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.

We study sequential bargaining between a proposer and a veto player. Both have single‐peaked preferences, but the proposer is uncertain about the veto player's ideal point. The proposer cannot commit to future proposals. When players are patient, there can be equilibria with Coasian dynamics: the veto player's private information can largely nullify proposer's bargaining power. Our main result, however, is that under some conditions there also are equilibria in which the proposer obtains the high payoff that he would with commitment power. The driving force is that the veto player's single‐peaked preferences give the proposer an option to “leapfrog,” that is, to secure agreement from only low‐surplus types early on to credibly extract surplus from high types later. Methodologically, we exploit the connection between sequential bargaining and static mechanism design.

A proposer requires a veto player’s approval to change a status quo. Proposer is uncertain about Vetoer’s preferences. We show that Vetoer is typically given a non-singleton menu, or delegation set, of options to pick from. The optimal set balances the extent of compromise with the risk of a veto. We identify conditions for certain delegation sets to emerge, including “full delegation”: Vetoer can choose any action between the status quo and Proposer’s ideal action. By contrast to expertise-based delegation, Proposer gives less discretion to Vetoer when their preferences are more (likely to be) aligned.