Question

The  following  payoff  table  shows  the  profit  for  a  decision  problem  with  2  states  and  3  decision

alternatives. Determine the range of the probabilities for state s1 for which it would be optimal to choose

decision d1.

States	d1	d2	d3
s1	8	4	10
s2	10	12	-2

Answer 1

Let the probability of 's1' = p and that for 's2' is (1 - p).

For the expected value of 'd1' to become more than that of 'd2',

8p + 10(1 - p) >= 4p + 12(1 - p)
or, -2p + 10 >= -8p + 12
or, 6p >= 2
or, p >= 1/3

Also, for the expected value of 'd1' to become more than that of 'd3',

8p + 10(1 - p) >= 10p + (-2)(1 - p)
or, -2p + 10 >= 12p - 2
or, 14p <= 12
or, p <= 12/14 or 6/7

So, the required range of 'p' is [1/3, 6/7]