Solving Zero-Sum Games
and the Minimax Theorem
Zero-Sum Games Refresher
Limited to two-person games
u1(a1,a2) + u2(a1,a2) = 0
Examples: Matching Pennies, Rock-Paper-Scissors
Zero-Sum Games Refresher
H
T
H
T
1,-1 -1,1
-1,1 1,-1
Matching Pennies
R
P
S
R
STANFORD UNIVERSITY CS 224M, Fall 2009 Practice Questions
Questions 1 True or False 2 Short Answers 3 Extensive-Form Games 4 Game Theory 5 VCG Total Points /12 /12 /16 /25 /15 /80
Name of Student: SUID: The Stanford University Honor Code: I attest that I
CS224M Homework #4
1
CS224M, Fall 2009-10 Homework #4 : Mechanisms & Auctions
Handout #6
Note: Homework #4 is due on November 16, 2009 in class. This problem set carries less weight than the previous assignments, and the maximum score for this assignment
CS224M Homework #3
1
CS224M, Fall 2009-10
Homework #3 : Multi-Agent Learning & Protocols
for Agents
Handout #4
Note: Homework #3 is due on November 2, 2008 in class
1. [25 points] (Fictitious Play) Consider the normal form game given in Table 1. Player 1
CS224M Homework #2
1
CS224M, Fall 2009-10
Homework #2 :
Handout #2
Note: Homework #2 is due on October 19, 2009 in class
1. [25 points] (Repeated Games) Consider the 2-player normal form game in Table 1.
A
B
C
D
L
(0, 0)
(0, 0)
(1, 4)
(2, 5)
R
(3, 1)
(3,
CS224M Homework #1
1
CS224M, Fall 2009-10
Homework #1 : Game Theory
Handout #1
Note: Homework #1 is due on October 05, 2009 in class
1. [25 points] (2-Player Normal Form Game)
Consider the following game:
Player 1
T
M
B
L
8,8
0,8
2,-12
Player 2
C
5,6
8,2
Computing Nash Equilibrium in
Normal-Form Games:
Complexity and Algorithms
using material generously shared by
Costis Daskalakis
and
Kevin Leyton-Brown
as well as excerpts from
Multiagent Systems: Algorithmic, Game Theoretic, and
Logical Foundations
by Y.
Equilibrium?:
Strong
Fight
Not
Fight
1, -2
-1,2
Not
2,-1
0,0
Weak
Fight
Not
Fight
-2,1
-1,2
Not
2,-1
0,0
p
1-p
Equilibrium?:
Strong
Fight
Not
Fight
1, -2
-1,2
Not
2,-1
0,0
Weak
Fight
Not
Fight
-2,1
-1,2
Not
2,-1
0,0
p2/3
1-p
Player 2: Fight = p(-2)+(1-p)(
Bayesian Games
An Example
Definitions
Applications
Bayesian Games
Players N=cfw_1,.,n
Action sets: Ai
A = A1 x A2 x An
Type sets: i
= 1 x 2 x n
Payoff/Utility functions: ui : A x R
Our Conflict Example
Players N=cfw_1,2
Action sets: Ai = cfw_Fig
Incomplete Information
Most settings involve incomplete
information:
Bidding in auctions
Regulating an industry (e.g., investment banks)
How should a political candidate campaign?
What price should a firm charge?
Incentives
Bayesian Games
An Exampl
Indefinite Repetition
Probability p that the game continues next period,
probability (1-p) that it ends.
Cooperate
Defect
Cooperate
3,3
0, 5
Defect
5,0
1, 1
Equilibrium?
One equilibrium: defect in every period,
regardless of what has happened in the past
Outline
Comparative statics of mixed equilibria
Multiplicity of Nash equilibrium Intersection
of best response correspondences
Existence of equilibrium
Nash Equilibrium
Every game that we have looked at so far has
an equilibrium
Is this a general pro
CS224m: Problem Set 4
Leon Lin
June 6, 2012
Problem 1
(a) Consider the case where there are 2 voters, V1 and V2 and 3 candidates, A, B, and C . Let the voters
have the following preferences:
V1 : B A C
V2 : A C B
V3 : C A B
If each voter declared their pr