166 JOURNAL OF THE SOCIETY OF COSlVIETIC CHEMISTS more or less likely than the other. This being so, environments are assumed equally likely and strategies accordingly measured by the average outcome across the environments. Thus we have Xx N 11.5/3, X2 N 11/3, Xa • 11/3 and Xx is accordingly chosen. There are many other ways of making this selection. Each policy could be viewed in terms of its worst possible outcome, in which case X2 would be chosen since it has the best worst outcome. This procedure of choice is the pessimist's criterion of Wald. Of course, one could equally well make the choice on the basis of the very best possible outcome, or a mixture of the best and worst outcomes. One selection criterion of interest to us here is that due to Savage- minimax regret criterion. The selection is made on the basis of the policy with the minimum maximum regret where, for each environment, regret is measured as the difference between what was obtained and the best that could have been obtained had the environment been known. For the above problem, we have for the regrets: Regret Good Average Poor Max regret 3.5 3.5 1 4 0 3 The chosen strategy is here Xa since it has the minimum maximum regret. Now suppose initially market prices of oe2 and oe1 are considered. If the decision-maker were to choose on the basis of Savage's criterion, the price would be fixed at oe2. However, we have just seen using the same criterion to choose between oe2, oe1 and oe1.50, the choice is oe1. This irrational behaviour is caused by the criterion concerned being non-transitive. Transitivity, sometimes referred to as coherence, is a very important concept in decision analysis. A decision-maker's preferences are transitive if, when A is preferred to B, and B preferred to C then A is preferred to C. With this condition unsatisfied, the decision-maker has no best choice between A, B and C. Few business men would be content to make decision under conditions of uncertainty. Most likely they would attempt to obtain further informa- tion as to the prevailing environment. This leads to a situation of risk. Decisions under risk In the case of decision-making under risk, the decision-maker assigsn probabilities to each of the environments indicating his degree of belief that

DECISION ANALYSIS 167 each particular one will prevail. The probabilities may either be objective or subjective in nature. Consider the situation of a marketing manager whose firm is contem- plating a special promotion for a new product during October and must decide upon the promotion now. Assuming his objective is profit maximi- zation over a specified period of time, there are two alternatives open to him X• approve the promotion, X: reject the promotion. It has been calculated the production and promotion costs will be oe100 000. No objective information is available concerning customer reaction, but it is assumed customer reaction can be measured on a three-point scale say, 'very favourable', 'favourable' and 'unfavourable'. The manager summa- rizes his feelings about the various categories in the following table. Very Profit favourable Favourable Unfavourable oe Zx Z• Za X•. 250 000 60 000 -- 100 000 X,, 0 0 0 p(Z) 0.4 0.3 0.3 p(Z) is a probability (subjective in this case) giving a measure of the mana- ger's personal degree of belief the various environments will prevail. It is suggested a decision should here be made on the basis of expected outcome, that is for each policy Xi, the choice is based upon x, In the present case, therefore, Xx ~ oe250 000 x 0.4 + oe60 000 x 0.3 - oe100 000 x 0.3 = oe88 000 X•~ oe0 x 0.4+ oe0 x 0.3 + oe0 x 0.3 =oe0 and policy Xx is chosen since it has the greatest expected value oe88 000. It is assumed here the decision-maker has no overpowering objection to the possible loss of oe100 000. Of course, the decision-maker will never actually get oe88 000 in any single case. Clearly his gain is fixed as one of oe250 000, oe60 000, moe100 000 or oe0. The interpretation of this result is: if the situation is as represented in the above table, and the identical decision has to be made not once but on numerous independent occasions, then the expected average return from policy Xx is oe88 000.