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Structural Instability of the Core

McKelvey, Richard D. and Schofield, Norman (1984) Structural Instability of the Core. Social Science Working Paper, 535. California Institute of Technology , Pasadena, CA. (Unpublished)

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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.

Item Type:Report or Paper (Working Paper)
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Additional Information:Revised. Original dated to July 1984. Published as McKelvey, Richard D., and Norman Schofield. "Structural instability of the core." Journal of Mathematical Economics 15.3 (1986): 179-198.
Group:Social Science Working Papers
Series Name:Social Science Working Paper
Issue or Number:535
Record Number:CaltechAUTHORS:20170908-145130668
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:81270
Deposited By: Jacquelyn Bussone
Deposited On:19 Sep 2017 21:46
Last Modified:03 Oct 2019 18:40

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