Question

QUESTION 2 Given is a Decision Tree Diagram. The Payoffs 1-14 are given in the table below. Answer questions a, b, and c. 1/3 1/3 Payoff 1 PayOff 2 PayOff 3 1/3 4 0.30 PayOff 4 0.50 PayOff 5 Alternative A 0.20 PayOff 6 9 1/3 1/3 PayOff 7 PayOff 8 2 1/3 PayOff 9 Alternative B 5 0.30 Payoff 10 0.50 (8 Payoff 11 0.20 PayOff 12 1/2 6 PayOff 13 1/2 PayOff 14 1 3 5 7 8 9 10 11 12 13 14 Payoff $ 2 7 4 4 6 4 3 6 2 -7 4 3 7 -2 4 7 5 a) The value at node 4 is b) The value at node 8 is (in 1 decimal place) c) The best course of action or decision is to select alternative

Answer 1

a. The value at node 4 = Max (Value at node 1, Pay off 4)

Value at node 1 = (1/3)*Pay off 1+(1/3)*Pay off 2+(1/3)*Pay off 3 = (1/3)*(Pay off 1+Pay off 2+Pay off 3) = (1/3)*(3+7+6) = 5.33

Pay off 4 = 4

Value at node 4 = Max (5.33,4) = 5.33

As node 4 is a square, it is a decision node. Hence a decision needs to be taken at this mode. This is pay off decision, hence the decision will be in favor of the maximum payoff.

Node 1 is a chance node. So, the expected payoff is calculated considering the probability of each payoff and adding them.

b. Value at node 8 = 0.3*Value at node 5 + 0.5*Pay off 11 + 0.2*Value at node 6

Value at node 5 = Max (Value at node 2, Pay off 10)

Value at node 2 = (1/3)*(Pay off 7+Pay off 8+Pay off 9) = (1/3)*(-7+4+3) = 0

Pay off 10 = 7

Value at node 5 = Max (0,7) = 7

Pay off 11 = (-2)

Value at node 6 = Max (Pay off 12, Value at node 3)

Value at node 3 = (1/2)*(pay off 13 + pay off 14) = (1/2)*(7+5) = 6

Pay off 12 = 4

Value at node 6 = Max (4,6) = 6

Value at node 8 = 0.3*Value at node 5 + 0.5*Pay off 11 + 0.2*Value at node 6 = 0.3*7 + 0.5*(-2) + 0.2*6 = 2.3

c.To find the best course of action, the values of each alternative has to be found.

Alternative A = value at node 7

Alternative B = value at node 8

Value at node 8 is already found which is 2.3

Value at node 7 = 0.3*Value at node 4 + 0.5*Pay off 5 + 0.2*Pay off 6

Value at node 4 is already calculated in part a, which is 5.33

Pay off 5 = 2

Pay off 6 = 4

Value at node 7 = 0.3*5.33+0.5*2+0.2*4 = 3.39

Alternative A = 3.39

Alternative B = 2.3

The best course of action is to select alternative A, as it has higher payoff as compared to alternative B.