Decisions under uncertainty

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Decisions under uncertainty

• A Different look at Utility Theory

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Overview
The author says that economic decisions made
under uncertainty are essentially gambles. Lets
first look at some gambles, and then come back to
decisions under uncertainty. Initially, our
gamble will be to flip a fair coin (one where the
probability of a head is .5 and the probability
of a tail is .5). The payout will depend on
which side of the coin is showing when the coin
lands at rest.
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Examples
Facts about several gambles with a fair
coin Gamble 1 heads means you win 100 and
tails means you lose 0.50 Gamble 2 heads means
you win 200 and tails means you lose 100 Gamble
3 heads means you win 20000 and tails means
you lose 10000 Note what a person would lose on
each gamble. Many people would say the loses in
gambles 2 and 3 make them uneasy and they
wouldnt take those gambles. But, some folks out
there might take gambles 2 and 3.
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Digress the mean
What is the mean or the average of the numbers 4
and 6? You probably said 5 and you are right.
This could be written (46)/2 (4/2) (6/2)
(1/2)4 (1/2)6, where in this last form you see
each number multiplied by ½. In this context the
mean is said to be a simple weighted average,
with each value weighted by ½. What would the
weights be if we wanted the average of 4, 5, and
6? 1/3! In general with n numbers the weight is
1/n. In other situations we may look at a
weighted average (not simple), though the weights
are found in a different way.
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Back to example
The expected value of a gamble is a weighted
average of the possible payout values and the
weights are the probabilities of occurrence of
each payout. We talk about EVi as the expected
value of gamble i. EV1 .5(100) .5(-0.50) 50
0.25 49.75. (Notice when you lose the loss
is subtracted out.) EV2 .5(200) .5(-100)
100 50 50 EV3 .5(20000) .5(-10000)
10000 5000 5000
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Example
In our example the EV for each gamble is
positive. The EV is the highest for gamble 3.
But, remember we said not many folks would
probably like it because of the uneasiness they
would feel by losing the 10000. A couple of guys
named Von Neumann (both names are just the
persons last name) and Morgenstern created a
model we now call the expected utility model to
deal with situations like this. They indicated
folks make decisions based not on monetary
values, but based on utility values. Of course
the utility values are based on the monetary
values, but the utility values also depend on how
people view the world.
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Expected Utility
Say we observe a person always buying chocolate
ice cream over vanilla ice cream when both are
available and both cost basically the same, or
even when chocolate is more expensive and always
when chocolate is the same price or cheaper. So
by observing what people do we can get a feel for
what is preferred over other options. When we
assign utility numbers to options the only real
rule we follow is that higher numbers mean more
preference or utility. Even when we have
financial options we can study or observe the
past to get a feel for our preferences.
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Expected Utility Theory is a methodology that
incorporates our attitude toward risk (risk is a
situation of uncertain outcomes, but
probabilities are known) into the decision making
process.
It is useful to employ a graph like this in our
analysis. In the graph we will consider a rule
or function that translates monetary values into
utility values. The utility values are our
subjective views of preference for monetary
values. Typically we assume higher money values
have higher utility.
Utility value
Monetary value
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In general we say people have one of three
attitudes toward risk. People can be risk
avoiders, risk seekers , or indifferent toward
risk (risk neutral).
Utility value
Utility values are assigned to monetary values
and the general shape for each type of person is
shown at the left. Note that for equal
increments in dollar value the utility either
rises at a decreasing rate(avoider), constant
rate or increasing rate.
Risk avoider
Risk indifferent
Risk seeker
Monetary value
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Here we show a generic example with a risk
avoider. Two monetary values of interest are,
say, X1 and X2 and those values have utility
U(X1) and U(X2), respectively
Utility
U(X2) U(X1)

