Transcript Lecture 4

2.2 Choice under Uncertainty:
Lotteries and Risk Aversion
Outline
2.1 Basic concepts: Preferences and utility
2.2 Choice under uncertainty: Lotteries and risk
aversion
2.3 Value of information: Decision trees and
backward induction
2.2 Choice under uncertainty: Lotteries and risk
aversion
In some individual decision problems, the final
outcome of the decision does not depend only
on my choice. It might also depend on events
that are unpredictable
Examples:
Buying a lottery ticket
Blackjack
Financial investment
Launching a new product
Where does the uncertainty come from?
Nature (this is the type of uncertainty that we are going to study now)
The result of my actions (the outcome) depends on the state
of nature, which is random
Other individuals (we will postpone this for few weeks)
The result of my actions depends both on what I do and
what others do.
Example
I have to choose one of the following two
alternatives (for free !)
•
a
$100 for sure
•
b
To flip a coin. If heads comes up I win
$200, otherwise I win nothing
What do I prefer ?
Example
States of Nature
Tails (50%)
a
$100
$100
b
$200
$0
Actions
Heads (50%)
Outcomes
What do we know about
these uncertain events ?
• We know all possible outcomes (that depend on the states
of nature)
• We do not know what actual outcome will be
• We have a probability measure of the likelihood of each
outcome
Without knowing the possible
outcomes or/and their probabilities,
making choices would be a rather
difficult task
Example
A box contains 90 balls, 30 are red while the remaining 60 balls
are either green or yellow (in an unknown ratio)
30 balls
60 balls
Example (continued)
Which one of these two alternatives do
you prefer ?
•
a
You win 100$ if you draw a red ball
•
b
You win 100$ if you draw a green ball
Example (continued)
And now, which one of these two alternatives do you prefer
?
•
c
You win 100$ if you draw a red or yellow ball
•
d
You win 100$ if you draw a green or yellow ball
Example (explanation)
Let R, G, and Y denote the number of Red, Green, and
Yellow balls respectively.
If a were preferred to b in the first choice, it must be because the
individual believes that
R>G
If d were preferred to c in the second choice, it must be because the
individual believes that
G+Y>R+Y
That is,
G>R
Contradiction ! !
Example (explanation)
This example is a well known case of paradoxical
behavior because of the lack of information
regarding the probabilities of the different
outcomes.
It is known as the Ellsberg Paradox, after Daniel
Ellsberg (born April 7, 1931) a former American
military analyst employed by the RAND Corporation
who precipitated a national political controversy in
1971 when he released the Pentagon Papers, a
top-secret Pentagon study of government
decision-making about the Vietnam War, to
The New York Times and other newspapers.
(from Wikipedia)
Basic Concepts
Actions: My possible choices of an alternative
States of Nature: The possible random events
Outcomes: That depend on both my actions and on
the state of nature
Probabilities: The probability of each state of
nature
Probabilities
A “probability” is a measure of the likelihood of the occurrence
of a random event
The higher the probability, the more likely is the event
If the probability is 1, then the event will occur for sure
(certain event)
If the probability is 0, then the event will not occur for sure
(impossible event)
Thus, the probability is always between 0 and 1
The sum of the probabilities of all the possible events is
always 1 (something will occur for sure)
Example
Imagine that you have a huge garden and you are
considering which type of trees you what to plant
there. You live near the sea so the probability of
snowing is relatively low (25%).
Example
States of Nature
Actions
Snows (25%)
Outcomes
Does not
Snow (75%)
Apples
$100
$40
Oranges
-$20
$140
Apricots
$80
$60
How do we represent it?
