Conference Programme
MONDAY Sept 15
Roberto Cominetti (Universidad
de Chile)
Recursive Optimal Transport and Fixed-Point Iterations for Nonexpansive
Maps
A popular method to compute a fixed point for a non-expansive
map $T:C\to C$ is the successive average iteration originally
proposed by Krasnoselskii and Mann $$x_{n+1}=(1-\alpha_{n+1})x_n+\alpha_{n+1}Tx_n.
\nleq (KM)$$ We establish an unexpected connection between $(KM)$
and optimal transport through a recursive formula that estimates
the distance between iterates $\|x_m-x_n\|\leq d_{mn}$. The recursive
optimal transport $d_{mn}$ induces a metric on the integers that
allows to characterize the rate of convergence of $(KM)$.As a
result, we completely settle Baillon and Bruck's conjecture for
the rate of convergence of the fixed point residuals, namely,
for every non-expansive map in any normed space the following
estimate holds with $\kappa=1/\sqrt{\pi}$ $$\|x_n-Tx_n\|\leq \kappa\frac{\mbox{diam}({}C)}{\sqrt{\sum_{i=1}^n\alpha_k(1-\alpha_k)}}.$$
The analysis exploits another surprising connection with discrete
probability and combinatorics, related to the Gambler's ruin for
a sum of non-homogeneous Bernoulli trials. We will discuss theextent
to which the constant $\kappa=1/\sqrt{\pi}$ is sharp.
Deniz Dizdar (University of Montréal)
Two-sided investments and matching with multi-dimensional types
and attributes
I study settings in which heterogeneous buyers and sellers, characterized
by cost types, must invest in attributes before they compete for
partners in a frictionless, continuum assignment market. I define
Cole, Mailath and Postlewaite's (2001a) notion of ex-post contracting
equilibrium in a general assignment game framework. Ex-ante efficient
investment and matching can always be supported in equilibrium.
The main part of the paper sheds light on what enables and what
precludes coordination failures resulting in mismatch of agents
(from an ex-ante perspective) and/or pairwise inefficient investments.
A kind of technological multiplicity is the key source of potential
inefficiencies. Absence of technological multiplicity rules out
pairwise inefficient investments, and it heavily constrains mismatch
in multi-dimensional environments with differentiated agents.
An example with simultaneous under -and over- investment shows
that even extreme exogenous heterogeneity may not suffice to rule
out inefficient equilibria in environments with technological
multiplicity.
Guillaume Carlier (Paris)
A Monge-Kantorovich approach to multivariate quantile regression
The aim of this talk is to present a way to extend the quantile
regression method of Koenker and Bassett to the multivariate setting.
A variant of the classical optimal transport problem with an additional
mean-independence constraint plays a crucial role in the analysis.
We shall also revisit the classical univariate case in the light
of this variational point of view. Joint work with Victor Chernozhukov
(MIT) and Alfred Galichon (Sc. Po, Paris).
Ismael Mourifié (Toronto)
Marriage matching with peer effects
Co-Authors:Aloysius Siow
This paper proposes a new static empirical marriage matching
function (MMF), the Log Odds MMF. A special case includes marriage
matching with peer effects. It also includes the Choo Siow frictionless
transferable utility MMF, a MMF with frictional transfers, the
Dagsvik Manziel non-transferable utility MMF and Chiappori, Salanie
and Weiss MMF. All these cases are empirically testable. Properties
of this Log Odds MMF are presented. A new existence and uniqueness
proof of the marriage distribution is provided.
Scott Kominers (Harvard)
Generalized Matching Market Design
In recent years, new theoretical discoveries have shown how to
generalize two-sided matching algorithms to incorporate contract
negotiation and complex market structures. In this talk, I survey
these "generalized matching" results, highlighting places
where deeper mathematics might be useful in improving our understanding.
Specific topics include weakend substitutability conditions, fixed-point
characterizations of stable outcomes, and solution concept correspondences.
TUEDSAY Sept 16
Eduardo Azevedo (University of Pennsylvania)
Perfect Competition in Markets with Adverse Selection (paper)
Policy makers and economists typically consider adverse selection
an important problem in many markets, such as insurance where
buyers can be more likely to need more insurance. Governments
typically respond to this perceived problem with complicated regulations,
which include mandates, but also interventions like subsidies,
risk adjustment, and regulation of contract characteristics. Even
though these complex interventions seem to be aimed at affecting
product characteristics, most economic models are quite limited
in determining what types of contracts arise in different markets.
