Decision Theory

D.J. White

• Publisher: Routledge
• Year: 2018
• eBook ISBN: 9781351523325
• Topic: Social Sciences
• Subtopic: Sociology
• Index: Social Sciences

All of human life may be seen as a process of decision-making, but it is only in recent years and in response to the needs of the large and complex organizations characterizing our society that this process has been subjected to scientifi c scrutiny. Out of this scrutiny-undertaken by a wide range of professionals in economics, administration, management, statistics, psychology, engineering, computer science, operations research, and systems analysis-there has begun to emerge a body of theory that has profound implications for improving practical decision-making. This book is the fi rst to bring together all the various aspects of decision theory into one cohesive treatment.

CHAPTER 1

Some Concepts and their Interpretation

It has become fairly evident, as a result of work carried out in the general areas of decision, both from assessments of published work and from contact with actual decision-makers (and here I used the word ‘decision-maker’ in its commonly accepted sense), that, contrary to certain scepticism held by certain people, it is absolutely essential, by the very nature of the work and by pragmatic aspects, to consider the semantical side of our subject. This we shall do throughout, in relationship to uncertainty, value, etc., but in this section we shall be primarily occupied with the concept of ‘decision’. As a result of the analysis it will be necessary to consider carefully the retention of the term ‘decision-maker’ as generally used. If it seems inadvisable to modify its usage, then we shall term our particular usage of decision ‘logical’ or ‘mathematical’ decision.

An essential pre-requisite of an occurrence of decision is the existence of a motivating state of ambiguity. Such a state may contain propositions whose truth is known and some whose truth is unknown. We may have ‘partial’¹ ambiguity, although one term ‘ambiguity’ will suffice. Smith (64) takes the view that decision is simply the resolution of ambiguity, although the resolution of ambiguity may be achieved without deciding, in any logical sense. Thus in a problem

, the ambiguity can be stated in the form ‘which act shall we take’? The choice of, say, a, is said to resolve this ambiguity. We may only wish to resolve some of the ambiguities. Thus, if the truth values of certain informational propositions are not known, we may still wish to select an investment opportunity without removing the informational ambiguities ! We may not know the operating cost and yet still choose the investment.

Should the word ‘ambiguity’ be ambiguous, it is worth noting that the usual usage, in the sense of ‘ambiguity of meaning of a proposition’, is consistent with ‘selection from a set of alternatives’ (our usage) if a particular meaning corresponds to one alternative. Thus if we say ‘c(a) = ·5’, c may refer to a cost calculated by any of several methods, and unless we know the method the proposition is ambiguous. Giving a meaning to the proposition is equivalent to asserting that a certain method was used in calculating c(a).

The above paragraph has much more than an academic content for, later on, we shall see that in analyzing our primary decisions (e.g. whether to accept a certain investment proposition) we have to make secondary decisions, which are equivalent to selecting certain statements as being true. The truth or falsity of these statements can have a significant bearing on the appropriateness of any decision based on them and hence their analysis is just as important as the analysis at the primary decision.

One noticeable occurrence is the use of ‘decision’ as being synonymous with ‘action’ or ‘alternative’. The use of ‘the set ox alternative decisions’ is quite common, when ‘the set of alternative actions’ is meant. This may be of little significance in practice, but from our point of view we can afford no ambiguity of expression. The set of alternative actions identifies the ambiguity and the resolution of this ambiguity constitutes the decision process and culminates in the decision.

Another noticeable occurrence is the use of the term ‘decision’ when the term ‘choice’ is meant. Let us quote Dunlop (26), viz:

‘There can be choice without decision ; but there cannot be decision without choice.

‘Throughout economics, that miserable puppet, the economic man, is allowed to choose but seldom if ever to decide. His choice, or choosing, is determined by his ‘preference’, just as choice of a strategy (not always clearly, in use, distinguished from move) is like collusion determined by rules, frequency ratio probabilities and self interest in the "Theory of Games". Colloquially, and more trivially, we encounter "take your choice", "make your choice", in each of which "decision" and an indefinite article can be substituted. If auxiliaries are to be necessary, we shall see clearly that for both decision and proof, the appropriate infinitive is "to arrive".’

These statements are extremely relevant to a theory of decision, and the key to our problem lies in the very last phrase, viz: ‘to arrive’. Logical decision requires a linking of the state of ambiguity to the act of selection by a set of identifiable unambiguous cognitive operations. Certainly the operations must be unambiguous themselves, otherwise we cannot carry out the process of arriving in a deductive sense. As we shall see later, the problem of ‘identifiability’ is severe and raises questions not only of the possibility of identifying operations but also of whether they are cognitive or not. Logically speaking, unless they are cognitive, then the decision-maker can hardly be said to be deciding. It is to be expected in practice that resolution of specific ambiguities will be achieved by a composite mixture of decision and choice, and, in fact, this has become obvious from encounters with decision-makers.

A problem situation begins with the recognition of a primary ambiguity, e.g. ‘shall we add to our production capacity?’ In resolving this problem, secondary problems are generated, e.g. ‘what information shall we get?’, ‘how thorough should we be in our analysis?’ Quite obviously, over a time period, between the recognition" of the primary problem and its final resolution, we can get a very complex mixture of secondary, and lower order ambiguities. The resolution of a problem may, therefore, involve a composite mixture of selection processes, some of which are cognitive and identifiable (which we term ‘pure decision’) and some of which are not so (which we term ‘pure choice’). In general, the gap between the primary ambiguity and the primary selection will be covered by a set of selection processes, some being pure decision and some pure choice.

