从双层规划的角度看道德风险理论的一阶条件方法

本文首先对双层规划的一个特殊例子即道德风险模型中使用的一阶条件方法（FOA）做简要的梳理，然后提出一种更为一般的使FOA有效的原则与方法。新方法主要依赖于代理人对委托人设置的目标的最优反应映射是否存在不动点，这个性质不要求原问题与用一阶条件放松以后的问题之间的约束集等价，从而也不要求代理人的期望效用对行动具有全局凹性。在新方法下，可以用较为简单的方法证明FOA在以下两种情形之一有效，即如果分布函数是概率分布的凸组合或者分布函数来自某些特殊的指数族分布。

On the first order approach for bilevel programming: moral hazard case

We revisit the first-order approach (FOA) in a classical setting of moral hazard model with multi-dimensional signal. After providing formal justification of Lagrangian duality, we reformulate the issue of validiting the FOA as an issue of the existence of a fixed point of the agent's best reaction to the principal's targeted effort level. Therefore, it is unnecessary to show the validity based on the global concavity of the agent's expected under a subclass of monotone contract. The new method allows the relaxation of several requirements of previous approaches. We generalize some results of Sinclair-Desgagne (1994) and Conlon (2009a) to validate the FOA for either the mixture probability model without the likelihood ratio order, or certain exponential family distributions with a bounded likelihood ratio.