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

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 $\beta$-approximate contract (resp. menu of contracts) of $I^S$ can in turn be converted to a $(\beta -\epsilon)$-approximate contract (resp. menu of contracts) of $I^M$. The reduction is in time polynomial in the input size and $\log(\frac{1}{\epsilon})$; moreover, when $\beta = 1$ (i.e., the given single-parameter solution is exactly optimal), the dependence on $\frac{1}{\epsilon}$ 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 $\Theta(n)$ where $n$ is the number of actions, answering the major open question of [Guruganesh et al. 2021] for the single-parameter case.