Advanced Microeconomics II - Problem set 1
Due date: classes on April 1
The following state space is given:
S = {(sun, wind), (rain, wind), (sun, no wind), (rain, no wind)}
It is assumed that these states exhaust all conceivable possibilities and that they do not overlap
(they are disjoint).
The set of possible consequences C contains the following elements:
w1 = windsurfing in the sun
w2 = windsurfing in the rain
w3 = sitting at the coast and waiting for the wind
b1 = sitting at the beach in the sun and no wind
b2 = sitting at the beach in the sun and wind
b3 = go home because of the rain
Suppose you are the decision maker, you went for vacations and first day in the morning before
you know the weather for today you have to decide between two possible courses of actions:
• go to the windsurfing spot (windsurf in short)
• go to the beach (beach in short)
These courses of actions can be thought of as acts, i.e. the functions from states into consequences:
States of nature
windsurf
beach
(sun, wind)
w1
b2
(rain, wind)
w2
b3
(sun, no wind)
w3
b1
(rain, no wind)
w3
b3
You immediately realize that you have the following ordinal preferences:
w1 ∼ b1 w2 b2 w3 ∼ b3
Additionally, you satisfy all the axioms of Schmeidler (1989) Choquet Expected Utility (i.e. the
Anscombe-Aumann (1963) axioms where Independence is replaced with a weaker Comonotonic
Independence). Suppose that there are two sources of uncertainty:
• subjective: the choice of states of nature above
• objective: some probability experiments - see below
Suppose that you are asked a series of questions, what do you like more:
• a lottery (with objective probability p) involving the best and the worst windsurfing consequence (w1 , p; w3 , 1 − p);
• or a sure middle windsurfing consequence w2 .
After a series of such questions, the probability value p∗ = 0.6 was found s.t.:
(w1 , p∗ ; w3 , 1 − p∗ ) ∼ w2
Similarly, you were asked a series of questions involving beach consequences, and finally the
probability value p∗∗ = 0.9 was found s.t.:
(b1 , p∗∗ ; b3 , 1 − p∗∗ ) ∼ b2
Problem 1. Can you infer from the above what are the utility function values for all the consequences b1 , b2 , b3 , w1 , w2 , w3 . If you can, are they unique?
You were also reading a lot about the frequency of rain and sun in this area at this time of
the year. You obtained the following information:
• The probability of ”sun”1 is ≥ 0.3
• The probability of ”rain” is not smaller than 0.4
• The probability of ”wind” is not smaller than 0.5
• The probability of ”no wind” is not smaller than 0.2
• The probability that state (sun,wind) does not occur is not smaller than 0.6
• The probability that the state (rain, no wind) does not occur is not smaller than 0.9
Problem 2. Find a capacity for this problem. Is it a convex capacity?
Problem 3. After you have found the capacity, find the core of the game defined by this capacity.
Calculate a Choquet Expected Utility for the lottery ”windsurf”, calculate the minimum over
probabilities in the core of the Subjective Expected Utility and compare them - you can use Excel
solver to find a minimum over probabilities in the core.
Problem 4. Repeat questions in Problem 2-Problem 4 for the lottery ”beach”.
Problem 5. After you did all that, can tell me which acts will be preferred by the decision maker:
”windsurf” or ”beach”? If I offered you now the following lottery with objective probabilities
(w1 , 0.28; w3 , 0.72) how would it be ranked compared to the acts ”windsurf” and ”beach”?
1
To be precise ”sun” is the sum of the first and third state in the table. Similarly for other terms.
2