X1 X2
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Expected Utility
In expected utility theory we want to focus on
wealth values and utility values. Gambles will
lead to adjustments in wealth. Lets call W the
initial wealth which can always be retained if no
gamble is taken, and call the wealth at a loss X1
W loss, and the wealth at a win X2 W
win. The expected value of wealth from a gamble
is then (p1 is the probability of a loss and p2
is the probability of a win) EV p1X1
p2X2 Note we called the expected value of a
gamble EV and I now have the expected value of
wealth with a gamble being EV. EV will mostly
stand for expected value of wealth, unless
otherwise stated.
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Expected Utility
The following is what needs to be considered to
get the expected utility of a gamble 1) Start
with a persons initial wealth W, 2) If the
gamble is taken identify X1 and X2, 3) Assign to
each wealth value in 2) the respective utility
value U1, and U2 and in a graph connect the U1
and U2 values with a chord. 4) Calculate the
expected value of wealth with the gamble EV. 5)
Calculate the expected utility of the gamble as
EU where EU p1U1 p2U2, and find the value on
the chord above the EV.
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Utility
U2 U1
EU

X1 EV X2
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Example continued
Say utility is assigned by the function U
sqrt(wealth) and a person has initial wealth
10000. Then for the three gambles we had before
the EUs would be EU1 .5sqrt(9999.5)
sqrt(10100) 100.248 EU2 .5sqrt(9900)
sqrt(10200) 100.247 EU3 .5sqrt(0)
sqrt(30000) 86.603 So in terms of EUs the
preferred order of gambles for this person is
gamble 1, then 2, and the 3. When we looked at
EVs the order was 3, 2, and 1. So expected
utilities of gambles may have a different rank
ordering than when looking at the EVs.
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Fair gambles
A fair gamble is one where the expected value of
the gamble is zero, i.e., p1(-loss) p2(win)
-p1(loss) p2(win) 0. This implies that the
expected value of wealth with the gamble is equal
to the value of wealth when not gambling at all,
which you might call your certain wealth. For
fair gambles EV p1(W loss) p2(W win)
p1W p2W p1(loss) p2(win) (p1p2)W 0 W,
since p1p21.
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Risk averse fair gamble
In this graph I have the generic view of a risk
lover. With the fair gamble we have the EV and
the EU is on the chord above the EV. If the
person does not gamble wealth will be W and the
utility there is just read off the utility
function here as Uw (note a risk averter has
diminishing marginal utility of wealth.)
Utility
Uw
EU
X1
X2
wealth
EV W
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Risk Averse fair gamble
For the fair gamble we again know EV W, but for
a risk averse person Uw gt EU. Thus we can
conclude risk averse folks will not accept fair
gambles. On the next slide you can see I
thickened part of the horizontal axis and the
chord connecting the two points on the utility
function associated with the wealth values under
the gamble. The probabilities of the gamble
could be changed (and the gamble would no longer
be fair) and the only way the person would accept
the gamble over having the certain wealth W is if
the EV was greater than W. So, a risk averse
person may gamble, but it has to be at favorable
odds.
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Risk averse fair gamble
If the EV of a gamble is above W (and is no
longer a fair gamble, but a favorable one), then
the person will end up on the chord segment that
has not been thickened and thus only then have
EUgtUw.
Utility
Uw
EU
X1
X2
wealth
EV W
W
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Risk Seeker fair gamble
In this graph I have the generic view of a risk
seeker. With the fair gamble we have the EV and
the EU is on the chord above the EV. If the
person does not gamble wealth will be W and the
utility there is just read off the utility
function here as Uw (note a risk seeker has
increasing marginal utility of wealth.
U
EU
Uw
W
X1
X2
EV W
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Risk Seeker fair gamble
For the fair gamble we again know EV W, but for
a risk seeker person Uw lt EU. Thus we can
conclude risk seeker folks will always accept
fair gambles. On the next slide you can see I
thickened part of the horizontal axis and the
chord connecting the two points on the utility
function associated with the wealth values under
the gamble. The probabilities of the gamble
could be changed (and the gamble would no longer
be fair) and the only way the person would NOT
accept the gamble over having the certain wealth
W is if the EV was less than W. So, risk
seeker may NOT gamble, but it has to be at
unfavorable odds.
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Risk Seeker fair gamble
If the EV of a gamble is below W (and is no
longer a fair gamble, but an unfavorable one),
then the person will end up on the chord segment
that has not been thickened and thus only then
have EUltUw. .
U
EU
Uw
W
X1
X2
EV W
W
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Risk Neutral fair gamble
U
The risk neutral person is indifferent between a
fair gamble and not gambling at all. If odds are
switched to favorable gambles will be favored and
if switched to unfavorable gambles will not be
taken.
Uw EU
X1
EV W
X2
W
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Certainty Equivalents
In a more general sense we could talk about
gambles that are fair or unfair. The certainty
equivalent of a gamble will be the sum of money
or wealth for which the individual would be
indifferent between the certain sum and the
gamble. We will examine these certainty
equivalents for folks with risk aversion,
neutrality and risk seeking prefrences.
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Risk averse certainty equivalent
Utility
The decision maker may have an option that is
certain. If so, the EU is simply the utility
along the utility curve (I called it Uw before).
So in this diagram we see that any sure bet
greater than Y has an expected utility greater
than the expected utility of the risky option.
U2 U1
EU

