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1

Equilibrium stacks for a non-cooperative game defined on a product of staircase-function continuous and finite strategy spaces

Romanuke Vadim

Journal of Mathematics and Applications

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2023

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Vol. 46

79--114

EN

A method of solving a non-cooperative game defined on a product of staircase-function strategy spaces is presented. The spaces can be finite and continuous as well. The method is based on stacking equilibria of “short” non-cooperative games, each defined on an interval where the pure strategy value is constant. In the case of finite non-cooperative games, which factually are multidimensional-matrix games, the equilibria are considered in general terms, so they can be in mixed strategies as well. The stack is any combination (succession) of the respective equilibria of the “short” multidimensional-matrix games. Apart from the stack, there are no other equilibria in this “long” (staircase-function) multidimensional-matrix game. An example of staircase-function quadmatrix game is presented to show how the stacking is fulfilled for a case of when every “short” quadmatrix game has a single pure-strategy equilibrium. The presented method, further “breaking” the initial staircase-function game into a succession of “short” games, is far more tractable than a straightforward approach to solving directly the “long” non-cooperative game would be.

2

Time-unit shifting in 2-person games played in finite and uncountably infinite staircase-function spaces

Romanuke Vadim

Journal of Mathematics and Applications

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2022

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Vol. 45

133--179

EN

A computationally efficient and tractable method is presented to find the best equilibrium in a finite 2-person game played with staircase-function strategies. The method is based on stacking equilibria of smaller-sized bimatrix games, each defined on a time unit where the pure strategy value is constant. Every pure strategy is a staircase function defined on a time interval consisting of an integer number of time units (subintervals). If a time-unit shifting happens, where the initial time interval is narrowed by an integer number of time units, the respective equilibrium solution of any “narrower” subgame can be taken from the “wider” game equilibrium. If the game is uncountably infinite, i. e. a set of pure strategy possible values is uncountably infinite, and all time-unit equilibria exist, stacking equilibria of smaller-sized 2-person games defined on a rectangle works as well.

3

Zero-sum games on a product of staircase-function finite spaces

Romanuke Vadim

Journal of Mathematics and Applications

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2021

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Vol. 44

75--91

EN

A tractable method of solving zero-sum games defined on a product of staircase-function finite spaces is presented. The method is based on stacking solutions of “smaller” matrix games, each defined on an interval where the pure strategy value is constant. The stack is always possible, even when only time is discrete, so the set of pure strategy possible values can be continuous. Any combination of the solutions of the “smaller” matrix games is a solution of the initial zero-sum game.

4

Finite approximation of continuous noncooperative two-person games on a product of linear strategy functional spaces

Romanuke Vadim

Journal of Mathematics and Applications

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2020

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Vol. 43

123--138

EN

A method of the finite approximation of continuous non-cooperative two-person games is presented. The method is based on sampling the functional spaces, which serve as the sets of pure strategies of the players. The pure strategy is a linear function of time, in which the trend-defining coefficient is variable. The spaces of the players’ pure strategies are sampled uniformly so that the resulting finite game is a bimatrix game whose payoff matrices are square. The approximation procedure starts with not a great number of intervals. Then this number is gradually increased, and new, bigger, bimatrix games are solved until an acceptable solution of the bimatrix game becomes sufficiently close to the same-type solutions at the preceding iterations. The closeness is expressed as the absolute difference between the trend-defining coefficients of the strategies from the neighboring solutions. These distances should be decreasing once they are smoothed with respective polynomials of degree 2.

5

Szacowanie cen kontraktów opcyjnych opartych na indeksie temperatury powietrza HDD

Iwaszczuk N., Łamasz B., Wzorek A.

Rudy i Metale Nieżelazne Recykling

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2014

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R. 59, nr 8

407--414

PL

Głównym celem pracy było znalezienie modelu wyceny derywatów pogodowych z funkcją wypłaty zależną od temperatury powietrza w Polsce. Zdecydowano się na analizę danych ze stacji meteorologicznej znajdującej się w porcie lotniczym Kraków-Balice. Analiza obejmuje lata 2004-2013. Przy opracowywaniu wzoru na średnią dzienną temperaturę, a także przy opisie dynamiki zmian temperatury, wykorzystano głównie rozwiązania zaproponowane przez P. Alatona, B. Djehiche’a i D. Stillbergera w 2002 roku. Wybór ten wydaje się słuszny, ze względu na podobny charakter warunków termicznych w Szwecji (lotnisko Bromma, którego dotyczy artykuł Alatona i in.) i w Polsce. Chodzi tu przede wszystkim o sezonowość temperatury, możliwość wyróżnienia czterech pór roku, a także zjawisko powrotu temperatury do średniej. Ostatnia z tych własności jest potwierdzeniem prawidłowości wyboru procesu Ornsteina-Uhlenbecka do opisu dynamiki zmian temperatury powietrza. Brakujące wartości zmiennych występujących we wzorach wyznaczono na podstawie analizy historycznej i przy wykorzystaniu znanych metod statystycznych. Szczegółowy przegląd modeli matematycznych, dokonany przez autorów artykułu, pozwolił na wybór i zastosowanie jednego z nich do wyceny opcji wystawianych na najpopularniejsze indeksy temperaturowe. Wykorzystano model oparty na konstrukcji równoważnej miary martyngałowej, która pozwala wyeliminować przy wycenie opcji pogodowych wpływ tzw. rynkowej ceny ryzyka. Wyznaczone w przykładzie praktycznym ceny opcji dotyczą miesięcy zimowych, w których konstrukcja indeksu pogodowego HDD przyjmuje prostszą postać.

EN

The main aim of this work was to find a weather derivatives pricing model with the function of payment depending on the air temperature in Poland. It was decided to analyze the data from the meteorological station located at the airport Krakow - Balice. The analysis covers the period 2004-2013. In developing the model for the average daily temperature as well as the description of the dynamics of changes in temperature used mainly solutions proposed by P. Alaton'a, B. and D. Djehiche Stillberger'a in 2002. This choice appears to be legitimate, due to the similar nature of the thermal conditions in Sweden (Bromma airport, which the article Alaton'a et al.) and Poland. It is primarily about the seasonality of temperature, the ability to distinguish the four seasons, and the phenomenon of the return temperature to medium. The latter property is a confirmation of the correctness of the selection process of Ornstein - Uhlenbeck to describe the dynamics of changes in air temperature. Missing values of the variables in the model were determined based on historical analysis and using the well-known statistical methods. A detailed review of mathematical models, made by the authors of the paper, allowed for the selection and use one of them to the valuation of options issued on the most popular indexes temperature. In the following study uses a model based on the construction of an equivalent martingale measure. Designated in a practical example the price of the option, refer to the winter months, in which the construction of weather index HDD takes a simpler form.