# Workshop “Optimization, Transport and Equilibrium in Economics” 2009 Abstracts and Papers

**Abstracts**

**A. Blanchet (U. Toulouse): “Regular and singular points in the parabolic obstacle problem and application to American options”**

**D. Bosc (Polytechnique): “Numerical Approximation of the Brenier Map”**

**Yann Brenier (CNRS FR 2800 Nice, France): “Competition and rearrangement theory”**

Abstract: We describe a simple competition model based on rearrangement theory. A vectorial (multicriteria) version is directly linked to optimal transportation theory and can also be related to convection theory in fluid mechanics.

**Giuseppe Buttazzo (U. Pisa): “Optimal pricing problems with transportation costs”**

Abstract: We consider an optimization problem in a given region where an agent has to decide the price of a product at every point. The customers know the pricing pattern and may shop at any place, paying the cost of the product and additionally a transportation cost. Two models will be considered: in the first one the agent operates everywhere, in the second one the agent operates only in a subregion. For both models a mathematical framework is provided and an existence result is given for a pricing strategy which maximizes the total profit of the agent. Some possible extensions and developments are discussed, as for instance the case of Nash equilibria when more agents operate on the same market.

**C. Decker (U. Toronto): “When do preferences uniquely determine the number of marriages between different types in the Choo-Siow matching model? Sufficient conditions for a unique equilibrium” (joint w. R. McCann and B. Stephens)**

In a transferable utility context, Choo and Siow (2005) introduced a marriage matching function which defines the gains generated by a marriage between agents of prescribed types in terms of the observed frequency of such marriages within the population, relative to the number of unmarried individuals of the same types. This definition is scale independent. Left open in their work was the question of whether, for a given population whose frequency of types is known, this gains matrix captures all of the additional statistical information used to define it. In a joint work with Robert McCann and Benjamin Stephens, we resolve this question in the affirmative, assuming the norm of the gains matrix (viewed as an operator) to be less than one.

**J. Donaldson (LSE): “Asymmetric Information in Hedonic Markets”**

Abstract: I study the role of information in a market for single, indivisible production good, by building on recent work on hedonic markets by Ekeland and by Chiappori, McCann, and Nesheim. I generalize the equilibrium concept to include signals and beliefs. For each price, a solution to the classical Monge problem defines a pure cheap-talk pooling equilibrium match. In contrast, if the sellers’ types are unobservable, then a separating equilibrium admits a complete information characterization. Efficiency is lost in both cases, but equilibrium measures remain equivalent to surplus-maximizing measures for appropriately modified surplus functions.

**I. Ekeland (UBC): “Optimal Transportation and the Structure of Cities”**

**A. Figalli (Polytechnique): “The optimal partial transport problem”**

Abstract: Given two densities of mass f and g, a variant of the classical optimal transport problem consists in transporting a fixed fraction of the mass of f onto g as cheaply as possible. First of all, we will see how one can prove in this setting existence and uniqueness of an optimal transport map. Moreover we will see that the issue of the regularity of this map translates into the study of a Monge-Ampere equation in a domain with free boundary, and in this way local regularity of the optimal transport map and $C^1$ regularity of the free boundary can be proven. In contrast with the classical case, this regularity result is optimal: we can construct two smooth densities on the real line, respectively supported on an interval, such that the optimal map is not.

**P.-N. Giraud (ENSMP): “Hubbert peak oil and Hotelling rent revisited by a simulation model”**

**O. Guéant (Dauphine): “A reference case for mean field games”**

**C. Jimenez (U. Brest): “Movement of crowds and transport with a convex obstacle”**

**J.-M. Lasry (Dauphine): “Oil production, and strategic interaction with substitutes: an Mean-Field Game approach”**

**P.-L. Lions (Dauphine): “Mean-Field Games”**

**Bertrand Maury (Universite Paris-Sud): “Crowd evacuation models of the gradient flow type”.**

We are interested in modeling the evacuation of a building in an emergency situation. We propose a class of models based on the following considerations : individuals tend to minimize their own dissatisfaction, and the global instantaneous behaviour results from a balance between fulfillment of individualistic tendencies and respect of the congestion constraint. This modelling approach can be carried out microscopically (each individual is represented by a disc) and macroscopically (the population is described by a diffuse density). The microscopic model has a natural gradient flow structure. We will describe how the Wasserstein setting, which in some way consists in following people in their motion (in a Lagrangian way) makes it possible to extend the gradient flow structure to the macroscopic model, and provides a natural framework for this type of unilateral evolution problem. We will present numerical strategies to solve those problems, and illustrate their behaviour in some standard situations. In particular we will emphasize the fact that the steepest descent principle on which the model is based is likely to lead to highly unoptimal evacuation in practice.

**R. McCann (U. Toronto): “When is multidimensional screening a convex program?” (joint with Alessio Figalli and Young-Heon Kim)**

Paper and abstract available **here**.