Nature branches
0.25
Decision branches
Snows
apples
oranges
Nature
apricots
0.75
Does not snow
Snows
(0.25)
apples
Doesn’t snow
(0.75)
Snows
oranges
100
40
(0.25)
-20
Doesn’t snow
140
(0.75)
Snows
apricots
(0.25)
Doesn’t snow
(0.75)
80
60
This might make the analysis easier
–
Make sure you have enought space
–
Think well what comes first, second…
–
Ordering matters
–
Write down the probabilities
Situations like the one depicted are called
“Choice over lotteries”
Example
Represent the following situation as a lottery.
You want to open a new small business, but you are affraid of
the crisis not being yet over. According to the experts' forecasts,
the probability that the crisis continues next year is of 20%. In
such case, if you open your business you will lose $80,000. If, on
the contrary, the situation improves (80% probability), your
business will earn $50,000. If you decide not to open the
business, you will not lose neither make any money.
0.8
Open
Don't Open
0.2
0.8
0.2
$50,000
-$80,000
0
0
Alternatively . . .
0.8
0.2
Open
Don't Open
0
$50,000
-$80,000
Compounded Lotteries
Sometimes we can play one lottery after the other.
Imagine that you are playing the following lottery with
a friend. She has two “real” lottery tickets, one is for
Christmas and the other one is for 6th of January.
She will flip a coin (it has probability ½ of tails and ½ of
heads), if the outcome is heads than you get the ticket
for Christmas. If it is tails you will get the other one.
Both lotteries have the same probability of winning,
0.001. The prize of the first one (Christmas) is
100,000 and of the second one is 50,000.
Represent it
0.5
L2
0.5
L3
L1
6th of January
Christmas
100,000
0.001
0.001
50,000
0.999
0
L3
L2
0.999
0
100,000
0.001
H: 0.5
0.999
0
L1
0.001
50,000
T: 0.5
0.999
0
Probability review
If we flip two coins, we have four possible results
Probability review
If two events, A and B, are independent (the occurrence of
one of them has nothing to do with the occurrence of the
other), then
p( A and B ) = p(A) x p(B)
Example
If we flip to coins
p( 2 heads ) = p( heads in 1st and heads in 2nd ) = ½ x ½ = ¼
Probability review
If two events, A and B, are incompatible (when one occurs the
other can not occur, and vice versa), then
p( A or B ) = p(A) + p(B)
Example
If we flip to coins
p( exactly 1 head ) =
= p( heads in 1st and tails in 2nd or tails in 1st and heads in 2nd ) =
=½ x½ +½ x½ =¼+¼ =½
Probability review
Hence, in terms of probability,
“and” is equivalent to “x”
p( A and B ) = p(A) x p(B)
“or” is equivalent to “+”
p( A or B ) = p(A) + p(B)
Back to our example . . .
100,000
0.001
H. 0.5
0.999
0
L1
0.001
50,000
T: 0.5
0.999
0
Can it be reduced (simplified) ?
What are the outcomes?
At the end I can have 0, 50,000 or 100,000
So instead or drawing the big tree I can reduce it.
How?
Can it be reduced (simplified) ?
P(100,000) = p(Heads and win) = 0.5 x 0.001 = 0.0005
P(50,000) = p( Tails and win) = 0.5 x 0.001 = 0.0005
P(0) = p( Heads and lose or Tails and lose) =
= 0.5 x 0.999 + 0.5 x 0.999 = 0.999
Check ! P(100,000) + P(50,000) + P(0) = 1
OK ! !
So, at the end we have that this . . .
100,000
0.001
H. 0.5
0.999
0
L1
0.001
50,000
T: 0.5
0.999
0
Reduces to this
100,000
0.0005
0.0005
50,000
L1
0.999
0
This reduction is only possible for the case of
individuals that do not obtain any special “benefit”
from gambling
Although that might be the case in some situations
(Las Vegas), it would not be “rational” in most
business and/or economics scenarios
What does this reduce to ?
1
1
1/3
1/4
1/3
1/3
3/8
1
2
3/8
3
1/4
1
3/8
3/8
2
3
Reduced equivalent lottery
1
1/2
1/4
2
1/4
3
How do we choose ?
