# Probability and Game Theory Course Syllabus

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1 Probability and Game Theory Course Syllabus DATE ACTIVITY CONCEPT Sunday Learn names; introduction to course, introduce the Battle of the Bismarck Sea as a 2-person zero-sum game. Monday Day 1 Pre-test for assessment. Lecture: Review of Set Theory (Instructor - I) Lecture: Enumeration of sets (Challenge Problem #1: generalize addition principle to the property of inclusion/exclusion) Lecture: Permutations and Combinations including many examples (I & TA) -Sets, elements, subsets, universal sets, empty sets. -Set operations: union, intersection, complements, disjoint sets. - Properties of these operations: commutative, associative, distributive laws; DeMorgan s Laws. Venn diagrams - Addition principle. - Multiplication principle, Cartesian product of sets, power sets - Enumeration when order of selection matters and when it does not. Selection with and without replacement. - multinomial coefficients - Recursion/Induction (Challenge Problem #2: Soldiers in a field) (Challenge problems #4,5,6 on partitions of integers) Work quietly on first 2 written assignments. (I & TA) - partitions of integers - Sets and enumeration.

2 Tuesday Day 2 Lecture: binomial theorem and set partitions - Binomial coefficients; Pascal s triangle and properties. (Challenge problem #3: generalize the binomial theorem to a multinomial theorem.) Lecture: Introduction to probability theory; Uniform probability models - Multinomial coefficients and enumeration of set partitions - Experiments, outcomes, sample spaces, events - Probability Models, probability of a union of events - Uniform sample spaces, using enumeration to compute probabilities. - Infinite Sample Spaces Challenge problems: Lemon Candy Problem, Putnam Problem, Buffon s Needle. Lecture: Birthday problem (TA) Lecture: Expected values/examples - Probability of a complement - Expected values; Independent Events - Binomial theorem, probability. Return and go over assignments 1 & 2. Work quietly on assignments 3 & 4.

3 Wednesday Day 3 Quiz I on sets and enumeration Work quietly on assignments. (Challenge problems #7-9: enumeration of compositions; challenge problem on Ramsey theory, Mr. Spock logic problem) Lecture: Matrix algebra - Ordered partitions (compositions) - Matrices, scaling, addition of matrices, dot product of vectors, Matrix multiplication. - Matrix determinants and inversion (2x2 case only), solving systems of linear equations. Return and go over Quiz I Class Discussion: Introduction to game theory the resolution of the Bismarck Sea Battle Video Zero-sum Games from the For All Practical Purposes series of educational mathematics videos. - Game trees, game matrices, dominant strategies, minimax techniques and saddle points, higher order dominance. - Expected values and matrix algebra. Work quietly on assignment #5 (TA)

4 Thursday Day 4 Work quietly on assignment #5 (TA) Class Discussion: Strictly Determined Games, Continued. - More on dominance, saddle points and minimax techniques. -movement diagrams. - Value of a game, fair games, translating a game. Saddle points are equivalent and interchangeable. Lecture: Introduction to Non-strictly Determined Games - Introduction to non-strictly determined games. Repeated play and mixed strategies. - Mixed strategies, probability vectors as strategies - Expected payoffs as matrix multiplication. - Expected Value Principle (when the opponent s strategy is known) - method of equalizing expectation (when the opponent s strategy is not known) 2x2 case only; - Probabilities & binomial coefficients Challenge problem Grid Problem Lecture: Introduction to Linear Programming. Video Linear Programming from the For All Practical Purposes series of Educational Videos. - Graphical Solutions, feasible points and optimal points, corner point theorem, marginal values. - Solving Strictly Determined Games, Dominance Work quietly on assignments #5 & 6 Read Chapters 1-2 of text. (TA)

5 Friday Day 5 Review assignments 4 & 5 (TA & I) Quiz II on binomial theorem, probability and expected value. Lecture: Non-strictly Determined Games, Continued - Equalizing expectation in 2xn case or the mx2 case. Inactive strategies. - Equalizing expectation in the 3x3 case; Williams theorem that for any game, the solution is the same as that of some square subgame. - Williams method of oddments (2x2 case) Lecture: More on Linear Programming: marginal values and GLP software demos - Sensitivity Analysis; Standard Form problems - Monte Carlo Techniques in Probability Theory Sunday Class Activity: Enact the Grid problem Work quietly on assignments #6 - #7 Read Chapter 3 of text - Zero-sum games; strictly determined and nonstrictly determined games.

