Ranking multidimensional alternatives and uncertain prospects

Abstract

We introduce a ranking of multidimensional alternatives, including uncertain prospects as a particular case, when these objects can be given a matrix form. This ranking is separable in terms of rows and columns, and continuous and monotonic in the basic quantities. Owing to the theory of additive separability developed here, we derive very precise numerical representations over a large class of domains (i.e., typically not of the Cartesian product form). We apply these representations to (1) streams of commodity baskets through time, (2) uncertain social prospects, (3) uncertain individual prospects. Concerning (1), we propose a finite horizon variant of Koopmans's (1960) [25] axiomatization of infinite discounted utility sums. The main results concern (2). We push the classic comparison between the ex ante and ex post social welfare criteria one step further by avoiding any expected utility assumptions, and as a consequence obtain what appears to be the strongest existing form of Harsanyi's (1955) [21] Aggregation Theorem. Concerning (3), we derive a subjective probability for Anscombe and Aumann's (1963) [1] finite case by merely assuming that there are two epistemically independent sources of uncertainty.

Bayesian Decision Theory and Stochastic Independence

“…The first adapts a result already obtained in Mongin and Pivato (2015) [16]. We develop it here because it neatly exemplifies how Bayesian decision theory can approach stochastic independence.…”

“…This theorem is Corollary 1(c) in Mongin and Pivato (2015) [16], but recast autonomously and in a different formal language, so as to facilitate the discussion of stochastic independence.…”

Fair social decision under uncertainty and belief disagreements

“…where {φ is (·| p)} i∈I,s∈S is a family of continuous, increasing and real-valued functions (see Aczel and Maksa 1996;Mongin and Pivato 2015;Zuber 2016). The significance of this representation theorem for us is that the weak separability conditions imply the so-called strong separability condition, which is the most useful necessary and sufficient condition for (social) preferences to be additively separable (see, for instance, Debreu 1960;Fishburn 1970).…”

“…The compatibility between the ex ante Pareto principle and the ex post Pareto principle can be achieved only if (1) the individuals and the planner follow the SEU axioms (Savage 1972) and (2) they have the same subjective beliefs (Mongin and Pivato 2015). This is a very stringent conclusion.…”

“…In other words, inequality aversion and uncertainty aversion of the 'social observer' are two sides of the same coin. 1 However, it has been shown (Mongin and Pivato 2015;Zuber 2016) that once we impose both approaches to be commutative with each other, there is no possibility of using any other method of social aggregation aside from utilitarianism, which excludes not only any concern of fairness but also any concern of uncertainty. This impossibility result thus forces us to choose between the two approaches if we want a sufficiently sharp and meaningful social welfare criterion.…”