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Department of Mathematical Sciences

# MATH3071 Decision Theory III

Decision theory concerns problems where we have a choice of decision, and the outcome of our decision is uncertain (which describes most problems!). Topics for the course typically include the following (contents may vary a bit from year to year):

1. Introduction to the ideas of decision analysis.
2. Decision trees - how to draw and how to solve.
3. Representing decision problems using influence diagrams.
4. Quantifying rewards as utilities - informal ideas, formal construction, relevance to statistical analysis and multi-attribute utility. Von Neumann-Morgenstern theory of maximum expected utility.
5. Alternative decision criteria and reflection on normative versus descriptive theories (including prospect theory).
6. Applications, e.g. in health, industry and insurance.
7. Bargaining problems: Nash theory for collaborative games.
8. Group decisions and social choice: Arrow's theory on social welfare functions, Harsanyi's utilitarianism, further developments.
9. Game theory: two-person zero-sum games, brief discussion of more complex games and applications in e.g. biology.
10. Further topics related to recent developments in research and applications.

## Outline of Course

Aim: To describe the basic ingredients of decision theory, for individuals and for groups, and to apply the theory to a variety of interesting and important problems.

### Term 1

• Introduction to Decision Analysis:Decision trees, sequential decisions; uncertainties and values, solution by backward induction; perfect information and cost of information; representation by influence diagrams.
• Utility: Von Neumann - Morgenstern theory of maximum expected utility; utility of money and risk aversion; multi-attribute utility; relevance to statistical analysis.
• Alternative theories: Alternative decision criteria, prospect theory.
• Applications: Some examples of applications in e.g. health, industry and insurance.

### Term 2

• Bargaining: Nash theory for collaborative games; alternative theories.
• Group Decisions and Social Choice: Arrow's theory on social welfare functions; Harsanyi's theory on utilitarianism; alternative theories and recent developments.
• Game Theory: Two-person zero-sum games; brief discussion of non-constant sum games and other more complex games, and of applications in e.g. biology.
• Further topics: Selection of topics related to recent developments in research and applications.

### Prerequisites

For details of prerequisites, corequisites, excluded combinations, teaching methods, and assessment details, please see the Faculty Handbook.