**J.-M. Morel (ENS Cachan): “Topics in Landscape evolution modeling”**

Abstract: In this talk I’ll present a PDE basic model for landscape evolution that stems from discussions with Giuseppe Buttazzo, Filippo Santambrogio, and simulations and numerical schemes devised by Alexander Chen. This tentative basic model is a system of three PDE’s. Its goal is to give the simplest formalization of the joint three phenomena that build landscape, namely erosion, transport, and sedimentation. The main question is whether this system can explain some features of landscape evolution, such as the formation of valleys and canyons.

**Pascal Mossay (Henley Business School, U. Reading): “On Spatial Equilibria in a Social Interaction Model” (joint work with Pierre PICARD, University of Manchester)**

Abstract: Social interactions are at the essence of societies and explain the gathering of individuals in villages, agglomerations, or cities. We study the emergence of multiple agglomerations as resulting from the interplay between spatial interaction externalities and competition in the land market. As opposed to Beckmann’s original framework, agents get dispersed across several cities distributed along a circle. Spatial equilibrium configurations involve a high degree of spatial symmetry in terms of city size and location, and can be Pareto-ranked.

**Ludger Ruschendorf (U. Freiburg): “Optimal mass transportation in risk measures and dependence orderings”**

**A. Pratelli (Pavia-Madrid): “Which planar convex set has the longest minimal bisecting chord?”**

Abstract: We will discuss an old open problem which is extremely easy to state: for any convex set in the plane, there are of course infinitely many bisecting chords (i.e. chords which divide the set in two parts of the same area), and by trivial compactness there is a shortest one. The problem is to determine, among all the sets with fixed area, which one has the longest minimal chord. At first glance, it may seem that the solution should be the disk, but it is known since years that it is not so, and the solution is not known. We will give the solution in a suitable class of convex sets, and discuss what can be said in the general case.

**Eugene Stepanov (U. St Petersburg): “New results on classical Steiner problem and its relationship with mass transportation”.**

Abstract. A classical Steiner (minimal connection) problem will be discussed in a general setting and topological as well as some geometrical regularity properties of its solutions will be studied. Finally, we will relate this problem to some mass transportation problems

**B. Salanie (Columbia U.): “Matching with trade-offs” (joint with A. Galichon)**

Abstract: We investigate in this paper the theory and econometrics of optimal matchings with competing criteria. The surplus from a marriage match, for instance, may depend both on the incomes and on the educations of the partners, as well as on characteristics that the analyst does not observe. Even if the surplus is complementary in incomes, and complementary in educations, imperfect correlation between income and education at the individual level implies that the social optimum must trade off matching on incomes and matching on educations. We characterize, under mild assumptions, the properties of the set of feasible matches, and of the socially optimal match. Then we show how data on the covariation of the types of the partners in observed matches can be used to test that the observed matches are socially optimal. Under optimality, our procedure also provides an estimator of the parameters that define social preferences over matches. We illustrate our approach on data from the June 1995 CPS.

**M. Scarsini (LUISS): “Repeated Congestion Games with Local Information” (joint with Tristan Tomala)**

Abstract: Congestion game is a widely used model to represent interactions of many agents who share common and limited resources. In this framework, agents or players act independently and choose which resource to use. The cost of using a resource is an increasing function of the total number of agents who use it, thus the name congestion. Typical applications are road traffic and internet routing. When agents act independently and selfishly, the system is expected to reach an equilibrium, that is a situation where each choice is optimal form an individual standpoint, taking the behavior of the rest of the society as fixed. From a social perspective this outcome can be quite bad. The equilibrium might well fail to maximize the social welfare. It might even be the case that agents unanimously prefer another outcome to the equilibrium, but fail to reach it, in absence of a coordination device.

In this talk we examine conditions under which efficiency can be obtained in equilibrium by repeating the game. The difficulty in doing so resides in the fact that each player cannot observe the moves of all other players but only of the ones with whom she shares a path.

**W. Schachermayer (U. Vienna): “Optimal and better transport plans”**

Abstract: We consider the Monge-Kantorovich transport problem in a purely measure theoretic setting, i.e. without imposing continuity assumptions on the cost function. It is known that transport plans which are concentrated on c-monotone sets are optimal, provided the cost function c is either lower semi-continuous and finite, or continuous and may possibly attain the value ∞. We show that this is true in a more general setting, in particular for merely Borel measurable cost functions which are finite almost everywhere on an open set. In a previous paper Schachermayer and Teichmann considered strongly c-monotone transport plans and proved that every strongly c-monotone transport plan is optimal. We establish necessary and sufficient conditions on c-monotone transport plans to be strongly c-monotone.

**E. Stepanov (St. Petersburg): “New results on classical Steiner problem and its relationship with mass transportation”**