If we flip a coin 1000 times, and we get $1 with heads
and lose $1 with tails . . . How much will we earn at
the end ? (approximately)
Most likely, we well get around 500 heads and 500 tails.
That is, we will earn $500 and lose $500. At the end, we
expect to make around $0
How much would you pay to play this lottery ?
How do we choose ?
If we flip a coin 1 time, and we get $1 with heads
and lose $1 with tails . . . How much will we earn at
the end ? (approximately)
In this case the final result is no so clear.
The expectation is taken as a rough estimate of what is the
value of such lottery based on what would be the earnings if
the lottery were played a large number of times.
It helps to decide among different alternatives
Expectation of a lottery
If a lottery L gives prizes x1, x2, x3, . . . with probabilities
p1, p2, p3, . . . respectively, then the expectation (or expected
value) of L, denoted E(L), is computed as
E(L) = p1 · x1 + p2 · x2 + p3 · x3 + · · ·
Ex: In the case of the coin (1$ if heads, -1$ if tails)
E(L) = ½ · ( 1 ) + ½ · ( -1 ) = 0
How much would you pay to play this lottery ?
The garden
0.25
100
E(apples) = 0.25 x 100 + 0.75 x 40 = 55
0.75
40
apples
oranges
0.25
0.75
-20
E(oranges) = 0.25 x (-20) + 0.75 x 140 = 100
140
apricots
0.25
80
0.75
E(apricots) = 0.25 x 80 + 0.75 x 60 = 65
60
The paradox of St. Petersburg
The gamble is to flip a coin several times until you obtain heads for
the first time.
If heads appear on the 1st round you win 2$ and the game is over.
If heads appear on the 2nd round you win 4$ and the game is over.
If heads appear on the 3rd round you win 8$ and the game is over.
And so on, each round doubles the pot
½
2
½
4
½
½
8
½
...
½
How much would you pay
to play this gamble ?
What is the Expectation ?
Heads
Tails and Heads
Tails and Tails and Heads
Tails and Tails and Tails and Heads
. . . and so on . . .
Win
Win
Win
Win
That is,
Win 2 with probability 1/2
Win 4 with probability ½ x ½ = 1/4
Win 8 with probability ½ x ½ x ½ = 1/8
Win 16 with probability ½ x ½ x ½ x ½ = 1/16
. . . and so on . . .
2
4
8
16
What is the Expectation ?
E(L) = 1/2 x ( 2 ) + 1/4 x ( 4 ) + 1/8 x ( 8 ) + · · · =
= 1
+
1
+ 1
+ · · · = ∞ !!
Hence, we should use another criteria to value lotteries
Because the Expectation might produce counterintuitive
results
Because it is a purely mathematical computation that does
not take into account the individual's preferences and
attitude towards risk
The expectation is just a convenient way of summarizing a
lottery
Summary:
• In many individual decision problems, the final outcome of the decision
does not depend only on my choice. It might also depend on events that
are random (uncertain or unpredictable) These situations are like lotteries
• We must know all the possible outcomes and their probabilities;
otherwise making choices would be a rather difficult task
• In each situation we must identify:
Actions: My possible choices of an alternative
States of Nature: The possible random events
Outcomes: That depend on both my actions and on the state of nature
Probabilities: The probability of each state of Nature
• Trees are convenient ways of representing decision problems under
uncertainty (lotteries)
Summary:
• The probability rules:
“and” is equivalent to “x”
p( A and B ) = p(A) x p(B)
“or” is equivalent to “+”
p( A or B ) = p(A) + p(B)
are useful for working with probabilities
• The expectation of a lottery,
E(L) = p1 · x1 + p2 · x2 + p3 · x3 + · · ·
is a convenient way of summarizing the value of a lottery
• But the expectation is a purely mathematical computation that does
not take into account the individuals’ preferences and attitudes towards
risk. It might produce counterintuitive results