DECISION ANALYSIS WITH SAMPLE INFORMATION

Transcription

1 DECISION ANALYSIS WITH SAMPLE INFORMATION In the previous section, we saw how probability information about the states of nature affects the expected value calculations and therefore the decision recommendation. To mae the best possible decision, the decision maer sometimes cannot simply rely on preliminary or prior probabilities. He may need to see additional information about the states of nature. This new information may be used to revise or update the prior probabilities so that more accurate probability estimates are obtained in order to mae the final decision. Additional information is usually obtained by performing designed experiments. Examples of methods that are used to update the state-of-nature probabilities are raw material sampling, product testing and maret research. Revised probabilities are nown as posterior probabilities (as opposed to prior). Information obtained through research or experimentation is referred to as an indicator. The experiment consists of taing a statistical sample so that the new information is nown as sample information. Let I 1 favourable maret research report and I unfavourable maret research report We denote the probability of a state of nature s, given an indicator I, as P ( s I ). This is the conditional probability that s will occur given that the outcome of the maret research study is indicator I. Since test mareting and expert opinions are never 100% accurate, we cannot say, for example, that, if the maret is favourable, then the revised estimate of the probability of buying an expensive system is 1. Test mareting is prone to sampling error and experts are not always right. This inability of indicators to predict the correct states of nature can be quantified by using the conditional probabilities P ( I s ). For example, given that the state of nature turns out to be favourable, what is the probability that the maret research will result in a favourable report? To carry out the analysis, we need conditional probabilities for all indicators, given all the states of nature, that is, P I 1 s ), P I s ), P I 1 s ) and P I s ). ( ( 1 ( 1 Let us assume that the following estimates are available for the conditional probabilities: Maret research State of nature Favourable ( I 1 ) Unfavourable ( I ) Favourable maret ( s 1 ) P ( I1 s1) P ( I s1 ) Unfavourable maret ( s ) P ( I1 s ) 0. 5 P ( I s ) ( Note that these estimates provide a reasonable degree of confidence in the maret research study. The reason for a high figure of 0.5 is that, when potential customers hear about using an expensive and sophisticated system, their enthusiasm may lead them to overstate their real interest in it. However, when they hear about the cost of using such a system, their responses may change very quicly to a No, than you!.

2 Once the indicator reliabilities are determined, the posterior probabilities can be found by using Bayes Theorem: NOTE A B) A B) B) There is a direct relationship between P I s ) and P I s ). From definition, ( ( I s I s ) ) and s ) s I s I ) ). I ) The numerators of the right-hand-side expressions being the same, we can write P I s ) s ) s I ) I ). ( We can now write the prior and posterior probabilities concerning I 1 for Carl and Betty s problem in the following table: State of nature ( s ) Prior P ( s ) Conditional P ( I 1 s ) Favourable maret ( s 1 ) Unfavourable maret ( s ) From these, we can find the probabilities based on indicator I 1: P ( s I1) I1 s1 ) s1) (0.90)(0.4) P s I ) I s ) s ) (0.5)(0.6) 1 ( and also the probability of indicator I 1: P I ) I s ) + s I ) ( Now, we can also find the posterior probabilities P I 1 s ) and P I 1 s ) : ( 1 s1 I1) 0.36 P ( s1 I1) and I ) s I1) 0.15 P ( s I1) I ) (

3 Example All these probabilities can now be summarise in the following table: States of nature Prior Conditional Joint Posterior s P s ) P I s ) P I s ) s I ) ( ( 1 ( 1 1 s s The same can be done for I : P ( I 1 ) 0.51 States of nature Prior Conditional Joint Posterior s P s ) P I s ) P I s ) s I ) ( ( ( s s P ( I ) 0.49 Developing a small driving range for golfers of all abilities has long been a desire of John Jenins. John, however, believes that the chance of a successful driving range is only about 40%. A friend of John has suggested that he conduct a survey in the community to get a better feeling of demand for such a facility. There is a probability of 0.9 that the research will be favourable if the driving range facility will be successful. Furthermore, it is estimated that there is a probability of 0.8 that the maret will be unfavourable if indeed the facility will be unsuccessful. John would lie to determine the chances of a successful driving range given a favourable result, from the mareting survey. Solution Maret research State of nature Favourable ( I 1 ) Unfavourable ( I ) Driving range successful ( s 1 ) P ( I1 s1) 0. 9 P ( I s1 ) 0. 1 Driving range unsuccessful ( s ) P ( I1 s ) 0. P ( I s ) 0. 8 States of nature Prior Conditional Joint Posterior s P s ) P I s ) P I s ) s I ) ( ( 1 ( 1 1 s s P ( I 1 ) 0.48

