Download Abstracts of the talks in May 2013 GaO: Minicourses: Alain

Abstracts of the talks in May 2013 GaO:
Minicourses:
Alain Bensoussan (International Center for Decision and Risk Analysis
School of Management, University of Texas – Dallas):
Control and Nash games with mean field effects
Abstract. Mean field theory has raised a lot of interest in the recent years, see in
particular Lasry-Lions [11],[12],[13], Gueant-Lasry-Lions [8], Huang-CainesMalhamé[9],[10], Buckdahn-Li-Peng [7]. There are a lot of applications. In
general the applications concern approximating an infinite number of players
with common behavior by a representative agent. This agent has to solve a
control problem perturbed by a field equation, representing in some way the
behavior of the average infinite number of agents. The state equation is
modified by the expected value of some functional on the state. We first review
in this presentation the linear-quadratic case. This has the advantage of getting
explicit solutions. In particular this leads to the study of Riccati equations. We
discuss two approaches. One in which the agent considers the mean field term
as external, and an equilibrium occurs when this mean field term coincides with
the average of his/her own action. The problem reduces to a fixed point. In
another one, the mean field is a functional of the state and therefore the agent
can influence it by his/her own decision. When there is no control, there is no
difference between the two approaches. However, with control the two
approaches are not equivalent. In particular, the fixed point approach leads to
non-symmetric Riccati equations, which have no control interpretation. They
raise interesting mathematical problems of their own. For nonlinear
nonquadratic problems, the approach which has been explored is the
endogeneous one. The control of the representative agent can influence the
mean field term, which is the average of the agent’s state. Dynamic
pogramming approach fails, because of the so called time inconsistency effect.
Fortunately, the stochastic maximum principle can be applied. The adjoint
variables are solutions of stochastic backward differential equations, with mean
field terms. In the approach of Lasry-Lions, the starting point is a Nash
equilibrium game for a very large number of players. In principle, the problem
can be treated by Dynamic Programming. The Bellman equation becomes a
system of nonlinear partial differential equations, for which the techniques of
[2] can be considered. When the number of players becomes infinite, and all of
them are identical, then going to the limit, one obtains an Hamilton-Jacobi
Bellman equation, with mean field term. The mean field term is reminiscent of
the coupling with other players, which existed before going to the limit. We
compare the various approaches, and their interprations as control problems. In
Lasry-Lions approach the limit is obtained thanks to ergodic theory, which
means that the limit control problem is an ergodic control problem, with mean
field effect. There is a different and interesting approach which also leads to
similar types of P.D.E with mean-field terms. The state equation is the
Chapman Kolmogorov equation, describing the probability measure of
the state. It is the dual control problem. Then, the Bellman equation can be
interpreted as a necessary condition of optimality for the dual problem. To
generate mean-field terms, it is sufficient to consider objective functions which
are not just linear in the probability measure, but more complex. This approach
has a different type of application. In the traditional stochastic control problem,
the objective functional is the expected value of a cost depending on the
trajectory. So it is linear in the probability measure. This type of functional
leaves out many current considerations in control theory, namely situations
where one wants to take into consideration not just the expected value, but
also the variance. This case occurs often in Risk Management. Moreover, one
may be interested by several functionals on the trajectory, even though one is
satisfied with expected values. If one combines these various expected values in
a single pay-off, one is lead naturally to mean-field problems. They are
meaningful even without considering ergodic theory, i.e. long term behavior.
