Obviously Strategy-Proof Multi-Dimensional Allocation and the Value of Choice

A principal must allocate a set of heterogeneous tasks (or objects) among multiple agents. The principal has preferences over the allocation. Each agent has preferences over which tasks they are assigned, which are their private information. The principal is constrained by the fact that each agent has the right to demand some status-quo task assignment. I characterize the conditions under which the principal can gain by delegating some control over the assignment to the agents. Within a large class of delegation mechanisms, I then characterize those that are obviously strategy-proof (OSP), and provide guidance for choosing among OSP mechanisms.

💡 Research Summary

The paper investigates a multi‑dimensional task (or object) allocation problem in which a principal must assign heterogeneous tasks to a finite set of agents. The principal cares about minimizing total social cost, which is a known function of the agent‑task pair (e.g., observable performance weights π_j(i)). Each agent privately values the tasks they receive through a cost vector c_j(i) and can demand a status‑quo allocation σ_j (the “outside option”) at the interim stage, imposing a participation constraint.
The author introduces “trading mechanisms,” a class of delegation mechanisms where the principal first offers each agent a menu of trading sets (subsets of feasible task bundles). Agents choose a trading set, and a pre‑specified selection rule P maps the profile of chosen sets into a feasible allocation that respects every agent’s choice. The status‑quo endowment η (typically σ) must belong to each agent’s menu, guaranteeing the right to revert to the outside option.
Two central questions are addressed. First, when does granting agents choice improve the principal’s objective relative to the status‑quo? Theorem 1 shows that if any Bayesian incentive‑compatible (BIC) mechanism can achieve a lower expected social cost than σ, then an obviously strategy‑proof (OSP) trading mechanism can achieve the same improvement. Thus, the restriction to OSP imposes no loss of optimality under the mild conditions identified, and the “value of choice” is robust to the stronger OSP requirement.
Second, what is the structure of OSP trading mechanisms? Theorem 2 provides a complete characterization: every OSP trading mechanism must be a “polarized‑ray” mechanism. Concretely, for each agent either (i) the menu is a singleton containing all allocations that are unambiguously preferred to the outside option (no choice), or (ii) the menu consists of a collection of “rays” in ℝ⁺^I, each ray having one endpoint at the status‑quo bundle σ_j and extending in a direction that improves the agent’s utility. The set of rays must satisfy a polarization condition—any two rays are either nested or disjoint—ensuring that the agent’s optimal choice is obvious at every decision node. As a corollary, the menu size for any agent is bounded by I + 1 (Corollary 2). Moreover, agents need only know the geometric form of their own menu; details of the selection rule or other agents’ menus are irrelevant, granting the mechanism strong informational robustness.
The paper also discusses computational aspects. While identifying the optimal OSP mechanism remains challenging, the restriction to polarized‑ray mechanisms simplifies the design space. By studying a large‑market (continuum) limit, the author derives a dual formulation that is more tractable, offering guidance for approximate optimal designs in practice.
Overall, the contribution is threefold: (1) it delineates clear, mild conditions under which delegating choice to agents yields a socially beneficial reduction in cost; (2) it fully characterizes the class of OSP mechanisms suitable for this setting, revealing that they must take the polarized‑ray form with bounded menu size; and (3) it demonstrates that such mechanisms are robust to misspecification of agents’ preference distributions and are computationally approachable via large‑market relaxations. The results bridge the gap between the theoretically optimal (but often intractable) Bayesian designs and practically implementable, strategy‑proof mechanisms for high‑dimensional allocation problems such as hospital case assignment, child‑protective services, tax collection, or refugee placement.