Problem #1:
On page 70, Plous discusses the responses subjects give to choices
between options A and B, and then options C and D. On page 71, he compares combined choices A & D with
combined choices B & C. So he
analyzes a total of six possible choices in all, four simple and two
combinations.
a) For each of these six compute its expected utility,
assuming that utility is determined by the dollar amount.
EUA = (1)($240) = $240
EUB = (.25)(-$1000) + (.75)(0) = $250
EUC = (1)(-$750) = -$750
EUD = (.75)(-$1000) + (.25)(0) = -$750
EUA & D = (.75)(-$760) + (.25)($240) = -$510
EUB & C = (.75)(-$750) + (.25)($250) = -$500
b) Now using the Prospect Theory graph (use the horizontal axis to represent the dollar amounts and the vertical axis to represent units of subjective value), compare the various options.
We need to find the PT-values for $240, $250, $1000, -$750, -$760, and -$1000 using the PT-graph. We’ll use intervals of $250, so that each square along the x-axis will represent $250 (i.e., on the gains side, we’ll have $250, $500, $750, and $1000 at each square). Counting the number of squares needed to reach the curve, we get the following values from the PT-graph:
Val ($240) = 1.5 unit (approx) Val (-$750) = -6 units
Val ($250) = 1.5 unit Val (-$760) = -6 units
Val ($1000) = 3.5 units Val (-$1000) = -6.5 units
Plugging these values into our expected value equations, we get the following expected values, where V stands for the subjective worth:
EVA = (1)(1.5) = 1.5
EVB = (.25)(3.5) + (.75)(0) = .875
EVC = (1)(-6) = -6
EVD = (.75)(-6.5) + (.25)(0) = -4.875
EVA & D = (.75)(-6) + (.25)(1.5) = -4.5 + .375 = -4.125
EVB & C = (.75)(-6) + (.25)(1.5) = -4.5 + .375 = -4.125
c) Does using the notion of value postulated by Prospect Theory explain the way people respond to these various options completely or do we still need to talk about attitudes toward probability and/or certain outcomes?
According to the original expected utility calculations, more people should have chosen B over A, and there was no difference between a choice of C and D. Using the PT values, A now has a higher expected value than B and D higher than C, explaining the results gathered in the study.
Problem #2:
On pages 74 and 75, Plous describes the jacket/calculator puzzle. Can one use the PT graph to explain why
people feel that the difference between $15 and $10 is greater than the
difference between $125 and $120?
If so, does this resolve all the puzzlement about the example? If no, how can we understand this bit of decision behavior
– or can we?
Using the PT graph, we can find the values for $15, $10, $120, and $125 by using intervals of $25 (you’ll need to add an extra square along the x-axis). Then, we’ll calculate the differences:
Val ($15) – Val ($10) = 1.2 – 1.0 = .2
Val ($125) – Val ($120) = 3.9 – 3.85 = .05
With the subjective values, we see why a greater difference is felt between $15 and $10 than between $125 and $120.
Problem #3: Early in the article assigned for this week, Tversky and Kahneman discuss an experiment concerning the outbreak of an unusual disease that is expected to kill 600 people and describes four options.
Imagine that the U.S. is preparing for the outbreak of an
unusual Asian disease, which is expected to kill 600 people. Two alternative programs to combat the
disease have been proposed. Assume
that the exact scientific estimate of the consequences of the programs are as
follows:
If
Program A is adopted, 200 people will be saved. [72 percent]
If
Program B is adopted, there is a 1/3 probability that 600 people will be
saved, and 2/3
probability that no people will be saved. [28 percent]
A second group of respondents was given the same cover
story as above with a different formulation of the alternative programs, as
follows:
If
Program C is adopted 400 people will die. [22 percent]
If
Program D is adopted there is a 1/3 probability that nobody will die, and
2/3 probability
that 600 people will die. [78 percent]
a) For each of these four compute its expected (dis)utility, assuming that utility is determined by the number of lives lost.