In this talk I will present a model of markets with adverse selection
where the characteristics of offered products are determined endogenously,
and that allows for realistic patterns of heterogeneity between
consumers. I will focus on what equilibrium in this type of market
looks like. From a theoretical perspective, this is related to
the literatures on "multidimensional screening," "competitive
screening," and "signaling games." Towards the
end of the talk I will discuss optimal government interventions
and their the relationship to policies used in practice. This
is joint work with Daniel Gottlieb.
Maria Gualdani (George Washington
University)
A price formation model: microscopic derivation, global well-posedness
and open problems
In financial markets, the evolution of prices is influenced by
the trading system and the nature of players.
In 2006 J.M. Lasry and P.L. Lions introduced in the contest of
Mean Field Games a partial differential equations based model,
where the price of the good enters in the model as a free boundary.
Since then this model has attracted a lot of attention due to
its original mathematical structure and interesting open problems.
In this talk we survey the existing literature, with particular
attention to microscopic derivation and global well-posedness.
Nizar Touzi (Ecole Polytechnique)
Sonia Jaffe (Harvard University)
Matching Markets with Taxation of Transfers
Coauthors: Scott Duke Kominers
We analyze the effects of taxation on outcomes in matching markets.
Taxes can make inefficient outcomes stable by causing workers
to prefer firms from which they receive high idiosyncratic match
utility, but at which they are less productive. In general, efficiency
can be non-monotonic in the tax. However, when agents on one side
of the market refuse to match without a positive transfer (wage),
increasing taxes always decreases efficiency. In addition to providing
a continuous link between canonical models of matching with and
without transfers, our model highlights a cost of taxation that
does not appear to have been examined previously.
Saeedeh Ketabi (University of Isfahan)
The network expansion problem with non-linear costs
Coauthors: Aashtiani, H. Z., Sharif University of Technology
In this paper the network expansion problem is studied, where
the new links’ capacities are designed, without improving
the existing link facilities. In this problem the summation of
two costs, the performance costs of existing and new links and
the construction costs of the new links are minimized. Generally,
the performance cost is a convex function of the flow and the
construction cost is a concave function of the new capacity. Therefore
the network expansion problem is to find the minimum of the difference
of convex functions over the linear constraints. Tuy in 1987 proposed
a method for the general case; in this method the problem is transformed
to a concave minimization over a convex feasible set, firstly.
Then at each iteration a relaxation of the problem is solved by
the branch and bound type method. In this paper the implementing
of the method for efficient solving the network expansion problem,
is discussed and the result is presented.
Alpár Richárd Mészáros
(University of Paris-Sud)
Variational approach to mean field games with density constraints
The introduction of the recent theory of mean field games is
due to J.-M. Lasry and P.-L. Lions. The basic idea behind this
theory is to use the mean field approach from statistical physics
to study differential games where the number of players is tending
to infinity.
In this talk we investigate some mean field game models under
density constraints. These constraints are very natural to suppose
because in many crowd motion models (with and without strategy
in the movement) one wants to avoid the congestion.
Our approach is a variational one, having its roots in the so-called
Benamou-Brenier dynamic formulation of optimal transportation.
Similar variational models for mean field games without density
constraints were studied recently by P. Cardaliaguet and his coauthors.
This talk is based on several ongoing joint works with F. Santambrogio
(University of Paris-Sud), F. J. Silva (University of Limoges)
and P. Cardaliaguet (University of Paris-Dauphine).
Monica G Cojocaru (University of Guelph)
Equilibrium in competitive help models in biological markets
Coauthors: Erin Wild, Univ. of Guelph
In this talk we show the emergence of equilibrium in models of
help competition between biological individuals. The model has
been introduced a few years back in the sociological context,
but we give here a full mathematical analysis of it pointing out
existence and properties of its equilibrium states in a dynamical
system context. We use both game theory, variational inequalities
and agent-based models to derive a complete picture of the model.
Last but not least, we give a flavour of how competitive help
can be placed in the context of a generalized Nash game.
Brendan Pass (University of Alberta)
Uniqueness and purity in multi-agent matching problems
Coauthors: Young-Heon Kim
This talk is focused on multi-marginal optimal transport, the
mathematical theory associated with multi-agent matching problems
under transferable utility; that is, matching problems where more
than two agents come together to form teams.
