# Lottery Contract

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```					                    Chapter 6
It is a truth very certain that when it is not
in our power to determine what is true we
ought to follow what is probable.—
Descartes

Decision Making and Risk:
Certainty Equivalents
1
and Utility
Decision Making Using
Certainty Equivalents
The certainty equivalent (CE) is the payoff
amount we would accept in lieu of under-
going the uncertain situation.

Shirley Smart would pay \$25 to insure her 1983
Toyota against total theft loss. CE = – \$25.

For \$1,000, Willy B. Rich would sell his Far-
Fetched Lottery rights. CE = \$1,000.

Lucky Chance would pay somebody \$100 to fill
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her Far-Fetched Lottery contract. CE = -\$100.
 A situation’s risk premium (RP) is the
difference its between expected payoff (EP)
and certainty equivalent (CE):
RP = EP - CE
 Shirley Smart’s car is worth \$1,000 and
there is a 1% chance of its being stolen.
Thus, going without insurance has
EP = (– \$1,000)(.01) + (\$0)(.99) = – \$10
RP = – \$10 – (– \$25) = \$15
3
 Playing the Far-Fetched Lottery has EP =
\$2,500. Thus,
 For Willy B. Rich,
RP = EP – CE = \$2,5000 – (\$1,000) = \$1,500
 For Lucky Chance,
RP = EP – CE = \$2,5000 – (– \$100) = \$2,600
 Different people will have different CEs
and RPs for the same circumstance.
 They have different attitudes toward risk.
4
Attitude Toward Risk
 People with positive RPs are risk averters.
 Lucky Chance has greater risk aversion than
Willy B. Rich, as reflected by her greater RP.
 We cannot compare Shirley’s risk aversion to
the others’ because circumstances differ.
 Risk averse persons have RPs that increase:
 When the downside amounts become greater.
 Or when the chance of downside increases.
 A risk seeker will have negative RP.
5  A risk neutral person has zero RP.
Maximizing Certainty Equivalent
 A plausible axiom:
Decision makers will prefer the act yielding
greatest certainty equivalent.
 A logical conclusion:
The ideal decision criterion is to maximize
certainty equivalent.
 Doing so guarantees taking the preferred action.
 But CEs are difficult to determine. One
approach is to discount the EPs.
 RP = EP – CE implies that CE = EP – RP.
6
Using Risk Premiums to Get
Certainty Equivalents
 Ponderosa Records president has the following
risk premiums, keyed to the downside.

 These were found by extrapolating from three
equivalencies (white boxes).
 Exact amounts are unknowable, but these values seem
7        to fit his risk profile.
Decision Tree Analysis with CEs
(Discounted Expected Payoffs)

