An Impossibility Theorem for Allocation Aggregation

Journal of Philosophical Logic:1-14 (forthcoming)
Abstract
Among the many sorts of problems encountered in decision theory, allocation problems occupy a central position. Such problems call for the assignment of a nonnegative real number to each member of a finite (more generally, countable) set of entities, in such a way that the values so assigned sum to some fixed positive real number s. Familiar cases include the problem of specifying a probability mass function on a countable set of possible states of the world (s=1), and the distribution of a certain sum of money, or other resource, among various enterprises. In determining an s-allocation it is common to solicit the opinions of more than one individual, which leads immediately to the question of how to aggregate their typically differing allocations into a single “consensual” allocation. Guided by the traditions of social choice theory (in which the aggregation of preferential orderings, or of utilities is at issue) decision theorists have taken an axiomatic approach to determining acceptable methods of allocation aggregation. In such approaches so-called “independence” conditions have been ubiquitous. Such conditions dictate that the consensual allocation assigned to each entity should depend only on the allocations assigned by individuals to that entity, taking no account of the allocations that they assign to any other entities. While there are reasons beyond mere simplicity for subjecting allocation aggregation to independence, this radically anti-holistic stricture has frequently proved to severely limit the set of acceptable aggregation methods. As we show in what follows, the limitations are particularly acute in the case of three or more entities which must be assigned nonnegative values summing to some fixed positive number s. For if the set V⊆[0,s] of values that may be assigned to these entities satisfies some simple closure conditions and (as is always the case in practice) Vis finite, then independence allows only for dictatorial or imposed (i.e., constant) aggregation. This theorem builds on and extends a theorem of Bradley and Wagner (Episteme, 9, 91–99 2012) and, when V={0,1}, yields as a corollary an impossibility theorem of Dietrich (Journal of Economic Theory, 126, 286–298 2006) on judgment aggregation
Keywords Allocation aggregation  Dictatorship  Judgment aggregation  Probability
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