Agreement with reservation of judgment under risk

Abstract: This paper studies preference aggregation under risk. In our model, each agent has an incomplete preference relation represented by a set of expected utility functions. The classical Pareto principle is silent on agreement involving indecisiveness. To examine the implications of respecting such agreement, we introduce the Paretian principle that can be applied when some individuals reserve their judgment. Our main result shows that, under this principle, for each combination of individuals' utility functions, there exists a corresponding social utility function constructed as a weighted sum of the individual ones. These aggregation rules guarantee natural properties that the standard Pareto principle fails to ensure.

• The paper presents the Pareto$^$ axiom as a novel method to aggregate incomplete individual preferences under risk.
• It utilizes convex cone separation theorems to characterize social utility as an affine-positive sum of individual utilities.
• The approach has practical implications for welfare analysis, mechanism design, and AI alignment in uncertain decision environments.

Agreement with Reservation of Judgment under Risk: A Technical Overview

Motivation and Problem Statement

This paper addresses a crucial gap in social choice under risk: how to aggregate preferences when individual agents may be indecisive due to incomplete preference relations, specifically within the expected multi-utility model. Traditional applications of the Pareto principle—for situations governed by von Neumann-Morgenstern preferences—do not generate any social guidance when individual judgments are incomplete. Yet, indecisiveness itself can represent a form of weak unanimous agreement that is normatively relevant for collective decision making.

The authors formalize a "dual" axiom to Pareto, termed Pareto$^$, which respects these forms of reservation. If every individual does not deem a lottery weakly better than another, then neither should the social preference. This principle reorients preference aggregation by giving formal status to collective non-affirmation, not just affirmation. The theoretical framework is situated in the multi-utility model as developed by [Dubra, Maccheroni, Ok 2004], where preference incompleteness reflects ambiguity or lack of introspective certainty.

Model and Formal Setting

The model is specified as follows: for a finite set of outcomes $Z$, individuals choose over the simplex of lotteries $\Delta(Z)$. Each agent $i$ has a closed, convex, non-singleton set $\mathcal{U}_i$ of utility functions representing their incomplete preferences. A prospect $l$ is weakly preferred to $l'$ if, for every $u_i \in \mathcal{U}_i$, the expected utility of $l$ (under $u_i$) exceeds that of $l'$. Reservation of judgment corresponds to situations where, for some pair, the sign of this difference is not constant over $\mathcal{U}_i$.

The social planner is treated analogously, with preference relation $\succsim_0$ and multi-utility set $\mathcal{U}_0$. The critical assumption is the existence of a "no-conflict" pair—outcomes $z_*,z^*$ such that every utility function in the model agrees on their strict ranking—thereby ensuring some degree of shared evaluation.

Main Results

A New Characterization: Pareto$^$ Principle

The paper's centerpiece is a full characterization of preference aggregation rules induced by their Pareto$^$ axiom. The main theorem establishes:

For any combination of individual utility functions $(u_1,\ldots,u_n)$, there must exist a corresponding social utility function $u_0$ that is an affine-positive sum of these:

$u_0 = \sum_{i\in N} \alpha_i u_i + \beta$

for some strictly positive weights $\alpha_i$ (not all zero) and affine shift $\beta$.

This sharply contrasts with [Danan and Harsanyi 2015], whose aggregation allows the social planner to disregard certain combinations of individual utility functions. The result here implies the social preference cannot discard any combination of utilities that all individuals consider plausible. Every tuple in the cross-product of individual multi-utility sets is necessarily reflected in the social set, ensuring agreement preservation even under indecisiveness.

Implications and Extensions

• Unanimity properties: If all agents share a particular utility function (modulo affine transformation), that utility function is necessarily present in $\mathcal{U}_0$.
• Consensus transfer: Should all individuals have identical uncertain preference sets, the planner’s set must coincide.
• Minimality: The paper analyzes weaker axioms, such as Non-Reversal and Pareto Incomparability, and shows that their aggregation consequences are strictly weaker; only Pareto$^$ ensures full respect for all combinations of plausible individual utilities.

Relationship to Literature

The theoretical contribution sits precisely at the intersection of classical aggregation (Harsanyi’s theorem), multi-utility choice ([Dubra et al., 2004]; [Danan et al., 2015]), and preference under ambiguity ([Bewley, 2002]; [Danan, Pivato, 2016; 2024]). The main result can be viewed as a comprehensive, dual extension of Harsanyi’s utilitarianism: all combinations of plausible utilities must be utilized, not merely those realized in actual individual choice.

Technically, the proofs leverage convex cone separation theorems to handle the richer structure introduced by incomplete preferences (as opposed to the conventional simplex geometry in utility space).

Practical and Theoretical Significance

• Welfare Analysis: The results provide foundational justification for social aggregation rules in environments where agents are unable to make complete preference comparisons (e.g., under risk or ambiguity).
• Robustness: The characterization ensures that no consensus is lost in social evaluation when individual indecision exists, eliminating the possibility for social planners to cherry-pick among individual utility specifications.
• Applicability in Mechanism Design: The principles derived are directly relevant for mechanisms or allocation rules under state or outcome uncertainty, where robust respect for agent indecision can be normatively required.

Potential Directions for Future Research

Future work may:

• Extend these results to dynamic settings or infinite outcome spaces.
• Study the computational complexity of aggregation with these strong unanimity requirements.
• Analyze aggregation under alternative representations of incomplete preferences, such as multi-prior ambiguity models.
• Probe implications for AI alignment, where aggregating the risk and uncertainty attitudes (often incomplete) of human overseers requires robust mechanisms that avoid arbitrary deletion of plausible perspectives.

Conclusion

This paper defines and characterizes aggregation rules for social preferences when individual judgments under risk are incomplete. By formalizing the Pareto$^$ axiom, the analysis ensures that all plausible combinations of individual utilities are respected in the social multi-utility set, closing a gap left by traditional interpretations of unanimity. These results enrich the theoretical foundation for welfare economics and mechanism design under incompleteness and suggest new lines of inquiry in settings where the aggregation of uncertainty and indecision is central.