Provisionally, we say that ambiguity Q, in a state of knowledge K. is resolved by a pure decision process if there is an identifiable unambiguous cognitive operation θ such that

θ has to be such that q follows deductively from (Q, K) via θ. It is not enough, for example, to say that ‘if act a is represented by [a₁, a₂, ... a_m] then θ is identical with forming the expression Σα_ia_i’. θ has to be completed by exhibiting that a formal rule exists, such as ‘choose the act with the. largest value of this expression’. The first form of θ establishes that Σα_ia_i is a sufficient index for choice, but it does not follow that the second form holds. Given, say a problem

¹, he may choose that act with the median value of

.

There is nothing to prevent θ generating a sequence of ambiguities Q₁, Q₂, Q₃ . . . as pointed out previously. Thus if a is location and size’ of warehouse. θ may required that operating costs (

, capital costs

and service factors

be evaluated first of all (thus posing a problem Q₁, the solution of which gives a¹, a², a³) and then that the final choice be made on the basis of the minimum value of

. θ can have many forms depending on how Q is initially presented. If θ is ambiguous or unidentifiable at any stage then the process is only partially decision. A better term for the common decision-maker is partial decision-maker, although this is not likely to gain acceptance in usage.

The selection of an action is not necessarily followed by its implementation. If, at the time of selection, any cognitive doubts arise as to whether it will be implemented, then it is not correct to say that the act has been selected. Good (30) introduces the concept of ‘the amount of deciding (although "selecting" would appear to be more appropriate) in mental event E in favour of act F’ where the probability of implementing the act serves the purpose of quantifying the amount of deciding. This has a direct bearing on probabilistic choice mentioned later, but this use of ‘deciding’ is not the same as ours and is more like ‘a measure of mental commitment to act F’ However, this may not tell the whole of the story, for a person may be absolutely convinced that he will implement a certain action but be prevented from doing so by unforeseen circumstances. If. between the selection and the implementation, he has cause to reconsider this selection, it simply triggers off another selection process.

Common language, in discussions of problems, makes use of the concepts of ‘preference’ and ‘choice’. So-called economic man is said to have ‘preferences’ for ‘bundles’ of commodities. A bundle may be denoted by a vector,

Such preferences are not necessarily decidable. Although a preference function describing choice may be derived, using input-output analysis techniques, the decision-maker does not necessarily know this function. As we shall see later on. the foundations of a pragmatic theory of decision have to centre on certain basic types of preference characteristics. Although these preferences may not be decidable, we can use them, and, indeed, recommend their usage, in the analysis of complex problems, of which they become component characterizations. We can use them to introduce decidability into complex problematic situations, although they may not, themselves, be decidable. If he does cognitively use a preference function, such as Σα_i, q_i, then we would ask how he derived this and whether or not it was decidable.

There are many instances of the use of linear decision functions in priority scheduling of production, and these decision functions are usually decidable by finding that one which optimizes production performance. There are, however, situations in which such rules are chosen, subjectively, by giving weights to each factor because of the subjective importance of each factor, e.g. such a case arises in giving priority to people on council house lists, or even to the selection of personnel.

If such a preference function is constructed by observing his choice behaviour, then, although this can now be used to decide his future choices, it does not mean that such a function was actually being used, cognitively, prior to this. This is certainly likely to be the case in personnel selection problems.

In order to illustrate the use of preference characteristics (cognitive or otherwise) in introducing decidability into problem resolution, let us consider a simple case, in which the commodities, [q_i] are to be given the interpretation of physical goods, and let the decisionmaker be given a quantity of money, x, and told to spend the lot on these commodities. Once the preference order for the q’s is known, the actual selection can be ‘decided’. In fact, if a value function v(q) has been determined, the selection is decided by solving the following problem, in which p_i is the price per unit of commodity i:

This shows that, although his preferences for the q’s may not be decidable such preference behaviour can be used in decision analyses, by using the preference functions as premises for decision in a different class of problem. This is an important point, for it is not meant to be implied, by ‘pure choice’ that the results are not usable in deductive propositions, only that such choice may not, itself, be deduced.

If, however, he is given the money, x, and not instructed to spend the lot, then, for some reason or other, he may treat money as a commodity. The commodity vector is then:

The selection can be then decided by solving the following problem, where v(q) is again inferred from observation of choice in terms of q:

A common difficulty occurring throughout the field of practical decision-making is the problem of determination of consequences, and this is discussed later in the report. It is, however, important to realize that...

Table of contents

• Cover Page
• Title Page
• Copyright Page
• Table of Content
• PREFACE
• 1. SOME CONCEPTS AND THEIR INTERPRETATION
• 2. THEORIES OF CHOICE, VALUE AND UNCERTAINTY
• 3. DECIDABILITY
• 4. SOME PRACTICAL CONSIDERATIONS IN DECISION THEORY
• 5. INFORMATION FOR DECISION
• 6. PRAGMATIC ASPECTS OF DECISION THEORY
• 7. MATHEMATICAL MODELS AND DECISION
• Index