Non-obvious manipulability in division problems with general preferences

In problems involving the allocation of a single non-disposable commodity, we study rules defined on a general domain of preferences requiring only that each preference exhibit a unique global maximum. Our focus is on rules that satisfy a relaxed form of strategy-proofness, known as non-obvious manipulability. We show that the combination of efficiency and non-obvious manipulability leads to impossibility results, whereas weakening efficiency to unanimity gives rise to a large family of well-behaved non-obviously manipulable rules.

💡 Research Summary

The paper studies the allocation of a single perfectly divisible good among a set of agents when each agent’s preference is only required to have a unique global maximum (a “peak”). This minimal domain assumption departs from the classic single‑peaked framework and captures realistic situations where utilities may have internal valleys, thresholds, or non‑monotonic sections (e.g., irrigation water with subsidy loss, fishing quotas with observation costs, electricity tariffs with notches).

The authors introduce the concept of non‑obvious manipulability (NOM), a relaxation of strategy‑proofness inspired by Troyan and Morrill (2020). A misreport is an “obvious manipulation” if (i) it yields a strictly preferred outcome for the agent compared with truth‑telling for some profile of the other agents, and (ii) every possible outcome under the misreport is strictly better than some possible outcome under truth‑telling. A rule is NOM if no such manipulation exists. NOM is strictly weaker than strategy‑proofness, allowing more rules while still ruling out manipulations that are easy to detect.

The paper first investigates the compatibility of efficiency (Pareto optimality), strategy‑proofness, and a modest fairness requirement called the equal‑division lower bound (no agent receives less than the equal share of the total endowment). Theorem 1 proves that no rule can satisfy all three simultaneously. The proof reduces the two‑agent case to a public‑good social‑choice problem and invokes the Barberà‑Peleg (1990) dictatorship result for continuous preferences; an induction argument extends the impossibility to any number of agents.

Relaxing strategy‑proofness to NOM while retaining own‑peak‑onlyness (the allocation depends on an agent’s report only through its peak) still yields strong negative results. Theorem 2 shows that with only two agents, any rule that is efficient, NOM, and own‑peak‑only must be dictatorial. For three or more agents, Theorem 3 demonstrates that any efficient, own‑peak‑only rule is bossy: an agent can change the allocation of others without affecting his own share. These impossibilities persist even under a minimal richness condition on the preference domain (for any two alternatives there exists a preference ranking one at the top and the other at the bottom).

To obtain positive results, the authors replace efficiency with unanimity: when the sum of all peaks equals the total endowment, each agent must receive his peak. Under unanimity, own‑peak‑onlyness, and the equal‑division guarantee (an agent whose peak equals the equal share receives exactly that share), they characterize a large family of rules called agreeable rules (Theorem 4). An agreeable rule works as follows: each agent is initially entitled to an equal share of the endowment; given the reported preference profile, the rule selects a coalition whose members’ peaks exactly sum to the portion of the endowment they collectively deserve. Coalition members receive their peaks, while all remaining agents receive the equal share. This construction guarantees NOM, unanimity, own‑peak‑onlyness, and the equal‑division guarantee.

However, some agreeable rules are still bossy. The paper therefore isolates the subclass of non‑bossy agreeable rules (Proposition 2). Non‑bossiness is achieved when the coalition‑selection mechanism always picks the maximal feasible coalition within a pre‑specified collection of nested coalitions. Proposition 3 presents a concrete parametric family of such rules: the parameter is a hierarchy of coalitions, and for any preference profile the rule selects the largest coalition in the hierarchy whose total peak matches the entitlement. These rules are transparent, satisfy all the desiderata, and avoid the bossiness pathology.

Section 5 extends the analysis to economies where agents have individual endowments and to peak‑responsive rules, which adjust allocations based on both the agent’s peak and his initial endowment. The same impossibility and possibility results carry over, showing the robustness of the framework.

In sum, the paper demonstrates that in a very general preference domain, the combination of efficiency and strong incentive compatibility is largely impossible, but by weakening efficiency to unanimity and focusing on the peak‑only information structure, one can construct a rich set of allocation mechanisms that are non‑obviously manipulable, fair (in the sense of equal‑division guarantees), and free of bossiness. The work broadens the literature on division problems beyond single‑peaked preferences and introduces non‑obvious manipulability as a useful design criterion for mechanisms operating under limited cognitive capabilities of agents.