(Non-)Commutative Aggregation

TL;DR

This paper critically examines the normative role of commutativity in belief aggregation, proposing a weakened version and exploring its implications for various aggregation rules and decision-making axioms.

Contribution

It introduces a weakened form of commutativity, analyzes its role in belief aggregation, and links it to classic decision-making axioms through a thought experiment.

Findings

Weakening of commutativity affects belief aggregation rules.

Equivalence between relaxations of commutativity and decision axioms.

Characterization of aggregation rules via dual-selves game.

Abstract

Commutativity is a normative criterion of aggregation and updating stating that the aggregation of expert posteriors should be identical to the update of the aggregated priors. I propose a thought experiment that raises questions about the normative appeal of Commutativity. I propose a weakened version of Commutativity and show how that assumption plays central roles in the characterization of linear belief aggregation, multiple-weight aggregation, and an aggregation rule which can be viewed as the outcome of a game played by "dual-selves," Pessimism and Optimism. Under suitable conditions, I establish equivalences between various relaxations of Commutativity and classic axioms for decision-making under uncertainty, including Independence, C-Independence, and Ambiguity Aversion.