Non-collegial simple games and the nowhere denseness of the set of preference profiles having a core

In recent papers, somewhat conflicting results on the generic emptiness of the core have been proven. Rubinstein (1979) has shown that the core is generically empty in the Kannai topology even without a restriction on the dimensionality of the alternative space. Schofield (1980) finds the core generically empty in the Whitney-C topology, but requires a dimensionality condition. This note explores the apparent conflict by proving a generalization of Rubinstein's result for the Whitney-C⁰ topology, and showing that the result does not extend to the Whitney-C¹ topology without a dimensionality condition.I would like to thank Kim Border and Richard McKelvey for their helpful discussion and comments. This research was supported in part by the National Science Foundation under Grant no. SES 78-2478781