4 DEVELOPING A DECISION STRATEGY A decision strategy is a rule to be followed by the decision maer. This rule recommends a particular decision based on whether the maret research report is favourable or unfavourable. We may now use a decision tree to find the optimal decision strategy for Carl and Betty s problem. From left to right, the tree shows the natural or logical order of the decision-maing process. First, they must decide whether or not to have a maret research study done and, after having obtained the maret research indicators, they must decide on which alternative to select and, finally, the state of nature will occur. The payoffs on the decision tree when no maret research is done are the same as those that we had originally in Table.1.. However, on the branches stemming from the decision to do the maret research, the cost of doing the research must be taen into account (here, we are going to assume that its cost is R 1 000). This means that all the payoffs related to maret research will be decreased by a thousand rands. Nodes 1, 3, 4 and 5 are decision nodes whereas nodes 6 to 14 are state-of-nature nodes. The probabilities for these branches must be found before a decision tree analysis can be carried out. These are posterior probabilities. Node is nown as an indicator node. Since the indicator branches are not under the control of the decision maer, but are determined by chance, the node is represented by a circle ust lie the state-of-nature node. Once the branch probabilities have been found, the expected value approach can be used to determine the optimal decision strategy. We then wor bacward through the decision tree and calculate the expected value at each state-of-nature node. The expected values are given in the tree below. We conclude that, if the maret research report is favourable, Carl and Betty should buy an expensive system (expected value of R ) and that, if the maret research report is unfavourable, they should not buy any system at all (expected value of R 1 000). Procedure for determining the optimal decision strategy 1. Draw the decision tree consisting of decision, indicator and state-of-nature nodes and branches that describe the sequential nature of the problem.. Calculate posterior probabilities to establish indicator and state-of-nature branch probabilities. 3. Wor bacwards through the decision tree by computing expected values at state-ofnature and indicator nodes. Hence determine an optimal decision strategy and its associated expected value.

5 Expensive system (d 1 ) [3 706.] 6 s1 I1 ) Maret unfavourable (s ) P ( s I1 ) Favourable report (I 1 ) I 1 ) 0.51 [3 706.] Less expensive system (d ) 3 [ ] 7 s1 I1 ) Maret unfavourable (s ) P ( s I1 ) [1 400] No system (d 3 ) [] 8 s1 I1 ) Maret unfavourable (s ) P ( s I1 ) Do maret research Unfavourable report (I ) I ) 0.49 [ 1000] 4 Expensive system (d 1 ) [ ] Less expensive system (d ) [ ] 9 10 P ( s1 I ) Maret unfavourable (s ) P ( s I ) s1 I ) Maret unfavourable (s ) P s I ) ( [1 400] 1 No system (d 3 ) [] 11 P ( s1 I ) Maret unfavourable (s ) P s I ) ( Expensive system (d 1 ) [ 800] s 1 ) 0.4 Maret unfavourable (s ) P ( s ) 0.6 Do no maret research [800] Less expensive system (d ) 5 [800] s 1 ) 0.4 Maret unfavourable (s ) P ( s ) 0.6 No system (d 3 ) [0] 14 s 1 ) 0.4 Maret unfavourable (s ) s )

6 Expected value of sample information From the above example, the expected value of the optimal decision strategy is R This is also nown as the expected value with sample information (EVwSI). Initially, when no maret research was carried out, the expected value was R 800 that value is the expected value without sample information (EVwoSI). The expected value of sample information (EVSI) is defined as the absolute value (modulus) of the difference between EVwSI and EVwoSI, that is Note EVSI EVwSI EVwoSI 1. The expected value without sample information (EVwoSI) is equal to the expected value without perfect information (EVwoPI). This is simply the expected value of the best decision based on prior probabilities.. It is very useful to determine the EVSI and EVPI in order to decide whether it is necessary to but additional information. It should be clear that the EVPI is the upper bound for the amount that one would be willing to pay for information. 3. The same equation handles both maximisation and minimisation problems since the absolute value function is used. Efficiency of sample information Efficiency of sample information is measured as a percentage of perfect information (which is obviously 100% efficient!). The formula is EVSI E 100 EVPI When the efficiency of sample information is high, we have almost perfect information so that additional information would not yield significantly better results. However, low efficiency ratings would lead the decision-maer to loo for other types of information.