Aggregating Credences into Beliefs: Agenda Conditions for Impossibility Results
This work addresses foundational issues in social choice and decision theory for researchers in judgment aggregation, but it is incremental as it builds on previous research to generalize results.
The paper tackles the problem of aggregating individual probabilistic beliefs into collective binary beliefs by identifying exact agenda conditions for impossibility theorems, demonstrating that path-connectedness and even-negatability lead to oligarchy results, negation-connectedness leads to triviality, and blockedness leads to impossibility when adding completeness and consistency.
Binarizing belief aggregation addresses how to rationally aggregate individual probabilistic beliefs into collective binary beliefs. Similar to the development of judgment aggregation theory, formulating axiomatic requirements, proving impossibility theorems, and identifying exact agenda conditions of impossibility theorems are natural and important research topics in binarizing belief aggregation. Building on our previous research on impossibility theorems, we use an agenda-theoretic approach to generalize the results and to determine the necessary and sufficient level of logical interconnection between the issues in an agenda for the impossibility theorems to arise. We demonstrate that (1) path-connectedness and even-negatability constitute the exact agenda condition for the oligarchy result stating that binarizing belief aggregation satisfying proposition-wise independence and deductive closure of collective beliefs yields the oligarchies under minor conditions; (2) negation-connectedness is the condition for the triviality result obtained by adding anonymity to the oligarchy result; and (3) blockedness is the condition for the impossibility result, which follows by adding completeness and consistency of collective beliefs. Moreover, we compare these novel findings with existing agenda-theoretic characterization theorems in judgment aggregation and belief binarization.