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Individuální studijní plán studentů doktorského studia na IES
(Vyplňte elektronicky, podepsaný formulář odevzdejte na sekretariát IES,
soubor uložte na Vaši webovou stránku)
Kontaktní údaje
Jméno a příjmení
Školní rok nástupu studia
Forma studia
Školitel
Téma doktorské práce
Pavel Doležel
2007/2008
Prezenční
Prof. RNDr. Ing. František Turnovec, CSc.
Three Essays on Operations Research in Economic Theory
Předpokládaný harmonogram zkoušek (kód/název/semestr)
2007/2008
ZS: ELBF-Economics and Law in Banking and Finance
ETPM-Economic Theory of Political Markets
LS: ELBFEconomics and Law in Banking and Finance
ETPM-Economic Theory of Political Markets
2008/2009
ZS: ETPM-Economic Theory of Political Markets
LS: ETPM-Economic Theory of Political Markets
2009/2010
ZS: ETPM-Economic Theory of Political Markets
LS: ETPM-Economic Theory of Political Markets
Státní doktorská zkouška
2010/2011
ZS: ETPM-Economic Theory of Political Markets
Malá obhajoba
LS: ETPM-Economic Theory of Political Markets
Velká obhajoba
Předpokládaný harmonogram výuky (kód/název/semestr)
2007/2008
ZS: Macroeconomics I
Microeconomics II
LS: Macroeconomics II
Microeconomics I
2008/2009
ZS: Microeconomics A I
Probability and Statistics I
LS: Probability and Statistics II
Microeconomics I
2009/2010
ZS: Microeconomics II
LS: Microeconomics I
Práce na disertaci
Synopse (1- 2 strany)
The Ph.D. thesis will consist of three essays dealing with the multi-index transportation problem and it’s
applications. The main topic will be the application of the general (continuous) multi-index MongeKantorovič transportation problem (MKTP) in the area of operations research, theory of choice and
theory of risk management.
The first essay „Operations Research in Risk Management“ will focus on the theory of risk
management, mainly on the statistical models predicting the probability of default, exposure at default
and the loss given default, as they are defined in the Bassel II Accord. This application can be widely
generalized in the area of statistical modelling, especially in general regression, validation techniques
and backtesting the developed models.
The second essay „Operations research in the theory of firm“ will consider the application of (MKTP) in
operations research, more specifically in the field of stochastic schedulling and planing. The stochastic
schedulling can be applied in project management, queuing theory, planning and production
optimization. This essay will be mainly focused on the discrete version of (MKTP).
In the third essay „Operations Research in the Choice Theory and Game Theory“ I will study the
application of (MKTP) in the mathematical economics, utility theory and choice theory, starting with what
is known, including the Richter’s nontransitive theory of choice and rational choice theory. I would like to
show and investigate the connections of multi-index transportation probelm to the game theory. The
main focus will be given to the closed preorders and continuous utility functions and special forms of
choice functions.
The first essay will be based on the following concept: Suppose we have a set of random samples of
the same length (so each sample is given by one probability distribution and the random variables
within the sample are independent to each other). The i-th variable of all the samples stands for the i-th
observation. Suppose, we have some output data vector of the same length as the random samples,
the i-th output is assigned to the i-th observation. We estimate the output data using the data we have,
using some kind of regression, either linear, polynomial, exponential, logit, probit, nonlinear,
generalized, regression and classification trees, stochastic processes prediction and many other
techniques. The question that arises as first in the area of fitting the data by a statistical model, is the
question of how well it fits the data. The second arising question is, what is the model’s predictive power
(how well it fits the newcomers and so, how it is robust). There are several methods of validating the
predictive power of statistical models, such is the coefficient of determination (which is very bad-mainly
due to its low robustness and dependancy on the number of predictors), adjusted coefficient of
determination (it overcomes the problem of the determination coefficient being dependant on the
number of predictors, but it is still very bad), the power curves, ROC (Operating Characteristics Curve)
and CIER (Condonal Information Entropy Ratio). However, in the credit risk modeling, it can be different
at the end (which is often considered to be the net profit), whether we overestimate the predicted value
of the output variable, or if we underestimate it and even in what level of the output we make the error
and how „big“ it is. On the other hand, we might face another problem of having niether discrete, nor
continuous probability distributions, because we must somehow solve the probelm of having many
probabilites of default or loss given defaults observations equal to zero and in the case of LGD also
equal to one. As is known, the proabilty that some value of absolutely continuous random variable will
be reached more than once is zero whenever the support is uncountably infinite. There is also a
problem of handling the estimated values as binary or categorical (banks usually use loss classes to
describe the level of loss the bank should assume it will have to face with some probability). I would like
to suggest another way of validating and backtesting the model using the (MKTP) which will be based
on the idea, that at first we estimate (or set) the cost of underestimating or overestimating the output
variable within the concept of net profit (I would like to study the situation, where the cost function which
forms the (MKTP) is defined using some of the data, which we use as predictors) and then we solve the
(MKTP) for the estimated cost function with given marginal distributions of the output variable and its
prediction. Once we have this done, we find the absolute value of the differences between the solution
of the (MKTP) and the joint probability distribution of the estimated and observed output variables. This
measure can be a good way how to backtest a model with respect to the cost function. In this essay I
will not study only this measure in practice and apply some sampling statistical techniques (bootstrap)
to make an estimation of its distribution, but I will also suggest how to use this to enter the estimation of
the output variable itself and study the effectiveness of such a suggestion and conditions for being
better than the traditional way of backtesting. In this essay I would like to mention very interesting theory
of Vapnik’s which is a rigorous way of eliminating much of what is in statistics considered normative
decission. I would like to incorporate the results of this theory to be applied together with the (MKTP), so
I at first choose the model using Vapnik’s theory for a few given families of functions and then apply all
the suggested preditive power measurements and the (MKTP) for backtesting and restructuring the
results of the prediction to better fit the (MKTP) result-at first randomly, then in some conducted way. I
would like to study extensively the relation between the family of functions chosen and the difference
between the optimal (MKTP) given joint probability distribution and the observed distribution and so the
ability to restructure the estimated values before we know the real observations.
