How can you create mathematical models for random courses of events that are affected by sudden events as well? A doctoral thesis by mathematician Rani Basna at Linnaeus University takes a new step forward in this field. But even though the thesis is the result of hard work, it has not been his toughest challenge during these years.
A doctoral thesis is very seldom understandable for a person with no prior knowledge of the subject. This is true about the subject mathematics in particular.
Rani Basna's thesis in mathematics is entitled "Mean Field Games for Jump Non-Linear Markov Process", which hardly makes it easier to understand for ordinary people. So what is the thesis really all about?
– My work may affect you if, for example, you are working at a company that is competing with other, similar companies and you are trying to maximize the payoff or minimize the costs, explains Rani Basna.
Mean field games
The importance of Rani's work is that it extends the classical results of mean field games to so-called jump processes and makes it possible to create models for these. Jump processes feature sudden events and exist almost everywhere in nature, Rani says.
Suppose that you have a number N of agents (you can also call them players or particles) and the dynamics of these agents represent a dynamical system consisting of N equations. In other words, they evolve in time and we describe their evolution as N stochastic differential equations. Now, suppose that each of these agents face an optimization problem, i.e. they all want to decide a strategy such that they maximize a payoff (or minimize a cost). Each agent should think of the other N-1 agents and also consider the possible strategies that they may use.
– This is a very complicated problem, because the agents then have a strong coupling. The mean field game methodology suggests that one can think of a weak coupling, i.e. a representative agent does not consider all the other agents. Instead, she or he only considers the overall influence of the agents, i.e. one takes the average effect of all the agents.
From here the term mean field came to the picture (you can see it as the mean value of the influence of all the agents). Then Rani studied a stochastic optimal control problem for a representative agent with the mean-field parameter involved. As a result, he proved that the strategies derived by the control problem constitute an approximate Nash equilibrium. An approximate Nash equilibrium is a situation where the N agents are almost satisfied by their strategies. They can at most increase their payoff by a very small amount if they deviate from the strategy derived by the mean-field method.
Rani was born in Syria and came to Växjö and Linnaeus University in 2008 with his wife Hiba. Both studied the Master Programme in Mathematics and Modelling; Rani specialized in Mathematical Statistics and Financial Mathematics. Two years ago, one plus one equaled three when they had a daughter during their doctoral studies. Hiba also presented her doctoral thesis at Linnaeus University recently.
To go for research studies in mathematics was not an odd choice, Rani thinks – he loves mathematics as well as fell in love with a mathematician.
– To raise new, challenging scientific questions and try to find answers to them, that is a fascinating process. By my opinion, this combination is the most exciting about research.
Toughest of all
During the four to five years that it takes to create a thesis, you will inevitably face different critical points. So did Rani. But the thesis in itself was not the toughest challenge to face.
– The most difficult one was the war in my country. I have lost many hours not being able to concentrate while people are dying at the same time somewhere in Syria. In particular in the beginning of the situation, when it was new feelings.