Review of Probability, Expected UtilityEconomics 302 - Microeconomic Theory II: Strategic Behavior

Instructor: Songzi Du

compiled by Shih En LuSimon Fraser University

January 13, 2015

ECON 302 (SFU) Lecture 2 January 13, 2015 1 / 11

Review of Probability

Probability is a number between 0 and 1 and is a (either subjective orobjective) measure of how likely an outcome happens.

Outcomes, mutually exclusive. For example, set of outcomes= {1, 2, 3, 4, 5, 6} for the roll of a dice. p(i) = 1/6 for i = 1, 2, . . . , 6.

Event is a subset of outcomes. For example, that the roll is even is anevent ({2, 4, 6}).

Probability of an event is the sum of probabilities of outcomes in thatevent.

Probabilities over all outcomes add up to 1.

ECON 302 (SFU) Lecture 2 January 13, 2015 2 / 11

Review of Probability

Probability is a number between 0 and 1 and is a (either subjective orobjective) measure of how likely an outcome happens.

Outcomes, mutually exclusive. For example, set of outcomes= {1, 2, 3, 4, 5, 6} for the roll of a dice. p(i) = 1/6 for i = 1, 2, . . . , 6.

Event is a subset of outcomes. For example, that the roll is even is anevent ({2, 4, 6}).

Probability of an event is the sum of probabilities of outcomes in thatevent.

Probabilities over all outcomes add up to 1.

ECON 302 (SFU) Lecture 2 January 13, 2015 2 / 11

Review of Probability

Probability is a number between 0 and 1 and is a (either subjective orobjective) measure of how likely an outcome happens.

Outcomes, mutually exclusive. For example, set of outcomes= {1, 2, 3, 4, 5, 6} for the roll of a dice. p(i) = 1/6 for i = 1, 2, . . . , 6.

Event is a subset of outcomes. For example, that the roll is even is anevent ({2, 4, 6}).

Probability of an event is the sum of probabilities of outcomes in thatevent.

Probabilities over all outcomes add up to 1.

ECON 302 (SFU) Lecture 2 January 13, 2015 2 / 11

Expected value

Suppose that the outcomes = {1, 2, . . . , n}. Probability of outcome iis p(i).

Suppose that we have a function that depend on the outcome: X (i).This is called a random variable.

The expected value of X is

E [X ] =n

i=1

X (i)p(i) = X (1)p(1) + X (2)p(2) + + X (n)p(n).

Expected value is a notion of average.

ECON 302 (SFU) Lecture 2 January 13, 2015 3 / 11

Expected value

Suppose that the outcomes = {1, 2, . . . , n}. Probability of outcome iis p(i).

Suppose that we have a function that depend on the outcome: X (i).This is called a random variable.

The expected value of X is

E [X ] =n

i=1

X (i)p(i) = X (1)p(1) + X (2)p(2) + + X (n)p(n).

Expected value is a notion of average.

ECON 302 (SFU) Lecture 2 January 13, 2015 3 / 11

Examples

What is the expected value of a roll of dice?

ni=1

i 16

= 1 16

+ 2 16

+ + 6 16

= 21 16

= 3.5.

If Alice gets a grade of 90 with probability 3/10, 80 with probability2/10, 70 with probability 4/10, and 0 with probability 1/10. What isher expected grade?

90 310

+ 80 210

+ 70 410

+ 0 110

= 27 + 16 + 28 = 71.

ECON 302 (SFU) Lecture 2 January 13, 2015 4 / 11

Examples

What is the expected value of a roll of dice?

ni=1

i 16

= 1 16

+ 2 16

+ + 6 16

= 21 16

= 3.5.

If Alice gets a grade of 90 with probability 3/10, 80 with probability2/10, 70 with probability 4/10, and 0 with probability 1/10. What isher expected grade?

90 310

+ 80 210

+ 70 410

+ 0 110

= 27 + 16 + 28 = 71.

