Question

Problem 3: (Application of the prospect theory probability weighting function): A famous implication of prospect theory is a preference for positively skewed prospects and an aversion to negatively skewed prospects. For example, Tversky and Kahneman (1992) found that people frequently preferred:

1. 1) Gaining $95 with certainty over a 95% chance of gaining $100 and a 5% chance of gaining $0.

2. 2) Losing $5 with certainty over a 5% chance of losing $100 and a 95% chance of losing $0.

3. 3) A 5% chance of gaining $100 and a 95% chance of gaining $0 over gaining $5 with certainty

4. 4) A 95% chance of losing $100 and a 5% chance of losing $0 over losing $95 with certainty.

These prospects can be represented as (?, ?; 0,1 − ?) where ? = 100 or ? = −100 and either ? = 0.05 or ? = 0.95. Using the probability weight w(?) = (1 − ?)? + ??, with 0 ≤ ? ≤ 1, and 0 ≤ ? ≤ 1 for the best outcome of a lottery, and the weight ?(?) = (1 − ?)(1 − ?) + ?? for the worst outcome of a lottery, and setting ? = 0.80, ? = 0.10, show that prospect theory predicts all four choices simultaneously. Use the weight ?(?) = 1 for a lottery that offers a single outcome with probability ? = 1. Assume a simple linear value function of the form ?(?) = ?.

Answer 1

Now, we have

$V(x,p;0,1-p) = w_x(p)v(x)+ w_0(1-p)v(0).\\ Since, v(x) = x \Rightarrow v(0)=0.\\ Thus, V(x;p) = w(x)x = [(1-\theta)\alpha+\theta p]x =(0.2(0.1)+0.8p)x.\\ V(x;p) = x(0.02+0.8p).\\$

1) V(95) = 95w(p) = 95(1) = 95; V(100,0.95; 0,0.05) = 100(0.02+0.8(0.95)) = 78.

Thus, 95 > 78 or Gaining $95 with certainty is preferred over a 95% chance of gaining $100 and a 5% chance of gaining $0.

2) V(-5) = -5; V(-100,0.05;0,0.95) = -100(0.02+0.8(0.05)) = -6.

Thus, -5 > -6 or Losing $5 with certainty is preferred over a 5% chance of losing $100 and a 95% chance of losing $0.

3) V(5) = 5; V(100,0.05;0,0.95) = 100(0.02+0.8(0.05)) = 6.

Thus 6 > 5 or A 5% chance of gaining $100 and a 95% chance of gaining $0 is preferred over gaining $5 with certainty.

4) V(-95) = -95; V(-100,0.95;0,0.05) = -100(0.02+0.8(0.95)) = -78.

Thus -78 > -95 or A 95% chance of losing $100 and a 5% chance of losing $0 is preferred over losing $95 with certainty.