Popular Admissibility and Parliamentary Stability in sequential Spatial Voting Games

Popular consultations and citizens’ assemblies can restrict the set of politically admissible alternatives before parliamentary decision-making takes place. We study a two-stage spatial voting model in which a first-stage admissibility restriction determines the domain over which a parliamentary voting game operates. The sequential core consists of admissible alternatives that no winning parliamentary coalition can block through another admissible alternative. At the coalition level, the no-blocking problem is a constrained Euclidean Pareto-efficiency problem. Applying the projection characterization of Carrizosa et al. (Operations Research Letters, 14(5):291–295, 1993.) represents the sequential core as the intersection of the Euclidean projections of the convex hulls of winning coalitions’ ideal points onto the admissible domain. This yields invariance to standard blocking conventions, a barycentric support interpretation, and preservation of unrestricted core points under projection, while localizing additional stability to the boundary of a full-dimensional admissibility domain. Our main existence results exploit the effective dimension of the projected no-blocking loci. When these loci are convex and lie in a common affine space of dimension r, an (r + 1)-fold intersection property of winning coalitions guarantees a nonempty sequential core. For symmetric quota rules, a quota above r/(r + 1) of the assembly is sufficient, and this bound is exact for the corresponding coalition-intersection property. Under the leading core-based admissibility rule, the first-stage domain is a polytope. Around almost every exterior point, a whole neighbourhood is projected affinely onto one proper face, so any parliamentary profile in that neighbourhood inherits the lower-dimensional stability condition. In the plane, this reduces the sufficient quota from two-thirds to a strict majority, even when the unrestricted majority core is empty.