8
How Good is the Analysis?
 This result is different from that of ordinary
back folding (Bayes decision rule).
 It specifically reflects underlying risk aversion.
 The result must be correct if CEs are right.
 The major weakness is the ad hoc manner
for getting the RPs, and hence the CEs.
 Many assumptions are made in extrapolating to
get the table of RPs.
 There is a cleaner way to achieve the same
9
thing using utilities.
Utility Theory
 Consider a set of outcomes, O1, O2, ..., On.
The following assumptions are made:
 Preference ranking can be done.
 Transitivity of preference: A is preferred to B
and B to C, then A must be preferred to C.
 Continuity: Consider Obetween. Take a gamble
between two more extreme outcomes; winning
yields Obest and losing Oworst. There is a win
probability q making you indifferent between
getting Obetween and gambling. Such a gamble is
10       called a reference lottery.
Utility Assumptions
 Continuity (continued):
 e.g., +\$1,000 v. Far-Fetched Lottery, you pick q.
 For Willy B. Rich, q = .5. (His CE was = +\$1,000.)
 For Lucky Chance, q = .9.
 If the win probability were .99, would you risk
+\$1,000 to gamble? What is your q?
 Substitutability: In a decision structure, you
would willingly substitute for any outcome a
gamble equally preferred.
 One outcome on Lucky Chance’s tree is +\$1,000;
she would accept substituting for it the Far-Fetched
11
Lottery gamble with .9 win probability.
Utility Assumptions and Values
 Increasing preference: Raising q makes any
reference lottery more preferred.
 Anybody would prefer the revised Far-Fetched
Lottery when two coins are tossed and just one head
will win the \$10,000. (The win probability goes
from .5 to .75.) You still might not like that gamble!
 Outcomes can be assigned utility values
arbitrarily, so that the more preferred always
gets the greater value:
u(Obest) = 10     u(Oworst)=0      u(Obetween)=50
 Willy has u(+\$10,000) = 500, u(-\$5,000) = 0 and
12         u(+\$1,000) = 250. These are his values only.
Utility Values
 Lucky has different values: u(+\$10,000) = 50,
u(-\$5,000) = -99, and u(+\$1,000) = 35.1.
 Like temperature, where 0o and 100o are
different states on Celsius and Fahrenheit
scales, so utility scales may differ.
 The freezing point for water is 0o C and the
boiling point 100o C. In-between states will
have values in that range, and hotter days will
have greater temperature values than cooler.
 So, too, with utility values. They will fall into
the range defined by the extreme outcomes,
Oworst and Obest. More preferred outcomes will
13     have greater utilities
Utility Values
 A reference lottery can be used to find the
utility for an outcome Obetween by:
 First, establish an indifference win probability
qbetween making it equally preferred to the
gamble:
 Obest with probability qbetween and
Oworst with probability 1 - qbetween
 Second, compute the lottery’s expected utility:
 u(Obest)(qbetween) + u(Oworst)(1 - qbetween)
 The above is u(Obetween).
14
Utility Values
 Using the Far-Fetched Lottery as reference:
Lottery           Willy       Lucky
Outcomes       Prob. Utility Prob. Utility
Obest (+\$10,000) q=.5 500 q=.9          50
Oworst (-\$5,000) 1 -.5       0  1 -.9 -99
Expected Utility: 250          35.1
Obetween (+\$1,000):   250           35.1

 The indifference q plays a role analogous to the
thermometer, reading attitude towards the
15       outcome similarly to measuring temperature.
The Utility Function
 Utility values assigned to monetary
outcomes constitute a utility function.
 From a few points we may graph the utility
function and apply it over a monetary range.
 Those points may be obtained from an
interview posing hypothetical gambles.
 Using u(+\$10,000)=100 and u(-\$5,000)=0
Shirley Smart gave the following equivalencies:
 A: +\$10,000 @ qA v -\$5,000 ≡ +\$1,000 if qA =.70
 B: +\$10,000 @ qB v +\$1,000 ≡ +\$5,000 if qB =.75
 C1: +\$1,000 @ qC1 v -\$5,000 ≡ -\$500 if qC1 =.70
16
 C2: +\$1,000 @ qC2 v -\$5,000 ≡ -\$2,000 if qC2 =.30
Shirley’s Utility Function
 Shirley’s utilities for the equivalent amounts are
equal to the respective expected utilities:
 u(+\$1,000) = u(+\$10,000)(.70) + u(-\$1,000)(1-.70)
= 100(.70) + 0(1-.70) = 70
 u(+\$5,000) = u(+\$10,000)(.75) + u(+\$1,000)(1-.75)
= 100(.75) + 70(1 - .75) = 92.5
 u(-\$500) = u(+\$1,000)(.70) + u(-\$5,000)(1-.70)
= 70(.70) + 0(1 - .70) = 49
 u(-\$2,000) = u(+\$1,000)(.30) + u(-\$5,000)(1-.30)
= 70(.30) + 0(1 - .30) = 21
 Altogether, Shirley gave 6 points, plotted on the
following graph. The smoothed curve fitting
17     through them defines her utility function.
Shirley’s Utility Function

