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Math 167: General Course Outline

Catalog Description

167. Mathematical Game Theory. (4) Lecture, three hours; discussion, one hour. Requisite: course 115A. Quantitative modeling of strategic interaction. Topics include extensive and normal form games, background probability, lotteries, mixed strategies, pure and mixed Nash equilibria and refinements, bargaining; emphasis on economic examples. Optional topics include repeated games and evolutionary game theory. P/NP or letter grading.

Outline update: D. Blasius, 5/02

NOTE: While this outline includes only one midterm, it is strongly recommended that the instructor considers giving two. It is difficult to schedule a second midterm late in the quarter if it was not announced at the beginning of the course.

Schedule of Lectures

Lecture Section Topics

1

0.1 (p. 3-8), 0.1.3, 0.2, 0.3, 0.4.1

Strategic Voting, Second Price Auction, Non-cooperative, Nash Equilibrium, Cournot Duopoly

2

1.1-1.3

Trees, Nim, Strategies

3

1.4-1.5

Zermelo's Algorithm, Binary Analysis of Nim, Begin Zermelo's Theorem

4

1.7-1.9

Zermelo's Theorem, Chess, Value of a Strictly Competitive Game, Subgame Perfect Equilibrium, Team Games, etc.

5

2.1

Review of Probability, Bayes Rule

6

2.2-2.3

Lotteries, Expectation, Game Values

7

2.4-2.5

Duel, begin Parcheesi

8

Exercises

Parcheesi, Do problems in Class (e.g. Monty Hall, ex. 2.6.26, hat problem)

9

3.1-3.2, 3.4

Preferences, Utility, Optimizing Utility

10

3.4

Von Neuman-Morgenstern Utility, examples

11

3.4-3.5

St. Petersburg Paradox, Risk Averse, Risk Loving

12

4.1

Payoff Functions via Expectation; Strategic Form of Duel, Bimatrices, Finding Pure Strategy NE's

13

4.6

Domination

14

5.2-5.3

Convexity, Supporting Lines, Cooperative Payoff Regions, Pareto Efficiency

15

Midterm

16

5.4-5.5

Bargaining Sets, (Generalized) Nash Bargaining Problems and Solutions, Methods of Computation

17

5.5

Nash Axioms, Nash's Theorem and Proof

18

6.2-6.4

Minmax & Maxmin, Security Strategies, Mixed Strategies

19

6.4

Mixed Strategy Payoffs, Computing Mixed Security Strategies via Maxmin Analysis (Examples)

20

6.4-6.6

Maxmin

21

6.7 or 6.8

Battleships or Inspection

22

7.1

Best Response (=Reaction Curve) Analysis of Bimatrix Games, Prisoner's Dilemma & Chicken

23

7.2

Relation of NE's to Maxmin Solutions of Associated Zero-sum Games and Pareto Optimality, Correlated Equilibria

24

Theorem that (p1, ..., pn) is an NE iff supp(pi) is contained in imax {pi(p1, ..., pi-1, - , pi+1, ..., pn)} for all i. Methods of computing Nash equilibria (2 player 2x3, 3x3 cases)

25

Computations, Word problems

26

7.2

Duopoly (Cournot, Stackelberg), Oligopoly, Perfect Competition

27

7.7

Sketch of Proof of Existence of NE

28

Review