Decision-making under conditions of uncertainty, Mathematics

Decision-Making Under Conditions of Uncertainty

With decision making under uncertainty, the decision maker is aware of different possible states of nature, but has insufficient information to assign any probabilities of occurrence to them. Variety of criteria have been proposed, each based upon certain attitude (whether optimistic or pessimistic) of the decision maker.

If the decision maker is optimistic, he believes that given the selection of any strategy, "nature" will act in a way which provides the greatest  reward. Thus, the decision maker identifies the maximum profit associated with the selection of each alternative and the maximum of these is chosen as the decision. This criteria is known as Maximax Criterion.

Strategy

SRs

SRs.

SRs.

Maximum of each alternative


A

20,000

2,000

1,000

20,000

B

30,000

27,000

- 16,000

30,000

C

40,000

15,000

- 40,000

40,000

Continuing with the same example, we can calculate the maximum of each alternative as shown in the above table and the maximum of these will give us the alternative to be chosen. Thus, in the above example, the maximum benefit is derived from the alternative C. The analyst believes that the nature will behave as per the state S1.

If the decision maker is pessimistic he believes that nature works against him and is convinced that whatever strategy is selected, nature will respond in such a way as to have only least possible profit. Thus, the decision maker first identifies the lowest profit associated with each decision alternative and he chooses that alternative which is the maximum of the above minimum profits. Thus, this criteria is called Maximin Criterion.

Strategy

S1

S2

S3

Minimum of each alternative


  A

20,000

2,000

1,000

 1,000

B

30,000

27,000

-16,000

-16,000

C

40,000

15,000

-40,000

-40,000

Thus, in the above example, the decision maker chooses to invest in portfolio A. Note this approach guarantees the minimum profit.

But, if the decision maker lies in between these extremes (neither optimistic nor completely pessimistic), but specifies his level of optimism by an 'index of optimism, a which is assigned a value between 0 and 1, both 0 and 1, inclusive, decision is made based upon weighted profits.

Weighted profits are calculated as

a (maximum profit for alternative) + (1 - a) (minimum profit for alternative).

The alternative which gives maximum of the weighted profits is the decision chosen by the decision maker. This approach is known as Hurwicz Criterion. Assume that the decision maker specifies the value of a as 0.6. He has a tendency towards optimism. The weighted profits are calculated as follows:

Strategy

Weighted Gains

Rs.


A

0.6 (20,000) + 0.4 (1,000) =

12,400

B

0.6 (30,000) + 0.4 (-16,000) =

11,600

C

0.6 (40,000) + 0.4 (-40,000) =

8,000

As per this, the best alternative is to invest in portfolio A.

Another approach known as Regret Criterion is adopted. This approach is based upon the regret one might have from making a particular decision. The regret is referred to as the opportunity loss or opportunity cost - a measure of magnitude of loss incurred by not selecting the best alternative.

The regret is measured by the difference between the maximum profit that we would have realized in case of known state of nature and the profit we realize. This criterion suggests that the decision maker should strive to minimize the largest regret. To use this criteria first we have to prepare the regret table as shown below.

Strategy

States: Gains in Investment Strategies

S1

S2

S3

A

B

C

40,000 - 20,000

40,000 - 30,000

40,000 - 40,000

27,000 - 2,000

27,000 - 27,000

27,000 - 15,000

1,000 - 1,000

1,000 - (-16,000)

1,000 - (-40,000)

 

Strategy/State

S1

S2

S3

Maximum Regret

A

20,000

25,000

0

25,000

B

 

10,000

0

17,000

17,000

C

0

12,000

41,000

41,000

To determine the regrets for the first column in the first table we assume state S1 occurs and determine the regret by choosing each of the alternatives. The magnitude of the regrets is the difference between the maximum profit, given the state, and the profit we realize choosing the alternative.

For column 1, the maximum profit equals Rs.40,000. Thus, the regrets are found by subtracting the profits realized for each alternative from Rs.40,000. From the above table we can say that if state S1 occurs and alternative C is chosen we would have no regret. Similarly, we have to find all other values and find the maximum for each alternative, i.e. maximum regret for each alternative as shown in the table. The alternative which gives us the minimum of these maximum regrets will be the decision. Thus, as far as this criteria is concerned it is best to invest in alternative B. It is also designed for the pessimist and assumes that the nature will respond, so as to maximize the level of regret for any alternative chosen.

Posted Date: 9/15/2012 2:12:39 AM | Location : United States







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