Question

Asset A pays $1,500 with certainty, while asset B pays $2,000 with probability 0.8 or $100 with probability 0.2. If offered the choice between assets A and B, a particular decision maker would choose asset A. Asset C pays $1,500 with probability 0.25 or $100 with probability 0.75, while asset D pays $2,000 with probability 0.2 or $100 with probability 0.8. If offered the choice between assets C and D, the decision maker would choose asset D. Show that the decision maker's preferences violate the independence axiom.

PLEASE HELP SHOW WORK PLEASE!!

Answer 1

First, let us compute the expected choice for each asset:

if   then

where, Z = $100 and p is the probability

For A:

E(A) = 1500 (1) + 100 (0)

= $1500

E(B) = 2000 (0.8) + 100 (0.2)

= 1600 + 20

= $1620

Hence, for B> A, E(B) > E(A), but he prefers A over B

E(C) = 1500 (0.25) + 100 (0.75)

= 375 + 75

= $450

E(D) = 2000 (0.2) + 100 (0.8)

= 400 + 80

= $480

Hence, for D>C, E(D) > E(C), and he prefers D over C.

Hence, if the decision maker is ready to take a greater risk, and prefers D over C then he must also take the greater risk in case of A and B, i.e. choose B over A.

If he goes according to the independence axiom, his preference should be:

D>C and B>A .......(Greater risk taker) or prefers $ 480 over $ 450 and $1620 over $1500

C>D and A>B ......(More of a safe player) i.e prefers $ 450 over $ 480 and $1500 over $1620

But here, the decision maker has chosen A>B and D>C, i.e he prefers A>B (a safe player here), while acts as a risk taker when he prefers D>C.Hence, it is a violation of the independence axiom.