# A Deterministic-Type Model of Consumer Choice

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Michael R. Hagerty (1980) ,"A Deterministic-Type Model of Consumer Choice", in NA - Advances in Consumer Research Volume 07, eds. Jerry C. Olson, Ann Abor, MI : Association for Consumer Research, Pages: 129-133.

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http://acrwebsite.org/volumes/9662/volumes/v07/NA-07

[This paper is based on a doctoral dissertation at the University of Illinois. Thanks are due to Seymour Sudman, Russ Belk, Jagdish Sheth and Peter Webb.]

This paper reviews several experiments for which many commonly used choice models cannot account, and suggests an alternative called the Situational Model (SM). SM assumes that choice is basically deterministic, but that situations can reliably alter preference. Some tests which can distinguish between SM and other choice models are outlined, including an experiment in variety-seeking.

CHOICE MODELS

Consumers are often observed to be inconsistent in choosing repeatedly among the same products. To explain this, one may either assume that choice conditions are changing, which affect preference (but that choice is fundamentally deterministic), or that individual choice cannot be exactly predicted and is stochastic. Bass (1974) has concluded that "It will never be possible to prove conclusively that behavior is fundamentally deterministic since it will never be possible to measure all of the variables which influence choice." While we agree that in a metaphysical sense the question cannot be resolved, we suggest that specific mathematical models of choice can be distinguished by empirical tests, and that some of these models may be more easily interpreted as "fundamentally deterministic,'' while others seem "fundamentally stochastic." This paper suggests a deterministic-type of choice model, and shows that its predictions differ from those of many probabilistic-type models when new products are entering the market or old ones are deleted.

The two choice models in most common use in marketing and economics seem to be Luce's Choice Axiom(Pessemier, Burger, Teach, and Tigert 1971; Silk and Urban 1978) and the First Choice Rule (Green and Srinivasan 1978). Luce's Choice Axiom (LCA) is a probabilistic model which simply predicts that when two products are in the market, the probability of a person's choosing product A over product B, defined as P(A;B), is equal to the preference for A, labeled a, divided by the sum of preferences for both products in the market: P(A;B) = a/a+b .

The LOGIT model (McFadden 1973) is also in common use, but it is a special case of LCA.

The First Choice Rule (FCR) is a deterministic model which predicts that a person always chooses the product which he ranks highest in overall preference. While this approximation is usually wrong for an individual subject, the model "assumes" that errors in prediction cancel each other when predicting aggregate choice. The next section shows that neither of these choice models can predict expected choice for certain new product entries.

COUNTER EXAMPLE TO PROBABILISTIC MODELS

It has long been known that most probabilistic models do not correctly predict certain cases in which brands are added or subtracted to the available alternatives. These counter examples are reviewed by Tversky (1972). The following example was stated originally by Debreu (1960) and was restated by Green and Srinivasan (1978).

Suppose that two substantially different modes of transportation are available to an individual: automobile (A) and bus (B). Suppose that for this person, his probability of choosing the auto, P(A;B), is .4. How would we expect the choice probabilities to change if a new mode of mass transit, C (which might be Rapid Transit with stops similar to the bus line), is introduced? First, since B and C are so similar, we expect P(B;C) = .5. Also, it seems reasonable that, since mass transit forms B and C are similar and therefore substitutable, the probability of choosing A, P(A;B,C), should remain at .4 (or decrease a little). However, the probability of choosing B, P(B;A,C), is expected to decrease substantially from .6 to .3, since C may be often interchanged for B. In summary, we expect that when the similar product C enters the market, P(A;B,C) = .4, P(B;A,C) = .3, and P(C;A,B) = .3.

How does this example violate LCA? LCA predicts that a set of real numbers, a, b, and c exist which can reproduce all of the choice probabilities. We can show that these numbers cannot exist. Under LCA, P(B;C) = b/b+c = .5 -> b = c. Also, P(A;B) = a/a+b = .4 -> a = 2/3 b. LCA thus predicts that P(A:B,C) = a/a+b+c = .25, contrary to our expectations of P(A;B,C) = .4.

This counter example also presents severe problems for FCR. FCR is usually applied by having subjects rank all possible products in preference. In this case, the products would be ranked as "A more probable than B," and "B about equal to C" when all three products are in the market. But when only two products A and B are in the market, this rank order predicts that A will still be most preferred. This is contrary to the example, where P(A;B) = .4 < P(B;A) = .6. Thus even if the errors in the deterministic FCR cancel over subjects, the very definition of what First Choice is can change depending upon what other products are considered in the set. (This example violates what is known as Independence from Irrelevant Alternatives.)