18
Using the Utility Function
 This utility function applies to the
Ponderosa decision.

19
Using the Utility Function
 Read the utility payoffs corresponding to
the net monetary payoffs.
 Apply the Bayes decision rule, with either:
 A utility payoff table, computing the expected
payoff each act.
 Or a decision tree, folding it back.
 The certainty equivalent amount for any act
or node may be found from the expected
utility by reading the curve in reverse.
 The following Ponderosa Records decision
20
tree was folded back using utility payoffs.
Decision Tree Analysis
with Utilities

21
Shape of Utility Curve and
Attitude Toward Risk
 The following shapes generally apply.

 The risk averter has decreasing marginal util-
ility for money. He will buy casualty insurance
and losses weigh more heavily than like gains.
 Risk seekers like some unfavorable gambles.
22      Risk neutrality values money at its face amount.
Important Utility Ramifications
 Hybrid shapes (like Shirley’s) imply shifting
attitudes as monetary ranges change.
 Regardless of shape, maximizing expected
utility also maximizes certainty equivalent.
 Therefore, applying Bayes decision rule with
utility payoffs discloses the preferred action.
 Primary impediments to implementation:
 Clumsiness of the interview process.
 Multiple decision makers.
 Attitudes change with circumstances and time.
23
Ratification of
Bayes Decision Rule
 Over narrow monetary ranges, utility curves
resemble straight lines.
 For a straight line, expected utility equals
the utility of the expected monetary payoff.
 Maximizing expected monetary payoff then
also maximizes expected utility. Thus:
 The Bayes decision rule discloses the preferred
action as long as the outcomes are not extreme.
 Managers can then delegate decision
making without having to find utilities.
24
Preferred actions will be found by the staff.
Using Utility Functions with
PrecisionTree
PrecisionTree can be used to evaluate
decision trees with with exponential and
logarithmic utility functions.
To get started, click on the name box of a
decision tree and the Tree Setting dialog box
appears, as shown next.

A              B
1      tree #1                  1
2                               0
25
Tree Settings Dialog Box   2. Select the
type of utility
(Figure 6-14)     function in the
Function line.
1. Check the
Here exponential
Use Utility
is chosen.
Function box.

3. Select the
risk coefficient,
R, in the R value
line. Here
10,000 is used.
4. Select
Expected Utility
in the Display
line. Other
options are
Certainty
Equivalent and
Expected Value.

5. Click OK.
26
Decision Tree with Exponential Utility
A
Function for R = 10,000 (Figure 6-15)
B                              C                             D                        E               F
1                                                                                                                       Success
80.0%      0.0
The2 optimal strategy is:                                                                                                           \$90,000      1
3                                                                                        Market nationally
FALSE
4
1. Not test market and to                                                                                        -\$50,000       -48
5                                                                                                                       Failure
20.0%     0.0
abort.
6                                                                                                                                      \$0   -244
7                                                         Favorable
50.0%
2. The corresponding
8                                                                             \$10,000               -1
9                                                                                                              TRUE           0
expected utility is 0.
10
Abort
\$0          -1
11                              Test market
FALSE
12                                            -\$15,000                 -2
13                                                                                                                       Success
20.0%    0.0
14                                                                                                                                 \$90,000    1
15                                                                                        Market nationally
FALSE
16                                                                                                            -\$50,000      -531
17                                                                                                                       Failure
80.0%     0.0
18                                                                                                                                      \$0   -664
19                                                         Unfavorable
50.0%
20                                                                                   \$0              -3
21                                                                                        Abort
TRUE           0
22                                                                                                                \$0          -3
23   Ponderosa Record Company
24                                       0
25                                                                                        Success
50.0%         0
26                                                                                                            \$100,000        1
27                                                         Market nationally
FALSE
28                                                                             -\$60,000             -201
29                                                                                        Failure
50.0%         0
30                                                                                                                  \$0      -402
31                              Don't test market
TRUE
32                                                    \$0               0
33         27                                              Abort
TRUE                 1
34                                                                                 \$0                 0

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Description: Lottery Contract document sample