Non-collegial simple games and the nowhere denseness of the set of preference profiles having a core (Q1072944)

In recent papers, somewhat conflicting results on the generic emptiness of the core have been proven. \textit{A. Rubinstein} [Econometrica 47, 511- 514 (1979; Zbl 0416.90006)] has shown that the core is generically empty in the Kannai topology even without a restriction on the dimensionality of the alternative space. \textit{N. Schofield} [J. Math. Econ. 7, 175-192 (1980; Zbl 0432.90010)] finds the core generically empty in the \(Whitney\)-C\({}^{\infty}\) topology, but requires a dimensionality condition. The present paper illuminates the apparent conflict between the results of Rubinstein and Schofield by restating Rubinstein's result in utility space and examining the effect on the nowhere denseness result of (1) shrinking the set of admissible preferences to those representable by smooth functions; (2) generalizing the absolute majority-rule game to the class of non-collegial simple games; and (3) varying the topology on the utility space. It is found that the nowhere denseness result is robust against the first two alterations: even if the dimension of the alternative space equals one, even if the game is not strong so that there are blocking coalitions, the set of utility profiles with nonempty core is nowhere dense in the \(Whitney\)-C\({}^ 0\) topology. Changes in the topology placed on the space of utilities, however, change the result: a restriction on the dimensionality of the alternative space is required before the nowhere denseness result follows.