Written examination in Computer and Systems Sciences
Department of Computer and Systems Sciences David Sundgren
Decision Support Methods BSM Friday 12 February to Monday 15 February 2016
The solutions must be complete and easy to follow. Formulations that are verbatim quotations from sources will not contribute to the points given to a solution, whether they are referenced or not, that is, wherevever you find a solution, formulate it in your own words and make sure that you understand what you write. For grade A a minimum of 41 points out of 45 possible are required, grade B requires at least 34 points, grade C requires at least 29 points, for grade D at least 25 points are required, for E at least 20 and for Fx at least 18 points are required.
1. Consider the event tree in Figure 1 below. What is the probability of the outcomes (a) F or G (b) C or D (c) E (d) H (e) E and H (f) E or H
0.1
C 0.5
0.4
0.7
A
0. 3
B
0.8 0.2
D E F G
0.5 0.5
H I
Figure 1: Event tree for Problem 1 (6 p.) 2. As eager stamp collector you become enthusiastic when you see a red three o¨re stamp bearing the image of king Oscar III. Since those stamps ought to be yellow and the red ones are misprints the red stamps are particularly valuable. But for the very reason that they
1
p
0.8 0.6 0.4 0.2 0 0
2
4
6
8
10
Outcome
Figure 2: Cumulative risk profiles of lotteries for Problem 7. are valuable the red stamps have been popopular to forge, in collector circles it is estimated that 25% of all red three ¨ore stamps are fake. On of the signs of forgery is that there is gum in the perforations since gum has been placed on the stamp after perforation. But sometimes the stamps are genuine anyway, in 10% of all real stamps you can see gum in the perforation. On the other hand 80% of all fake stamps have gummed perforation. The red three ¨ ore stamp you look at has gum in the perforation. What is the probability that it is genuine? (4 p.) 3. Give an example not present in the course material of when preferential independence is violated. (5 p.) 4. Working as decision analyst for a decision maker you try to determine her value function for the interval from e 5,000 to e 50,000 through the difference standard technique. By norm transition you have found that the unnormalised value function is such that v(e 10,000) = 2 and that v(e 25,000) = 3. Further, the decision maker gives the indifference statement (e 10,000 → e 25,000) ∼ (e 25,000 → e 50,000) The question is, what is the normalised value v(e 25,000)? 5. Consider this pair of lotteries: 1 1 3 1 1 A : apple, ; orange, , B : pineapple, ; grape, ; orange, 4 4 4 4 2 and C:
1 2 1 apple, ; grape, ; orange, 6 3 6
,D :
1 5 pineapple, ; grape, 6 6
(5 p.)
Show that to be consistent with the independence axiom of utility theory if A ≻ B then C ≻ D. (5 p.) 6. Choose a behavior that is not consistent with utility theory that you find more or less reasonable and discuss why it might and might not be justifiable. (5 p.) 7. Consider the two cumulative risk profiles in Figure 2. Can anything be said about the risk attitude of a person that prefers the lottery with the dashed cumulative risk profile compared to that of a person with the opposite preference? (5 p.) 2
8. Let A = (2, 0.1; 3, 0.5; 8, 0.4), B = (1, 1) and C = (4, 0.9; 6, 0.1) be three lotteries. Determine any stochastic dominance relations between them, does any of them dominate the others or is any any of them dominated by the others? (5 p.) 9. The Allais paradox can be described as follows: experiments have shown that most people prefer A over B and B ′ over A′ if the lotteries A, B, A′ , B ′ are defined as A = (2 000, 1), B = (10 000, 0.1; 2 000, 0.89; 0, 0.01) A′ = (2 000, 0.11; 0, 0.89), B ′ = (10 000, 0.1; 0, 0.9) We have seen in presentation 20 that to prefer A over B and B ′ over A′ violates the independence axiom for subjective expected utility theory, the so called “sure thing principle”. Determine if such preferences also violate the independence axiom of utility theory, namely that If a b holds for two lotteries, then it also has to hold for all lotteries c and all probabilities p, that p · a + (1 − p) · c p · b + (1 − b) · c. (5 p.)
3
