Succinct Ambiguous Contracts

2025-03-04

AI Summary

This paper studies the design of *succinct fuzzy contracts*—contracts composed of at most $k$ classical contracts—under information asymmetry, aiming to maximize the principal’s worst-case utility (per Dütting et al.’s model). We introduce the *shifted minimum-payment contract* framework, a divide-and-conquer structure that characterizes implementability of succinct contracts. We formally define and quantify the *succinctness gap*: the multiplicative utility loss incurred by restricting to $k$-contract menus, and prove a tight bound—omitting even a single contract can halve the principal’s utility. Computationally, we provide a polynomial-time algorithm when $k$ is close to $n$, yet show NP-hardness for any constant $k$ or $k = eta n$ with $eta < 1$. Finally, we establish tight upper and lower bounds on the succinctness gap, revealing a fundamental trade-off between contract simplicity and utility maximization.

Technology Category

• Multiagent Systems: Mechanism Design
• Game Theory and Economic Paradigms: Mechanism Design
• Reasoning under Uncertainty: Decision/Utility Theory

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Cryptocurrency and smart contracts
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Real-world contracts are often ambiguous. Recent work by D""utting et al. (EC 2023, Econometrica 2024) models ambiguous contracts as a collection of classic contracts, with the agent choosing an action that maximizes his worst-case utility. In this model, optimal ambiguous contracts have been shown to be ``simple"in that they consist of single-outcome payment (SOP) contracts, and can be computed in polynomial-time. However, this simplicity is challenged by the potential need for many classic contracts. Motivated by this, we explore emph{succinct} ambiguous contracts, where the ambiguous contract is restricted to consist of at most $k$ classic contracts. Unlike in the unrestricted case, succinct ambiguous contracts are no longer composed solely of SOP contracts, making both their structure and computation more complex. We show that, despite this added complexity, optimal succinct ambiguous contracts are governed by a simple divide-and-conquer principle, showing that they consist of ``shifted min-pay contracts"for a suitable partition of the actions. This structural insight implies a characterization of implementability by succinct ambiguous contracts, and can be leveraged to devise an algorithm for the optimal succinct ambiguous contract. While this algorithm is polynomial for $k$ sufficiently close to $n$, for smaller values of $k$, this algorithm is exponential, and we show that this is inevitable (unless P=NP) by establishing NP-hardness for any constant $k$, or $k=eta n$ for some $etain(0,1)$. Finally, we introduce the succinctness gap measure to quantify the loss incurred due to succinctness, and provide upper and lower bounds on this gap. Interestingly, in the case where we are missing just a single contract from the number sufficient to obtain the utility of the unrestricted case, the principal's utility drops by a factor of $2$, and this is tight.

Problem

• Explores succinct ambiguous contracts with limited classic contracts.
• Analyzes complexity and structure of optimal succinct ambiguous contracts.
• Quantifies utility loss due to succinctness with succinctness gap measure.

Innovation

• Succinct ambiguous contracts with k classic contracts
• Divide-and-conquer principle for optimal contracts
• Polynomial-time algorithm for k close to n