# Theories, Facts and the Analysis of Consumer Choice

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Raymond Dacey (1975) ,"Theories, Facts and the Analysis of Consumer Choice", in NA - Advances in Consumer Research Volume 02, eds. Mary Jane Schlinger, Ann Abor, MI : Association for Consumer Research, Pages: 671-678.

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http://acrwebsite.org/volumes/5790/volumes/v02/NA-02

This paper considers the role of selected public institutions and the mechanisms employed by these institutions. The approach is to characterize "consumers" as "decision-makers" in a formal way, to posit the general problem of "consumer choice" as a generalized decision problem, and to specify the role of inductive logics employed by these decision-makers in interpreting "facts" announced by producers together with "theories" announced by public institutions which regulate the producers' announcements.

Consider the consumer of household products. The F.T.C. specifies and enforces the use of measurement procedures to be applied to the contents of these products and the public disclosure of these applications. Alternatively, consider the investor in publicly offered corporate securities. Here the S.E.C. and the Financial Accounting Standards Board are among institutions which specify and enforce measurements of and disclosure about the securities.

In these and other analogous cases, the "governing" institution is imposing a "theory" upon the producers governed. This point is simply a variant of the well-known view that measurements (in general, observations) are "theory-laden." [See, e.R.. Hanson (1965). Ch. 1 and (1971). D. 17f. and Simon (1970). p. 17.] In this paper the disclosed measurements are regarded as "facts" to be interpreted by the decision-maker in light of the imposed theory. Together, these facts and theories bear upon the consumer's choice among alternative products.

It is in this context that the present paper considers the behavior of the consumer. Specifically, the consumer is viewed here as a decision maker in an uncertainty situation. The point of concern is the effect of the receipt of factual and theoretical information upon the consumer's behavior. The major finding is that given a specific and common form for the theory provided by a regulatory institution the consumer, though capable of inductive learning, will, under certain conditions, not revise his view of the world (i.e., his outcome mapping). Further, the conditions vary depending upon the consumer's "inductive philosophy." Put another way, the present analysis provides restrictions on inductive behavior and transmitted information sufficient to guarantee the applicability of classical Bayesian decision theory.

Consider an individual facing a decision problem D = <X,A> with state-space X and act-space A who possesses the personality P = <p,F,u> where:

p: X x A -> 0 is an outcome mapping,

bF: X -> [0,1] is a probability measure on X,

u: 0 -> Re is a utility function.

Such an individual resolves the problem a by selecting a*eA so as to maximize Eu(a) = Eu(p(x,a))b(x). The point of interest is the individual's reaction to learning something of relevance to p (and perhaps the states of nature).

The connection between outcome mappings and generalizations (i.e., "for all" statements) is as follows. An individual having accepted a generalization possesses the predictive theory whereby the resultant outcome of performing an act in a given state can be determined. [See Bezdek and Hannon (1974) for an example of an economic theory serving as an outcome mapping in a policy decision problem.] For example, given a Keynesian theory of economic behavior (or more accurately, the generalization expressing it), the effect of a tax increase in a state of high unemployment is easily determined. The generalization, i.e., the predictive theory, provides the outcome mapping.

It remains to characterize the individual decision maker's learning process, i.e., his inductive logic. For simplicity the range of the inductive logic is restricted to the decision problem D. Specifically, it is presumed that the individual has an inductive logic I = <P,P,U> where:

P is the class of weak and strong generalizations that yield outcome mappings on XxA,

P is a probability measure over the sentences of a language L, [F is P restricted to X.]

U is an expected (epistemic) utility function, defined in terms of P.

The individual's inductive logic selects, upon the receipt of new information e, a generalization qua outcome mapping so as to maximize U.

Given a personality P = <p,F,u> and an inductive logic I = <P,P,U>, both restricted to the decision problem D = <X,A>, the point of interest is the individual's reaction to a sentence e of relevance [On relevance see Carnap (1962), Ch. 6.] to p, i.e., a sentence e such that P(p|e) … P(p). The new sentence e may take either of two forms. First, e may consist of purely factual information to the effect that specific attributive constituents are instantiated by particular individual objects. Second, the new sentence e may be more revolutionary, including factual information and a theory T (a truth-functional combination of attributive constituents) introducing and relating a new concept, say Po, to the concepts P1, P2, ... ,Pk.