6 Monday Day 6 Work quietly on assignments #6 - #7 Lecture: Duality in Linear Programming and its economic interpretation. Lecture: Introduction to the Simplex Method. Sensitivity Analysis Class Discussion: Buffon s Needle Problem: how to set it up - Marginal values as decision variables - Linear Programming via simplex method. - Sensitivity Analysis - Infinite Sample Spaces - Rules for pivoting Lecture: Simplex Method continued. Return and go over Quiz II Class Discussion of the Buffon s Needle Challenge Problem Brief introduction to variable sum games - Introduction to trigonometric functions - ordered pair payoffs, movement diagrams, Nash equilibria - Solving linear programming problems algebraically. Go over assignment 5 Work quietly on assignments #7 - #8

7 Tuesday Day 7 Work quietly on assignments #7 - #8 (TA) Movie: A Beautiful Mind - The life of John Nash Class Discussion: The Jamaican Fishing Problem (TA) Make-up Quiz II Work quietly on assignments #8 & #9 (TA) Read Chapter 4 - Application of game theory to Anthropology - Games against Nature - Probability and expected values - Tree diagrams; information sets, games of partial information, backwards induction. Wednesday Day 8 Return make-up quizzes Lecture: Linear Programming: Solving minimization problems via duality. Lecture/Demo: Using Software to solve LP problems. Class Discussion: Guerilla Warfare & Missile Games (TA) Lecture: Linear Programming applied to game theory - Duality- solving dual problems via the simplex method - Working with Mathematica - Using Expected Values as payoffs and partitions as strategies. - Solving mxn games via simplex method - Proof of the minimax theorem Work quietly on assignments #8 - #10 (TA)

8 Thursday Day 9 Work quietly on assignments #8 - #10 (TA) Return and go over assignment #6; review of solving games via linear programming Lecture: Games in Extensive Form Lecture/Class Demo: Using Excel to solve LP problems and games. Return & go over assignment #7 -Game Trees, Information sets; Extensive and Normal Forms are equivalent; introduce Cuban Missile Crisis -Solving zero-sum games via linear programming - Why oddments works; why equalizing expectations does not work for strictly determined games; games with more than one optimal solution. Quiz III on Zero-sum Games Assignments #9 & #10 Read Chapter 7

9 Friday Day 10 Return and go over Quiz III Lecture: Applications to Business. Lecture: Games Against Nature (TA) - Games of partial information - Playing against non-rational opponents - Axioms for playing games against nature - Milnor s axioms Lecture: Variable Sum Games - Dominance, Nash equilibria - Payoff polygons, Pareto Optimality - Equalizing and prudential strategies - Iterated Games; Tit for Tat -Prisoner s dilemma; chicken Sunday Video: Prisoner s Dilemma from the For All Practical Purposes series of educational mathematics videos. Work on Assignments #9 - #10 Read Chapters 11 & 12 - Games Against Nature - Finding Nash Equilibria, SSS games.

10 Monday Day 11 Lecture: Variable Sum Games, Continued - Games solvable in the strict sense Lecture: Strategic Moves (threats and promises) - Communication between players in variable sum games Tuesday Day 12 Class tournament Wholesalers vs. Retailers. (I & TA) Lecture: 2x2 Ordinal Games Lecture: Ordinal games and the theory of moves. Applications to the Yom Kippur War. Assignments #11 & #12 Read Chapters 12 & 14 make-up quiz III (TA) Lecture: Theory of moves, cont. (I & TA) Movie: 13 Days Class discussion on Evolutionary Stable Strategies - Games of partial information; communication via actions. - Applications the superpowers arms race, Cuban missile crisis. - Nash equilibria, dominance, Prisoner s dilemma - Sequential games; non-myopic equilibria - Relaxing assumptions of game play: communication, sequential play, initial states -Strategic moves, applications of the prisoners dilemma - The Cuban missile crisis - Applications of game theory to biology - Evolutionary Stable Strategies Assignments #11 & #12 Read Chapters 14 & 15 (TA) - Ordinal Games, Theory of moves

11 Wednesday Day 13 Work quietly on homework (TA) Go over homework Class Discussion: n-person games (I ) Lecture: n-person games, continued - Prisoner s dilemma in Puccini s opera Tosca, battle of the sexes, etc. - n-person version of Prisoner s dilemma; Tragedy of the Commons - Coalitions, security levels, n-person prisoners dilemma; - Games in characteristic function form - n person Prisoners Dilemma - Applications to the football draft. Thursday Day 14 Friday Day 15 Catch up on all assignments Read Chapters (TA) - Jeopardy-type class tournament (TA) Study for final Exam (I & TA) Final Exam PBS video: A Brilliant Madness Fill out SPEs Return Exams Hand out Solutions to Challenge Problems Class Party (bring games) Closing Ceremony - review all topics -The life and work of John Nash

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