In the classical, two agent setting, the generalized Spence-Mirrlees
condition ensures uniqueness and purity of stable matches, under
a regularity condition on the first marginal. I will present an
analogous condition, developed jointly with Y.-H. Kim, which ensures
purity and uniqueness in the multi-agent framework. This condition
is much stronger than its two marginal counterpart, as I will
attempt to illustrate with a handful of examples.
Oksana Pichugina (Brock University)
Functional Representations of Combinatorial Sets and Applications
in Optimization
Coauthors: Sergey Yakovlev (Ukraine)
We consider a class of optimization problems on combinatorial
sets whose images in the Euclidean space are inscribed into a
sphere. This class includes Boolean Programming, Optimization
over permutations and other combinatorial configurations. It has
many real-world applications.
There are presented different representations of these combinatorial
sets as intersection of continuum ones such as intersection of:
a) a sphere and a combinatorial polyhedron, b) two or more surfaces.
These representations are used in several original approaches
to solving this class problems reducing the initial discrete problem
to series of continuous ones. Approach 1 - polyhedral-spherical
method for solving combinatorial problems, which uses a continuous
representation of the set as an intersection of a sphere and a
combinatorial polyhedron as well as an analytical description
of the polyhedron. Approach 2 - Penalty Method applicable to the
case of: a) differentiable target function, b) availability of
strict differentiable functional representation of the combinatorial
set and an explicit solution of a linear problem over the set.
It should be noted that in these approaches the Convex Optimization
is not always applicable, but in computational algorithms we can
essentially use the fact that all functions of functional representations
of discrete sets inscribed into a sphere can be considered as
convex since for any function defined over such a set there is
a convex extension of the function from the set into Euclidean
space.
Hanzhe Zhang (University of Chicago)
Stochastic Investments and Bidimensional Matching: Explaining
Marriage Age Patterns and the College Gender Gap
I construct an equilibrium investment-and-matching framework in
which people first make investments that yield stochastic returns
and then match based on men and women's realized wage and women's
reproductive fitness. The framework allows me to expand static
marriage market analyses to simultaneously study marital, educational,
and occupational choices. Namely, the model explains the evolution
of the relationships between marriage age and personal income.
The recent global phenomenon that more women than men go to college
naturally arises in the unique equilibrium.
WEDNESDAY Sept 17
Alfred Galichon (Center for Economic
Policy Research )
Connecting matching models with and without Transferable Utility,
1 , 2
This lecture aims at providing an empirical framework for matching
models with heterogeneity in tastes and general transfer technologies.
It is organized in two parts: 1. Generalized Entropy of Choice
and Capacity-constrained Discrete Choice. We first revisit the
literature on random utility models by emphasizing the role of
a proper generalization of the notion of entropy, defined using
Legendre transforms. The duality between the selection model and
the assignment model follows, as well as the duality between the
equilibrium characterization problem and the identification problem.
2. Equilibrium characterization and identification in matching
models. The previous theory is then applied to characterize equilibrium
and provide identification in matching models with imperfectly
transferable utility (ITU), including as special cases both the
transferable utility (TU) and nontransferable utility (NTU) models.
THURSDAY Sept 18
Lars Nesheim (University College London)
Minyi Huang (Carleton)
Yeon-Koo Che (Columbia University)(Paper)
Efficiency and Stability in Large Matching Markets
Authors: Yeon-Koo Che (Columbia University), and Olivier
Tercieux (Paris School of Economics)
We study efficient and stable mechanisms in matching markets
when the number of agents is large and individuals' preferences
are drawn randomly from a class of distributions allowing for
both common value and idiosyncratic components. In this context,
as the market grows large, all Pareto efficient mechanisms (including
top trading cycles, serial dictatorship, and their randomized
variants) are asymptotically payoff equivalent (``up to the renaming
of the agents''), yielding utilitarian upper bound in the limit.
If objects' priorities are also randomly drawn but agents' common
values for objects are heterogenous, then well-known mechanisms
such as deferred acceptance and top trading cycle mechanisms fail
either efficiency or stability even in the asymptotic sense. We
propose a new mechanism is asymptotically efficient, asymptotically
stable and asymptotically incentive compatible.
Keywords: Large matching market, Pareto efficiency, Stability,
Fairness, Payoff equivalence, Random graph theory.
Fuhito Kojima (Stanford University)
Stable Matching in Large Economies
Complementarities of preferences have been known to jeopardize
the stability of two-sided matching markets, yet they are a pervasive
feature in many matching markets. We revisit the stability issue
with such preferences in a large market. Workers have preferences
over firms while firms have preferences over distributions of
workers and may exhibit complementarity. We demonstrate that if
each firm's choice changes continuously as the set of available
workers changes, then there exists a stable matching even with
complementarity. Building on this result, we show that there exists
an approximately stable matching in any large finite economy.