The first case admits of two possibilities. Let p denote the outcome mapping selected by U given e, and let p be the initial outcome mapping. First, if p = p and F(x|e) … F(x) for at least one xeX, then the classical Bayesian expected utility framework is applicable and the individual resolves the "posterior" problem A by selecting aeA so as to maximize E[u(a)|(e] = Eu(p(x,a))F(x|e). However, if p … p, then the Bayesian framework is inapplicable. The second possibility involves an extension of the Bayesian notion of learning and yields certain interesting peculiarities that are discussed elsewhere. [See Dacey (1973a), (1973b).]

Let us turn now to the second, more interesting, form of e, i.e., e = e&T. The sentence e provides factual information while the theory T relates the "new" (monadic) predicate Po to P1,P2, ... , Pk. [There is no loss of generality in considering the introduction of only one new predicate. The ensuing analysis generalizes for any finite number of new predicates.] In order to incorporate the properties of e&T into the present analysis it is necessary to specify the properties of P and U.

Following Niiniluoto and Tuomela (1973) we take P to be a Hintikka-type probability measure defined parametrically on a two dimensional continuum with parameters l and %. l, as in Carnap's one dimensional continuum, [See Carnap (1952).] is a measure of disorder in the universe; % is an index of caution. [See Hintikka (1966).] The values of the parameters are specified by the individual, and once specified completely determine his probability measure P.[See Hintikka (1966), and Niiniluoto and Tuomela (1973).] As such P is an inductive probability measure and not a purely subjective measure, and thus U, determined in terms of P, is an (expected) inductive (i.e., epistemic) utility function. A reasonable form for U is

U(h|e) = P(h|e) - P(h),

where h is any statement in the domain of P and e is a sentence of factual information. [See Hintikka and Pietarenen (1966) for the initial presentation of U, and see Niiniluoto and Tuomela (1973), pp. 68-70 for a discussion of U.] So defined U is a measure of relevance and embodies an extension of the concept of substantive information. [On relevance see Carnap (1962). For a discussion of substantive information see Bar-Hillel (1964). p. 307.]

For e = e&T, U gives rise to two variants, namely U1(h|e&T) = P(h|e&T) - P(h),

and

U2(h|e&T) = P(h|e&T) - P(h|T).

The adoption of either U1 or U2 to the exclusion of the other reflects the individual's philosophical/methodological position. U1 seems fitted to the position of a methodological instrumentalist, whereas U2 seems natural for a scientific realist. [On the instrumentalist and realist properties of U1 and U2 see Niiniluoto and Tuomela (1973), p. 70. For an extensive discussion of realism and its relationship to instrumentalism, see Hooker (1974). In short an instrumentalist takes a theory to be merely a device of convenience in organizing data; i.e., a theory is a mere instrument. A realist takes a theory literally, adhering to the view that the theory (more or less accurately) describes reality.] The pair <U,U1> or <U,U2> constitutes the inductive theory of the individual.

Consider now the syntactical structure of T. The theory T constitutes a formal (truth-functional) link between Po and P1,P2, ... ,Pk. The simplest such link is a definition. Now definitions, for obvious reasons, should be eliminable and non-creative. [On eliminability and noncreativity of definitions see Suppes (1957), p. 153 and Simon (1970), p. 20.] However, if a theory T possesses these properties then it cannot be said to introduce a new concept. Rather, T, as a definition, merely provides a specification of the "new" concept Po in teras of P1,P2, ... ,Pk, such that Po serves only as a simplified, economical abbreviation for a (perhaps) complex truth-functional structure built out of P1,P2, ... , Pk. For example, the concept "calorie" (or "Btu") is an economical substitute for a rather complex conceptual structure involving "temperature," "pressure," and "volume." The question naturally arises: Can a theory T, if it is an eliminable, non-creative definition of Po in terms of P1,P2, ... ,Pk have any effect on the inductive selection of outcome mappings? The somewhat surprising answer is yes. [The role of theoretical concepts in scientific theorizing has spawned a sizeable literature and involves a number of philosophical issues. See the bibliography in Niiniluoto and Tuomela for references. [This is not to say that theoretical terms where introduced via an explicit definition have no import whatsoever. See Hintikka and Tuomela (1970), pp. 44-46.]]

