We study independent private values auction environments in which the auctioneer’s revenue depends non- linearly on bidders’ interim winning probabilities. Our framework accommodates heterogeneity among bidders and places no ad hoc constraints on the mechanisms available to the auctioneer. Within this general setting, we show that feasibility of interim winning probabilities can be tested along a unidimensional curve—the principal curve—and use this insight to explicitly characterize the extreme points of the feasible set. We then combine our results on feasibility and extremality to solve for the optimal auction under a natural regularity condition. We show that the optimal mechanism allocates the good based on principal virtual values, which extend Myerson’s virtual values to nonlinear settings and are constructed to equalize bidders’ marginal revenue along the principal curve. We apply our approach to the classical linear model, settings with endogenous valuations due to ex ante investments, and settings with non-expected utility preferences, where previous results were largely limited either to symmetric environments with symmetric allocations or to two-bidder environments.

Many transactions in the marketplace rely on hard (or verifiable) information about the underlying value of the intended exchange, typically through certification— housing, diamonds, bonds being cases in point. What is the class of Pareto efficient certifications for such scenarios? This paper studies the canonical monopolistic screening problem, and models certification as hard information produced through a test to be flexibly chosen pre-trade. It argues that Pareto efficient tests take a simple form—they produce certification with a partitional structure, often with one or two thresholds. This claim is shown to be true for both the linear trading model and the non-linear pricing model.

Multiple long run players play one amongst multiple possible stage games in each period. They observe and recall past play and are aware of the current stage game being played, but are uncertain about the future evolution of stage games. This setup is termed an uncertain repeated game. The solution concept requires that a subgame perfect equilibrium be played no matter what sequence of stage games realize. The feasible set of payoffs is then so large and complex that it is not obvious how to frame standard results such as the folk theorem, and further how to construct credible rewards and punishments that work irrespective of the future evolution of games. The main goal of the paper is to build such a language through two different perspectives— one in which the modeler has access to the true stochastic process but not the players and another in which there is simply maximal uncertainty; and then to construct credible dynamic incentives that work generally for uncertain repeated games. A complete characterization of equilibria is presented for large discount factors and various extensions to related models and results are discussed in detail.

We consider optimal joint nonlinear earnings taxation of couples. We use multi-dimensional mechanism design techniques to study this problem and show that the first-order approach – that restricts attention to only local incentive constraints – is valid for a broad set of primitives. Optimal taxes are characterized by the solution to a certain second-order partial differential equation. Using the Coarea Formula, we solve this equation for various conditional averages of optimal tax rates and identify key forces that determine the optimal tax rates; show how these rates depend on earnings of each spouse, correlation in spousal earnings, and redistributive objectives of the planner; compare optimal rates for primary and secondary earners; identify both the conditions under which simple tax systems are optimal and the sources of welfare gains from more sophisticated taxes when those conditions are not satisfied. Optimal tax rates for married individuals are increasing in correlation of spousal earnings but are lower than the tax rates for single individuals, and the marginal rates for one spouse increase (decrease) in the earnings of the other when both spouses have low (high) earnings.

There are many economic problems where the observable data consists of prices and selling times– airline tickets and hotel bookings are leading examples. This paper pro- vides a dynamic pricing model to help better understand such scenarios: A seller wants to sell to a buyer a good with a fixed date of consumption. The buyer’s value for it can change over time according to a Poisson process prior to the date. The seller posts a two-part tariff where the first part extracts the buyer’s surplus modulo self-selection rents, and the second part sequentially segments the market in the spirit of second degree price discrimination through a continuously increasing price path. The buyer always pays the first part of the tariff and then solves an optimal stopping problem– which price to accept for trade. The solution of this pricing problem, solved in closed form, is shown to implement the optimal deterministic contract. In the process, a novel dynamic mecha- nism design problem is operationalized where the standard relaxed approach generically fails. Alternate implementations through refund and subscription contracts are also pre- sented. The gains from randomization are explained through the channel of information acquisition, and the optimal contract for limited informational change modeled through a fixed number of Poisson arrivals is also solved.

Financial constraints preclude many surplus producing economic transactions, and inhibit the growth of many others. This paper models financial constraints as the interaction of two forces: the agent has persistent private information and is strapped for cash. The wedge between the optimal and efficient allocation, termed distortion, increases over time with each successive “bad shock” and decreases with each “good shock”. At any point in the contract, an endogenous number of “good shocks” are required for the principal to provide some liquidity and then eventually for the contract to become efficient. Efficiency is reached almost surely. The average rate at which contract becomes efficient is decreasing in persistence of shocks; in particular, the iid model predicts a quick dissolution of financial constraints. This speaks to the relevance of modeling persistence in dynamic models of agency. The problem is solved recursively, and building on the literature, a technical tool of finding the minimal subset of the recursive domain that houses the optimal contract is further developed.

Dynamic contracts typically allow the principal to relax future incentive constraints by backloading the agent’s information rents or asking the agent to post a bond upfront which is liquidated over time. An implicit modeling assumption at play there is that both the principal and agent have equal access to capital, captured by equal discount rates. This paper introduces unequal discounting in a canonical dynamic screening problem where the agent has Markovian private information and limited commitment. The backloading force is tempered by an inter-temporal cost of incentive provision. The optimal contract features cycles with infinite memory. The interaction of Marokvian information and unequal discounting introduces technical challenges by rendering the standard relaxed problem approach invalid for certain parameters. An approximately optimal and simple alternative is provided, where both terms are made formalized.

We study quitting games and define the concept of absorption paths, which is an alternative definition to strategy profiles that accommodates both discrete- time aspects and continuous-time aspects, and is parametrized by the total probability of absorption in past play rather than by time. We then define the concept of sequentially 0-perfect absorption paths, which are shown to be limits of ε-equilibrium strategy profiles as ε goes to 0. We establish that any quitting game that does not have simple equilibria (that is, an equilibrium where the game terminates in the first period or one where the game never terminates) has a sequentially 0-perfect absorption path. We finally identify a class of quitting games that possess sequentially 0- perfect absorption paths

Characterizing and explicitly computing equilibria of undiscounted dynamic games have been a challenge for many years. In this paper, we study quitting games, which are stopping games where the terminal payoff does not depend on the stage of termination. We adapt the recursive approach of Abreu, Pearce, and Staccheti (1990) to characterize a certain subset of the set of subgame-perfect ε-equilibrium payoffs. Our approach is based on the novel representation of strategy profiles through absorption paths, which was developed in Ashkenazi-Golan, Krasikov, Rainer, and Solan (2023), and our characterization focuses on absorption paths in which exactly one player randomizes between quitting and continuing at any point in time. We then adapt the results to larger classes of absorption paths. Since quitting games form a special case of both stopping games and stochastic games, our approach may be useful in studying more general classes of these games