Anyway, in all the previous considerations, the averaging approach reduces an
infinite number agent to a representative agent, who has a control problem to
solve, with an external effect, representing the averaged impact of the infinite
number of players. Of course, this framework relies on the assumption
that the players behave in a similar way. By construction, it eliminates the
situation of a remaining Nash equilibrium for a finite number of players, with
mean field terms. In most real problems of economics, there is not just one
representative agent and a large community of identical players, which impact
with a mean field term. There is the situation of several major players,
and large communities. So a natural question is to consider the problem of these
major players. They know that they can influence the community, and they also
compete with each other. So the issue is that of differential games,
with mean field terms, and not of mean field equations arising from the limit of
a Nash equilibrium for an infinite number of players. One way to recover this
system of nonlinear P.D.E. with mean field terms is to consider averaging
within groups. Each of them is composed of an homogeneous community, but
different communities are not identical. To recover the system of nonlinear
P.D.E. it is easier to proceed with the dual problems as explained above. One
can consider a differential game for state equations which are probability
distributions of states, and evolve according to Chapman-Kolmogorov
equations. One recovers nonlinear systems of P.D.E. with mean field terms,
with a different motivation. Another interesting feature of this approach is
that we do not need to consider an ergodic situation, as it is the case in the
standard approach of mean field theory. In fact,considering strictly positive
discounts is quite meaningful in our applications. This leads to systems of
nonlinear P.D.E. with mean field coupling terms, that we can study with a
minimum set of assumptions. The ergodic case, when the discount vanishes,
requires much stringent assumptions, as it is already the case when there is no
mean field term. We refer to Bensoussan-Frehse [BF1], [BF3] and BensoussanFrehse-Vogelgesang [BFV1], [BFV2] for the situation without mean field term.
Basically our set of assumptions remains valid and we have to incorporate
additional assumptions to deal with the mean field terms.
References
[1] Andersson,D.,Djehiche,B. (2011) A Maximum Principle for SDEs of Mean
Field Type, Applied Mathematics and Optimization 63, p.341-356
[2] Bensoussan,A., Frehse,J. (2002) Regularity Results for Nonlinear Elliptic
Systems and Applications,
Springer Applied Mathematical Sciences, Vol 151, 2002
[3] Bensoussan,A., Frehse,J. (1995) Ergodic Bellman Systems for Stochastic
Games in arbitrary Dimension, Proc. Royal Society, London, Mathematical and
Physical sciences A, 449, p. 65-67
[4] Bensoussan,A., Frehse,J. (2002) Smooth Solutions of Systems of
Quasilinear Parabolic Equations,
ESAIM: Control, Optimization and Calculus of Variations , Vol.8, p. 169-193
[5] Bensoussan,A., Frehse,J., Vogelgesang, J., (2010) Systems of Bellman
Equations to Stochastic Differential
Games with Noncompact Coupling, Discrete and Continuous Dynamical
Systems, Series A, 274, p. 1375-1390
[6] Bensoussan,A., Frehse,J., Vogelgesang, J., (2012) Nash and Stackelberg
Differential Games, Chinese Annals of Mathematics, Series B
[7] Buckdahn,R., Li, J.,Peng, SG., (2009) Mean-field backward stochastic
differential equations and related partial differential equations, Stochastic
Processes and their Applications, 119, p.3133-3154
[8] Guéant,O., Lasry,J.-M., Lions,P.-L.(2011) Mean Field Games and
applications, in A,R. Carmona et al.(eds), Paris-Princeton Lectures on
Mathematical Sciences 2010, 205-266
[9] Huang,M., Caines,P.E., Malhamé, R.P. (2007) Large-population costcoupled LQG problems with
nonuniform agents: individual-mass behavior and decentralized _−Nash
equilibria, IEEE Transactions on Automatic Control, 52 (9), 1560-1571
[10] Huang,M., Caines,P.E., Malhamé, R.P. (2007) An Invariance Principle in
Large Population Stochastic Dynamic Games, Journal of Systems Science and
Complexity, 20 (2), 162-172
[11] Lasry,J.-M., Lions,P.-L. (2006). Jeux à champ moyen I- Le cas
stationnaire, Comptes Rendus de l’Académie des Sciences, Series I, 343,619625.
[12] Lasry,J.-M., Lions,P.-L. (2006). Jeux à champ moyen II- Horizn fini et
contrôle optimal, Comptes
Rendus de l’Académie des Sciences, Series I, 343,679-684.
