# Deep Contract Design via Discontinuous Networks

arXiv: https://arxiv.org/abs/2307.02318 · HTML: https://mlsanity.com/paper/2307.02318/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 54 references that are in the map.
- First posted on arXiv 5 Jul 2023.

## Builds on (most cited first, 27 of 27)

- [Empirical Evaluation of Rectified Activations in Convolutional Network](https://mlsanity.com/paper/1505.00853/) (2015)
- [Understanding Deep Neural Networks with Rectified Linear Units](https://mlsanity.com/paper/1611.01491/) (2016)
- [How Do Classifiers Induce Agents To Invest Effort Strategically?](https://mlsanity.com/paper/1807.05307/) (2018)
- [Incomplete Contracting and AI Alignment](https://mlsanity.com/paper/1804.04268/) (2018)
- [Provable Robustness of ReLU networks via Maximization of Linear Regions](https://mlsanity.com/paper/1810.07481/) (2018)
- [Adaptive Contract Design for Crowdsourcing Markets: Bandit Algorithms for Repeated Principal-Agent Problems](https://mlsanity.com/paper/1405.2875/) (2014)
- [Optimal Auctions through Deep Learning: Advances in Differentiable Economics](https://mlsanity.com/paper/1706.03459/) (2017)
- [The Sample Complexity of Online Contract Design](https://mlsanity.com/paper/2211.05732/) (2022)
- [Simple versus Optimal Contracts](https://mlsanity.com/paper/1808.03713/) (2018)
- [Exact and Consistent Interpretation for Piecewise Linear Neural Networks: A Closed Form Solution](https://mlsanity.com/paper/1802.06259/) (2018)
- [Algorithmic Bayesian Persuasion](https://mlsanity.com/paper/1503.05988/) (2015)
- [Contracts under Moral Hazard and Adverse Selection](https://mlsanity.com/paper/2010.06742/) (2020)
- [The Sample Complexity of Up-to-varepsilon Multi-Dimensional Revenue Maximization](https://mlsanity.com/paper/1808.02458/) (2018)
- [The Complexity of Contracts](https://mlsanity.com/paper/2002.12034/) (2020)
- [Multi-Agent Contracts](https://mlsanity.com/paper/2211.05434/) (2022)
- [Learning Approximately Optimal Contracts](https://mlsanity.com/paper/1811.06736/) (2018)
- [Bayesian Agency: Linear versus Tractable Contracts](https://mlsanity.com/paper/2106.00319/) (2021)
- [Combinatorial Contracts](https://mlsanity.com/paper/2109.14260/) (2021)
- [Designing Menus of Contracts Efficiently: The Power of Randomization](https://mlsanity.com/paper/2202.10966/) (2022)
- [A Simple and Approximately Optimal Mechanism for an Additive Buyer](https://mlsanity.com/paper/1405.6146/) (2014)
- [A Duality-Based Unified Approach to Bayesian Mechanism Design](https://mlsanity.com/paper/1812.01577/) (2018)
- [A randomized gradient-free attack on ReLU networks](https://mlsanity.com/paper/1811.11493/) (2018)
- [Birds of a Feather Flock Together: A Close Look at Cooperation Emergence via Multi-Agent RL](https://mlsanity.com/paper/2104.11455/) (2021)
- [Approximate Optimality of Linear Contracts Under Uncertainty](https://mlsanity.com/paper/2211.06850/) (2022)
- [Mixture Selection, Mechanism Design, and Signaling](https://mlsanity.com/paper/1508.03679/) (2015)
- [On the Hardness of Signaling](https://mlsanity.com/paper/1402.4194/) (2014)
- [Solving Dynamic Principal-Agent Problems with a Rationally Inattentive Principal](https://mlsanity.com/paper/2202.01691/) (2022)

## Built on by (most cited first, 11 of 11)

- [GemNet: Menu-Based, Strategy-Proof Multi-Bidder Auctions Through Deep Learning](https://mlsanity.com/paper/2406.07428/) (2024)
- [Principal-Agent Reinforcement Learning: Orchestrating AI Agents with Contracts](https://mlsanity.com/paper/2407.18074/) (2024)
- [The Bandit Whisperer: Communication Learning for Restless Bandits](https://mlsanity.com/paper/2408.05686/) (2024)
- [Social Environment Design](https://mlsanity.com/paper/2402.14090/) (2024)
- [Learning to Negotiate via Voluntary Commitment](https://mlsanity.com/paper/2503.03866/) (2025)
- [Reward Shaping for (Inference-Time) Alignment: A Stackelberg Game Perspective](https://mlsanity.com/paper/2602.02572/) (2026)
- [Learning Truthful Mechanisms without Discretization](https://mlsanity.com/paper/2506.22911/) (2025)
- [Learning Fair and Preferable Allocations through Neural Network](https://mlsanity.com/paper/2410.17500/) (2024)
- [Learning Recommender Mechanisms for Bayesian Stochastic Games](https://mlsanity.com/paper/2505.22979/) (2025)
- [Scalable Principal-Agent Contract Design via Gradient-Based Optimization](https://mlsanity.com/paper/2510.21177/) (2025)
- [LLM Active Alignment: A Nash Equilibrium Perspective](https://mlsanity.com/paper/2602.06836/) (2026)

## Abstract (arXiv)

Contract design involves a principal who establishes contractual agreements about payments for outcomes that arise from the actions of an agent. In this paper, we initiate the study of deep learning for the automated design of optimal contracts. We introduce a novel representation: the Discontinuous ReLU (DeLU) network, which models the principal's utility as a discontinuous piecewise affine function of the design of a contract where each piece corresponds to the agent taking a particular action. DeLU networks implicitly learn closed-form expressions for the incentive compatibility constraints of the agent and the utility maximization objective of the principal, and support parallel inference on each piece through linear programming or interior-point methods that solve for optimal contracts. We provide empirical results that demonstrate success in approximating the principal's utility function with a small number of training samples and scaling to find approximately optimal contracts on problems with a large number of actions and outcomes.