BSc: Game Theory

Game Theory

• Course name: Game Theory
• Course number: R-01

Course Characteristics

Key concepts of the class

• Game Theory: Basics of the mathematical theory of games, including Nash Equilibrium, Mixed Strategies, and Evolutionary Game Theory
• Applications of Computer Programming: Implementation of Game Playing agents

What is the purpose of this course?

Game Theory is a powerful method to make predictive decisions about common business cases and acts as a foundational course to decision making in AI systems, such as Bayesian techniques and game trees and Monte Carlo Tree Search. As such the purpose of this course is to provide a solid foundation on the basic structures of mathematical games including the canonical 2 by 2 structures of the Prisoner’s Dilemma, Chicken, Hawk and Dove, and Battle of the Sexes. Then looks at more complicated business examples such as price setting, making creditable threats and promises. It also gives practical instruction on the application of computers in game playing, especially tournament play and development of decision making models.

Prerequisites

• Analytic Geometry And Linear Algebra II: real vector and matrix operations, convex hull and span.
• Probability: probability distribution and mean function.
• Calculus: extreme values of differentiable functions. Discrete Math: paths in directed acyclic weighted graphs.

- What should a student remember at the end of the course?

By the end of the course, the students should be able to give basic definitions of games

• Should be able to define Nash Equilibrium, Domination, Mixed v. Pure Strategies
• Should be able to define Evolutionary Stability
• Should be able to define a number of common agent types (always cooperate/defect, Tit-for-tat, Grudger)

- What should a student be able to understand at the end of the course?

By the end of the course, the students should be able to understand the basic

• A student should understand how game theory affects common daily situations, such as internet trade (as a PD)
• Should understand the role of Evolutionary Stable Strategies
• Should understand the history of tournaments as methods to evaluate agent play

- What should a student be able to apply at the end of the course?

By the end of the course, the students should be able to apply game theory to solve problems in limited real world cases.

• Program a finite state machine to play iterated games
• Apply both domination and Nash equilibrium to solve pure games
• Apply both domination, Nash equilibrium, and mixed strategies to solve mixed strategic games

Course evaluation

Course grade breakdown

		Proposed points
Labs/seminar classes	10
Interim performance assessment	40
Midterm and Exam	50

If necessary, please indicate freely your course’s features in terms of students’ performance assessment.

Grades range

Course grading range

		Proposed range
A. Excellent	90-100
B. Good	75-89
C. Satisfactory	60-74
D. Poor	0-59

This course has a required class element of practical work on course elements, the retake is not a substitute for practical knowledge, and it is inherently unfair that students who have not submitted these practical elements are graded the same as those who have accomplished course materials. In order to be eligible for the retake, a student is required to have submitted all course assignments, and have a passing grade on those elements. The lacking/failing elements, can be presented for this purpose up to 3 business days before the retake for evaluation. In the case of the resubmission of a failing element, a document should be attached noting which changes have been made to the assignment in order to lead to a passing mark. Lack of these elements in a passing state presented to the committee will be considered a failing grade for the retake.

The retake grade will count as the final course grade, with the first retake giving no more than a B as final grade, and the second retake giving no more than a C. The first retake will be a written exam submitted to the professor, and the second retake as an oral commission.

Resources and reference material

• Andrew McEachern, Game Theory : A Classical Introduction, Mathematical Games, and the Tournament
• Thomas Schelling, Strategy and Conflict
• William Poundstone, Prisoner’s Dilemma

Course Sections

The main sections of the course and approximate hour distribution between them is as follows:

Course Sections

Section	Section Title	Teaching Hours
1	Domination and Nash	16
2	Advanced strategics	16
3	Tournament and Agents	16

Section 1

Section title:

Domination and Nash

Topics covered in this section:

• 2 by 2 classical games (Chicken, Prisoner’s dilemma, Battle of the Sexes, coin flips)
• n by m games and methods of reduction
• Domination and Nash Equilibrium
• Game Tree Roll Out

