Computational complexity to verify the unstability of effectivity function

Abstract

Ann-person game is considered where each player has a preference order over a finite setA of possible alternatives and a rule for social choice is given in the form of an effectivity functionE. The effectivity function is called stable if for any combination of individual preference orders there exists a subset ofA called a core such that any alternative in the core cannot be ‘dominated’ by such individual preferences. It has been shown by Keiding (1985) that the effectivity functionE is stable if and only ifE does not generate any ‘Cycle’. This paper is concerned with the computational complexity of the problem (CYCLE) for determining whether or not a given effectivity function has a Cycle, We show that a familiar NPC problem SATISFIABILITY can be transformed into CYCLE through a polynomial time procedure. This, combined with the fact that CYCLE is an NP problem, implies NP-completeness of CYCLE, and therefore that of verifying the unstability of the effectivity function, thereby formally proving a previously unanswered conjecture.