A theory T if it is an explicit definition of Po in terms of P1,P2, ... . Pk has a limited inductive role.[This is not to say that theoretical terms where introduced via an explicit definition have no import whatsoever. See Hintikka and Tuomela (1970), pp. 44-46.] An explicit definition T is irrelevant with respect to e to any generalization g, i.e., P(g|e&T) - P(g|e) for all generalizations g. However, if T is a disjunction of explicit definitions, then T has inductive import. The notion of T as a disjunction of explicit definitions is pertinent to the discussion of the governance of measurement and disclosure, as in the instances cited earlier.

Society includes various institutions whose purpose is the pronouncement of definitional theories like T. For example, the National Bureau of Standards (together with the International Bureau) periodically provides definitional rules concerning the fundamental quantitative concepts of science. In the business world, the Securities and Exchange Commission together with the Financial Accounting Standards Board dictate acceptable accounting procedures for financial reporting by pronouncing definitional theories. Similarly, the Cost Accounting Standards Board dictates acceptable accounting procedures for those engaged in governmental contracting. Of more direct relevance to the theory of consumer behavior is the Federal Trade Commission, which pronounces definitional theories concerning consumer goods. Very rarely are these theories strictly explicit definitions. Typically, in order to allow some flexibility on the part of those imposed upon, the pronounced theories permit alternative definitions of a concept. For example, neither the concept of "watt" as relates to stereo equipment, nor the notion "octane" for gasoline is explicitly defined. Rather, the pronounced theory T consists of a disjunction T1vT2v ... v Tm, where each Ti is an explicit definition (of the new concept Po in terms of P1,P2, ... ,Pk). A user of T then has the option of employing a specific disjunction. A consumer may be aware of T but typically will not know the specific Ti used to generate the factual information supplied via advertising. [Note that if the user, here the consumer, is aware of the specific Ti being used by the producer, then Ti, since it is an explicit definition, is irrelevant to any sentence h, i.e., P(h|e&Ti) = P(h|e). On the general issue of the information content of advertising see Nelson (1970) and (1974). The latter announces that "Information is generated by advertising because of consumer power in the product market" (1974), p. 730. This is a weak position. Advertising becomes information when translated or interpreted via a definitional theory. Thus consumer power consists of the power to institute regulatory agencies capable of enforcing adherence by producers to definitional theories.] Our concern is with the effect of a knowledge of T upon the consumer/decision maker's selection of an outcome mapping. Specifically, our purpose is to examine the role of T in the following situation. Suppose, given only the factual sentence e the decision maker selects outcome mapping p*. However, given e&T, where T "governs" concepts reported on by e, he selects p. If e constitutes the decision maker's initial knowledge, then our concern is with the incremental impact of the definitional theory T. Note that acquisition of T can be seen as a reduction in ambiguity concerning the concept Po. Ambiguity is completely removed if the operative Ti is made known. Ambiguity can, if it leads to the absorption of p in place of p*, cause the expected utility framework to be inapplicable. Thus, we are also here concerned with the effects of an attempt to reduce ambiguity on the adequacy of a theory of consumer (i.e., decision making) behavior.

Consider now the formal properties of T as a disjunction of explicit definitions. A concept Po is said to be piecewise definable in terms of concepts P1,P2, ... ,Pk in a theory T if and only if T logically implies a finite disjunction of explicit definitions of Po in terms of P1,P2, ... ,Pk. [Niiniluoto and Tuomela (1973), p. 57.] Clearly, if T is itself such a disjunction, then it provides a piecewise definition of Po.

We now turn to a consideration of the role of the piecewise definitional theory T in the inductive choice of outcome mappings. More specifically, our interest is in a situation wherein the individual reverses his selection on the basis of T, i.e., a situation where given the evidence e and an inductive structure <U,Ui> he selects an outcome mapping p*, but given T&e and <U,Ui> he selects p where p … p*. Thus, our concern is with T's role relative to that of e in reversing the choice of an outcome mapping.

Two results due to Niiniluoto and Tuomela are of immediate interest. Let T = T1vT2 (where 'v' denotes 'or') be a piecewise definition of Po in terms of P1,P2, ... ,Pk

(A) If t = T1&T2 is incompatible with e, or if t is incompatible with the assumption of a non-empty universe, then P(g|e&T) = P(g|e).

