Meaningful Probability

Meaningful Probability We have established from the previous section that it would be useful to know how likely were the various market conditions in Table 3.3. Can we devise a measure of likelihood? In fact we, along with many other texts, have simply assumed that this is possible. In section 2.4.1 we supposed that the manager was able to say that there was a 60 per cent chance of a product launch being successful. But what does this statement mean? We suggested that this information was based upon the observation that 40 per cent of all new products failed in that market. The implication therefore is that probability estimates should be based upon some notion of the frequency of occurrence. The tossing of a coin is often given as an example, since we know that there is a 50:50 chance of a head or a tail from carrying out the test a number of times. From sample tests we can infer that the true probability is 1/2 or 50:50; indeed, the whole of statistics is built up from this process of inference. To some, estimating the success or otherwise of a new product is no different. If this view were accepted, then our only problem would be collecting relevant observations from the past. In fact, for many business problems it is not possible to collect sufficient relevant observations to infer a probability in this manner. In our product launch example, the similarity of the research reports on 'product launches' may only be superficial and the statistic spurious. For example, a manager may believe that most product failures are due to insufficient market research. If the new product being considered has been subjected to extensive research, the manager may reasonably believe that the chances of failure are much lower than 40 per cent. If the manager were to say that the chances of failure were in his estimate only 20 per cent, he could not be proven wrong in any sense. After all, if the product succeeds, both the research report and the manager have agreed that this would be the most likely outcome. If it fails, then who is to say that it was an event with only a 20 per cent chance or an event with a 40 per cent chance? A manager may wish to use past frequences, research reports, or his own ideas on the causes of the events, or some other method, or a mixture of these approaches, we cannot reject any of these bases for making the estimation. For this reason such probabilities are termed subjective probabilities - they are not necessarily based upon objective observations such as frequencies. The term subjective does not mean that the estimates are irrational. A manager will hopefully look at all the information available, and interpret it in roughly the right manner. The net result should be a probability estimate that is more likely to lead to a satisfactory decision than one that is restricted to past frequencies. However, as we have said, the matter cannot be proven one way or the other in view of the fact that we are predicting a single future event. This leads us to an important statement on the role of decision analysis: a decision technique that uses probabilities cannot in any sense determine objectively the 'best' option; analysis merely organizes the decision maker's opinions, and derives logical inferences and interpretations in a consistent manner such that the manager chooses the option with which he or she is happiest. Without analysis, the decision maker may have a number of relevant opinions about a problem but may not know how to apply them. After analysis, the manager would probably be happier about his choice of best option. It is now appropriate to turn to a criticism that is often made of probabilities. First let us present the correct argument. Suppose we are faced with two options, as in Figure and that we valued each pound equally, e.g. that would mean that we are just as happy at receiving the last £1 of the £120 in Option 2, as receiving the first £1. Let us also assume and we shall relax these assumptions later on) that the decision maker is a person who does not mind taking a gamble, i.e. he does not prefer Option 1 simply because one of the results has an 80 per cent chance as opposed to a 50 per cent chance. It seems intuitively reasonable to say from this that such a decision maker would be indifferent, or equally happy, between an 80 per cent chance of £100 and a 100 per cent chance of £80. In this way, we can compare the two options by converting all the returns to an equivalent 100 per cent probability. For Option 2, a 50 per cent chance of £120 is equivalent to a 100 per cent chance of £60, and a 50 per cent chance of £30 is equivalent to a 100 per cent chance of £15. Thus Option 2 is equivalent to a 100 per cent chance of £75. The manager concerned would therefore prefer Option 1 with an equivalent certain return of £80 as opposed to £75. This measure of 'equivalent certain return', or 'certainty equivalent' is not just an ordinal measure but is cardinal as well (see section 3.2 for an explanation of these terms). Therefore the difference of £5 between the two measures is a measure of the amount by which the decision maker feels better off. Put another way, he or she would be willing to pay up to £5 to be offered Option 1 rather than Option 2.