Existence of solutions to principal-agent problems with adverse selection under minimal assumptions

We prove an existence result for the principal-agent problem with adverse selection under general assumptions on preferences and allocation spaces. Instead of assuming that the allocation space is finite-dimensional or compact, we consider a more general coercivity condition which takes into account the principal's cost and the agents' preferences. Our existence proof is simple and flexible enough to adapt to partial participation models as well as to the case of type-dependent budget constraints.Comment: 22 page

Existence, Uniqueness, Concavity and Geometry of the Monopolist’s Problem Facing Consumers with Nonlinear Price Preferences

A monopolist wishes to maximize her profits by finding an optimal price menu. After she announces a menu of products and prices, each agent will choose to buy that product which maximizes his utility, if positive. The principal's profits are the sum of the net earnings produced by each product sold. These are determined by the costs of production and the distribution of products sold, which in turn are based on the distribution of anonymous agents and the choices they make in response to the principal's price menu. In this thesis, two existence results will be provided, assuming each agent's disutility is a strictly increasing but not necessarily affine (i.e., quasilinear) function of the price paid. This has been an open problem for several decades before the first multi-dimensional result obtained here and independently by N\"oldeke and Samuelson in 2015. Additionally, a necessary and sufficient condition for the convexity or concavity of this principal's (bilevel) optimization problem is investigated. Concavity when present, makes the problem more amenable to computational and theoretical analysis; it is key to obtaining uniqueness and stability results for the principal's strategy in particular. Even in the quasilinear case, our analysis goes beyond previous work by addressing convexity as well as concavity, by establishing conditions which are not only sufficient but necessary, and by requiring fewer hypotheses on the agents' preferences. Moreover, the analytic and geometric interpretations of a specific condition relevant to the concavity of the problem have been explored. Finally, various examples are given to explain the interaction between preferences of agents' utility and monopolist's profit which ensure statements equivalent to the concavity of the principal-agent problem. In particular, an example with quasilinear preferences on $n$-dimensional hyperbolic spaces is given with explicit solutions to show uniqueness without concavity. Similar results on spherical and Euclidean spaces are also provided. Additionally, the solutions of hyperbolic and spherical cases converge to those of Euclidean spaces as curvature goes to 0.Ph.D

Existence of solutions to principal–agent problems with adverse selection under minimal assumptions

International audienceWe prove an existence result for the principal-agent problem with adverse selection under general assumptions on preferences and allo-cation spaces. Instead of assuming that the allocation space is finite-dimensional or compact, we consider a more general coercivity condi-tion which takes into account the principal’s cost and the agents’ pref-erences. Our existence proof is simple and flexible enough to adapt to partial participation models as well as to the case of type-dependent budget constraints

Wasserstein Control of Mirror Langevin Monte Carlo

International audienceDiscretized Langevin diffusions are efficient Monte Carlo methods for sampling from high dimensional target densities that are log-Lipschitz-smooth and (strongly) log-concave. In particular, the Euclidean Langevin Monte Carlo sampling algorithm has received much attention lately, leading to a detailed understanding of its non-asymptotic convergence properties and of the role that smoothness and log-concavity play in the convergence rate. Distributions that do not possess these regularity properties can be addressed by considering a Riemannian Langevin diffusion with a metric capturing the local geometry of the log-density. However, the Monte Carlo algorithms derived from discretizations of such Riemannian Langevin diffusions are notoriously difficult to analyze. In this paper, we consider Langevin diffusions on a Hessian-type manifold and study a discretization that is closely related to the mirror-descent scheme. We establish for the first time a non-asymptotic upper-bound on the sampling error of the resulting Hessian Riemannian Langevin Monte Carlo algorithm. This bound is measured according to a Wasserstein distance induced by a Riemannian metric ground cost capturing the squared Hessian structure and closely related to a self-concordance-like condition. The upper-bound implies, for instance, that the iterates contract toward a Wasserstein ball around the target density whose radius is made explicit. Our theory recovers existing Euclidean results and can cope with a wide variety of Hessian metrics related to highly non-flat geometries