Y
EV
X1 X2
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Another Example Say Utility U square root of X,
where X is a dollar amount the person ends up
with, Then U(4) 2 and U(16) 4, for example.
Say a risky option will result in 4 50 of the
time and 16 50 of the time. The expected value
is 10 because .5(4) .5(16) 10 and the
expected utility is 3 because .5U(4) .5U(16)
.5(2) .5(4) 3. Now, if there is an option
that will pay more than 9 with certainty, than
the certain option is better. So, 9 is the
certainty equivalent of this uncertain gamble.
Lets see this on the next slide.
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U(x)
U(16)4
U(x)
EU 3 U(4)2
4 9 10 16 x
Any certain option above 9 gives a utility value
greater than the expected utility of the
uncertain option.
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Risk seeker certainty equivalent
In this graph I have the generic view of a risk
seeker. With the fair gamble we have the EV and
the EU is on the chord above the EV. Y is the
certainty equivalent of the gamble. Any certain
option above Y would be preferred to the gamble
shown by the risk seeker.
U
EU
Uw
W
X1
X2
EV
Y
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Reducing Risk
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In a previous section we mentioned that sometimes
we face an uncertain situation with regards to
monetary values. We saw 1) The expected monetary
value, EV, of the uncertain situation (what I
will now call a gamble) is the sum of some
numbers, where each number is a monetary value
multiplied by its probability of occurring, 2)
The expected utility of a gamble (what I will now
write as EU) is the sum of some numbers, where
each number is the utility of a monetary value
multiplied by its probability of occurring, 3)
The expected utility of a gamble does not occur
on the utility function (unless the person is
risk neutral), but on the chord or line segment
that connects utility values of each part of the
gamble and directly above the EV of the gamble.
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Say we have a risk lover and the gamble G leaves
Y1 p1 of the time and Y2 p2 of the time.
U
EV p1Y1 p2Y2 EU p1U(y1) p2U(Y2)
U(Y2)
EU
U(Y1)
Y
Y1 EMV Y2
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Say we have a risk avoider and the gamble G
leavesY1 p1 of the time and Y2 p2 of the time.
U
EMV p1Y1 p2Y2 EU p1U(y1) p2U(Y2)
U(Y2)
EU
U(Y1)
Y
Y1 EMV Y2
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• On the last few slides I show you generic cases
  of a risk lover and a risk avoider. You see a
  gamble with monetary values Y1 and Y2, with
  associated probabilities p1 and p2 (where p2 1-
  p1).
• I now want to show something we saw in the
  previous section, but I want to be more precise
  in my language.
• Sometimes we may have an opportunity that is
  known with certainty. The utility of the
  opportunity will be on the utility function for
  the individual and will be noted U(C).
• The decision rule for choosing between a gamble
  and a certain payoff is
• choose the certain option when U(C) gt EU(G),
  and
• choose the gamble when U(C) lt EU(G).
• Of course, when the two are equal the individual
  would be indifferent between the two.

