Efficient Reduction from Multi-Parameter to Single-Parameter Bayesian Contract Design

Core Concepts

There exists an efficient polynomial-time reduction from the general multi-parameter Bayesian contract design problem to the single-parameter Bayesian contract design problem. This reduction preserves approximation guarantees, implying that the two problems are computationally equivalent.

Abstract

The paper presents a fundamental algorithmic connection between two models of Bayesian contract design - the general multi-parameter setting and the single-parameter setting. The key insights are: The authors establish an efficient polynomial-time reduction from the general multi-parameter Bayesian contract design (BCD) problem to the single-parameter BCD problem. This reduction is (almost) approximation-preserving, meaning that any approximate solution to the single-parameter instance can be converted to an approximate solution to the original multi-parameter instance with a small additive loss. This reduction is somewhat surprising because in the closely related field of Bayesian mechanism design, a polynomial-time reduction from multi-parameter to single-parameter settings is believed to not exist. The authors' result demonstrates the intrinsic difficulty of addressing moral hazard in Bayesian contract design, regardless of being single-parameter or multi-parameter. As corollaries, the authors show that single-parameter BCD is computationally hard - it is NP-hard to compute a 1/K^(1-δ)-approximation to the optimal single contract, and NP-hard to compute a constant-factor approximation to the optimal menu of contracts, even when the agent's type distribution is regular. This resolves open questions posed in prior work. The authors also construct a class of single-parameter BCD instances that have a large Ω(n) gap between the principal's utility from the optimal menu of contracts and the optimal single contract, where n is the number of agent actions. This answers a major open question from prior work.

Stats

For any δ ∈ (0, 1], it is NP-hard to compute a 1/K^(1-δ)-approximation to the optimal single contract, where K is the number of agent types. For any constant ρ > 0, it is NP-hard to compute a ρ-approximation to the optimal menu of contracts.

Quotes

"The main result of this paper is an almost approximation-preserving polynomial-time reduction from the most general multi-parameter Bayesian contract design (BCD) to single-parameter BCD." "This efficient reduction is somewhat surprising because in the closely related problem of Bayesian mechanism design, a polynomial-time reduction from multi-parameter to single-parameter setting is believed to not exist."

Key Insights Distilled From

A Reduction from Multi-Parameter to Single-Parameter Bayesian Contract Design

by Matteo Casti... at arxiv.org 04-05-2024

https://arxiv.org/pdf/2404.03476.pdf

Deeper Inquiries

How do the structural properties of the multi-parameter and single-parameter settings differ in a way that enables this efficient reduction, despite the apparent complexity gap in Bayesian mechanism design

The structural properties of the multi-parameter and single-parameter settings differ in terms of the complexity of describing the agent's type. In the multi-parameter setting, the agent's type is characterized by multiple parameters that can affect both the agent's productivity and effort costs. This complexity arises from the need to consider a tuple of mn + n parameters to describe the agent's type fully. On the other hand, in the single-parameter setting, the agent's type is simplified to a single value that represents the agent's cost per unit of effort, with the assumption that all efforts are equally productive. The efficient reduction from the multi-parameter to the single-parameter setting is enabled by carefully designing the reduction algorithm to capture the essence of the agent's type in both settings. By constructing a single-parameter instance that captures the essential characteristics of the multi-parameter instance, the reduction algorithm can effectively translate solutions between the two settings. Despite the apparent complexity gap in Bayesian mechanism design, the reduction leverages the similarities in the underlying optimization problems to achieve an almost approximation-preserving transformation.

Can the techniques developed in this paper be extended to establish similar reductions between other classes of multi-parameter and single-parameter optimization problems in mechanism design

The techniques developed in this paper can potentially be extended to establish similar reductions between other classes of multi-parameter and single-parameter optimization problems in mechanism design. The key lies in identifying the fundamental structural properties and constraints that govern the optimization problems in each setting. By understanding how these properties interact and influence the design of optimal contracts, it may be possible to devise reduction algorithms that bridge the gap between different parameterizations. However, the extension of these techniques to other classes of problems would require a thorough analysis of the specific characteristics and constraints of the optimization problems involved. Each class of problems may have unique features that impact the feasibility and effectiveness of reduction techniques. Nonetheless, the general approach of constructing equivalent instances and designing efficient reduction algorithms could serve as a foundation for exploring reductions in other settings within mechanism design.

What are the key insights that allow the authors to construct single-parameter instances with a large gap between the optimal menu and single contract, and how can these insights be generalized to other settings

The key insights that allow the authors to construct single-parameter instances with a large gap between the optimal menu and single contract lie in the strategic design of the instance parameters. By carefully choosing the action costs, efficacy, and type distributions, the authors create instances where a menu of contracts incentivizes different agent types to take specific profitable actions, while a single contract can only incentivize a limited set of actions. This strategic parameter selection creates a scenario where the optimal menu of contracts significantly outperforms the optimal single contract in terms of the principal's utility. These insights can be generalized to other settings by focusing on the fundamental principles of contract design under moral hazard and adverse selection. By understanding how different parameters influence agent behavior and principal utility, one can design instances that highlight the advantages of using a menu of contracts over a single contract. The key lies in creating scenarios where the flexibility of a menu of contracts allows for more effective incentivization of diverse agent types, leading to superior outcomes compared to a single contract.