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

TL;DR

This work establishes an almost approximation-preserving polynomial-time reduction from the general multi-parameter Bayesian contract design problem to the single-parameter version, revealing a computational equivalence between the two extremes under both moral hazard and adverse selection. The reduction constructs a single-parameter instance with carefully weighted parameters and auxiliary components so that each agent type behaves as in the original multi-parameter instance, enabling conversion between β-approximate menus and contracts with an additive loss ε. As a byproduct, the paper resolves open questions: optimal single-parameter contracts are not APX-hard under regular distributions, and the principal's gap between menu and single-contract utility in the single-parameter setting can be Θ(n). The analysis also delineates conditions under which a single contract can be as effective as a menu, specifically when the production matrix $F$ has full rank and the limited-liability constraint is relaxed. Overall, the results deepen understanding of moral hazard in Bayesian contract design and provide a modular reduction framework to transfer algorithmic and hardness results across contract formats.

Abstract

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. That is, for any multi-parameter BCD instance $I^M$, we construct a single-parameter instance $I^S$ such that any $β$-approximate contract (resp. menu of contracts) of $I^S$ can in turn be converted to a $(β-ε)$-approximate contract (resp. menu of contracts) of $I^M$. The reduction is in time polynomial in the input size and $\log(\frac{1}ε)$; moreover, when $β= 1$ (i.e., the given single-parameter solution is exactly optimal), the dependence on $\frac{1}ε$ can be removed, leading to a polynomial-time exact reduction. 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. Our result demonstrates the intrinsic difficulty of addressing moral hazard in Bayesian contract design, regardless of being single-parameter or multi-parameter. As byproducts, our reduction answers two open questions in recent literature of algorithmic contract design: (a) it implies that optimal contract design in single-parameter BCD is not in APX unless P=NP even when the agent's type distribution is regular, answering the open question of [Alon et al. 2021] in the negative; (b) it implies that the principal's (order-wise) tight utility gap between using a menu of contracts and a single contract is $Θ(n)$ where $n$ is the number of actions, answering the major open question of [Guruganesh et al. 2021] for the single-parameter case.

Figures (3)

• Figure 1: The example of the construction procedure $I^{S}=\texttt{Single}(I^M,\epsilon)$, with only two types $\theta_1$ and $\theta_2$. We add one dummy outcome $\bar{\omega}$ and one action $\bar{a}$ with cost $0$. Best viewed in color.
• Figure 2: Comparisons of the exponential weights between action cost and expected payment.
• Figure 3: The flow of proof of Lemma \ref{['lemma:single-multi']}. From a given feasible contract $\bar{P}$ for instance $I^S$, we construct a contract $P$ for $I^M$ through a sequence of intermediary contracts $\breve{P}$ and $\breve{P}^*$.

Theorems & Definitions (32)

• Definition 2.1
• Definition 2.2: Incentive Compatibility (IC) myerson1982optimal
• Remark 2.1
• Theorem 3.1
• Lemma 3.2
• Lemma 3.3
• Theorem 4.1
• Corollary 4.2
• Corollary 4.3
• Corollary 4.4