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ISE 562; Dr. Smith
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Utility Concepts 2 Decision Theory
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ISE 562; Dr. first another EVSI example:
You are thinking about investing $1000 in one of 3 stocks, A, B, or C. The return on investment of these stocks depends on the probability that each company will release their latest product in the next 6 months.
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ISE 562; Dr. R = new product release
NR = no product release
The net returns on the investments after 1 year are estimated to be:
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With prior probabilities of product release:
P(R) P(NR)
= 0.70 = 0.30
ISE 562; Dr. have the option of buying an on-line
investment newsletter that costs $10/month that tracks whether the new product releases of companies are on schedule or not. Their track record is summarized by the following likelihoods. Let
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= on schedule
= late/not on schedule
ISE 562; Dr. (S| R) =0.85
P(S|NR) =0.10 P(L|R) =0.15 P(L|NR) =0.90
• What is the EVPI?
• What is the EVSI?
• What is your optimal decision strategy?
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ISE 562; Dr. Smith
• What is the EVPI?
Prob = 0.70 0.30
EV 140 250* 150
• EVPI=EV given PI – EV(a*) = .7(400)+.3(0) – 250 = 30
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ISE 562; Dr. Smith
• What is the EVSI?
= EV with SI – EV without SI
• Need to compute posterior probabilities • and the decision tree…
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ISE 562; Dr. =sched
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ISE 562; Dr. newsletter 256
250 Don’t buy
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.72 .28 .72
A .30 B .70
Note: tree does not include cost of sampling
ISE 562; Dr. Smith
• EVSI=EVwithSI–EVwithoutSI
• EVSI=256–250=6
• It appears that the value of additional information (newsletter) is positive. Now the EVSI must be compared to the cost of the information (cost of sampling, CS)
• ThatisENGS=EVSI–CS=6–10=-4
• Optimal strategy: don’t buy newsletter
since EVSI = 6 < CS = 10*
*Does not matter if cost for one month or year; still a negative ENGS
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ISE 562; Dr. to Utilities...
Utility function, U(X)
Maps decision attribute X to the interval [0,1] where U(worst case)=0.0 and U(best case)=1.0
Captures risk attitude
– Risk neutral
– Risk seeking
– Risk averse
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ISE 562; Dr. that if decision maker is risk neutral, the utility function can be written as:
U(X)=a+bX where X=return of money
• From definition of expected value:
E[U(X)]=E[a+bX]=E[a]+E[bX]=E[a]+bE[X]
• Expected value of constant = constant so,
E[U(X)]=a+bE[X] and E[X]=expected monetary value so,
• EU=a+bEMV b>0, so EU a maximum when EMV a maximum
If utility function is linear with respect to $, then EMV
and EU criterion are equivalent
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ISE 562; Dr. Smith
• Assessing utility function from decision maker interviews is labor-intensive
• Worthwhile for high-value or high-cost consequence decisions
• Provides precision beyond simple risk- neutral assumption
• If time, cost, or payoffs limited in
scope, are there simplifications
available?
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ISE 562; Dr.
Risk-neutral case (already discussed)
3-point utility function (only assess 0.50 certainty equivalent)
Use mathematical functions to represent utility
U(X)= aX+b (piecewise linear) U(X)=a-e-bX (exponential) U(X)=a log(X+b)
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Suppose we have investments with profits between $15k and $30k (more > less); we ask the decision maker, “Which would you prefer?”
ISE 562; Dr. .
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Pw=1/2 PL=1/2
Receive profit= 30k Receive profit= 15k
IF RATIONAL, SHOULD CHOOSE OPTION B
Receive profit= 15k for certain
ISE 562; Dr. Smith
Now bump the sure thing up to 1⁄4 the interval from worst value (15+ 1⁄4 (30 – 15)~ 18)
SUPPOSE THEY CHOOSE “B”
Receive profit= 18k for certain
Receive profit= 30k Receive profit= 15k
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ISE 562; Dr. Smith
They chose B (gamble) so next value between 18 and 30: (18 + 30)/2~ 24)
SUPPOSE THEY CHOOSE “B”
Receive profit= 24k for certain
Receive profit= 30k Receive profit= 15k
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ISE 562; Dr. Smith
• Chose gamble so next value between 24
and 30: (24 + 30)/2~ 27) Suppose they choose B
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Receive profit= 27k for certain
S 30 G 27 G 24
Receive profit = 30k G18 G15
Receive profit = 15k
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Receive profit = 28.5k for certain
Indiff 28.5 G 27
Receive profit= 30k Receive profit= 15k
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ISE 562; Dr. can now calculate the risk premium—the amount
the decision maker should be willing to pay to avoid the risk of the worst case if positive, or take the risk of getting the best case if negative; it is the difference between the expected value and the certainty equivalent (indifference point):
RP=EV – CE
For the example RP= 22.5 – 28.5 = -6
• The negative value indicates the DM is willing to pay to gamble; i.e., they are willing to pay $6,000 to take the risk of receiving a profit of
$30,000 (pay to play).
