It all depends on independence - Daniel Eckert; Frederik Herzberg (Bielefeld University, Institute of Mathematical Economics (IMW), 2009)
Eliaz (2004) has established a 'meta-theorem' for preference aggregation which implies both Arrow's Theorem (1963) and the Gibbard-Satterthwaite Theorem (1973, 1975). This theorem shows that the driving force behind impossibility theorems in preference aggregation is the mutual exclusiveness of Pareto optimality, individual responsiveness (preference reversal) and non-dictatorship. Recent work on judgment aggregation has obtained important generalizations of both Arrow's Theorem (List and Pettit 2003, Dietrich and List 2007a) and the Gibbard-Satterthwaite Theorem (Dietrich and List 2007b). One might ask, therefore, whether the impossibility results in judgment aggregation can be unified into a single theorem, a meta-theorem which entails the judgment-aggregation analogues of both Arrow's Theorem and the Gibbard-Satterthwaite Theorem. For this purpose, we study strong monotonicity properties (among them non-manipulability) and their mutual logical dependences. It turns out that all of these monotonicity concepts are equivalent for independent judgment aggregators, and the strongest monotonicity concept, individual responsiveness, implies independence. We prove the following meta-theorem: Every systematic non-trivial judgment aggregator is oligarchic in general and even dictatorial if the collective judgment set is complete. However, systematicity is equivalent to independence for blocked agendas. Hence, as a corollary, we obtain that every independent (in particular, every individually responsive) non-trivial judgment aggregator is oligarchic. This result is a mild generalization of a similar theorem of Dietrich and List (2008), obtained by very different methods. Whilst Eliaz (2004) and Dietrich and List (2008) use sophisticated combinatorial and logical arguments to prove their results, we utilize the filter method (cf. e.g. Dietrich and Mongin, unpublished) and obtain a much simpler and more intuitive derivation of our meta-theorem.
urn:nbn:de:hbz:361-14469
Keywords: 330; Judgment aggregation; Independence axiom; Monotonicity axiom; Oligarchy; Impossibility results; Non-manipulability; Partial rationality; Nutzenfunktion; Präferenztheorie; Aggregation; Soziale Wohlfahrtsfunktion; Unmöglichkeitstheorem; Theorie