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Ultra Efficient Contracts: Pushing the Boundaries of Tractable Contract Design

Michal Feldman, Liat Yashin

Abstract

We study the optimal contract problem in the \emph{combinatorial actions} framework of Dütting et al.~[FOCS'21], where a principal delegates a project to an agent who chooses a subset of hidden, costly actions, and the resulting reward is given by a monotone set function over the actions. The principal offers a contract that specifies the fraction of the reward the agent receives, and the goal is to compute a contract that maximizes the principal's expected utility. Prior work established polynomial-time algorithms for \emph{gross substitutes} rewards, while showing NP-hardness for general submodular rewards; subsequent work extended tractability to \emph{supermodular} rewards, demonstrating that tractable cases exist in both the substitutes and complements regimes. This left open the precise boundary of tractability for the optimal contract problem. Our main result is a polynomial-time algorithm for the optimal contract problem under \Ultra\ rewards, a class that strictly contains gross substitutes but is not confined to subadditive rewards, thereby bridging the substitutes and complements regimes. We further extend our results beyond additive costs, establishing a polynomial-time algorithm for \Ultra\ rewards and cost functions that are the sum of additive and symmetric functions. To the best of our knowledge, this is the first application of \Ultra\ functions in a prominent economic setting.

Ultra Efficient Contracts: Pushing the Boundaries of Tractable Contract Design

Abstract

We study the optimal contract problem in the \emph{combinatorial actions} framework of Dütting et al.~[FOCS'21], where a principal delegates a project to an agent who chooses a subset of hidden, costly actions, and the resulting reward is given by a monotone set function over the actions. The principal offers a contract that specifies the fraction of the reward the agent receives, and the goal is to compute a contract that maximizes the principal's expected utility. Prior work established polynomial-time algorithms for \emph{gross substitutes} rewards, while showing NP-hardness for general submodular rewards; subsequent work extended tractability to \emph{supermodular} rewards, demonstrating that tractable cases exist in both the substitutes and complements regimes. This left open the precise boundary of tractability for the optimal contract problem. Our main result is a polynomial-time algorithm for the optimal contract problem under \Ultra\ rewards, a class that strictly contains gross substitutes but is not confined to subadditive rewards, thereby bridging the substitutes and complements regimes. We further extend our results beyond additive costs, establishing a polynomial-time algorithm for \Ultra\ rewards and cost functions that are the sum of additive and symmetric functions. To the best of our knowledge, this is the first application of \Ultra\ functions in a prominent economic setting.

Paper Structure

This paper contains 37 sections, 41 theorems, 63 equations, 1 figure, 11 algorithms.

Table of Contents

  1. Introduction
  2. Gross substitutes rewards.
  3. Supermodular rewards.
  4. Ultra rewards.
  5. Our Contribution
  6. Additional Related Work
  7. Algorithmic contract design.
  8. Ultra set functions.
  9. Model and Preliminaries
  10. Combinatorial Contracts Model
  11. The combinatorial-action contract model.
  12. Contract design.
  13. Utilities and best responses.
  14. Value and Demand Oracles.
  15. Classes of Set Functions
  16. ...and 22 more sections

Key Result

Theorem 1

Let $f$ be an Ultra reward function, and let $c$ be a cost function that decomposes into the sum of an additive component and a monotone symmetric component. Then the optimal contract can be computed in polynomial time.

Figures (1)

  • Figure 1: The relationship between monotone submodular, Ultra, and GS functions. GS is precisely the intersection of submodular and Ultra set functions.

Theorems & Definitions (81)

  • Theorem : Beyond Additive Costs; Theorem 4.2
  • Theorem : Weakly well-layered Rewards; Appendix \ref{['sec:appendix-wwl-1']} Proposition \ref{['prop:optimal-contract-wwl-sym']}
  • Definition 2.1
  • Definition 2.2: well-layered (WL)
  • Definition 2.3: Exchange and Triplet properties
  • Theorem 2.4
  • Definition 2.5: Ultra
  • Theorem 2.6
  • Example 2.7: GS vs. Ultra demand queries
  • Lemma 2.8: Monotonicity lemma, DFG24combinatorial
  • ...and 71 more