GAMBLING: PROBABILITY AND DECISION

The course will use gambling to analyze the role of probability in decision making. The course will cover different theoretical and applied aspects of gambling and related topics. Several applications in different fields will be translated into a gambling language and analyzed with the tools of probability and decision theory.

Calculus, basic linear algebra, and basic probability. The course will be self-contained, but some knowledge of very basic game theory will help.

Knowledge and understanding: The aim of the course is to provide students with important mathematical instruments in probability, gambling and game theory in order to analyze casino games. Applying knowledge and understanding: Students will learn advanced instruments in probability and game theory useful to study lots of models in many fields in which randomness is involved. In particular students will apply these tools to casino games in order to understand how much profitable is a game, how is it possible to minimize the expected loss and to maximize the expected win. Criticism of judgment: At the end of the course, students will be aware of the relevant ingredients for the analysis of casino games. Indeed students will learn parameters used to discuss whether a game is advantageous or not. Moreover, given a specific game, students will be able to decide which wager is more convenient. Communication skills: Students will be stimulated to develop communication skills through written papers and during the exercises lectures. In particular, students will be in possession of a proper language to analyze casino games and to study different models in many fields. Learning skills: Students will have proper skills and knowledge to analyze casino games. Moreover they will be in possession of advanced probabilistic tools for studying many problems in different fields, such as economics, finance, sociology, informatics, biology and physics. This knowledge will make students more versatile, more prepared and more attractive for many relevant advanced programs.

The course will be mostly based on elementary topics in the theory of gambling, but at the same time all the arguments will be discussed in a rigorous way. The course has no pretense to exhaustiveness and in particular it has as main aim to give fundamental general tools to analyze casino’s games. In particular during the course, students will learn many important instruments in probability theory (such as martingales and Markov chains) useful for many applications also in other settings. The course starts by recalling the prerequisites in probability theory with particular attention to the notion of conditional expectation. Then martingales will be introduced, leading to the definition of the martingale system and to the principle of conservation of fairness. The heart of the course will be mainly concentrated on the notion of house advantage that will help the student to find procedures to optimize the expected profit in casino’s games. Kelly systems will be also analyzed in the framework of superfair games, giving a way to maximize the expected profit by simply betting a particular proportion of the available capital. Particular attention will be devoted also to Markov chains in order to speak about card shuffling and in particular to determine the number of shufflings needed to have a well mixed deck. During the course students will see also a quick review on game theory and particular attention we be given to utility theory. In this framework students will analyze the attitude towards risk of a gambler. Finally, many applications to casino’s games will be seen, such as Craps, Roulette, Slot Machines, Poker and general lotteries.

-Lecture Notes of the course. -Bollman, M. (2014). Basic Gambling Mathematics: The Numbers Behind The Neon. Stati Uniti: Taylor & Francis. -Ethier, S. N. (2010). The Doctrine of Chances: Probabilistic Aspects of Gambling. Germania: Springer Berlin Heidelberg. -Levin, D. A., Wilmer, E. L., Propp, J., Wilson, D. B., Peres, Y. (2017). Markov Chains and Mixing Times. Stati Uniti: American Mathematical Society. -Norris, J. R., Norris, J. R., Norris, J. R. (1998). Markov Chains. Regno Unito: Cambridge University Press. -Zamir, S., Maschler, M., Solan, E. (2013). Game Theory. Stati Uniti: Cambridge University Press. -Karlin, A. R., Peres, Y. (2017). Game Theory, Alive. Stati Uniti: American Mathematical Society.

Two lectures per week are devoted to frontal lectures. The third lecture will be practical (exercises).

The grade of the students will be established through a written exam, but students can also decide to improve the grade of the written exam though an oral exam. Students can carry with them a calculator, one paper with the relevant formulas (given by the teacher) and the table of the normal distribution. The exam is divided into two parts: a first part related to the contents of the first part of the course and a second part related to the second part of the course. The written exam can also take place as two distinct tests: - the first partial test, called first midterm test, takes place in the middle of the semester (the date will be announced in the first weeks of the course); - the second partial test, called second midterm, takes place during the last lecture of the course. In this case the final grade will be the average of the two midterms. If a student fails one of the two midterms, the lacking part can be recovered during the written exam by simply solving the exercises concerning the needed part. If a student decides to take the midterms, such a decision has to be communicated by email to the teacher. If a student wants to improve the grade of the written exam, it is possible to take the oral exam. Before the end of the course, a list of possible questions for the oral exam will be published on the page of the course.

A minimum grade of 27/30 and participation during the course.