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COMP30191 the Theory of Games and Game Models

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COMP30191 the Theory of Games and Game Models COMP30191 The Theory of Games and Game Models Andrea Schalk [email protected] School of Computer Science University of Manchester August 7, 2008 About this course This is an introduction into the theory of games and the use of games to model a variety of situations. While it is directed at third year computer science students it is also suitable for students of mathematics, and possibly of physics. It contains some proofs but does not require advanced knowledge of mathematics. The course goes beyond what is classically understood as game theory (the basics of which are covered in Sections 1 and 2) by making connections with computer programs for games such as Chess, and using games to model scenarios in sociology or biology. This course has been taught in the past as CS3191 and CS3192. What this course is about Games have been used with great success to describe a variety of situations where one or more entities referred to as players interact with each other according to various rules. Because the concept is so broad, it is very flexible and that is the reason why applications range from the social sciences and economics to biology and mathematics or computer science (games correspond to proofs in logic, to statements regarding the `fairness' of concurrent systems, they are used to give a semantics for programs and to establish the bisimulation property for processes). As such the theory of games has proved to be particularly fruitful for areas which are notoriously inaccessible to other methods of mathematical analysis. There is no set of equations which describes the goings-on of the stock-market (or if there is, it's far too complicated to be easily discoverable). Single transactions, however, can be described using (fairly simple) games, and from these components a bigger picture can be assembled. This is a rather different paradigm from the one which seeks to identify forces that can be viewed as the variables of an equation. Games have also been successfully studied as models of conflict, for example in biology as well as in sociology (animals or plants competing for resources or mating partners). In particular in the early days of the theory of games a lot of work was funded by the military. When playing games it is assumed that there is some sort of punishment/reward system in place, so that some outcomes of a game are better for a player than others. This is typically described by assigning numbers to these outcomes (one for each player), and it is assumed that each player wishes to maximise his number. This is what is meant when it is stipulated 1 that all players are assumed to behave rationally. Games are then analysed in order to find the actions a given player should take to achieve this aim. This is what is referred to as a game theoretic analysis, but it should be pointed out that there are other ways of looking at the behaviour of players. Sociologists, psychologists and political scientists, for example, are more likely to be interested what people actually do when playing various games, not in what they should be doing to maximize their gains. The only way of finding out about people's behaviour is to run experiments and watch, which is a very different activity from the one this course engages in. To give a practical example, assume you are given a coin and, when observing it being thrown, you notice that it shows heads about 75% of the time, and tails the remaining 25%. When asked to bet on such a coin, a player's chances are maximized by betting on heads every single time. It turns out, however, that people typically bet on heads 75% of the time only! Economists, on the other hand, often are interested in maximizing gains under the as- sumption that everybody else behaves `as usual', which may lead to different results than if one assumes that all players play to maximize their gains. Provided the `typical' behaviour is known, such an analysis can be carried out with game-theoretic means. Depending on the size of the game in question, this analysis will take different forms: Games which are small enough allow a complete analysis, while games which consist of a great many different positions (such as Chess or Go) can not be handled in that way. In this course we examine games of different sizes and appropriate tools for analysing them, as well as a number of applications. Organization This course has a webpage at http://www.cs.man.ac.uk/~schalk/3192/index.html. where all information not contained in the notes can be found. These notes form the foundation for the course, and students are expected to work through them in their own time. To help them with this task there will be one question sessions per week which serve two purposes, to give students the opportunity to ask questions regarding the material they were set • to read in the past fortnight; to give the lecturer the opportunity to introduce the more technical concepts before • setting the reading for the coming fortnight. Part of the notes is a collection of exercises that aim to prepare students for solving the `problem solving' part of the exam. There are unassessed as well as assessed exercises, and the latter form the assessed course work. The unassessed exercises come in a part (a) and a part (b)|the purpose of the second part is to provide additional examples for practice during the revision period. In weeks A student learning is supported by examples classes. In these the students get help with solving the exercises, including the assessed course work. These examples 2 classes take place in the lecture slots with the lecturer and demonstrators present to ensure an instructor/student ration of about 1 to 10. In the last week of term the assessed course work is marked. Marking takes place as follows: Students present their workings of the assessed exercises and a demonstrator or lecturer asks them to explain these workings. If the students can do so satisfactorily, and if their workings do not contain more than minor errors, then they will get full marks for an attempted exercise. Exercises where students cannot give a coherent explanation of what they have done will result in zero marks for that exercise. The purpose of these measures is to avoid students merely copying solutions. The timetable for scheduled activities is as follows: When What Where Who Week 1 Introduction Mo 4.00pm in 1.1 All Weeks 2-5, 7-12 Question sessions Mo 4.00pm in 1.1 All Weeks A (3, 5, 8, 10) Examples classes We 9.00am in 2.19 Surnames A{L We 10.00am in 2.19 Surnames M{Z Week 12 Assessment We 9.00am in 2.19 Surnames A{L We 10.00am in 2.19 Surnames M{Z Set reading. Week 1 2 3 4 5 6 Theme Games Strategies Equilibria Non 0-sum Mixed strats Solving Pages 1{14 14{22 23{33 34{42 42{48 48{59 Week 7 8 9 10 11 12 Theme Models Evolution I Evolution II Medium games Large games Catch-up Pages 60{75 76{88 88{97 98{110 111{130 The timetable for the examples classes is as follows: Examples Class Week Exercises Assessed Exercises 1 3 1{7 1(a), 1(b), 1(c), 2 2 5 7{13 1(b), 1(c), 3(a) 3 8 13{18 1(d), 3(a), 3(b), 4 4 10 19{24 4, 5 The fifth examples class in week 12 is reserved for marking the assessed exercises. The assessed course work provides 20% of the final mark for the course, with 80% coming from an exam. All the material in the notes is examinable, including the exercises. Past exam papers are available online, and links are provided at the course's webpage. On the same webpage feedback on previous years' exam performance can be found. The reason for changing the format from the traditional lecture course is the experience I have gained over seven years of teaching this course. The material is fairly abstract, and to learn such material, an active mode of learning is required, where students try to solve exercises by themselves (rather than just `consuming' what is being presented by somebody else). Or, as somebody else put it, mathematics is not a spectator sport. Watching me explain it all in lectures would only be a small first step towards understanding the material|my expectations are made much clearer in the new format. I used to have the more outgoing 3 students asking for additional meetings in the second half of term, whereas the rest never received any help in a smaller group setting. Again the new format gives that chance to every student on the course. I would appreciate the readers' help in order to eliminate any remaining errors and to improve the notes. If you spot something that seems wrong, or doubtful, or if you can think of a better way of explaining something then please let me know by sending email to [email protected]. I will keep a list of corrigenda available on the course's webpage. I would like to thank David MacIntosh, Isaac Wilcox, Robert Isenberg, Roy Schestowitz, Andrew Burgess, Xue Qing Huang, Cassie Wong, Florian Sturmer,¨ Mark Daniel, Steve Hughes, Kevin Patel, and Mousa Shaikh-Soltan from previous years' courses for helping me improve the course material.
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