Towards a General Theory of Arbitrary Decisions

Many recent investigations of Computational Decision Theory focused on the problem of making decisions in presence of more than one single criterion. The so called multiple criteria decision theory, in particular, deals with such problems, in depth. In general these investigations tend to be specifically quantitative, and withal in those approaches that look at purely qualitative models, the solution of such problems is a procedure that looks for optimal tradeoff among criteria. In this paper we argue that multiplicity of criteria really exists only when the criteria are strictly incompatible, namely when they provide methods to evaluate decisions that cannot be merged together. We study the problem of making decisions when at least two admissible evaluation criteria are strictly incompatible. Decisions made in such context are named arbitrary. The notion of arbitrary is examined as opposite to the notion of mechanical; we nevertheless argue that such decisions are not irrational. An arbitrary decision is good, generally speaking, whenever it makes sense to provide a justification for it, namely an explanation of the reasoning process that brought us to such a decision. The notion of justification is the core notion of all the theory developed in this paper. A justification should be the explanation of the reasoning process that has brought to the decision we indeed justify. We model justification by means of the notion of perfect decision. A decision is perfect whenever all the applicable criteria select that decision as optimal. A justification is, in the model we present here, a subproblem of the originally posed one for which the decision is perfect. In other words, a justification of a decision is a reduction of a decision problem to a subproblem for which the decision is the best one, namely such that all the criteria agree on it. We analyze the notion of arbitrariness as foundational for a theory of purely qualitative decisions, provide a formal model of such decisions, and study the computational complexity of the problems of exhibiting a justification.