Lecture 7: Mixed strategies and expected
payoffs
As we have seen for example for the Matching pennies game or the
Rock-Paper-scissor game, sometimes game have no Nash
equilibrium. Actually we will see that Nash equilibria exist if we
extend our concept of
Homework 8
Exercise 1: Consider an assymetric version of the Cournot duopoly game where the
marginal cost for firm 1 is c1 and the marginal cost for firm 2 is c2 .
1. If 0 < ci < 12 P0 what is the Nash equilibrium?
2. if c1 < c2 < P0 but 2c2 > P0 + c1 , w
Week 5: Expected value and Betting systems
Random variable A random variable represents a measurement in a random experiment. We usually denote random variable with capital letter X, Y, . If S is the sample
space of the experiment then to each i S the ran
Chapter 7: Proportional Play and the Kelly Betting
System
Proportional Play and Kellys criterion: Investing in the stock market is, in effect,
making a series of bets. Contrary to bets in a casino though, one would generally believe
that the stock market
Chapter 1: Probability models and counting
Part 1: Probability model
Probability theory is the mathematical toolbox to describe phenomena or experiments
where randomness occur. To have a probability model we need the following ingredients
A sample space
Chapter 10: Mixed strategies Nash equilibria,
reaction curves and the equality of payoffs theorem
Nash equilibrium: The concept of Nash equilibrium can be extended in a natural
manner to the mixed strategies introduced in Lecture 5. First we generalize th
Homework 6
Exercise 1: Suppose that you run the Monte-Carlo algorithm to compute 10000 times
and observe 7932 points inside the circle. What is your estimation for the value of ?
Using Chebyshev describe how accurate your estimation of , the answer should
Homework 5
Exercise 1: The standard deviation (X) of a random variable is the square root of the
variance Var(X)
p
(X) = Var(X) .
and it characterizes the spread of the random variable X. If a random variable X has
expected value and standard deviation ,
Homework 7
Exercise 1: The snowdrift game: Two drivers are caught in a snowstorm and a big
snowdrift blocks the road. To go home they have to clear the path. The fairest solution
is for them to clear the path together. If one simply refuses to do it, the
Chapter 4: Gamblers ruin and bold play
Random walk and Gamblers ruin. Imagine a walker moving along a line. At every
unit of time, he makes a step left or right of exactly one of unit. So we can think that his
position is given by an integer n Z. We assum
Homework 4
Exercise 1:
1. At a certain casino card game which has paying odds of 1 to 1, your probability of
winning each game is p = 0.494. You walk into the casino with $25 dollars with
the goal to get $500. Compute the probability for you to succeed if
Chapter 6: Variance, the law of large numbers and
the Monte-Carlo method
Expected value, variance, and Chebyshev inequality. If X is a random variable
recall that the expected value of X, E[X] is the average value of X
Expected value of X :
E[X] =
X
P (X
Chapter 9: Nash equilibrium for monopolies and
duopolies
We discuss here an application of Nash equilibrium in economics, the Cournots
duopoly model. This is a very classical problem which in fact predates modern game
theory by more than a century.
Supply
Week 8: Basic concepts in game theory
Part 1: Examples of games
We introduce here the basic objects involved in game theory. To specify a game ones
gives
The players.
The set of all possible strategies for each player.
The payoffs: if each player picks
Homework 8
1. The snowdrift game: Two drivers are caught in a snowstorm and a big snowdrift
blocks the road. To go home they have to clear the path. The fairest solution is for
them to clear the path together. If one simply refuses to do it, the other dri
Lecture 8: Mixed strategies Nash equilibria
and reaction curves
Nash equilibrium: The concept of Nash equilibrium can be
extended in a natural manner to the mixed strategies introduced in
Lecture 5. First we generalize the idea of a best response to a
mix
Lecture 12: Combinatorics
In many problems in probability one needs to count the number of
outcomes compati- ble with a certain event. In order to do this we
shall need a few basic facts of combinatorics
Permutations: Suppose you have n objects and you ma
Lecture 14: Bayes formula
Conditional probability has many important applications and is the
basis of Bayesian approach to probability:
Consider events B1, B2, Bn
which are pairwise disjoint and B1 B2 Bn These events are
called the hypotheses.
The probab
Lecture 1: Introduction and examples of
games
We introduce here the basic objects involved in game theory. To
specify a game ones gives
The players.
The set of all possible strategies for each player.
The payoffs: if each player picks a certain strateg
Lecture 2: Game Theory
Example:
Battle of the sexes
Robert and Chelsea are planning an event of entertainment. Above
all they value spending time together but Robert likes to go to the
game while Chelsea prefers to go to the ballet. They both need to
deci
Lecture 3: Dominated strategies and their
elimination
Let us consider a 2-player game, the players being named Robert
(the Row player) and Collin (the Column player). Each of the
players has a number of strategies at his disposal. In the examples
in lectu
Lecture 4: Nash equilibrium
Nash equilibrium: The mathematician John Nash introduced the
concept of an equi- librium for a game, and equilibrium is often
called a Nash equilibrium. They provide a way to identify
reasonable outcomes when an easy argument b
Lecture 5: Symmetric game
Symmetric game and social dilemma
Many of the examples in the previous lectures can be thought as
describing social dilemma. These are games played by members of
some group, who have the same incentives and interests although
the
Lecture 6: Nash equilibrium in economics:
monopolies and duopolies
We discuss here an application of Nash equilibrium in economics,
the Cournots duopoly model. This is a very classical problem
which in fact predates modern game theory by more than a
centu
Homework 9
1. Consider an assymetric version of the Cournot duopoly game where the marginal
cost for firm 1 is c1 and the marginal cost for firm 2 is c2 .
(a) If 0 < ci < 21 P0 what is the Nash equilibrium?
(b) if c1 < c2 < P0 but 2c2 > P0 + c1 , what is
Chapter 3: Linear Difference equations
In this chapter we discuss how to solve linear difference equations and give some
applications. More applications are coming in next chapter.
First order homogeneous equation: Think of the time being discrete and tak