[u-indep-1]
Class 18 - MEASURING A UNIVARIATE UTILITY FUNCTION
Review
TreePlan is a software to compute the returns and graph them on a decision tree. A
trial version and user’s manual can be downloaded from the following website;
http://www.treeplan.com/academic.htm
A demonstration of the software is attached for your reference.
Figure 1-A strictly convex utility function
Certainty equivalent
A certainty equivalent of a lottery is an amount y^ such that the DM is indifferent
between the lottery and the amount y^ for certain. Therefore, y^ is defined by
v(y^) = E[v( ỹ )], or y^ = v-1{E[v( ỹ )]},
where ỹ is the uncertain outcome of a lottery. Notice that the certainty equivalent
is not the same as the expected return P y1+(1&P) y2 except for a risk-neutral DM.
By way of notation, it is common to write the certainty equivalent y^ in terms of the
loss and win in a lottery, y1 and y2 respectively:
P y1 ¾ (1&P)(y2) - y^ .
The certainty equivalent of the convex utility function is overlaid in Figure 1, so
is the 0–1 normalization for v common among utility functions.
For a convex function (or a risk-prone DM), y^ > P y1 + (1&P)(y2). The reverse is
true for a risk averse DM.
Quartile method to determine univariate utility function
Suppose we set v(y1) = 0 and v(y2) = 1, where y1 = ymin = 0 and y2 = ymax in Figure
1. Keeping the same probability of win and loss, the point y^ is now placed
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between the loss and win amounts of the lottery, and another certainty equivalent
defined as:
P(y1) ¾ (1&P)(y^) - y^'.
Then we find yet another certainty equivalent by examining the interval between ^y
and y2, resulting in y^".
P(y^)) ¾ (1&P)(y2) - y^".
The result is v(y^') = (1&P)2 and v(y^") = (1&P)(1+P), as shown below:
E(v(y^)) = P1 v(y1) + P2 v(y2) = P(v(0)) + (1&P) v(ymax) = (P)(0) + (1&P)(1) = 1&P
E(v(y^') = P1 v(y1) + P2 v(y^) = P(v(0)) + (1&P) v(y^) = (P)(0) + (1&P)(1&P) = (1&P)2
E(v(y^")) = P1 v(y^) + P2 v(y2) = (P)(1&P) + (1&P)(1) = (1&P)(1+P)
Subsequent points are obtained by substituting y^'and y^" in the binary lottery for y1,
y^. and y2 in turn. The process can be repeated as often as desirable or practical to
sketch out the full function v(y).
We illustrate this process in Figure 2. It can be seen that the process is particularly
simple for P = 0.5—a probability most people can associate with the common
experience of coin flipping. The certainty equivalents so obtained also divide the
utility range into halves, quarters, and so on. For this reason, this method is often
called the fractile method.
Figure 2 - The fractile method (Steps 1, 2, and 3)
Example
Figure 3 - A utiltiy surface for two attributes
The three attributes—cost, time-to-completion, and effectiveness—have the
following ranges:
cost
20-40 (less is preferred)
time
10-30 (less is preferred)
effectiveness
40-60 (more is preferred).
The following information is used to specify the conditional single-attribute
utility-functions using the quartile method. For example, 0.5(20) r 0.5(40) means
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a lottery in which there is a 50-50 chance of obtaining a score of 20 or 40 (Figure
3.20, P.235).
Lotteries over C, given T = 10, E = 60 and T = 30, E = 40,
0.5(20) r 0.5(40) ~ 30
0.5(30) r 0.5(40) ~ 35
0.5(30) r 0.5(20) ~ 25.
Lotteries over T, given C = 40, E = 40 and C = 20, E = 60,
0.5(10) r 0.5(30) ~ 20
0.5(20) r 0.5(30) ~ 25
0.5(20) r 0.5(10) ~ 15
Lotteries over E, given C = 40, T = 30 and C = 20, T = 10,
0.5(40) r 0.5(60) ~ 50
0.5(50) r 0.5(40) ~ 45
0.5(50) r 0.5(60) ~ 55.
Figure 4 - Univariate utility functions for cost, time and effectiveness
EXERCISE:
Please sketch the conditional univariate utility function for effectiveness.
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[convex-fractile-indep-1]
A strictlyy convex utilityy function
Fractile method (Step 1)
Fractile method (Step 2)
ŷ´
Fractile method (Step 3)
ŷ´
A ut
utility
ty surface
su ace for
o ttwo
o att
attributes
butes
v (y1max,y2max)
(y1max,y
y2max)
y2
(y1min,y2min)
V (y1min,y2min)
y1
Conditional univariate utility functions
Univariate utility functions for cost
Univariate utility functions for time
C:\crs notes\graph-paper
Univariate utility functions for effectiveness