Review of Probability, Expected Utility
Economics 302 - Microeconomic Theory II: Strategic Behavior
Instructor: Songzi Du
compiled by Shih En Lu
Simon Fraser University
January 11, 2016
ECON 302 (SFU)
Lecture 2
January 11, 2016
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Review of Probability
Probability is a number between 0 and 1 and is a (either subjective or
objective) 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.
ECON 302 (SFU)
Lecture 2
January 11, 2016
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Review of Probability
Probability is a number between 0 and 1 and is a (either subjective or
objective) 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 an
event ({2, 4, 6}).
Probability of an event is the sum of probabilities of outcomes in that
event.
ECON 302 (SFU)
Lecture 2
January 11, 2016
2 / 11
Review of Probability
Probability is a number between 0 and 1 and is a (either subjective or
objective) 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 an
event ({2, 4, 6}).
Probability of an event is the sum of probabilities of outcomes in that
event.
Probabilities over all outcomes add up to 1.
ECON 302 (SFU)
Lecture 2
January 11, 2016
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Expected value
Suppose that the outcomes = {1, 2, . . . , n}. Probability of outcome i
is p(i).
Suppose that we have a function that depend on the outcome: X (i).
This is called a random variable.
ECON 302 (SFU)
Lecture 2
January 11, 2016
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Expected value
Suppose that the outcomes = {1, 2, . . . , n}. Probability of outcome i
is 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
X
X (i)p(i) = X (1)p(1) + X (2)p(2) + · · · + X (n)p(n).
i=1
Expected value is a notion of average.
ECON 302 (SFU)
Lecture 2
January 11, 2016
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Examples
What is the expected value of a roll of dice?
ECON 302 (SFU)
Lecture 2
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Examples
What is the expected value of a roll of dice?
n
X
i·
i=1
1
1
1
1
1
= 1 × + 2 × + · · · + 6 × = 21 × = 3.5.
6
6
6
6
6
If Alice gets a grade of 90 with probability 3/10, 80 with probability
2/10, 70 with probability 4/10, and 0 with probability 1/10. What is
her expected grade?
ECON 302 (SFU)
Lecture 2
January 11, 2016
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Examples
What is the expected value of a roll of dice?
n
X
i·
i=1
1
1
1
1
1
= 1 × + 2 × + · · · + 6 × = 21 × = 3.5.
6
6
6
6
6
If Alice gets a grade of 90 with probability 3/10, 80 with probability
2/10, 70 with probability 4/10, and 0 with probability 1/10. What is
her expected grade?
90 ×
3
2
4
1
+ 80 ×
+ 70 ×
+0×
= 27 + 16 + 28 = 71.
10
10
10
10
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Lecture 2
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Independence
We may have two unrelated events. They are called independent
events. For example, the rolls of two dice; the weather in Vancouver
and the weather in Miami; the birth dates of two strangers, etc.
If two events are independent, knowing one event has no bearing on
the probability of the other event.
ECON 302 (SFU)
Lecture 2
January 11, 2016
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Independence
We may have two unrelated events. They are called independent
events. For example, the rolls of two dice; the weather in Vancouver
and the weather in Miami; the birth dates of two strangers, etc.
If two events are independent, knowing one event has no bearing on
the probability of the other event.
Probability of two independent events = probability of first
event × probability of second event.
ECON 302 (SFU)
Lecture 2
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Independence
Alice goes to class with probability 1/2 (and skip class with
probability 1/2). Bob goes to class with probability 1/3 (and skip
class with probability 2/3). Assume Alice and Bob act independently.
What is the probability that both show up in class?
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Lecture 2
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Independence
Alice goes to class with probability 1/2 (and skip class with
probability 1/2). Bob goes to class with probability 1/3 (and skip
class 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?
ECON 302 (SFU)
Lecture 2
January 11, 2016
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Independence
Alice goes to class with probability 1/2 (and skip class with
probability 1/2). Bob goes to class with probability 1/3 (and skip
class 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?
ECON 302 (SFU)
Lecture 2
January 11, 2016
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Independence
Alice goes to class with probability 1/2 (and skip class with
probability 1/2). Bob goes to class with probability 1/3 (and skip
class 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 11, 2016
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Choice Under Uncertainty
You probably have learned about preferences and utility functions
over certain outcomes.
When preferences are complete and transitive, they can be
represented by a utility function.
(I.e. There exists a utility function u such that A % B if and only if
u(A) ≥ u(B).)
ECON 302 (SFU)
Lecture 2
January 11, 2016
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Choice Under Uncertainty
You probably have learned about preferences and utility functions
over certain outcomes.
When preferences are complete and transitive, they can be
represented by a utility function.
(I.e. There exists a utility function u such that A % B if and only if
u(A) ≥ u(B).)
But life is full of uncertainty! You don’t know for sure how good (or
how bad) the economy will be in two years. You don’t know for sure
what other people will do.
You have to make decision under uncertainty.
Goal: represent preferences over uncertain outcomes.
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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) is
the probability that outcome i occurs. (We must have
p(1) + p(2) + ... + p(n) = 1).
ECON 302 (SFU)
Lecture 2
January 11, 2016
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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) is
the probability that outcome i occurs. (We must have
p(1) + p(2) + ... + p(n) = 1).
Example: you get a grade of A with probability 30%, B with
probability 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)
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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
X
u(i)p(i) = p(1)u(1) + p(2)u(2) + · · · + p(n)u(n),
i=1
which is the L’s expected utility.
Expected utility gives a preference over lotteries.
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Expected Utility
Example: You prefer A over B over F. Assigning utility of 2 to A, 1 to
B, and 0 to F would represent your preferences over these certain
outcomes.
ECON 302 (SFU)
Lecture 2
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Expected Utility
Example: You prefer A over B over F. Assigning utility of 2 to A, 1 to
B, and 0 to F would represent your preferences over these certain
outcomes.
But suppose you prefer {B for sure} over {a 90% chance of A and a
10% chance of F}. With the above utilities, does expected utility
represent your preferences over lotteries?
ECON 302 (SFU)
Lecture 2
January 11, 2016
10 / 11
Expected Utility
Example: You prefer A over B over F. Assigning utility of 2 to A, 1 to
B, and 0 to F would represent your preferences over these certain
outcomes.
But suppose you prefer {B for sure} over {a 90% chance of A and a
10% chance of F}. With the above utilities, does expected utility
represent your preferences over lotteries?
So it’s important to assign the right intensity of utility to each
outcome — not just the order (ordinal utility), but the size matters
(cardinal utility).
ECON 302 (SFU)
Lecture 2
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Axioms for Expected Utility
Given preferences over lotteries, it turns out that it’s not always
possible to find a set of utilities over outcomes such that expected
utility represents the said preferences over lotteries.
Just as you needed assumptions on preferences over outcomes to
build a utility function representing them, you need assumptions on
preferences over lotteries to build an expected utility function
representing 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 Oskar
Morgenstern, the founding fathers of game theory.
Expected utility makes preference over lotteries easy to work with.
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