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One Action Too Many: Inapproximability of Budgeted Combinatorial Contracts

Michal Feldman, Yoav Gal-Tzur, Tomasz Ponitka, Maya Schlesinger

TL;DR

The paper studies budgeted multi-agent contract design with combinatorial actions under a broad class of objectives (BEST). It proves an information-theoretic inapproximability for submodular rewards when budgets are present, even with demand oracles, and identifies gross-substitutes rewards as a tractable frontier enabling constant-factor approximations under any budget. It also presents an FPTAS for additive reward structures, extending tractability to special cases and showing a three-way barrier among reward structure, budgets, and action richness. The work leverages best-response properties, demand bundles, and a downsizing framework to connect different objectives and budgets, offering a unified view of when constant-factor or near-optimal contracts are feasible. The results sharpen the boundary between tractable and intractable contract design in multi-agent, combinatorial-action scenarios, with clear implications for mechanism design under budget constraints.

Abstract

We study multi-agent contract design with combinatorial actions, under budget constraints, and for a broad class of objective functions, including profit (principal's utility), reward, and welfare. Our first result is a strong impossibility: For submodular reward functions, no randomized poly-time algorithm can approximate the optimal budget-feasible value within \textit{any finite factor}, even with demand-oracle access. This result rules out extending known constant-factor guarantees from either (i) unbudgeted settings with combinatorial actions or (ii) budgeted settings with binary actions, to their combination. The hardness is tight: It holds even when all but one agent have binary actions and the remaining agent has just one additional action. On the positive side, we show that gross substitutes rewards (a well-studied strict subclass of submodular functions) admit a deterministic poly-time $O(1)$-approximation, using only value queries. Our results thus draw the first sharp separation between budgeted and unbudgeted settings in combinatorial contracts, and identifies gross substitutes as a tractable frontier for budgeted combinatorial contracts. Finally, we present an FPTAS for additive rewards, demonstrating that arbitrary approximation is tractable under any budget. This constitutes the first FPTAS for the multi-agent combinatorial-actions setting, even in the absence of budget constraints.

One Action Too Many: Inapproximability of Budgeted Combinatorial Contracts

TL;DR

The paper studies budgeted multi-agent contract design with combinatorial actions under a broad class of objectives (BEST). It proves an information-theoretic inapproximability for submodular rewards when budgets are present, even with demand oracles, and identifies gross-substitutes rewards as a tractable frontier enabling constant-factor approximations under any budget. It also presents an FPTAS for additive reward structures, extending tractability to special cases and showing a three-way barrier among reward structure, budgets, and action richness. The work leverages best-response properties, demand bundles, and a downsizing framework to connect different objectives and budgets, offering a unified view of when constant-factor or near-optimal contracts are feasible. The results sharpen the boundary between tractable and intractable contract design in multi-agent, combinatorial-action scenarios, with clear implications for mechanism design under budget constraints.

Abstract

We study multi-agent contract design with combinatorial actions, under budget constraints, and for a broad class of objective functions, including profit (principal's utility), reward, and welfare. Our first result is a strong impossibility: For submodular reward functions, no randomized poly-time algorithm can approximate the optimal budget-feasible value within \textit{any finite factor}, even with demand-oracle access. This result rules out extending known constant-factor guarantees from either (i) unbudgeted settings with combinatorial actions or (ii) budgeted settings with binary actions, to their combination. The hardness is tight: It holds even when all but one agent have binary actions and the remaining agent has just one additional action. On the positive side, we show that gross substitutes rewards (a well-studied strict subclass of submodular functions) admit a deterministic poly-time -approximation, using only value queries. Our results thus draw the first sharp separation between budgeted and unbudgeted settings in combinatorial contracts, and identifies gross substitutes as a tractable frontier for budgeted combinatorial contracts. Finally, we present an FPTAS for additive rewards, demonstrating that arbitrary approximation is tractable under any budget. This constitutes the first FPTAS for the multi-agent combinatorial-actions setting, even in the absence of budget constraints.

Paper Structure

This paper contains 35 sections, 30 theorems, 63 equations, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. Our Results
  3. Inapproximability for Submodular Instances.
  4. Constant-Factor Approximation for Gross Substitutes Instances.
  5. Discussion: A Three-Way Barrier to Approximation.
  6. FPTAS for Special Cases.
  7. Our Techniques
  8. Inapproximability for Submodular Instances.
  9. Constant-Factor Approximation for Gross Substitutes Instances.
  10. Related Work
  11. Combinatorial Contracts.
  12. Linear Contracts.
  13. Contracts under Budget Constraints.
  14. Other Contractual Models.
  15. Model and Preliminaries
  16. ...and 20 more sections

Key Result

Theorem 1

For the class of instances with submodular $f$, any BEST objective $\varphi$ (including profit, reward, and welfare), any budget $B \in (0,1)$, and any approximation guarantee $K : \mathbb{N} \to [1,\infty)$, any randomized algorithm can achieve a $K(n)$-approximation with respect to $\varphi$ under

Figures (1)

  • Figure 1: The three vertices of the triangle represent the dimensions along which the settings we consider differ: (i) the structure of $f$ (general submodular vs. gross substitutes), (ii) the presence of budget constraints, and (iii) the type of the agents' action space (binary vs. combinatorial). Any pair of properties admits a constant-factor approximation, as indicated along each edge of the triangle. The figure illustrates that the impossibility arises only from the combination of all three properties: the interior inapproximability region corresponds to submodular $f$ with budgets and combinatorial actions simultaneously. $(\star)$ All results shown hold for all BEST objectives, except for the $O(1)$-approximation for the multi-agent combinatorial-actions setting without budget constraints of multimulti, which holds only for profit maximization.

Theorems & Definitions (58)

  • Theorem 1: Inapproximability for Submodular Instances; \ref{['thm:multi-multi-inapprox']}
  • Theorem 2: Constant-Factor Approximation for Gross Substitutes Instances; \ref{['cor:gsapprox']}
  • Theorem 3: FPTAS for Additive Instances; \ref{['thm:fptas']}
  • Theorem 4: FPTAS for Single-Agent Instances; \ref{['thm:single-agent-fptas']}
  • Definition 2.1: Subset Stability, Definition 3.2 of multimulti
  • Lemma 2.2: Doubling Lemma, Lemma 3.3 of multimulti
  • Definition 2.3: Objectives in the Multi-Agent Combinatorial-Actions Model
  • Definition 2.4: BEST Objectives
  • Theorem 3.1: Inapproximability for Submodular Instances
  • Corollary 3.2: Exponential Lower Bound on Expected Approximation Ratio
  • ...and 48 more