ECON 302 (SFU) Lecture 2 January 13, 2015 4 / 11

Examples

What is the expected value of a roll of dice?

ni=1

i 16

= 1 16

+ 2 16

+ + 6 16

= 21 16

= 3.5.

If Alice gets a grade of 90 with probability 3/10, 80 with probability2/10, 70 with probability 4/10, and 0 with probability 1/10. What isher expected grade?

90 310

+ 80 210

+ 70 410

+ 0 110

= 27 + 16 + 28 = 71.

ECON 302 (SFU) Lecture 2 January 13, 2015 4 / 11

Independence

We may have two unrelated events. They are called independentevents. For example, the rolls of two dice; the weather in Vancouverand the weather in Miami; the birth dates of two strangers, etc.

If two events are independent, knowing one event has no bearing onthe probability of the other event.

Probability of two independent events = probability of firstevent probability of second event.

ECON 302 (SFU) Lecture 2 January 13, 2015 5 / 11

Independence

We may have two unrelated events. They are called independentevents. For example, the rolls of two dice; the weather in Vancouverand the weather in Miami; the birth dates of two strangers, etc.

If two events are independent, knowing one event has no bearing onthe probability of the other event.

Probability of two independent events = probability of firstevent probability of second event.

ECON 302 (SFU) Lecture 2 January 13, 2015 5 / 11

Independence

Alice goes to class with probability 1/2 (and skip class withprobability 1/2). Bob goes to class with probability 1/3 (and skipclass with probability 2/3). Assume Alice and Bob act independently.What is the probability that both show up in class?

What is the probability that at least one of them skips class?

What is the probability that exactly one of them shows up in class?

What is the probability of 4 Heads from 4 tosses of a coin?

ECON 302 (SFU) Lecture 2 January 13, 2015 6 / 11

Independence

Alice goes to class with probability 1/2 (and skip class withprobability 1/2). Bob goes to class with probability 1/3 (and skipclass with probability 2/3). Assume Alice and Bob act independently.What is the probability that both show up in class?

What is the probability that at least one of them skips class?

What is the probability that exactly one of them shows up in class?

What is the probability of 4 Heads from 4 tosses of a coin?

ECON 302 (SFU) Lecture 2 January 13, 2015 6 / 11

Independence

Alice goes to class with probability 1/2 (and skip class withprobability 1/2). Bob goes to class with probability 1/3 (and skipclass with probability 2/3). Assume Alice and Bob act independently.What is the probability that both show up in class?

What is the probability that at least one of them skips class?

What is the probability that exactly one of them shows up in class?

What is the probability of 4 Heads from 4 tosses of a coin?

ECON 302 (SFU) Lecture 2 January 13, 2015 6 / 11

Independence

Alice goes to class with probability 1/2 (and skip class withprobability 1/2). Bob goes to class with probability 1/3 (and skipclass with probability 2/3). Assume Alice and Bob act independently.What is the probability that both show up in class?

What is the probability that at least one of them skips class?

What is the probability that exactly one of them shows up in class?

What is the probability of 4 Heads from 4 tosses of a coin?

ECON 302 (SFU) Lecture 2 January 13, 2015 6 / 11

Choice Under Uncertainty

You probably have learned about preferences and utility functionsover certain outcomes.

When preferences are complete and transitive, they can berepresented by a utility function.

(I.e. There exists a utility function u such that A % B if and only ifu(A) u(B).)

But life is full of uncertainty! You dont know for sure how good (orhow bad) the economy will be in two years. You dont know for surewhat other people will do.

You have to make decision under uncertainty.

Goal: represent preferences over uncertain outcomes.

ECON 302 (SFU) Lecture 2 January 13, 2015 7 / 11

Choice Under Uncertainty

You probably have learned about preferences and utility functionsover certain outcomes.

When preferences are complete and transitive, they can berepresented by a utility function.

(I.e. There exists a utility function u such that A % B if and only ifu(A) u(B).)

But life is full of uncertainty! You dont know for sure how good (orhow bad) the economy will be in two years. You dont know for surewhat other people will do.

You have to make decision under uncertainty.

