A linear algebra approach to non-transitive expected utility

The purpose of this paper is twofold. One of the aims is to give a view of P. C. Fishburn's non-transitive expected utility through geometric linear a lgebra. The other is to give an application of these ideas to ii social cho ice problem. Actually, the problem under consideration arises from the theo ry of dynamical systems of structures in a society as studied in [All] and [Om1]. A variation of this model as presented in [Om2] leads to the problem of how macro structures of a society form coalitions. Their possible irrat ional behaviour may be treated using non-transitive expected utility.