Optimal design of distributed decision policies can be a difficult task, illustrated by the famous Witsenhausen counterexample. In this paper we characterize the optimal control designs for the vector-valued setting assuming that it results in an internal state that can be described by a continuous random variable which has a probability density function. More specifically, we provide a genie-aided outer bound that relies on our previous results for empirical coordination problems. This solution turns out to be not optimal in general, since it consists of a time-sharing strategy between two linear schemes of specific power. It follows that the optimal decision strategy for the original scalar Witsenhausen problem must lead to an internal state that cannot be described by a continuous random variable which has a probability density function.
翻译:以著名的Witsenhauseen反例为例,对分布式决策政策的最佳设计可能是一项困难的任务。在本文中,我们描述矢量估值设置的最佳控制设计,假设它产生一个内部状态,可以用具有概率密度函数的连续随机变量来描述。更具体地说,我们提供一个依靠我们先前的经验性协调问题结果的精灵辅助外框。这个解决方案在总体上并不理想,因为它包括两个特定力量线性计划之间的时间共享战略。因此,最初的 scalar Witsenhausesen问题的最佳决定战略必须导致一个无法用具有概率密度函数的连续随机变量来描述的内部状态。