(B) If an individual is cautious and e reports on a large sample, then P(g|e) - P(g) = P(g|e&T) - P(g|T). [Niiniluoto and Tuomela (1973), pp. 72, 75-76.]

Conditions (A) and (B) together with the (necessary) conditions for decision reversal have interesting consequences.

First, consider decision reversal from p* to p for a <U,U1> individual. Presume that given e alone he prefers p* to p, i.e., U(p*|e) > U(^p|e). This is equivalent to (RI) P(p*|e) - P(p*) > P(p|e) - P(p). Presume also that given e&T he prefers p to p*, i.e., U1(p|e&T) > U1(p*|e&T). But this is (RII) P(p|e&T) - P(p) > P(p*|e&T) - P(p*).

Now suppose the antecedent of (A) holds. Then P(p*|e&T) = P(p*|e) and P(p|e&T) = P(p|e). These equations, together with (RI) and (RII) imply a contradiction. Thus, if t is incompatible with e, or if t is incompatible with the assumption of a non-empty universe, then reversal from p* given e to p given e&T is impossible.

Now, consider p* - p decision reversal for a <U,U2> individual. Given e alone he prefers p* to p, i.e., U(p*|e) > U(p|e). This is equivalent to (RI). Given e&T he prefers p to p*, i.e., U2(p|e&T) > U2(p*|e&T). This is (RIII) P(p|e&T) - P(p|T) > P(p*|e&T) - P(p*|T). Together with two minor assumptions, [The two further assumptions are | [P(p|e) - P(p)] - [P(p|e&T) - P(p|T)] | < | [P(p|e) - P(p)] - [P(p|e) - P(p)]| and | [P(p|e&T) - P(p)] - [P(p|e) - P(p)] | < | [P(p|e&T) - P(p)] - [P(p|e&T) - P(p)] |, both of which are virtually guaranteed by the meaning of =.] the antecedent of (B), (RI) and (RIII) imply a contradiction. Thus, if the individual is cautious and e reports on a large sample so that P(p*|e) P(p*) = P(p*|e&T) - P(p*|T) and P(p|e) - P(p) = P(p|e&T) - P(p|T), and two minor assumptions are valid, then reversal from p* to p is impossible for a <U,U2> individual.

The conditions which block p* - p reversal for a<U,U1> individual are strikingly different from the analogous conditions for a <U,U2> individual. For the former, the mere incompatibility of t with e is sufficient. However, this same condition causes no difficulty for a <U,U2> individual. On the other hand, if a <U,U2> individual employs a factual report e about the observation of a large number of individual objects and is himself a cautious individual, then his reversal from p* and p is impossible. Furthermore, the conditions sufficient to block a <U,U2> individual from a p* - p reversal do not have the same power for a <U,U1> individual. Thus, the different inductive forms characterized by (U,U1> and <U,U2> inductive theories are quite distinct, and what constitutes an impossibility for one has little or no effect on the other.

As noted earlier, if p* = p and F(x) … F(x|e) for at least one xeX, then the classic Bayesian posterior analysis is applicable to the resolution of the problem A. Thus, for an instrumentalist, a piecewise definition T = T1vT2 is,relative to factual evidence, irrelevant to the selection of an outcome mapping as long as t = T1&T2 is incompatible with e. However, for a realist, a piecewise definition T is irrelevant only if the observation report e concerns a large number of individual objects and the individual is cautious. If these conditions are met, then classical Bayesian analysis will characterize the individual's behavior.

The foregoing analysis provides conditions sufficient to guarantee that a decision maker will maintain the expected utility model as his decision making framework after absorbing a piecewise definition T. The conditions are only sufficient and not necessary; furthermore, they are not the only sufficient conditions. Together with various restrictions on probability assignments conditions similar to the antecedents of (A) and (B) yield further conditions sufficient to block p* - p reversal. These further conditions are not treated here. Their existence is of course of great relevance to any one conducting experiments on the effect of information on Bayesian decision makers.

Blocking outcome mapping reversal is of special interest to those conducting controlled economic experiments, as in the preceding papers delivered by Kagel and Lloyd. One controls an experiment expressly to avoid non-economic influences on the behavior of the subjects. Information introduced as a catalyst can have such an effect. If, however, the information incorporates the "correct" kind of piecewise definitions, then the undesired outcome mapping shift can be avoided.

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