Near-Optimal Mechanisms for Resource Allocation Without Monetary Transfers

TL;DR

The paper studies sequential resource allocation among $n$ strategic agents without monetary transfers, quantifying how quickly the central planner’s welfare approaches the first-best under both finite horizon $T$ and infinite-horizon discounting with factor $\gamma$. It introduces the promised-utility framework and two geometric lemmas to characterize the achievable utility regions, yielding universal rates of $\mathcal{O}(\sqrt{1-\gamma})$ (or $\mathcal{O}(1/\sqrt{T})$) and, under structural conditions on utility distributions, faster rates up to $\mathcal{O}(1-\gamma)$ (or $\mathcal{O}(1/T)$). The results unify finite-horizon and infinite-horizon settings and provide constructive mechanisms that realize these rates, with faster convergence in smooth-full-information or highly interconnected utility profiles and slower rates for discrete distributions. The analysis offers deep insights into how planner flexibility to reward or penalize agents, while managing social welfare costs, governs the speed of convergence to the first-best and informs practical design of near-optimal, moneyless allocation mechanisms. These tools and findings have implications for scheduling, public service provisioning, and distributed resource management where monetary transfers are undesirable or infeasible.

Abstract

We study the problem in which a central planner sequentially allocates a single resource to multiple strategic agents using their utility reports at each round, but without using any monetary transfers. We consider general agent utility distributions and two standard settings: a finite horizon $T$ and an infinite horizon with $γ$ discounts. We provide general tools to characterize the convergence rate between the optimal mechanism for the central planner and the first-best allocation if true agent utilities were available. This heavily depends on the utility distributions, yielding rates anywhere between $1/\sqrt T$ and $1/T$ for the finite-horizon setting, and rates faster than $\sqrt{1-γ}$, including exponential rates for the infinite-horizon setting as agents are more patient $γ\to 1$. On the algorithmic side, we design mechanisms based on the promised-utility framework to achieve these rates and leverage structure on the utility distributions. Intuitively, the more flexibility the central planner has to reward or penalize any agent while incurring little social welfare cost, the faster the convergence rate. In particular, discrete utility distributions typically yield the slower rates $1/\sqrt T$ and $\sqrt{1-γ}$, while smooth distributions with density typically yield faster rates $1/T$ (up to logarithmic factors) and $1-γ$.

Figures (6)

• Figure 1: Illustration of the two scenarios (a) and (b) from \ref{['lemma:trivial_case']} which correspond to cases in which there is already an optimal allocation strategy for $\gamma=0$ (resp. $T=1$) in the infinite-horizon (resp. finite-horizon) setting. Figure (a) corresponds to the scenario in which agent $i$ dominates all other agents. Figure (b) corresponds the scenario in which agent $i_1$ and $i_2$ have non-zero utility mass at $0$ and the support of $\alpha_iu_i$ for $i\in\{i_1,i_2,i_3\}$ follow a hierarchy.
• Figure 2: Illustration of \ref{['lemma:ball_in_region_no_assumptions']} (left) and \ref{['lemma:prove_upper_bounds']} (right) with $2$ agents. (1) On the left, if the gray ball $B(\boldsymbol{x}, r)$ is within the full-information utility set $\mathcal{U}^\star$ with a margin $\delta$ satisfying Eq \ref{['eq:constraint_safe_boundary']}, then $B(\boldsymbol{x}, r)$ is contained in $\mathcal{U}_\gamma$. The region $\mathcal{U}^\star$ here corresponds to agents having uniform utility distributions on $[0,1]$. (2) On the right, the ball $B(\boldsymbol{x}, r)$ separates the region $\mathcal{U}^\star$ in two connected components, the one $\mathcal{C}$ closest to the boundary being included within a margin $\delta$. If $\delta$ satisfies the constraints from \ref{['lemma:prove_upper_bounds']}, then the gray region is outside $\mathcal{U}_\gamma$ (note that the region is slightly smaller than the connected component $\mathcal{C}$ (corresponding to the dash-dotted ball). The region $\mathcal{U}^\star$ here corresponds to agents having uniform utility distributions on $\{1/6,5/6\}$.
• Figure 3: Proof of \ref{['thm:universal_lower_bound']} in the standard case $0<U_i<\mathbb{E}[u_i]$ for all $i\in[n]$. For any $\boldsymbol{U}\in\mathcal{U}^\star$, the dotted region satisfies $\prod_{i\in[n]}[0,U_i]\subset \mathcal{U}^\star$. We can then fit a ball $B(\boldsymbol{x}, r)$ as in the figure with $r=\delta=\sqrt{C\frac{1-\gamma}{\gamma}}$. These satisfy Eq \ref{['eq:constraint_safe_boundary']} for $1-\gamma$ sufficiently small. Hence $B(\boldsymbol{x},r)\subset \mathcal{U}_\gamma$ and as a result $d(\boldsymbol{U},\mathcal{U}_\gamma)=\mathcal{O}(\sqrt{1-\gamma})$.
• Figure 4: Example of graph for which each node has an incoming and outcoming edge but does not admit a partition into strongly connected components.
• Figure 5: Illustration of how the gluing strategy achieves faster convergence rates when the frontier of the full-information region $\mathcal{U}^\star$ is close to flat. The figure depicts a region $\mathcal{U}^\star$ with an exaggerated piece-wise affine boundary for illustration purposes. The dashed gray region corresponds to the region that can be shown to be achievable in $\mathcal{U}_\beta$ using \ref{['lemma:ball_in_region_no_assumptions']} directly, which is limited by the maximum-radius ball tangent to the rgion of optimal utility vectors that can be fit within $\mathcal{U}^\star$ (represented with a dotted circle). The necessary margin $\delta_0$ is inversely proportional to the radius of the corresponding black ball as per \ref{['eq:constraint_safe_boundary']}. The gluing strategy effectively increases this radius as shown in the red arc, resulting in a smaller margin $\delta_2$. To avoid the region from stepping outside of $\mathcal{U}^\star$, we can glue the corresponding region with the blue balls which can be proved to be within $\mathcal{U}_\beta$ directly using \ref{['lemma:ball_in_region_no_assumptions']}. This results in the extra red-dashed region which can then be proved to be within $\mathcal{U}_\beta$. Note that since the radius of blue balls is necessarily smaller than that of the black ball, we have $\delta_1\geq \delta_0$. The main benefit of the gluing strategy is that at the second layer of the construction, the effective radius of the boundary of the new red-dashed region has been artificially increased. The gluing strategy can be used with an arbitrary number of layers to reach exponential convergence rates to the boundary of $\mathcal{U}^\star$.

Theorems & Definitions (31)

• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 5
• Lemma 6
• Theorem 7
• Theorem 8
• Corollary 9
• Theorem 10
• Theorem 11