We characterize the set of extreme points of monotonic functions that are either majorized by a given function f or themselves majorize f and show that these extreme points play a crucial role in many economic design problems. Our main results show that each extreme point is uniquely characterized by a countable collection of intervals. Outside these intervals the extreme point equals the original function f and inside the function is constant. Further consistency conditions need to be satisfied pinning down the value of an extreme point in each interval where it is constant. We apply these insights to a varied set of economic problems: equivalence and optimality of mechanisms for auctions and (matching) contests, Bayesian persuasion, optimal delegation, and decision making under uncertainty.

中文翻译：

---

极值点和专业化：经济应用

我们描述了单调函数的一组极值点，这些函数要么被给定的函数f 主化，要么它们本身主化f，并表明这些极值点在许多经济设计问题中起着至关重要的作用。我们的主要结果表明，每个极值点都以可计数的区间集合为唯一特征。在这些区间之外，极值点等于原始函数f并且在函数内部是常量。需要满足进一步的一致性条件，以确定每个区间中的极值点的值，其中它是恒定的。我们将这些见解应用于一系列不同的经济问题：拍卖和（匹配）竞赛机制的等效性和最优性、贝叶斯说服、最优授权和不确定性下的决策。