What forms of evaluation were used to test students’ performance in this section?

|a|c| & Yes/No
Development of individual parts of software product code & 0
Homework and group projects & 1
Midterm evaluation & 1
Testing (written or computer based) & 1
Reports & 0
Essays & 0
Oral polls & 1
Discussions & 1

Typical questions for ongoing performance evaluation within this section

1. What is an externality in PD?
2. Give the Nash Equilibrium of an example 2 by 2 game
3. Give the Domination in an example n by m game
4. What is the payoff matrix for a given game

Typical questions for seminar classes (labs) within this section

1. List and example the set of externalities of PD
2. Worked examples of Nash Equlibrium
3. Worked examples of Domination
4. Given the payoff matrix for a given game, what is the outcome of play.

Test questions for final assessment in this section

1. List and example the set of externalities of PD
2. Worked examples of Nash Equilibrium and Domination
3. Given the payoff matrix for a given game, what is the outcome of play.
4. A button is put before you. If you don’t press the button you get a low passing grade on this question. If you press the button and less than half the class presses the button you get a high passing grade for this question. If more than half the class presses the button then those who press the button get a failing grade for this question. Do you press the button?

Section 2

Section title:

Advanced strategics

Topics covered in this section:

• Multiple player and random player games
• Higher level Strategic planning including - Shelling’s theory of credible threats and promises
• Introduction to Evolutionary Game theory

What forms of evaluation were used to test students’ performance in this section?

|a|c| & Yes/No
Development of individual parts of software product code & 1
Homework and group projects & 1
Midterm evaluation & 0
Testing (written or computer based) & 1
Reports & 0
Essays & 0
Oral polls & 0
Discussions & 1

Typical questions for ongoing performance evaluation within this section

1. Using Domination and Nash Equilibrium find a mixed strategy solution for a given game.
2. How does this game differ if we only allow for pure strategies rather than mixed?
3. What is the difference between Evolutionary Stable Strategies and Dominator Theory
4. What is the ESS for an Iterated Prisoner’s Dilemma?

Typical questions for seminar classes (labs) within this section

1. Why does IPD not have a clear Nash Equilibrium and why should we use a Evolutionary Stable Strategies
2. Show the finite state representation for TFT
3. Demonstrate the Outcome of a population of half ALLC and half ALLD
4. Demonstrate the Best Response method on an example matrix

Test questions for final assessment in this section

1. Simulate the IPD, does the ESS occur?
2. When the time-line of an ESS is extended but we include the restriction of a finite space, what happens to the equilibriumum
3. Demonstrate the Best Response method on an example matrix

Section 3

Section title:

Tournament and Agents

Topics covered in this section:

• Agent players
• Computer Tournaments and Evolutionary Tournaments

What forms of evaluation were used to test students’ performance in this section?

|a|c| & Yes/No
Development of individual parts of software product code & 1
Homework and group projects & 1
Midterm evaluation & 0
Testing (written or computer based) & 1
Reports & 1
Essays & 0
Oral polls & 1
Discussions & 1

Typical questions for ongoing performance evaluation within this section

1. Define the meaning of a lock box.
2. Define the meaning of a nice strategy.
3. How can we make a strategy more cooperative by changing its structure?
4. Give a listing of IPD agents and a short description of their ruleset

Typical questions for seminar classes (labs) within this section

1. Program a finite state machine for IPD which implements a Lock Box
2. What properties does a finite state machine have (i.e. is it nice)
3. If a state machine is nice - what does it’s transitions matrix look like?
4. How can a 3 and 4 state lockbox reach cooperation?

Test questions for final assessment in this section

1. Show a working finite state machine IPD model of Trifecta.
2. Create a Tournament agent to compete against your classmates for a given game.
3. Given a set of players for a IPD model, what is the most likely equilibrium outcome, explain.