Course detail

# Game Theory

The course deals with Mathematical game theory which is oftenly called the Theory of interactive decision making. The game theory became a popular tool for analysing of intelligent entities in many situations of competition or cooperation. This theory is being commonly applied in area of control, economic models, psychology, sociology, foreign affairs, evolutionary biology and informatics too. By computer science point of view, the game theory is an extension to artificial intelligence with algorithms of decision making, competing and bargaining. This also relates to multi-agent approaches. Games will be treated as models of real or fictitious situations with attributes of intelligence and competition. Students will go through basic terminology of games by the mechanism of their playing (sequential, strategic), by distribution of payoffs in a game (zero/nonzero sum games), by possible cooperation of players (cooperative, non-cooperative) and also by state of information in a game (complete/incomplete information). After the introduction, the games will be extended with possible repetition of moves (repeated games) and its effect to players behavior. In the second part of the semester, we will pay attention to game applications, mechanism design, auctions, social choice, economic and market models and others.

Learning outcomes of the course unit

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.

Prerequisites

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.

Co-requisites

Not applicable.

Recommended optional programme components

Not applicable.

• Straffin, P.D.: Game Theory and Strategy, The Mathematical Association of America, 2003
• Gibbons, R.: Game Theory for Applied Economists, Princeton University Press, 1992
• Osbourne, M.J., Rubinstein, A.: A Course in Game Theory, MIT Press, 1994

• různí autoři: Classics in Game Theory, edited by Harold W. Kuhn, Princetown University Press, 1997
• 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
• Dresher, M.: The Mathematics of Games of Strategy, Theory and Applications, Dover Publications, 1981
• McCarty, N., Mierowitz, N.: Political Game Theory: An Introduction, Cambridge University Press, 2007
• různí autoři: Algorithmic Game Theory, edited by Noam Nisan, Cambridge University Press, 2006
• Osbourne, M.J., Rubinstein, A.: A Course in Game Theory, MIT Press, 1994
• Fudenberg, D., Tirole, J.: Game Theory, MIT Press, 1991
• Dorfman, R., Samuelson, P.A., Solow, R. M.: Linear Programming and Economic Analysis, Dover Publications, 1986
• 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
• Kreps, D.: Game Theory and Economic Modelling, Oxford University Press, 1990
• von Neumann, J.,  Morgenstern, O.: Theory of Games and Economic Behavior, Princeton University Press, 1944
• Mailath, G., Samuelson, L.: Repeated Games and Reputations, Oxford University Press, 2006
• Krishna, V.: Auction Theory, Elsevier, 2002
• 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
• Rasmunsen, E.: Games and Information, Blackwell Publishing, 2007

Planned learning activities and teaching methods

Not applicable.

Assesment methods and criteria linked to learning outcomes

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

Language of instruction

Czech

Work placements

Not applicable.

Course curriculum

Syllabus of lectures:
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.

Syllabus - others, projects and individual work of students:
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.

Aims

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.

Specification of controlled education, way of implementation and compensation for absences

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.

Classification of course in study plans

• Programme IT-MGR-2 Master's

branch MPV , any year of study, winter semester, 4 credits, elective
branch MGM , any year of study, winter semester, 4 credits, elective
branch MIS , any year of study, winter semester, 4 credits, elective
branch MBS , any year of study, winter semester, 4 credits, elective
branch MIN , any year of study, winter semester, 4 credits, compulsory-optional
branch MMI , any year of study, winter semester, 4 credits, elective
branch MMM , any year of study, winter semester, 4 credits, compulsory
branch MSK , 1. year of study, winter semester, 4 credits, compulsory-optional
branch MBI , 2. year of study, winter semester, 4 credits, compulsory

#### Type of course unit

Lecture

26 hours, optionally

Teacher / Lecturer

Syllabus

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.

Project

13 hours, compulsory

Teacher / Lecturer

Syllabus

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.