[13] Lasry,J.-M., Lions,P.-L. (2007). Mean Field Games, Japanese Journal of
Mathematics 2(1),229-260.
E Kalai and E Shmaya:
Learning and Stability in Repeated Games of Incomplete
Information
Abstract: A game is Bayesian, if players have private information about their
own types, describing information that is known to themselves but not to their
opponents. A Bayesian equilibrium is a Nash equilibrium of a Bayesian game.
Due to their broad applicability, for several decades Bayesian equilibria have
been a major subject of study among researchers and users of game theory.
Two established observations about Bayesian equilibia are the following:
1. Learnability. In playing a Bayesian repeated game with initially drawn
.xed private types, the play of any Bayesian equilibrium must converge to the
play of a Nash equiliblium in which the initially-private types are common
knowledge. In other words, all esential private information is learned with
time (see Kalai and Lehrer Econometrica 1993).
2. Robustness. The Bayesian equilibria of one-shot Bayesian games become highly robust as the number of players increases. More speci.cally, equilibrium strategies remain at equilibrium even if the game is altered to allow for
repeated revisions of choices, observatility of choices and information leakage,
delegation of choices and more (see Kalai Econometrica 2004).
While the properties above have strong implications in economics, political
science, computer science and other .elds, their applicability is limited because
of restrictive assumptions, for examples that the individual types are statistically
independent.
A recent study of the presenters deals with the following:
3. Learning and stability in Bayesian repeated games with many
players. The mathematics of Bayesian equilibria induces strong learnability
and stability properties when one combines repetitions with many players. As
a result, important applications in the areas mentioned above can be obtained
in a model of many-players dynamic game studied by Kalai and Shmaya.
However, exact detailed analysis of Bayesian repeated large games is highly
intracktable for the players and other analysts of the game. This di¢ culty
is overcome by the use of a new notion of compressed equilibrium, which is
manageble and has the following attractive properties: (A) It provides a good
approximation for Bayesian equilibria when the number of players is large,
without requiring knowledge of the number of players, the number of
repetitions and other parameters. (B) It shows that the number of unstable
periods of play, in which players learn about initially unkown parameters, is
bounded (See Kalai and Shmaya 2013).
The lectures provide details on items 1-3 above.
Sorin Solomon: The Importance of Being Discrete.
The emergence of complex macroscopic collective phenomena from simple
elementary microscopic laws
Abstract: The lectures will revolve around the emergence of macroscopic
complex self-organized adaptive systems out of many elementary objects with
very simple interactions. We will start with very simple solvable theoretical
models and then validate their predictions by agent based simulations and by
confrontation with empirical data.
In particular we will expose the very surprising macroscopic effects of
microscopic random autocatalytic interactions. The simplest example is a
system with randomly diffusing agents A and X submitted to the "reactions":
proliferation: A+ X -> A +X+X (with rate r) death : X-> . (with rate m)
In conditions (a r < m) in which their analysis by continuum methods
(differential equations of the logistic type acting on their space-time continuous
densities a and x) dx/dt = r a x - m x + difussion x
predict a space uniform, time decaying asymptotic state
x~ exp t(ar-m) , we show that system has a very lively dynamics.
This is due to the spontaneous emergence of collective adaptive collective X
objects formed spontaneously around the local very rare, very large fluctuations
in the A density which are amplified by the exponential dynamics to
macroscopic X collective objects. In turn those objects behave as to follow and
take advantage (for their own survival) of the fortuitous A fluctuations.
Since the differential equations version of the system above has been used
extensively in the last 200 years, the consequences of the discreteness impinge
onto a wide range of applications in biology, economics, etc.
We will discuss some of the applications for prognosis and policy steering:
sustainability, globalization, wealth distribution, financial fluctuations, crises
and recovery, emergence of cooperation, etc.