EUA = (1)(200) = 200 saved or -400 lost
EUB = (1/3)(600) + (2/3)(0) = 200 saved or -400 lost
EUC = (1)(-400) = -400 lost
EUD = (2/3)(-600) + (1/3)(0) = -400 lost
b) Now using the Prospect Theory graph above (using the horizontal axis to represent number of lives and the vertical axis to represent units of subjective value), compare the various options.
Using the graph with intervals of 200, we get the following values:
Val (200) = 1.5 Val (-400) = -4.5
Val (600) = 3.25 Val (-600) = -6
We can now calculate new expected values:
EVA = (1)(1.5) = 1.5
EVB = (1/3)(3.25) + (2/3)(0) = 1.08
EVC = (1)(-4.5) = -4.5
EVD = (2/3)(-6) + (1/3)(0) = -4
c) Does using the notion of value postulated by Prospect Theory explain the way people respond to these various options completely or do we still need to talk about attitudes toward probability and/or certain outcomes?
The expected values calculated using the subjective values from the PT graph indicate that people favor Program A and Program D, which corresponds with the results gathered in the study (72% chose A over B and 78% chose D over C).
Problem #4:
On page 99, Plous discusses four options and describes people’s
reaction to them.
a) Is Rational Choice Theory helpful in understanding their responses? Explain briefly.
We can calculate the following expected utilities:
EUA = (1/1000)($5000) = $5
EUB = (1)($5) = $5
EUC = (1/1000)(-$5000) = -$5
EUD = (1)(-$5) = -$5
According to Rational Choice Theory, there is no difference between A and B and between C and D.
b) Does the Prospect Theory account of value (and the above graph) explain their responses? Explain briefly.
These numbers are difficult to graph accurately but we know ahead of time that the Prospect Theory of value won't explain these decisions because they do NOT follow the pattern of preferring sure gains over probable gains and probable losses over sure losses. Quite the contrary!
c) Does the Prospect Theory of decision weights and the graph on Plous, p. 98 explain their responses? Explain briefly.
Using the PT decision weights, we know that people tend to overweigh small probabilities. Thus, we might substitute 1/10 or 1/100 for 1/1000 in our expected value equations, thus making the expected utility of A slightly larger than $5 while that of C comes out to be more negative than -$5. By overweighting the small probabilities, the calculated difference between A and B and between C and D now fits the observed behavior.
d) Is it also necessary to postulate a “certainty effect” in this case? Explain briefly.
In choosing between C and D, risk is eliminated completely for D. This “certainty effect” could thus explain why D is favored over C but it won't explain why A is favored over B.
Problem #5:
Compare the following two situations:
Situation 1:
Would you volunteer to receive a vaccine that would be 50% effective in
preventing a disease that is expected to afflict 20% of the population?
Situation 2:
There are two mutually exclusive and equally probable strains of the
disease, each of which is expected to affect 10% of the population. The vaccine offers complete protection
against one strain and none against the other.
In one particular study (which gave more details about
the situations), about 40% of the subjects said Yes to Situation 1, but only
57% of the respondents opted for the vaccine in Situation 2.
a) From the perspective of Rational Choice Theory, how do the two situations differ?
According to Rational Choice Theory, there is no difference between the two situations because your 20% risk of contracting a disease is reduced to 10% in each case.
b) Does the Prospect Theory of value help explain the different responses to these situations? Explain.
PT values will not help,
because there are no values to consider in these situations.
c) Does the Prospect Theory of weights help explain the different responses? Explain.
PT weights will also not help, because the probabilities are the same in each case.
d) This illustrates the phenomenon of “pseudocertainty”. Briefly explain.
In Situation 2, it appears that the risk of contracting a disease is eliminated although in fact you will still have a 10% chance of contracting a disease with the vaccine. See Plous, pp. 100-101 for further discussion of this problem.