We apply our analysis to show the existence of stable matchings
in probabilistic and time-share matching models with a finite
number of firms and workers.
Qingmin Liu (Columbia University)
Gabriel Penagos (Universidad Javeriana)
A Martingale Approach for Portfolio Allocation with Stochastic
Volatility and Jumps
A market model composed of a risky asset and a riskless bond
is
considered. The risky security satisfies a stochastic differential
equation which
includes a jump component with lognormal amplitude change. Volatility
is
assumed stochastic and following a mean reverting process. Investors
objective
is to maximize the expected utility on terminal wealth, hence,
the optimal
allocation rule is derived through the use of martingale and duality
techniques.
Weights on assets are found along with the expressions for market
price of risk,
market price of volatility risk and the market price of jump risk.
The results
are applied to market data, therefore, the conditional characteristic
function
associated with the market model is calculated and the first four
cumulants
are derived. Exact expressions for the mean, standard deviation,
skewness and
excess kurtosis are obtained. Model parameters are estimated by
means of
a distance minimization using a discretization of the empirical
characteristic
function"
Duality techniques means that I change the primal problem into
a dual problem using the Legendre -
Fenchel transform (convex conjugate) in order to find the Equivalent
Martingale Measure. The, now, stochastic control problem (a PDE
given by the infinitesimal generator) is to find the optimal control
(price of risk) when including jumps (apart from brownian motion).
FRIDAY Sept 19
Alexander Kolesnikov (Higher School
of Economics)
TBA
Arnaud Dupuy (CEPS/INSTEAD)
Migration in China: To work or to wed?
Authors: Arnaud Dupuy, Alfred Galichon, and Liping Zhao
Why do people migrate? In this paper we study the trade-o s between
migrating to work and migrating to wed. To this aim, we develop
a marriage matching model in which men and women, are initially
distributed over various locations, i.e. were born and raised
in various locations. To each location corresponds a marriage
market and a labor market. Men and women can choose to stay at
their current location and enter the local marriage market and
labor market or migrate to a diff erent location and enter the
marriage market and labor market at that location. Migration induces
additional costs but may also generate benefi ts in the form of
better labor market perspectives in the destination's labor market.
These costs and benefi ts are specifi c to each man and each woman.
Our model encompasses the classical matching model a la Becker
(1973) and Shapley and Shubik (1972) with the hedonic model a
la Rosen (1974). We bring the model to the data and use data from
China. Preliminary results indicate that improving mobility may
lead to large welfare gains.
Marc Henry (Pennsylvania State University)
Identifying multi-attribute hedonic models
Authors: Victor Chernozhukov, Alfred Galichon and Marc Henry
This paper derives conditions under which preferences and technology
are nonparametrically identified in hedonic equilibrium models,
where products are differentiated along more than one dimension
and agents are characterized by several dimensions of unobserved
heterogeneity. With products differentiated along a quality index
and agents characterized by scalar unobserved heterogeneity, single
crossing conditions on preferences and technology provide identifying
restrictions. We develop similar shape restrictions in the multi-attribute
case and we provide identification results from the observation
of a single market. We thereby extend identification results in
Matzkin (2003) and Heckman, Matzkin, and Nesheim (2010) to accommodate
multiple dimensions of unobserved heterogeneity.
Filippo Santambrogio (Université
Paris-Sud)
Urban equilibria and displacement convexity
I will present some classical equilibrium models for the distribution of the
population in a urban area where agents want to balance between
a cost for land, which depends on the local population density
(i.e. agents prefer to be as spread as possible), and an accessing
cost (where, on the contrary, if they are too spread they are
too far). The equilibrium condition can be seen to be equivalent
to be a critical point for a certain global functional (which
is not, in general, equal to the total social utility), but the
corresponding optimization problem is often non-convex as soon
as interaction energies are present. In this case multiple equilibria
could be observed, and not all equilibria are optimizers.
Yet, in some cases convexity strikes back, via the notion of displacement
convexity introduced by McCann in 1997.
This notion, taken from optimal transport theory, will turn out
to be very important even when the equilibrium and optimization
problems do not explicitly involve optimal transport notion.
I will discuss general cases where multiple equilibria can or
cannot occur, using this notion of convexity, following a joint
work with A. Blanchet (Toulouse 1, France) and P. Mossay (Reading,
UK).
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