Game Theory

• 26 hrs lectures
• 26 hrs projects

• 60 pts final exam (written part)
• 40 pts projects

The THE course is going to give the students certain education in area of rational strategic decision making in conflict situations, to learn them creating models of such situations, to analyze the situations through the models and in some cases to predict future evolution of the modeled systems. The course extends the education of artificial intelligence with strategic decision making. Applications and use will be oriented to the computer science (control, decisions, safety and security, games, networking) and also to social sciences like economics, sociology and political sciences.
Students will get a wide knowledge of game theory and a plenty of its applications in engineering and social sciences. When passed the course, the students will be able to create a simple model of given game situation and predict its probable future evolution.
In more general level, the study of rational decision making give a certain skills of problem analysis, selecting possible strategies and actions leading to its solving, assigning some utility to the strategies and finally, to accept a best decision in that situation. Mathematical game models also present clearly solutions to many problems in every day life. Moreover, the course introduces and plenty of applications of the computer science to natural and social sciences.

Students should have a basic knowledge of discrete mathematics, algebra and mathematical analysis, as they are basic tools to describe the studied problems. Basics of artificial intelligence and computer modelling are also required.

• Cesa-Bianci, N., Lugosi, G.: Prediction, Learning, and Games, Cambridge University Press, 2006
• Shubik, M.: Game Theory in the Social Sciences: Concepts and Solutions, MIT Press, 1984
• McCarty, N., Mierowitz, N.: Political Game Theory: An Introduction, Cambridge University Press, 2007
• Osbourne, M.J., Rubinstein, A.: A Course in Game Theory, MIT Press, 1994
• Fudenberg, D., Tirole, J.: Game Theory, MIT Press, 1991
• Schelling, T. S. : The Strategy of Conflict, Harvard Press, 1980
• Dugatkin, L., Reeve, H.: Game Theory and Animal Behavior, Oxford University Press, 1988
• Morrow, J.: Game Theory for Political Scientists, Princeton University Press, 1994
• Mailath, G., Samuelson, L.: Repeated Games and Reputations, Oxford University Press, 2006
• Gintis, H.: Game Theory Evolving, Princeton University Press, 2000
• Miller, J.: Game Theory at Work, McGraw-Hill, 2003
• Straffin, P.D.: Game Theory and Strategy, The Mathematical Association of America, 2003
• Hespanha, J. P.: Noncooperative Game Theory: An Introduction for Engineers and Computer Scientists, Princeton University Press, 2017

1. Introduction, history of game theory, motivations to its study, theory of choice, basic terminology, basic classification of games, information in a game.
2. Two player games with zero-sum payoffs: concept, saddle point, minimax theorem.
3. Two player games with nonzero-sum payoffs: concept,  strategy dominance, Nash equilibrium in pure and mixed strategies, basic algorithms to find the Nash equilibrium.
4. Mathematical methods in nonzero-sum games: proof of Nashe's lemma of equilibrium existence in games with finite sets of strategies, algorithms to compute the equilibrium, graphical solution to games, linear programming.
5. Sequential game with perfect/imperfect information: concept, applications, Stackelberg equilibrium, backward induction.
6. Cooperative games and bargaining: presumptions for possible cooperation, bargaining in nonzero-sum games, Nash  bargaining solution.
7. Repeated games: concept (finite/infinite number of repetitions), solution. Applications of repeated games.  Effect of repetitions to players behavior.
8. Mechanism design: introduction to Mechanism design. Choice under uncertainty.
9. Social choice, public voting: Arrow's paradox, mechanisms of voting.
10. Auctions: study of rationality in auctions (mechanism with money). Business applications.
11. Correlated equilibrium: effect of correlation to rational behavior, definition of correlated equilibrium and its relation to Nash equilibrium. Computing of correlated equilibria, applications.
12. Evolutionary biology: strategic behavior in population of many entities, evolutionary stable strategy, case studies in the nature.
13. Applications in economics and engineering: basic solution of oligopoly in analytic and numerical manner, nontrivial case study and its analysis. Application of game theory in computer networks. Applications in psychology, sociology and foreign affairs.

Students will be given an individual project to solve. The project is going to be one of these areas:

• Study - detail reading of given scientific paper and its analysis.
• Implementation - implementation of a given algorithm.
• Applications - a case-study and its model.

individual project, final exam

Individual project and final exam. The final exam has two alternativies.
The minimal number of points which can be obtained from the final exam is 20. Otherwise, no points will be assigned to a student.

Students have to release a individually made project which will be evaluated at least with one half of points (20 points from 40).