McKelvey, Richard D. and Schofield, Norman
(1986)
*Structural instability of the core.*
Journal of Mathematical Economics, 15
(3).
pp. 179-198.
ISSN 0304-4068.
http://resolver.caltech.edu/CaltechAUTHORS:20171113-170122722

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## Abstract

Let σ be a q-rule, where any coalition of size q, from the society of size n, is decisive. Let w(n,q)= 2q-n+1 and let W be a smooth ‘policy space’ of dimension w. Let U(W)N be the space of all smooth profiles on W, endowed with the Whitney topology. It is shown that there exists an ‘instability dimension’ w∗(σ) with 2≦w∗(σ)≦w(n,q) such that: 1. (i) if w≧w∗(σ), and W has no boundary, then the core of σ is empty for a dense set of profiles in U(W)N (i.e., almost always), 2. (ii) if w≧w∗(σ)+1, and W has a boundary, then the core of σ is empty, almost always, 3. (iii) if w≧w∗(σ)+1 then the cycle set is dense in W, almost always, 4. (iv) if w≧w∗(σ)+2 then the cycle set is also path connected, almost always. The method of proof is first of all to show that if a point belongs to the core, then certain generalized symmetry conditions in terms of ‘pivotal’ coalitions of size 2q−n must be satisfied. Secondly, it is shown that these symmetry conditions can almost never be satisfied when either W has empty boundary and is of dimension w(n,q) or when W has non-empty boundary and is of dimension w(n,q)+1.

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Additional Information: | © 1986, Elsevier Science Publishers B.V. (North-Holland). Received July 1984, final version accepted June 1986. The contribution of the first author is supported, in part, by NSF grant SES-84-09654 to the California Institute of Technology, and that of the second author is based on work supported by NSF grant SES-84-18295 to the School of Social Sciences. University of California at Irvine. Presented at the Panel on Social Choice Theory, the Annual meeting of the Public Choice Society, New Orleans, February 21-23, 1985. Formerly SSWP 535. | ||||||||||||

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Persistent URL: | http://resolver.caltech.edu/CaltechAUTHORS:20171113-170122722 | ||||||||||||

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Deposited By: | Jacquelyn Bussone | ||||||||||||

Deposited On: | 15 Nov 2017 23:34 | ||||||||||||

Last Modified: | 15 Nov 2017 23:34 |

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