Goal: represent preferences over uncertain outcomes.

ECON 302 (SFU) Lecture 2 January 13, 2015 7 / 11

Notation and Terminology

Suppose a situation has n possible outcomes, labeled 1, 2, ..., n.

A lottery [p(1), p(2), ..., p(n)] is a list of probabilities, where p(i) isthe probability that outcome i occurs. (We must havep(1) + p(2) + ... + p(n) = 1).

Example: you get a grade of A with probability 30%, B withprobability 40%, F with probability 30%.

Define outcome 1 as A, outcome 2 as B, and outcome 3 as F.

The lottery is then [0.3, 0.4, 0.3].

ECON 302 (SFU) Lecture 2 January 13, 2015 8 / 11

Notation and Terminology

Suppose a situation has n possible outcomes, labeled 1, 2, ..., n.

A lottery [p(1), p(2), ..., p(n)] is a list of probabilities, where p(i) isthe probability that outcome i occurs. (We must havep(1) + p(2) + ... + p(n) = 1).

Example: you get a grade of A with probability 30%, B withprobability 40%, F with probability 30%.

Define outcome 1 as A, outcome 2 as B, and outcome 3 as F.

The lottery is then [0.3, 0.4, 0.3].

ECON 302 (SFU) Lecture 2 January 13, 2015 8 / 11

Expected Utility

Suppose outcome 1 gives you utility u(1), outcome 2 u(2), and so on.What is the utility of lottery L = [p(1), p(2), ..., p(n)]?

Natural answer:

E[u] =n

i=1

u(i)p(i) = p(1)u(1) + p(2)u(2) + + p(n)u(n),

which is the Ls expected utility.

Expected utility gives a preference over lotteries.

ECON 302 (SFU) Lecture 2 January 13, 2015 9 / 11

Expected Utility

Example: You prefer A over B over F. Assigning utility of 2 to A, 1 toB, and 0 to F would represent your preferences over these certainoutcomes.

But suppose you prefer {B for sure} over {a 90% chance of A and a10% chance of F}. With the above utilities, does expected utilityrepresent your preferences over lotteries?

So its important to assign the right intensity of utility to eachoutcome not just the order (ordinal utility), but the size matters(cardinal utility).

ECON 302 (SFU) Lecture 2 January 13, 2015 10 / 11

Expected Utility

Example: You prefer A over B over F. Assigning utility of 2 to A, 1 toB, and 0 to F would represent your preferences over these certainoutcomes.

But suppose you prefer {B for sure} over {a 90% chance of A and a10% chance of F}. With the above utilities, does expected utilityrepresent your preferences over lotteries?

So its important to assign the right intensity of utility to eachoutcome not just the order (ordinal utility), but the size matters(cardinal utility).

ECON 302 (SFU) Lecture 2 January 13, 2015 10 / 11

Expected Utility

Example: You prefer A over B over F. Assigning utility of 2 to A, 1 toB, and 0 to F would represent your preferences over these certainoutcomes.

But suppose you prefer {B for sure} over {a 90% chance of A and a10% chance of F}. With the above utilities, does expected utilityrepresent your preferences over lotteries?

So its important to assign the right intensity of utility to eachoutcome not just the order (ordinal utility), but the size matters(cardinal utility).

ECON 302 (SFU) Lecture 2 January 13, 2015 10 / 11

Axioms for Expected Utility

Given preferences over lotteries, it turns out that its not alwayspossible to find a set of utilities over outcomes such that expectedutility represents the said preferences over lotteries.

Just as you needed assumptions on preferences over outcomes tobuild a utility function representing them, you need assumptions onpreferences over lotteries to build an expected utility functionrepresenting them.

There are four required assumptions (i.e., axioms): completeness,transitivity, continuity, and independence.They are fairly reasonable and intuitive.

Axioms for expected utility are due to John von Neumann and OskarMorgenstern, the founding fathers of game theory.

Expected utility makes preference over lotteries easy to work with.

ECON 302 (SFU) Lecture 2 January 13, 2015 11 / 11