The Threshold Decision Making

We investigate the following problem of construction of a social decision rule. Given a set of n agents, each agent evaluates alternatives from a finite set X using complete and transitive preferences, and we look for a complete and transitive social preference over the alternatives. This kind of aggregation has been considered in many publications, beginning with the seminal work by Arrow [6]. In order to solve the problem, two ways have been in general proposed. Arrow’s kind of axiomatics can be described as the local aggregation [1]; in other words, the aggregation is done on the basis of pairwise comparisons. Another way is to use certain non-local procedures, e.g., positional rules, for which axiomatic approaches have been developed rather poorly [8, 9]. On the other hand, an application of the Borda rule is often not adequate, since any summation of ranks has a compensatory nature: a low evaluation of some alternative by one agent can be compensated by a high evaluation of another agent. Thus, if we would like to take carefully into account low evaluations of alternatives, the Borda rule or its counterparts cannot be used. We present an axiomatic construction of the new aggregation procedure called the ‘threshold rule’. The axioms used are Pairwise compensation, Pareto domination, Non-compensatory threshold and Contraction ([2]–[5], [7]). The Pairwise compensation axiom means that if all agents but two evaluate two alternatives equally, and the two agents put ‘mutually inverse’ grades, then the two alternatives have the same rank in the social decision. The Pareto domination axiom states that if the grades of all agents for one alternative are not less than for the second alternative and the grade of at least one agent for the first alternative is strictly greater than that of the second one, then in the social ranking the first alternative has a higher rank than the second one.