Question

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 1/3 PayOff 1 Payoff 2 PayOff 3 4 0.30 PayOff 4 0.50 PayOff 5 Alternative A 0.20 PayOff 6 1/3 1/3 1/3 PayOff 7 PayOff 8 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

Payoff	1	2	3	4	5	6	7	8	9	10	11	12	13	14
$	5	-2	5	2	9	2	5	-5	3	4	4	-3	7	6

  
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

The value at node1 = (1/3 * payoff1) + (1/3 * payoff2) + (1/3 * payoff3)

= (1/3 * 5) + (1/3 * -2) * (1/3 * 5) = 8/3 = 2.7

The value at node2 = (1/3 * payoff7) + (1/3 * payoff8) + (1/3 * payoff9)

= (1/3 * 5) + (1/3 * -5) + (1/3 * 3) = 1

The value at node3 = (1/2 * payoff13) + (1/2 * payoff14)

= (1/2 * 7) + (1/2 * -6) = 13/2 = 6.5

The value at node4 = maximum value amongst node1 and payoff4

= max[2.7, 2]

= 2.7

value at node5 = maximum value amongst node2 and payoff10

= max[1, 4]

= 4

value at node6 = maximum value amongst node3 and payoff12

= max[6.5, -3]

= 6.5

value at node7 = (0.3 * node4) + (0.5 * payoff5) + (0.2 * payoff6)

= (0.3 * 2.7) + (0.5 * 5) + (0.2 * 2) = 1.1

value at node8 = (0.3 * node5) + (0.5 * payoff11) + (0.2 * node6)

= (0.3 * 4) + (0.5 * 4) + (0.2 * 6.5) = 4.5

value at node9 = maximum value amongst node7 and node8

= max[1.1, 4.5]

= 4.5

hence

a) The value at node 4 is 2.7

b) The value at node 8 is 4.5

c) The best course of action or decision is to select alternative B since it has highest value i,e 4.5