We characterize the extreme points of the set of incentive-compatible mechanisms for screening problems with linear utility. Our framework subsumes problems with and without transfers, such as monopoly pricing, principal-optimal bilateral trade and barter exchange, delegation and veto bargaining, or belief elicitation via proper scoring rules. In every problem with one-dimensional types, extreme points admit a tractable description. In every problem with multi-dimensional types, extreme points are dense in a rich subset of incentive-compatible mechanisms, which we call exhaustive mechanisms. Building on these characterizations, we derive parallel conclusions for mechanisms that can be rationalized as (uniquely) optimal under a fixed objective. For example, in the multi-good monopoly problem, mechanisms that uniquely maximize revenue for some type distribution are dense among all incentive-compatible and individually rational mechanisms. The proofs exploit a novel connection between menus of extreme points and indecomposable convex bodies, first studied by Gale (1954).

翻译：我们刻画了具有线性效用的筛选问题中激励相容机制集合的极值点。我们的框架涵盖了包含转移支付与不包含转移支付的各类问题，例如垄断定价、委托人最优的双边交易与物物交换、授权与否决议价，或通过严格评分规则进行的信念诱导。在所有一维类型问题中，极值点均具有易于处理的描述。在所有多维类型问题中，极值点在激励相容机制的一个丰富子集中稠密，我们称该子集为穷举机制。基于这些刻画，我们针对可在固定目标下被（唯一）最优化的机制得出了平行的结论。例如，在多商品垄断问题中，对于某些类型分布能唯一最大化收益的机制在所有激励相容且个体理性的机制中是稠密的。证明利用了极值点菜单与不可分解凸体之间的新颖联系，该联系最早由Gale（1954）研究。