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ISE 562; Dr. Smith
• Chose gamble so next value between 27
and 30: (27 + 30)/2~ 28.5)
NOW SUPPOSE THEY ARE INDIFFERENT BETWEEN A AND B
THE CERTAINTY EQUIVALENT OF THE GAMBLE IS 28.5
THE EXPECTED VALUE OF THE GAMBLE IS .50(15)+.50(30) = 22.5
ISE 562; Dr. utility function for profit
RP=EV – CE <0
Risk seeking
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RP (negative)
Risk premium of -6 to try and get 30
28.5 30 CE
ISE 562; Dr. the decision maker had been indifferent at a CE of $17,000:
Then RP= 22.5 – 17 = +5.5
S 30 S 27 S 24
Indiff 17 G 15
• The positive value indicates the DM wants to avoid the risk of only receiving 15).
• In this case the decision maker would be willing to pay $5,500 to avoid the chance of receiving a profit of only $15,000.
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ISE 562; Dr. utility function for profit
RP=EV – CE >0
Risk averse
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RP (positive)
Risk premium of 5.5 to try and avoid 15
22.5 30 EV
ISE 562; Dr. Smith
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ISE 562; Dr. : Risk attitude
Suppose we have the following utility function:
U( x ) x2 20 x 100
Identify the risk attitude.
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ISE 562; Dr. the risk premium, RP=EV-CE where
if RP < 0 then risk seeking if RP > 0 then risk averse if RP = 0 then risk neutral
Need to compute CE and EV…
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ISE 562; Dr. 1: find the CE
CE: from expected utility EU=pU(bestcase) + (1-p)U(worstcase) = 1⁄2 U(100) + 1⁄2 U(20)
= 1⁄2 (10000) + 1⁄2 (400) = 5200
So CE=inverse of utility function at 5200 U(x)=5200=x2 so x= 72.1
So CE=72.1
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ISE 562; Dr. 2: find the EV
EV: from expected value
EV=p(bestcase) + (1-p)(worstcase)
= 1⁄2 (100) + 1⁄2 (20)
= (50) + (10) = 60
So EV = 60
And RP=EV-CE= 60 – 72.1 = – 12.1 Since RP<0, utility function is risk seeking
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ISE 562; Dr. P=EV-CE= 60 – 72.1 = – 12.1
Since RP<0, utility function is risk seeking
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20 EV=60 100
ISE 562; Dr. 2: using G and L
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=U(best case) – U(EV) =U(100) – U(60) =10000 – 3600
=U(EV) – U(worst case) =U(60) – U(20)
=3600 – 400
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ISE 562; Dr. 2: using G and L
If G-L > 0 risk seeking If G-L < 0 risk averse If G-L = 0 risk neutral So
G – L =6400 – 3200 = 3200 > 0 so risk seeking
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ISE 562; Dr. =6400 L=3200
Since G-L>0, utility function risk seeking 10,000 U(best)
400 U(worst)
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ISE 562; Dr. nother example: Risk attitude
Suppose we have the following utility function: x16
U(x) 20 16 x 36 general case :
U(x) xa axb ba
Identify the risk attitude.
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ISE 562; Dr. the risk premium, RP=EV-CE
if RP < 0 then risk seeking if RP > 0 then risk averse if RP = 0 then risk neutral
Need to compute CE and EV…
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ISE 562; Dr. E: from expected utility
EU=pU(bestcase) + (1-p)U(worstcase) = 1⁄2 U(36) + 1⁄2 U(16)
= 1⁄2 (1) + 1⁄2 (0) = 0.50
So CE=inverse of utility function at 0.50 U(x)=0.50= x 16 solving for x= 21
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ISE 562; Dr. V: from expected value
EV=p(bestcase) + (1-p)(worstcase) = 1⁄2 (36) + 1⁄2 (16)
= (18) + (8) = 26
So EV = 26
And RP=EV-CE= 26 – 21 = +5
Since RP>0, utility function is risk averse
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ISE 562; Dr. P=EV-CE= 26 – 21 = +5
Since RP>0, utility function is risk averse
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16 CE=21 EV=26 36
ISE 562; Dr. 2: using G and L
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=U(best case) – U(EV) =U(36) – U(26)
=1 – 0.707
=U(EV) – U(worst case) =U(26) – U(16)
=0.707 – 0
ISE 562; Dr. 2: using G and L
If G-L > 0 risk seeking If G-L < 0 risk averse If G-L = 0 risk neutral So
G – L =0.293 – 0.707 = – 0.414 < 0 so risk averse
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ISE 562; Dr. ’s work a real example:
• Suppose you are going to buy a car
with 10 possible models to consider.
• The range of mpg is 28 to 42 across
the 10 vehicles.
• We want the utility function for mpg.
• I need a volunteer!
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ISE 562; Dr. Smith
“Which would you prefer?”
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Receive mpg = 28 for certain
To overhead display
Receive mpg = 42 G
Receive mpg = 28k Utility Function
ISE 562; Dr. theory is based on a set of axioms
of coherence for rational decision making. Satisfaction of these axioms implies maximization of expected utility as the decision making criterion.