The second essay will be focused on a different field of (MKTP) application, which is the stochastic
scheduling and planning, within the thory of firm. In this essay I will study the interrelations between
multi-index transportation problem and the theory of matroids and apply some results of multi-index
assignment problems arising in the theory of firm.
In the third essay I will investigate the (MKTP) in the choice theory, matematical economics and game
theory. In this essay I will study the closed preorders and continuous utility functions used in
microeconomic theory, which eenables microeconomists to use constrained optimization techniques
such as Kuhn-Tucker conditions. I will investigate the choices and decisions made in various economic
situations and economic agents behavior from the theoretical point of view. I will describe briefly the
theory of rational behavior and then study some generalistions of this theory and it’s consequences. I
will relax the transitivity assumption as is described in Richter’s work. Because the decission making is
closely studied in game theory, I will investigate also the game theory and its concepts, such as the
Nash equilibria, Pareto optimality and the stochastic games and their connection to (MKTP). It can for
example solve the problem of creating the pay-off function in certain games so that the game has
certain properties. When we have a situation of multiple players with the same set of actions, facing the
same pay-off function and the game is not sequential and each player know what is the total payoff over
all possible actions, but don’t know what state will be exactly reached due to other players choices, we
can a priori choose such pay-off function, so that the expected pay-off of all players is maximal possible.
There are many other applications of the (MKTP) in the field of game theory and I will investigate some
of them and propose some further extensions. I will apply the conclusions in the theory of political
science.
In all three essays I will use theoretial concepts from different parts of mathematical theories. Due to the
complexity of the (MKTP) I have to put in at least the basics of the measure theory to define the (MKTP)
and describe the duality theory and explicit solutions, I also need to include basics of the functional,
stochastic (to introduce quadratic variation and Brownian motion and its applications in financial
modelling) and real calculus, simple algebra (at least to handle matrices and describe some interesting
concepts, such as matroids and their connection to (MKTP)), theory of neural networks to be able to
build some of the algorithms, theory of genetic and evolutionary algorithms, theory of probability and
stochastic processes (which will be the most important one), statistics, Vapnik’s theory of statistical
learning which can be placed somewhere between the classical statistics and neural networks theory,
theory of deterministic and stochastic schedulling, combinatorial optimization, linear and nonlinear
programming and optimization method (SIMPLEX algorithm at least) and the wide range of theory of
network flows and connected graph theory. I will never use more theory than I need and that will be
indispensable to understand what is going on in the different applications. Therefore I omit many
definitions and theorems, that I will use, but I always try to put reference of where these results can be
found.
Within the work I would like to at least in one of the essays to create some algorithms, that solves the
real problems and implment it in VBA (Visual Basics for Applications) and then compare them from the
point of view of the time effectivity and solution effectivity.
The conclusions of the three essays should contain theoretical extensions of the theory, new
applications and algorithms to solve the problems more effectively and/or faster and suggestions for
modelling and backtesting the scoring models in risk management, suggestions for creating systems
within the framework of game theory to attain the most effective pay-offs and avoid adverse behavior of
individuals and finaly suggestions for the firms to better conduct the production and schedulling
problems.
Základní literatura
S. T. Rachev and L. Ruschendorf. Mass Transportation Problems: Volume I: Theory (Probability and its
Applications). Springer; 1 edition, 1998.
S. T. Rachev and L. Ruschendorf. Mass Transportation Problems: Volume II: Applications (Probability
and its Applications). Springer; 1 edition, 1998.
E. Aarts and J. Lenstra. Local Search in Combinatorial Optimization.
Princeton University Press, 2003.
T. S. Arthanari and Y. Dodge. Mathematical Programming in Statistics.
Willey, 1981.