So far, the counter example is merely what we __expect__ to happen upon entry of a similar product. These expectations have been confirmed by Becker, De Groot, and Marschak (1963) and by Tversky (1972) for choice among similar bets. It seems likely that violations of LCA and FCR would also occur with entries of new __consumer__ products.

Two implications of this counter example should be noted. First, the inherently probabilistic model of LCA cannot predict certain new product entries. Tversky (1972) further shows more generally that no probabilistic model which obeys simple scalability [Models that possess simple scalability account for choice by assuming that every object can be described by a __single__ number. More formally, simple scalability holds if and only if there exists a scale u defined on all t choice alternatives and functions F_{n} in n arguments, 2 < n < t, such that for any A = {x,...z}, P(x,A) = F_{a}[u(x),...,u(z)], where each F_{a} is strictly increasing in the first argument and strictly decreasing in the remaining a-1 arguments. LCA possesses simple scalability, where Fa = u(x)/Eyeau(y) .] can predict this and other counter examples. Hence, if one espouses the view that choice is inherently probabilistic and at the same time wishes to predict these counter examples, then he is severely limited in the choice model he selects. The model which Tversky (1972) proposed to satisfy both of these requirements, called Elimination by Aspects (EBA), is quite complex both in computation and in estimation. Perhaps this is why we have found only one tentative application of EBA in seven years (McFadden, 1975).

The second implication of this counter example is that market share after new product entry may be incorrectly predicted by techniques such as conjoint measurement or dollar-metric analysis which use FCR or LCA. The extent of error in the predicted aggregate market share will depend both on the extent of error in individual choice predictions and on the distribution of errors in the population. Clearly, though, at least some distributions can cause incorrect market share predictions (e.g., if everyone has preferences identical to the person described in this counter example, then market share will be in error as much as individual choice predictions are). The actual error in prediction in a real new product entry is unknown, since we have found no conjoint studies which measure actual market share after product introduction.

The next section describes an inherently deterministic model which can account for this counter example, and which has some implications different from EBA, LCA, and FCR.

THE SITUATIONAL MODEL

Suppose (a) that a consumer in a given situation can at least rank order his preferences for all products in the market, (b) that he always chooses his most preferred product, and (c) that varying situations may change his preference order for the products. These assumptions will be called the Situational Model (SM). We first show how it predicts the counter example correctly, then list some of its properties.

Recall that our counter example assumes that the consumer chooses auto transit (A) 40% of the time, and public transit (B) 60%. It is possible to hypothesize that the trips taken by auto were motivated by some aspect of the situation (e.g., time was an important factor). In empirical studies it is always possible to test this hypothesis by manipulating the situation experimentally. However, in the absence of evidence we need hypothesize only two types of situations which affect the consumer's choice of transportation. These situations may be displayed as:

In Situation 1, which occurs 40% of the time, A is preferred to B and C, which are tied since they are so s/m-ilar. In Situation 2, which occurs 60% of the time, B and C are preferred over A.

The probabilities of choice follow from these situations. For example, P(A;B) is just the proportion of time in which A is preferred to B. C may be ignored in this case, since it is not in the choice set. Noting that A is always preferred to B in Situation 1, and never in Situation 2, then

P(A;B) = (1)(40%) + (0)(60%) = .4.

Similarly, the probability of choosing A when both B and C are in the market, P(A;B,C), may be computed from the two situations as

P(A;B,C) = (1)(40%) + (0)(60%) = .4,

consistent with our expectations. Finally, P(B;A,C) is computed as .3, since B is never preferred in Situation 1, but is preferred half the time in Situation 2 (since it is tied for first place with C in Situation 2). The reader may convince himself that under the Situational Model

P(A;C) = (1) 40% + (0) 60% = .4,

and P(B;C) = (2) 40% + (2) 60% = .5,

consistent with our expectations in the counter example. The Situational Model can account not only for this counter example to LCA, but for ail of the other counter examples reviewed by Tversky (1972).

As it happens, the Situational Model is formally identical to one proposed by Block and Marshak (1960) called the Random Utilities (RU) model. They proved that RU obeys regularity, [Regularity is the property that the probability of choosing an object cannot be increased by adding another object to the choice set.] but that it can violate most other properties, including Weak Stochastic Transitivity (WST). [WST is an extension of the deterministic transitivity property, which is, if A is preferred to B, which is preferred to C, then A is preferred to C. WST is, if P(A;B) $ .5 and F(B;C) $ .5, then P(A;C) $ .5.] Because most choice experiments confirm WST, however, RU was apparently not considered further as a choice model. The interpretation of SM differs from RU in that SM assumes that various preference rankings which may occur are due to manipulable and measurable situations, rather than to random unknowable factors. Thus a partial knowledge of which situations are operative or what their relative frequency is can strengthen the general predictions of RU. In the extreme, when all situations are controlled, RU would become a completely deterministic model predicting that the most preferred object is always chosen.