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ISE 562; Dr. Smith
1. Completeness Axiom. Existence of a
preference ordering. Given any 2 payoffs X1 and X2, the decision maker can decide whether X1 preferred to X2; X2 preferred to X1; or indifference between X1 and X2
In other words, the decision maker can make up their mind (not indecisive)
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ISE 562; Dr. Smith
2. Transitivity of preferences. If the
decision maker prefers X1 to X2 and X2 to X3, then they should prefer X1 to X3. (same for indifference)
In other words, the decision maker has a logical preference ordering (logically consistent)
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ISE 562; Dr. Smith
3. Continuity of preferences. If the decision
maker prefers X1 to X2 and X2 to X3, then a value of p can be found such that pX1 and (1-p)X3 is preferred to X2; and another value of p* can be found such that X2 is preferred to p*X1 and (1-p*)X3; and another value of p’ such that the DM is indifferent between X2 and p’X1 and (1-p’)X3
In other words, an indifference point can be found between the ranges of X1 and X2; X2 and X3; and X1 and X3. (X1 cannot be so good that the transitivity of X1 to X3 is violated (X2>pX1 and (1-p)X3).
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ISE 562; Dr. ontinuity (cont). We can always find a value of p, such that if X1 > X2 > X3, then:
That is, L(X1,X3,p) = X2; we can find a value of p that produces indifference between the lottery and X2.
0.99 $20 EV = 19.85
0.01 $20 EV = 5.15
0.99 $5 Take the $10
EV ? 10 =? 10
Where would you switch?
0.01 $5 Take the gamble
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Example: you can receive $20 (=X1), $10 (=X2), or $5 (=X3). So X1 > X2 > X3. Continuity states a value of p exists where you will be indifferent. Consider:
ISE 562; Dr. Smith
4. Independence. If the decision maker
prefers X1 to X2 and X3 is some other
payoff, then any pX1 and (1-p)X3 is
preferred to the same pX2 and (1-
pX1+(1-p)X3 > pX2+(1-p)X3
In other words, the decision maker’s preference between X1 and X2 is independent of other consequences like X3 using the same probability, p.
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ISE 562; Dr. Smith
4. Independence example 1
X1=receive $20; X2= receive $20 (X1 = X2); let X3= receive
$12. X1 is equal to X2 so lottery A is equivalent to the same lottery with X2 and X3; e.g.,
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Receive 20=X1
EV= 1⁄2 (20)+ 1⁄2 (12)= 16
PL=1/2 Receive 12=X3
Receive 20=X2
EV= 1⁄2 (20)+ 1⁄2 (12)= 16
Receive 12=X3
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ISE 562; Dr. Smith
4. Independence example 1
X1=receive $20; X2= receive $10 (X1 > X2); let X3= receive
$12. X1 is preferred to X2 and the lottery A is preferred to the same lottery with X2 and X3; e.g.,
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A preferred to B and X1 preferred-> X2 independent of X3
Receive 20=X1
EV= 1⁄2 (20)+ 1⁄2 (12)= 16
PL=1/2 Receive 12=X3
Receive 10=X2
EV= 1⁄2 (10)+ 1⁄2 (12)= 11
Receive 12=X3
ISE 562; Dr. Smith
4. Independence example 2
X1=car costs $16,000; X2= $20,000; X3=$22,000 X1 is preferred to X2 and the lottery A is preferred to
the same lottery with X2 and X3
Receive car with X1=16,000
EV= 1⁄2 (16000)+ 1⁄2 (22000)= 19000
Receive car with X3=22,000
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Receive car with X2=20,000
EV= 1⁄2 (20000)+ 1⁄2 (22000)= 21000
Receive car with X3=22,000
A preferred to B and X1 preferred-> X2 independent of X3
ISE 562; Dr. Smith
5. Substitutability. If the decision maker is
indifferent between X1 and X2, then they may be substituted for each other as payoffs in any decision making problem.
In other words, equally preferable payoffs can be substituted for each other in any decision making problem.
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ISE 562; Dr. Smith
6. Monotonicity. If the decision maker
prefers consequence X1 to X2, then pX1 and (1-p)X2 is preferred to qX1 and (1-q)X2 if and only if p>q.
In other words, preference for a lottery involving 2 consequences is determined by the probabilities and the consequences.
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ISE 562; Dr. Smith
• Do the axioms hold?
• Big question in fields of decision
theory, economics, and psychology
• Under what conditions are they
• What happens if they are violated?
• How can such problems be mitigated?
• We will address this later in the course during the “Decision biases” lectures.
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ISE 562; Dr. Smith
What if the axioms hold?
Decision maker preferences can be described by a utility function under the axioms of utility.
Subjective judgments about uncertain quantities can be described by a probability distribution satisfying the 3 axioms of probability
The rational decision maker should make decisions by maximizing expected utility.
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Utility and decision making implications Payoffs at terminating branches of decision tree may have more than one attribute
Can use multiattribute utility to represent decision maker preferences for different attributes.
• Additive multiattribute decision model
• Group decision rules based on utility These topics will be addressed soon
ISE 562; Dr. Smith
• Multiplicative multiattribute decision model Group decision making
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