Sufficient conditions for strategy-proofness

Determine sufficient conditions under which a belief aggregation rule f: Δ(S)^I → Δ(S) is strategy-proof in the public portfolio-choice domain where agents share a common twice-continuously differentiable, strictly concave von Neumann–Morgenstern utility index u, the feasible set of state-contingent outcomes is X = {x ∈ ℝ_+^S : π·x = w}, society selects x(q) ∈ X by maximizing ∑_{s∈S} u(x_s) q_s for aggregate belief q ∈ Δ(S), and each agent i with belief p_i ranks aggregate beliefs by q R(p_i) q′ if and only if ∑_{s∈S} p_{i,s} u_s(q) ≥ ∑_{s∈S} p_{i,s} u_s(q′).

Background

Strategy-proofness in this setting requires that no agent can benefit, according to their expected utility preferences over aggregate beliefs induced by the public portfolio choice, by misreporting their belief to alter the chosen aggregate belief. The paper establishes that monotonicity is necessary for strategy-proofness and that in this domain monotonicity is equivalent to recursive invariance.

However, existing sufficiency results (e.g., richness condition R1 ensuring monotonicity suffices for strategy-proofness) do not apply because the domain here lacks the required richness: preferences arise from a fixed common vNM index and a non-product budget set, which prevents separability-based arguments. Identifying tractable sufficient conditions tailored to this domain remains unresolved.