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Q3. For simplicity assume only two states of the world S = {S1, S2}. Suppose a consumer evaluates the AA acts as an average of the best possible payoffs across t subjective states and ignores the actual distribution of probabilities. Formally, the value of an act f is V(f) = 0.5 (max{z: f(s₁)(z) > 0} + max{z: f(s₂)(z) >0}), where f(s)(z) is a probability of prize z in states and max{z: f(s)(z) > 0} is the greatest prize possible (with some positive probability) in state s. Consider the following example of how such consumer evaluates acts f and g below: 0.2 100 S₁ f 0.8 10 S2 1.0 90 0.5 (0 S₁ g S₂ 10.5 0.3 30 20 0.7 50 The highest payoff of act f in s₁ is 100 and the highest payoff of act f in s₂ is 90, so the value of act f will be V(f) = 0.5(100 +90) = 95. At the same time, the highest value of act g in s₁ is 30 and the highest value of act g in s₂2 is 50, so the value of act g is V(g) = 0.5(30+50) = 40. Note that this is only an example, and in what follows you have to consider all possible acts. (a) Does this preference satisfy monotonicity? (b) Does this preference satisfy the independence axiom? If your answer is yes, you have to prove the statement. If your answer is no, you have to give a counter-example.

Q3. For simplicity assume only two states of the world S = {S1, S2}. Suppose a consumer evaluates the AA acts as an average of the best possible payoffs across t subjective states and ignores the actual distribution of probabilities. Formally, the value of an act f is V(f) = 0.5 (max{z: f(s₁)(z) > 0} + max{z: f(s₂)(z) >0}), where f(s)(z) is a probability of prize z in states and max{z: f(s)(z) > 0} is the greatest prize possible (with some positive probability) in state s. Consider the following example of how such consumer evaluates acts f and g below: 0.2 100 S₁ f 0.8 10 S2 1.0 90 0.5 (0 S₁ g S₂ 10.5 0.3 30 20 0.7 50 The highest payoff of act f in s₁ is 100 and the highest payoff of act f in s₂ is 90, so the value of act f will be V(f) = 0.5(100 +90) = 95. At the same time, the highest value of act g in s₁ is 30 and the highest value of act g in s₂2 is 50, so the value of act g is V(g) = 0.5(30+50) = 40. Note that this is only an example, and in what follows you have to consider all possible acts. (a) Does this preference satisfy monotonicity? (b) Does this preference satisfy the independence axiom? If your answer is yes, you have to prove the statement. If your answer is no, you have to give a counter-example.

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
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Q3. For simplicity assume only two states of the world S = subjective states and ignores the actual distribution of probabilities. Formally, the value of an act f is {S1, s2}. Suppose a consumer evaluates the AA acts as an average of the best possible payoffs across th V(f) = 0.5 (max{z: f(s1)(z) > 0} + max{z : f(s2)(z) > 0}), where f(s)(z) is a probability of prize z in state s and max{z : f(s)(z) > 0}is the greatest prize possible (with some positive probability) in state s. Consider the following example of how such consumer evaluates acts f and g below: 0.2 0.8 1.0 0.5 0.5 0.3 0.7 00 01 90 30 20 50 The highest payoff of act f in s1 is 100 and the highest payoff of act f in s2 is 90, so the value of act f will be V(f) = 0.5(100 + 90) = 95. At the same time, the highest value of act g in s1 is 30 and the highest value of act g in s2 is 50, so the value of act g is V(g) = 0.5(30 + 50) = 40. Note that this is only an example, and in what follows you have to consider all possible acts. (a) Does this preference satisfy monotonicity? (b) Does this preference satisfy the independence axiom? If your answer is yes, you have to prove the statement. If your answer is no, you have to give a counter-example.
Transcribed Image Text:Q3. For simplicity assume only two states of the world S = subjective states and ignores the actual distribution of probabilities. Formally, the value of an act f is {S1, s2}. Suppose a consumer evaluates the AA acts as an average of the best possible payoffs across th V(f) = 0.5 (max{z: f(s1)(z) > 0} + max{z : f(s2)(z) > 0}), where f(s)(z) is a probability of prize z in state s and max{z : f(s)(z) > 0}is the greatest prize possible (with some positive probability) in state s. Consider the following example of how such consumer evaluates acts f and g below: 0.2 0.8 1.0 0.5 0.5 0.3 0.7 00 01 90 30 20 50 The highest payoff of act f in s1 is 100 and the highest payoff of act f in s2 is 90, so the value of act f will be V(f) = 0.5(100 + 90) = 95. At the same time, the highest value of act g in s1 is 30 and the highest value of act g in s2 is 50, so the value of act g is V(g) = 0.5(30 + 50) = 40. Note that this is only an example, and in what follows you have to consider all possible acts. (a) Does this preference satisfy monotonicity? (b) Does this preference satisfy the independence axiom? If your answer is yes, you have to prove the statement. If your answer is no, you have to give a counter-example.
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