Questions of consensus among Bayesian investigators from two perspectives are discussed: (1) Inference: What are the agreements in posterior probabilities that result from increasing shared data? (2) Decisions: What are the shared (strict) preferences of Bayesian decisionmakers? When do their shared preferences support coherent compromises? Concerning topic (1), results about asymptotic consensus and certainty with increasing evidence are reported. The findings deal with extending a conclusion of Blackwell and Dubins from pairs of agents to larger communities. The discussion is in a setting rather more general than that of existing results, as i.i.d. data is not required, nor are conditional probability given only non-null events restricted. Increasing evidence creates (almost sure) certainty of the truth and, depending upon the size of the community, it leads to varieties of concensus for conditional (posterior) distributions. Concerning topic (2), the authors report results on the shared agreements of two Bayesian decision makers who have some differences in their probabilities for events and some differences in their utilities for outcomes. The present results are couched in a setting where acts are ‘horse lotteries,’ with ‘coherence’ defined by axiomatic restrictions on preferences over such acts. Subject to a weak Pareto condition, it is established that the only coherent preference schemes are the two agents’ preferences themselves. Moreover, stronger Pareto conditions exclude even these extreme solutions. This negative finding is immune to familiar maneuvers with Arrow’s ‘impossibility’ theorem. In particular, it is insensitive to restrictions on the domain of (pairs of) agents considered. And it is unaffected by allowing ‘interpersonal utilities’ precluded by Arrow’s multipartite ‘independence of Irrelevant Alternatives’ clause. Thus, the answers to the two sets of questions are very different.