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Back on the slides I have some vertical dashed
lines. I put them there on purpose. I want you
to think of the location as values of a certain
payoff, I now call C, and then we can see that
U(C) EU. The payoff C is called the certainty
equivalent of the gamble.
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Say we have a risk avoider and the gamble G
leaves Y1 p1 of the time and Y2 p2 of the time.
U
The risk premium, rp, of a gamble is the EV of
the gamble minus the certainty equivalent of the
gamble. rp EV - C and will always be positive
for a risk avoider. The risk premium for a risk
lover will be negative and it will be zero for a
risk neutral person.
U(Y2)
EU
U(Y1)
Y
Y1 EV Y2
C
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Say Y2 is value of property if no fire and Y1 is
the value of the property with a fire. The EV
p1Y1 p2Y2. EU p1u(Y1) p2U(Y2)
A Gamble of no fire insurance
U
U(Y2)
EU
C is the certainty equivalent of the gamble.
U(Y1)
Y
Y1 EV Y2
C
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If a person buys insurance it changes the risky
situation into a certain situation. If Y2 - C
fee paid for insurance the individual will have C
with certainty. To see this we note If no fire
the individual has Y2 - fee C, and If fire the
individual gets restored to Y2 and has still paid
the fee so the certain property value is
C. SOOOOOO Y2 - C is really the maximum fee the
person would pay for insurance and they would
like to pay less. Y2 C is called the
reservation price for insurance.
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Lets take the point of view of the insurance
company - and we do not have to look at the graph
here. They pay claim of Y2 - Y1 p1 of the time
and they pay 0 p2 of the time for an expected
claim of p1(Y2 - Y1) p2(0) p1(Y2 - Y1) This
is called the actuarially fair insurance premium
- meaning this is the minimum they have to charge
to be able to pay out all the claims. Now look in
the graph - here is an amazing result Y2 - EV
Y2 - (p1Y1 p2Y2) Y2(1 - p2) -p1Y1 p1(Y2 -
Y1), so Y2 - EV is the actuarially fair premium
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Review Y2 - EMV least insurance company will
charge, Y2 - C Most person will pay, (Y2 - C) -
(Y2 - EMV) EMV - C is the room the person and
the insurance company have to negotiate for the
insurance. Before we said EMV - C was the risk
premium and now we see it is the most the person
would pay over the actuarially fair premium to
insure against the gamble. Now insurance
companies pay out claims and pay employees and
electricity and other admin. expenses. The
company has to get some of EMV - C to pay these
expenses.
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People wont buy the insurance if the insurance
company needs more than EMV - C to cover its
other expenses because the person would have more
utility without it in that case. Next lets look
at how information can be beneficial in reducing
risk.
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U
situation without information situation with
information
Y
Y1 C EMV Y2
U
Y
Y1 c EMV Y2
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On the previous slide I show two graphs. Both
have the same utility function for an individual.
The top graph is a situation where the
individual has no information and the bottom
graph shows what happens when more information is
obtained. Note more information may not eliminate
risk, but it can reduce it. Lets study an
example to show context. Say an individual can
buy a painting and if it is a real master
painting the wealth of the individual will be Y2.
If the painting is a fake the individual will
lose some of his expenditure because the painting
is no big deal - wealth is Y1. We see the
certainty equivalent of the gamble is C.
Presumably the individual will buy the painting
if the certainty equivalent of the gamble is
better than his wealth by not buying the painting
at all.
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Now say the person can hire a painting expert to
see if the painting is a fake or not. If the
expert says the painting is a fake then you will
not buy it and will not lose on the low end. But
if it is a real painting you will have the same
high end wealth because you will buy the
painting. So information from an expert in this
case gives you the same high end value but makes
your low end value better than without
information. But, the expert is going to want to
charge you for the information. How much should
you pay? Since C is the certainty equivalent
with information and C is the certainty
equivalent without the information, the person
would pay up to C - C for the information and
the utility of the person would be improved.