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Partial Allocations in Budget-Feasible Mechanism Design: Bridging Multiple Levels of Service and Divisible Agents

Georgios Amanatidis, Sophie Klumper, Evangelos Markakis, Guido Schäfer, Artem Tsikiridis

TL;DR

The paper advances budget-feasible mechanism design by allowing partial allocations in procurement. It introduces a deterministic, truthful framework for both a multi-level (k-level) model with concave, per-agent valuations and a divisible-agent model with concave, separable valuations, achieving constant-factor approximations: $(2+\sqrt{3})$ for indivisible multi-level services and $(4+2\sqrt{3})$ for the divisible setting, with a sharp 2-approximation in the linear-valuation special case. The core technique is a backward greedy approach that starts from the fractional optimum and truncates levels while preserving truthfulness via Myerson-style payments; large-market analyses show improvements as market largeness grows. The results establish a separation between divisible and indivisible settings and offer practical, polynomial-time mechanisms applicable to procurement with fractional or multi-level service offerings. Collectively, the work broadens the applicability of budget-feasible mechanisms to more flexible procurement scenarios and lays groundwork for further improvements under additional feasibility constraints.

Abstract

Budget-feasible procurement has been a major paradigm in mechanism design since its introduction by Singer (2010). An auctioneer (buyer) with a strict budget constraint is interested in buying goods or services from a group of strategic agents (sellers). In many scenarios it makes sense to allow the auctioneer to only partially buy what an agent offers, e.g., an agent might have multiple copies of an item to sell, they might offer multiple levels of a service, or they may be available to perform a task for any fraction of a specified time interval. Nevertheless, the focus of the related literature has been on settings where each agent's services are either fully acquired or not at all. The main reason for this, is that in settings with partial allocations like the ones mentioned, there are strong inapproximability results. Under the mild assumption of being able to afford each agent entirely, we are able to circumvent such results in this work. We design a polynomial-time, deterministic, truthful, budget-feasible $(2+\sqrt{3})$-approximation mechanism for the setting where each agent offers multiple levels of service and the auctioneer has a discrete separable concave valuation function. We then use this result to design a deterministic, truthful and budget-feasible $O(1)$-approximation mechanism for the setting where any fraction of a service can be acquired and the auctioneer's valuation function is separable concave (i.e., the sum of concave functions). For the special case of a linear valuation function, we improve the best known approximation ratio for the problem from $1+φ$ (by Klumper & Schäfer (2022)) to $2$. This establishes a separation between this setting and its indivisible counterpart.

Partial Allocations in Budget-Feasible Mechanism Design: Bridging Multiple Levels of Service and Divisible Agents

TL;DR

The paper advances budget-feasible mechanism design by allowing partial allocations in procurement. It introduces a deterministic, truthful framework for both a multi-level (k-level) model with concave, per-agent valuations and a divisible-agent model with concave, separable valuations, achieving constant-factor approximations: for indivisible multi-level services and for the divisible setting, with a sharp 2-approximation in the linear-valuation special case. The core technique is a backward greedy approach that starts from the fractional optimum and truncates levels while preserving truthfulness via Myerson-style payments; large-market analyses show improvements as market largeness grows. The results establish a separation between divisible and indivisible settings and offer practical, polynomial-time mechanisms applicable to procurement with fractional or multi-level service offerings. Collectively, the work broadens the applicability of budget-feasible mechanisms to more flexible procurement scenarios and lays groundwork for further improvements under additional feasibility constraints.

Abstract

Budget-feasible procurement has been a major paradigm in mechanism design since its introduction by Singer (2010). An auctioneer (buyer) with a strict budget constraint is interested in buying goods or services from a group of strategic agents (sellers). In many scenarios it makes sense to allow the auctioneer to only partially buy what an agent offers, e.g., an agent might have multiple copies of an item to sell, they might offer multiple levels of a service, or they may be available to perform a task for any fraction of a specified time interval. Nevertheless, the focus of the related literature has been on settings where each agent's services are either fully acquired or not at all. The main reason for this, is that in settings with partial allocations like the ones mentioned, there are strong inapproximability results. Under the mild assumption of being able to afford each agent entirely, we are able to circumvent such results in this work. We design a polynomial-time, deterministic, truthful, budget-feasible -approximation mechanism for the setting where each agent offers multiple levels of service and the auctioneer has a discrete separable concave valuation function. We then use this result to design a deterministic, truthful and budget-feasible -approximation mechanism for the setting where any fraction of a service can be acquired and the auctioneer's valuation function is separable concave (i.e., the sum of concave functions). For the special case of a linear valuation function, we improve the best known approximation ratio for the problem from (by Klumper & Schäfer (2022)) to . This establishes a separation between this setting and its indivisible counterpart.
Paper Structure (20 sections, 23 theorems, 58 equations, 1 figure, 6 algorithms)

This paper contains 20 sections, 23 theorems, 58 equations, 1 figure, 6 algorithms.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Further Related Work
  4. Model and Preliminaries
  5. Basic Model
  6. k-Level Model
  7. Divisible Agent Model
  8. Fractional k-Bounded Knapsack Problem
  9. A Budget-Feasible Mechanism for Multiple Levels of Service
  10. A Truthful Greedy Mechanism
  11. Ensuring the Budget-Feasibility of Sort-&-Reject
  12. Sort-&-Reject for Large Markets
  13. Mechanisms for Divisible Agents
  14. A Constant Factor Approximation Using Sort-&-Reject
  15. An Improved Mechanism for Linear Valuation Functions
  16. ...and 5 more sections

Key Result

Theorem 2.1

A monotone non-increasing allocation rule $\mathbf{x}(\mathbf{b})$ admits a payment rule that is truthful and individually rational if and only if for all agents $i \in N$ and all bid profiles $\mathbf{b}_{-i}$, we have $\int_{0}^{\infty} x_{i}(z,\mathbf{b}_{-i}) dz < \infty$. In this case, we can t

Figures (1)

  • Figure 1: Illustration of the critical payments of agent $i$.

Theorems & Definitions (43)

  • Theorem 2.1: archer01myerson81
  • Remark 2.1
  • Theorem 3.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • ...and 33 more