F. Bock. An algorithm for solving traveling-salesman and related network
optimization problems. Operations Research. Society of America,
October 1958. The fourteen National Meeting.
B. Korda. Učebnice lineárníıho programování. SNTL, 1962.
C. Villani. Topics in Optimal Transportation (Graduate Studies in Mathematics, Vol. 58). American
Mathematical Society, 2003.
R. L. Rardin. Optimization in Operations Research. Prentice Hall; 1st edition, 1997.
The list will be updated.
Harmonogram prací
2008/2009
ZS: Working on disertation thesis, finishing the first part of the first essay named „Operations Research
in Risk Management“, which will deal with the application and theory of the general transportation
problem in credit risk modelling, validating and becktesting the models.
LS: Working on the disertation thesis, finishing the second part of the first essay „Operations Research
in the Risk Management“, which will deal with the application of the general transportation problem in
the validation of statistical scoring models. Working on the GAUK 66409 grant „Ekonomická teorie voleb
a nová volební historie ČR“ recieved for 2009-2010 and on the GAČR 402/09/1066 grant “Political
economy of voting: Rational voter theory and models of strategic voting and manipulation”. Finishing the
paper dealing with the efficiency of voting systems intended for submittion in AUCO.
2009/2010
ZS: Working on the disertation thesis, finishing the first part of the second essay named „Operations
Research in the Theory of Firm“ dealing with the theory of stochastic schedulling. Finishing the work on
GAUK 66409 grant for the year 2009 and on the GAČR 402/09/1066 grant with outputs prepared for
publication.
LS: Working on the disertation thesis, finishing the second part of the second essay named „Operations
Research in the Theory of Firm“ dealing with the applications of Monge-Kantorovič problem in the
stochastic schedulling and its applications. Working on GAUK 66409 grant conclusions and preparing
the final version of grant paper. Preparing for the state doctoral exams.
2010/2011
ZS: Working on disertation thesis, finishing the first part of the third essay named „Operation Research
in the Choice Theory and Game Theory“ delaing with the applications of Monge-Kantorovič Problem in
the theory of choice and its applications. Finishing the GAUK 66409 grant. Preparing for the small
defense of disertation thesis.
LS: Working on disertation thesis, finishing the second part of the third essay named „Operation
Research in the Choice Theory and Game Theory“ delaing with the applications of Monge-Kantorovič
Problem in the game theory. Preparing for the final defense of the disertation thesis.
Předpokládaná publikace výsledků
2008/2009
ZS: Sending a contribution to AUCO with a review of recently published book „Voting, Power and
Freedom“.
LS: Sending a contribution to AUCO with assumed title „On the Efficiency of Voting in the IC, IAC and
GIAC models“.
2009/2010
ZS: Sending a contribution to IES WP with assumed title „Monge-Kantorovič Problem in the Theory of
Firm-Applications“.
Sending a contribution to AUCO with assumed title „Gerrymandering in the Czech Parliamentary
Elections“.
Sending a contribution to AUCO with assumed title „New voting history of Czech Republic“.
LS: Sending a contribution to EconLit Journal with assumed title „Monge-Kantorovič Problem in the
Theory of Firm-Applications“.
Sending a contribution to AUCO with assumed title „Rational, Semi-rational and Irrational Voter in the
Czech Elections“.
2010/2011
ZS: Sending a contribution to EconLit Journal with assumed title „Monge-Kantorovič Problem in the
Choice Theory-Applications“.
LS: Sending a contribution to IES WP with assumed title „Monge-Kantorovič Problem in Game TheoryApplications“.
Sending a contribution to EconLit Journal with assumed title „Monge-Kantorovič Problem in Game
Theory-Applications“.
Sending a contribution to AUCO with assumed title „Geometry of Voting“.
Konkretizace studijního plánu pro 1. rok studia
Výuka
Účast na doktorských seminářích
Začátek práce na disertaci
Předpokládané zkoušky
Další aktivity
Starting the work on disertation thesis
Grant request: Grant request to the Grant Agency of UK with the topic „Multi-Index Monge-Kantorovič
Transportation Problem in Credit Risk Modelling“.
Publications:
ZS: Sending a contribution to IES WP with assumed title „Monge-Kantorovič Problem in Credit Risk
Management“.
LS: Sending a contribution to EconLit Journal with assumed title „Monge-Kantorovič Problem in Credit
Risk Management“.
LS: Sending a contribution to IES WP with assumed title „Application of Monge-Kantorovič Problem in
the Choice Theory“.
Participation on doctoral seminars: ELBF (Monge-Kantorovič Transportation Problem in Credit Risk
Management), ETPM (Application of Monge-Kantorovič Transportation Problem to Choice and Game
Theory)
Teaching:
ZS: Microeconomics II, Macroeconomics I
LS: Microeconomics I, Macroeconomics II
Other activities:
ZS, LS: Attending at least 50% of the doctoral students defenses.
Vyjádření školitele
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podpis školitele
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podpis studenta