Possibility or impossibility of non-dictatorial strategy-proof belief aggregation beyond two states

Ascertain whether non-dictatorial, non-constant strategy-proof belief aggregation rules f: Δ(S)^I → Δ(S) exist, or prove they are impossible, in the public portfolio-choice domain with common von Neumann–Morgenstern utility u and linear budget constraint X = {x ∈ ℝ_+^S : π·x = w}, particularly for |S| ≥ 3 where the range of f would exceed one dimension; in other words, resolve whether there are strategy-proof rules with more than one-dimensional range under these domain restrictions or whether an impossibility theorem analogous to Zhou’s applies.

Background

Classical results for pure public goods (e.g., Zhou’s impossibility theorem) imply severe limits on strategy-proofness, such as at most a one-dimensional range, which in probability aggregation translates to feasibility primarily in the two-state case (where median rules can work).

The domain considered here is stricter than in those classical results: agents have a fixed common vNM utility, beliefs uniquely determine preferences over aggregate beliefs, and the feasible set is not a product set, invalidating separability-based possibility results. Whether these restrictions allow for strategy-proof belief aggregation rules beyond two states (or force impossibility) remains unknown.