3M03_Merged_Notes

# 3M03_Merged_Notes - Game Theory Notes 2 Table of Contents 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

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Game Theory Notes

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2 Table of Contents 1. Background Stuff 3 2. Overview 5 3. Extensive Form 6 4. Strategies 10 5. Preferences 12 6. Normal Form 13 7. Efficiency 16 8. Beliefs 17 9. Mixed Strategies 18 10. Expected Utility 19 11. Dominance 20 12. Best Response 22 13. Nash Equilibrium 23 14. Mixed-Strategy Nash Equilibrium 27 15. Existence of Nash Equilibrium 30 16. Cournot Model 32 17. Hotelling Model 34 18. Bertrand Model 35 19. Weaknesses of Nash Equilibrium 40 20. Subgames 41 21. Subgame-Perfect Nash Equilibrium 42 22. Stackelberg Model 43 23. Limit Capacity 44 24. Advertising 47 25. Price Guarantees 48 26. Strictly Competitive Games 50 27. Equivalent Nash Equilibria 51 28. Security Strategies 52 29. Parlour Games 54 30. One-Card Poker 55
3 Background Material for Introductory Game Theory Calculus As with most standard microeconomic theory, game theory typically assumes that an agent behaves optimally (from its own point of view). In formal modelling, this frequently involves taking a derivative of some objective function (the thing the agent cares about) with respect to some choice variable(s) (the thing(s) the agent directly affects) and setting that derivative equal to zero (in hopes of characterizing the maximum of the objective function). The bottom line is that you want to have a solid grounding in introductory calculus for this course. Economics You need suprisingly little background in economics for this course. It helps to be familiar with utility and the way economists think about efficiency. Set Theory Set theory is used in game theory largely for compact notation. A set is an unordered collection of things. The things that make up the set are the elements of the set, sometimes called the members of the set. A set can consist of a collection of anything, even other sets. For the purpose of this course, sets will normally consist of strategies. Notation: A = {1, 5, 7} means the set A has elements 1, 5, and 7. Curly brackets are normally used to enclose the elements of a set and the elements are separated by commas. (Round brackets are used for vectors for which the order of the elements matters.) The order of elements doesn't matter: If B = {5, 1, 7} then A = B and we can also write B = {1, 5, 7}. Repeated elements shouldn't be there: Writing A = {1, 5, 5, 7} is bad. Not really that it's wrong--its most likely effect is to confuse the reader. Sets can be members of sets: If C = {2, 3} and D = {A, C} then D = {{1, 5, 7}, {2, 3}}. The 'nesting' of elements within sets within sets matters: If E = {1, 5, 7, 2, 3} then D E. {{1}} {1}. The empty set: There is something known as the 'empty set', denoted φ (the Greek letter phi, sometimes written as ϕ ) which has no members. It's also sometimes called the 'null set'.

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4 Additional notation: means 'is a member of'. e.g. 1 A, A D means 'is not a member of. e.g. 1 C is an operator denoting the union of two sets. e.g. E=A C, {1, 5} {5, 7}=A is an operator denoting the intersection of two sets. e.g. C {2, 4}={2}, A C= φ means 'is a subset of'. e.g. A E, A A means 'is a proper subset of'. e.g. A E × is an operator denoting the cross product of two or more sets.
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## This note was uploaded on 09/05/2008 for the course ECONOMICS ECON 3M03 taught by Professor Brucejames during the Fall '08 term at McMaster University.

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3M03_Merged_Notes - Game Theory Notes 2 Table of Contents 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

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