Oligarchies in Judgment Aggregation
Klaus Nehring · 2006
In the general framework of abstract binary Arrowian aggregation introduced by Wilson (1975), we characterize the aggregation problems in which the only Arrowian aggregators are oligarchic. 1 In a range of different settings, social choice theory has produced a large number of impossibility results driven by Arrow-like independence conditions. Frequently, these impossibilities take the form of a dictatorship, as in Arrow’s classical contribution. Early papers by Wilson (1975) and Rubinstein and Fishburn (1986) attempted setting and providing sufficient combinatorial conditions that yielded a version of Arrow’s original result as a special case. Under a natural monotonicity assumption, the first characterization in this framework was obtain by Nehring and Puppe (2002,2005) (stated there in a property-space framework in the context of an analysis of strategy-proof social choice); by dropping this monotonicity assumption, this result was subsequently generalized by Dokow and Holzman (2005). Dictatorial impossibilities cover only a fraction of the important (near-)impossibility results in the literature. Frequently, the structure of the aggregated space immediately gives rise to minimal possibilities in the form of a unanimity or, more generally, an oligarchic rule. For example, in the aggregation of partial rather than weak orders, one can consistently aggregate individual preferences by asserting a preference between two alternatives at the social level if and only if a fixed group of agents (the “oligarchy”) asserts this preference. In this case, the interesting possibility question is whether there exist Arrowian aggregation rules different from these obvious, yet rather degenerate ones. Quite often, as in the case of partial orders, see Gibbard (1969), the answer is negative, leading to an oligarchic impossiblity result. Other instances occur in the aggregation of equivalence relations (Mirkin 1975, Fishburn-Rubinstein 1986) and in judgment aggregation (Nehring-Puppe 2005b, Dietrich-List 2006, Dokow-Holzman 2006). In this note, we offer a general characterization when such oligarchic impossibilities arise, providing a counterpart to the characterization of dictatorial impossibilities in