For a relatively pedagogical review see:
Intermitency and Localization
G Yaari, D Stauffer, S Solomon, in Encyclopedia of Complexity and Systems
Science, edited by R Meyers, 4920-4930 Springer,
2009. http://arxiv.org/abs/0802.3541
A very early popular New Scientist cover article is:
http://www.sorinsolomon.net/shum.cc/~sorin/ordinary/ordinary.html
George Zaccour:
Differential Games and Applications
Abstract: The objective of this mini course is to introduce the participants to
differential games theory and some of its applications in economics and
marketing. The first two lectures introduce the basics of the theory, and the next
two deal with applications in environmental economics (international
environmental agreements, taxation of emissions, forest management, etc.) and
in marketing (pricing, advertising, conflict and cooperation in marketing
channels, etc.).
1. Introduction to noncooperative differential games
2. Introduction to cooperative differential games
3. Selected applications in environmental economics
4. Selected applications in marketing
Lectures:
S. Gaubert (INRIA and CMAP, Ecole Polytechnique,
[email protected] http://www.cmap.polytechnique.fr/~gaubert):
Nonlinear Perron-Frobenius methods applied to zero-sum games
Abstract: Nonlinear Perron-Frobenius theory deals with self-maps of a cone that
are order-preserving or nonexpansive in certain metrics associated to this cone.
Such maps arise when considering the Shapley operators of zero-sum games,
sometimes up to certain transformations (log or exp-glasses), or when
considering ows on the cone of positive definite matrices arising from decision
problems with a linear quadratic structure. We shall give here an overview of
results on zero-sum games obtained by Perron-Frobenius techniques. This
covers in particular asymptotic results on value iteration, including DenjoyWolf type results (convergence of the normalized value iteration to a boundary
point), certificates of growth (Collatz-Wielandt type theorems), existence results
for the mean payoff per time unit under algebraic (definability) assumptions, or
geometric convergence results obtained by nonlinear spectral theory techniques.
This survey is based on joint works with Vigeral (arXiv:1012.4765), Bolte
and Vigeral (arXiv:1301.1967), and Akian and Nussbaum (arXiv:1201.1536).
Musa Mammadov. Turnpike theorems for time-delays systems
Abstract: There is a significant body of research on optimal control problems
for time delay systems, including the existence of optimal control and necessary
conditions of optimality. However, the problem of qualitative behaviour of
optimal solutions has not been approached in a systematic way. This is mainly
due to the presence of time delay that highly complicates the behaviour of
solutions. In this talk, an optimal control problem involving a special class of
nonlinear systems with time delay is considered. This class of systems appears
as a mathematical model in several important applications in economics,
biology and medicine. A turnpike theorem is established which states that under
some conditions all optimal solutions converge to a unique equilibrium.
Markus Kirkilionis: Complex Games: Formal Extensions of Agent-Based
Modelling Approaches in Economy
Abstract: We introduce first a mathematical framework which will be a triple
M=(S,H,R), with S for systems components, like the basic players in a game, H
for hierachy, i.e. the structural organisation of the model, like club membership
of the players, and R being the transition rules, or single events that update the
system state. This structure will be defined to be sufficient to describe
state evolution of the system. On top of this structure, which is a mathematical
formalisation of stochastic agent-based models, the typical game theoretic
settings, like a set of strategies for each player, will be introduced. The basic
evolution of the system is described by the corresponding master equation. We
discuss how to derive from this framework some basic differential games which
have been investigated in the literature, thus giving them a new interpretation by
an embedding approach.
Wei Yang (Department of Mathematics and Statistics University of Strathclyde
Glasgow): Sensitivity analysis for HJB equations with applications to coupled
backward-forward systems
Abstract: In this talk, we analyse the dependence of the solution of the
Hamilton-Jacobi-Bellman equations on a functional parameter. This sensitivity
analysis not only has the interest on its own, but also is important for the mean
field games methodology, namely for solving a coupled backward-forward
system. We show that the unique solution of a Hamilton-Jacobi-Bellman
equation and its spacial gradient are Lipschitz continuous uniformly with
respect to a functional parameter. As an application of our results, verifiable
criteria for a feedback regularity condition are presented.