S. Weinberger, On the Topological Social Choice Model, Journal of Economic Theory 115. (2004), 377-384 (a review)
• Baigent, N., Chapter 18 - Topological theories of social choice. Editors: K. Arrow, A. Sen, K. Suzumura, in Handbook of Social Choice and Welfare, Elsevier, (2011), Volume 2, 301-334. (a review) Category:social choice theory

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