从多参数到单参数贝叶斯契约设计的归约
A Reduction from Multi-Parameter to Single-Parameter Bayesian Contract Design
AI总结：
本文提出一种近似保持的多项式时间归约，将多参数贝叶斯契约设计（BCD）转化为单参数BCD，并借此证明单参数BCD最优契约设计的NP困难性及契约菜单与单一契约间Θ(n)的效用差距。
摘要：
本文的主要结果是一个近似保持的多项式时间归约，将最一般的多参数贝叶斯契约设计（BCD）归约到单参数BCD。也就是说，对于任何多参数BCD实例$I^M$，我们构造一个单参数实例$I^S$，使得$I^S$的任何$β$-近似契约（对应地，契约菜单）反过来可以转换为$I^M$的$(β-ε)$-近似契约（对应地，契约菜单）。该归约的时间复杂度是输入大小和$log(\ rac{1}ε)$的多项式；此外，当$β=1$（即给定的单参数解是精确最优的）时，对$\ rac{1}ε$的依赖可以被消除，从而得到多项式时间的精确归约。这种高效的归约有些令人惊讶，因为在密切相关的贝叶斯机制设计问题中，人们认为不存在从多参数到单参数设置的多项式时间归约。我们的结果表明，无论单参数还是多参数，在贝叶斯契约设计中处理道德风险问题具有内在的困难性。作为副产品，我们的归约回答了算法契约设计近期文献中的两个开放问题：(a) 它表明即使当代理人的类型分布是正则的，单参数BCD中的最优契约设计也不在APX中，除非P=NP，从而否定地回答了[Alon et al. 2021]的开放问题；(b) 它表明委托人使用契约菜单和单一契约之间的（阶数上的）紧效用差距是$Θ(n)$，其中$n$是行动的数量，回答了[Guruganesh et al. 2021]关于单参数情况的主要开放问题。
英文摘要：
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.
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