Aggregating Credences into Beliefs: Agenda Conditions for Impossibility Results

TL;DR

This paper explores the logical conditions under which impossibility theorems in belief aggregation occur, identifying specific agenda properties that lead to oligarchies, triviality, or impossibility results.

Contribution

It generalizes previous impossibility results using an agenda-theoretic approach, pinpointing exact logical conditions for belief aggregation impossibilities.

Findings

Path-connectedness and even-negatability lead to oligarchy results.

Negation-connectedness results in triviality when adding anonymity.

Blockedness causes impossibility when adding completeness and consistency.

Abstract

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.