Structural instability of the core

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.