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Equal-Pay Contracts

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

TL;DR

This work studies equal-pay and nearly-equal-pay contracts in multi-agent combinatorial-action environments, addressing both computational tractability and fairness. It establishes a complete algorithmic map across reward families, proving constant-factor approximations for submodular rewards, and sharp hardness results (no PTAS for GS, no constant-factor for XOS) that also hold for unconstrained contracts. The paper introduces the price of equality as a rigorous metric of efficiency loss due to fairness, showing tight bounds of $ ilde{O}ig( rac{ obreak extstyle obreak obreak obreak }{ obreak obreak }ig)$ for XOS with combinatorial actions and $ ilde{ ext{Ω}}( rac{ obreak extstyle obreak obreak }{ obreak obreak })$ lower bounds for additive and subadditive cases, respectively. Its methods hinge on agent-aware demand-query reductions, surrogate submodular objectives, and stability lemmas that convert equilibria into near-optimal equal-pay contracts, yielding insights that extend to BEST objectives and broader fairness constraints. Overall, the results delineate when equal-pay contracts are practically viable and quantify the cost of fairness for a broad class of rewards in algorithmic contract design.

Abstract

We study multi-agent contract design, where a principal incentivizes a team of agents to take costly actions that jointly determine the project success via a combinatorial reward function. While prior work largely focuses on unconstrained contracts that allow heterogeneous payments across agents, many real-world environments limit payment dispersion. Motivated by this, we study equal-pay contracts, where all agents receive identical payments. Our results also extend to nearly-equal-pay contracts where any two payments are identical up to a constant factor. We provide both algorithmic and hardness results across a broad hierarchy of reward functions, under both binary and combinatorial action models. While we focus on equal-pay contracts, our analysis also yields new insights into unconstrained contract design, and resolves two important open problems. On the positive side, we design polynomial-time O(1)-approximation algorithms for (i) submodular rewards under combinatorial actions, and (ii) XOS rewards under binary actions. These guarantees are tight: We rule out the existence of (i) a PTAS for combinatorial actions, even for gross substitutes rewards (unless P = NP), and (ii) any O(1)-approximation for XOS rewards with combinatorial actions. Crucially, our hardness results hold even for unconstrained contracts, thereby settling the corresponding open problems in this setting. Finally, we quantify the loss induced by fairness via the price of equality, defined as the worst-case ratio between the optimal principal's utility achievable by unconstrained contracts and that achievable by equal-pay contracts. We obtain a bound of $Θ(\log n/ \log \log n)$, where $n$ is the number of agents. This gap is tight in a strong sense: the upper bound applies even for XOS rewards with combinatorial actions, while the lower bound arises already for additive rewards with binary actions.

Equal-Pay Contracts

TL;DR

This work studies equal-pay and nearly-equal-pay contracts in multi-agent combinatorial-action environments, addressing both computational tractability and fairness. It establishes a complete algorithmic map across reward families, proving constant-factor approximations for submodular rewards, and sharp hardness results (no PTAS for GS, no constant-factor for XOS) that also hold for unconstrained contracts. The paper introduces the price of equality as a rigorous metric of efficiency loss due to fairness, showing tight bounds of for XOS with combinatorial actions and lower bounds for additive and subadditive cases, respectively. Its methods hinge on agent-aware demand-query reductions, surrogate submodular objectives, and stability lemmas that convert equilibria into near-optimal equal-pay contracts, yielding insights that extend to BEST objectives and broader fairness constraints. Overall, the results delineate when equal-pay contracts are practically viable and quantify the cost of fairness for a broad class of rewards in algorithmic contract design.

