One Action Too Many: Inapproximability of Budgeted Combinatorial Contracts

2025-11-25

AI Summary

This paper studies contract design for combinatorial actions in budget-constrained multi-agent principal–agent settings, where the objective jointly optimizes principal profit, agent rewards, and social welfare. Methodologically, it integrates submodular and gross-substitutes reward modeling with value and demand query oracles, rigorously distinguishing computational complexity between budgeted and unbounded regimes. The key contributions are: (i) a strong inapproximability lower bound—under submodular rewards, no randomized polynomial-time algorithm can approximate the optimal budget-feasible contract within any finite factor; (ii) for gross-substitutes rewards, a deterministic constant-factor approximation algorithm; and (iii) for additive rewards, the first fully polynomial-time approximation scheme (FPTAS). These results precisely delineate the tractability frontier for budgeted combinatorial contract design.

Technology Category

• Multiagent Systems: Mechanism Design
• Game Theory and Economic Paradigms: Mechanism Design
• Search and Optimization: Combinatorial Optimization

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

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 extit{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.

Problem

• Investigating budget-constrained combinatorial contract design with multiple agents
• Establishing computational hardness for approximating submodular reward functions
• Identifying tractable cases for gross substitutes and additive reward functions

Innovation

• Budgeted combinatorial contracts with gross substitutes rewards
• FPTAS for additive rewards in multi-agent settings
• Hardness proof for submodular rewards with budget constraints