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Games, Choice and Optimization

This module runs in alternate years: 2013-14, 2015-16 and so on

Prerequisites: Calculus 1: Single Variable; Algebra 1: Techniques and Applications

BUEM022S6 (30 credits)

Aims

This module introduces the relatively modern mathematical areas of Game Theory, Choice Theory and Linear Programming, and the relationships between them. The theory and techniques of each area is developed, and applications of them to economics, business and politics are explored.

Teaching and Assessment

Teaching for this module will take place throughout the year, with eight evenings of lectures in each of the Autumn and Spring Terms and two evenings of revision and consolidation in the Summer Term.

Of the final course mark, 80% is based on a three-hour exam in the summer term and the other 20% is from assessed coursework.

Coursework will consist of short, problem based assignments. You will have around three weeks to complete each one.

The examination in the summer term will consist of 8 short (5 mark) questions, which are compulsory, and 4 long (20 mark) questions from which candidates must answer two.

Syllabus

Linear Programming
Expressing a problem using a linear programme, graphical methods, the simplex algorithm, duality, sensitivity analysis.

Individual Choice
Menus, reflexivity, completeness, transitivity, utility functions, continuity, rationality, choice under uncertainty.

Social Choice
Social choice procedures, plurality, the Hare system, the Borda count, sequential pairwise voting, approval voting, dictatorships, Pareto condition, Condocet’s criterion, monotonicity, independence of irrelevant alternatives, social welfare functions, Arrow type theorems.

Static Games
Matrix form, movement diagrams, saddle points, mixed strategies, zero-sum games, graphical methods for solving games with one player having two strategies, von Neumann’s minimax theorem, using linear programming to find the solution of a game, nonzero-sum games, the prisoner’s dilemma, chicken, battle-of-the-sexes, Nash’s theorem and Nash equilibria, Pareto optimality, graphical representation of a two player game.

Dynmanic Games
Game trees, backwards induction, imperfect information, patent races, bank runs, the Leontieff wage-labour negotiations, the duopoly problem–classical model, Cournot model, Stackelberg model, collaboration and side payments.

Voting Games
Proper and strong games, monotonicity, weighted games, weighted games, swap-robustness, trade-robustness, power indices, bloc voting, qualified majority voting in the EU Council of Ministers.

Learning Outcomes

On successful completion of this module a student will be expected to have:

Subject Specific

  • knowledge and understanding of, and the ability to use, mathematical and/or statistical techniques;
  • knowledge and understanding of a range of results in mathematics;
  • an appreciation of the need for proof in mathematics, and the ability to follow and construct mathematical arguments;
  • awareness of the use of mathematics and/or statistics to model problems in the natural and social sciences, and the ability to formulate such problems using appropriate notation;
  • an understanding of the importance of assumptions and an awareness of where they are used and the possible consequences of their violation;
  • an appreciation of the power of generalization and abstraction in the development of mathematical theories;
  • knowledge and understanding of a range of modelling techniques, their conditions and limitations, and the need to validate and revise models;
  • a deeper knowledge of some particular areas of mathematics;

Intellectual

  • the ability to comprehend conceptual and abstract material;
  • the ability to develop a logical and systematic approach to problem solving;

Practical

  • problem-solving skills, including the ability to assess problems logically and to approach them analytically;
  • highly developed quantitative skills;
  • the ability to transfer knowledge and expertise from one context to another;

Personal and Social

  • the ability to learn independently using a variety of media;
  • the ability to work independently with patience and persistence;
  • time-management and organizational skills;
  • general IT skills, including word processing and spreadsheets;
  • good communication skills, including the ability to write coherently;
  • the ability to complete a sustained and substantial task;
  • the ability to complete work in a limited time period.
Department of Economics, Mathematics and Statistics, Birkbeck, University of London, Malet St, London WC1E 7HX.