Abstract

We study multi-agent contract design, where a principal incentivizes a team of agents to take costly actions that jointly determine the project success via a combinatorial reward function. While prior work largely focuses on unconstrained contracts that allow heterogeneous payments across agents, many real-world environments limit payment dispersion. Motivated by this, we study equal-pay contracts, where all agents receive identical payments. Our results also extend to nearly-equal-pay contracts where any two payments are identical up to a constant factor. We provide both algorithmic and hardness results across a broad hierarchy of reward functions, under both binary and combinatorial action models. While we focus on equal-pay contracts, our analysis also yields new insights into unconstrained contract design, and resolves two important open problems. On the positive side, we design polynomial-time O(1)-approximation algorithms for (i) submodular rewards under combinatorial actions, and (ii) XOS rewards under binary actions. These guarantees are tight: We rule out the existence of (i) a PTAS for combinatorial actions, even for gross substitutes rewards (unless P = NP), and (ii) any O(1)-approximation for XOS rewards with combinatorial actions. Crucially, our hardness results hold even for unconstrained contracts, thereby settling the corresponding open problems in this setting. Finally, we quantify the loss induced by fairness via the price of equality, defined as the worst-case ratio between the optimal principal's utility achievable by unconstrained contracts and that achievable by equal-pay contracts. We obtain a bound of , where is the number of agents. This gap is tight in a strong sense: the upper bound applies even for XOS rewards with combinatorial actions, while the lower bound arises already for additive rewards with binary actions.
Paper Structure (51 sections, 43 theorems, 109 equations, 3 figures, 1 table, 5 algorithms)

This paper contains 51 sections, 43 theorems, 109 equations, 3 figures, 1 table, 5 algorithms.

Table of Contents

  1. Introduction
  2. Multi-Agent Combinatorial-Actions Model.
  3. Our Results
  4. Algorithms and hardness.
  5. Price of Equality.
  6. Beyond Profit Maximization.
  7. Our Techniques
  8. Demand Queries Are Agent-Agnostic.
  9. Constant-Approximation for Submodular Rewards, for Equal-Pay Contract (Theorem \ref{['thm:constant_apx_submodular']}).
  10. Hardness of Approximation for XOS Rewards, for Equal-Pay and Unconstrained Contracts (\ref{['thm:no_constant_xos']}).
  11. No PTAS for GS Rewards, for Equal-Pay and Unconstrained Contracts (\ref{['thm:no-ptas-gs']}).
  12. Price of Equality (\ref{['thm:price']}).
  13. Related Work
  14. Combinatorial Team Contracts.
  15. Other Combinatorial Models.
  16. ...and 36 more sections

Key Result

Lemma 2.3

Suppose $f$ is submodular. Let $S$ be a subset-stable action set with respect to a contract $\boldsymbol{\alpha}$, such that $S_i = \emptyset$ for all $i$ with $\alpha_i=0$. Then any equilibrium $S^\dagger{}$ with respect to $2 \boldsymbol{\alpha}$ fulfills $f(S^\dagger{}) \geq \frac{1}{2} f(S)$.

Figures (3)

  • Figure 1: Illustration of the definition of informative demand queries. The large rectangle represents the set of all agents, and the dashed circle represents the hidden subset of agents $G$. The smaller rectangle depicts the set $L(\boldsymbol{p})$ for a given price vector $\boldsymbol{p}$. The red region corresponds to $L(\boldsymbol{p}) \setminus G$, while the green region corresponds to $L(\boldsymbol{p}) \cap G$. A demand query $\boldsymbol{p}$ is informative if the total number of agents in the red and green regions is at most $2\ell^4$, and the green region alone contains at least $\ell$ agents.
  • Figure 2: The structure of the buckets used in \ref{['alg:pof_xos']}. Nodes represent agents ordered from left to right by increasing $\alpha_i^\star$. The text inside each node denotes the label assigned to its index. Blue boxes indicate the buckets, and the text above each box shows the label assigned to that bucket by the algorithm.
  • Figure 3: The coverage function $f$ that exemplifies an unbounded price of equality.

Theorems & Definitions (86)

  • Definition 2.1: Equal-Pay Contract
  • Definition 2.2: Subset Stability, Definition 3.2 of multimulti
  • Lemma 2.3: Modified Doubling Lemma
  • Lemma 2.4: multimulti
  • Definition 2.5: Dropout Stability dutting2025black
  • Lemma 2.6: Scaling-for-Existence Lemma dutting2025black
  • Lemma 2.7: duetting2022multi
  • Theorem 3.1
  • Theorem 3.2: Reduction to no agent is large / single agent (equal-pay), multimulti
  • Theorem 3.3: multimulti
  • ...and 76 more