In summary, SM is an inherently deterministic model which can account for all of the counter examples raised against LCA and other probabilistic models employing simple scalability. The inherently probabilistic model which has received the most attention recently which can also account for the counter examples is EBA. The next section examines how these opposing models may be distinguished.

CRITICAL LINES OF EVIDENCE

Three lines of critical evidence are identified which may distinguish SM from EBA. First, the Situational Model predicts that when all situations are held constant, choice is deterministic. On the other hand, EBA and other probabilistic choice models predict that choice is always stochastic. Bass (1974) relied on this line of evidence exclusively. Second, SM predicts that when situations do vary, violation of WST is possible. EBA instead predicts that such a violation is impossible. Third, SM would predict that the "stochastic" choices collected by Tversky (1972) in support of EBA were actually due to reliable context effects in the experimental sequence. Probabilistic models instead predict that context effects, even if they exist, could not account for Tversky's results.

Finally, we pursue a fourth line of evidence which is not strictly critical, but which could make SM easier to apply. This is a theory of satiation -- of how preferences change reliably even when no apparent situations change.

Deterministic Choice

Bass, Pessemier, and Lehmann (1972) controlled most situations in an experiment in which subjects were allowed to choose once a day at no cost one soft drink out of eight offered. The results did not support deterministic models, since only 55% of the choices could be predicted by the subject's stated favorite brand. However, the novelty of the situation may have caused more switching than would occur in more "steady state" conditions. Best (1978) threw out the first two weeks of data in a similar experiment to control for the novelty effect, but added choices made in the morning to those made in the afternoon, thus introducing possible changes in the situation (e.g., in the morning one might select a soda to keep him awake, but in the afternoon he might prefer a taste-quencher). Best found that 61% of the choices were predicted by the subject's stated favorite brand. Work is underway to eliminate both of these possible compounds. Unless the proportion of choices accounted for by first choice is considerably higher, an inherently probabilistic model seems more appropriate for consumer choice.

One reason why we hesitate to accept these two reported experiments is that psychological studies of choice have yielded results more consistent with the deterministic hypothesis. For example, Mosteller and Nogee (1951) allowed subjects to choose the gamble which they most preferred to play for money. Subjects were quite consistent in their choices except when the payoff was in a narrow band (a two to three-cent band out of a range of 17 cents), near the subject's indifference point. Davidson, Suppes, and Siegel (1957) in a similar but better controlled experiment found no inconsistency at all in choices. Even experimenters who espouse probabilistic theories (e.g., Tversky, 1972; Tversky and Russo, 1969) must use stimuli which are very close to the subject's indifference point to get inconsistent preferences.

To summarize these results from psychology, most stimuli presented in experiments are chosen consistently, contrary to probabilistic views. Of course, trying to __prove__ that choice is consistent for all stimuli when conditions are held constant is like trying to prove that a frictionless body moves forever in the absence of other forces. While the magnitude of the situational effects may be decreased (as frictional effects can be decreased), it is impossible to completely control them. Hence, this section taken alone cannot "prove" the deterministic assertion, but can only show that it is not inconsistent with present knowledge. Final acceptance of SM depends upon the other criteria considered below.

Violation of WST

Hagerty (1979) has derived the conditions under which SM violates WST. He has demonstrated that if the frequencies of situations affecting preference are uniformly and independently distributed, violations of WST are expected about 20% of the time. Thus these violations are expected to be relatively rare under SM unless situations are carefully selected. If it can be reliably demonstrated that even one experiment yields violations of WST, it would constitute evidence against EBA, since EBA requires properties even stronger than WST. We are searching for such an experiment by trying to manipulate context effects or satiation effects.

Context Effects

If the results of Tversky's (1972) experiment supporting EBA are shown to be reliable context effects, then a probabilistic model would not be needed. Context effects are here used to denote any hitherto ignored effects, such as sequence of trials, variety-seeking alternation of choice, or position effects. While these effects are expected to be small, the amount of inconsistency to be explained in this experiment is also small. (Recall from the last section that Tversky used only stimuli to which subjects were nearly indifferent.) Work is progressing on the effect of stimulus sequence on choice.

Variety-Seeking Behavior

Variety-seeking is a commonly suggested but rarely investigated effect which may account for the inconsistency in Tversky's (1972) experiment (i.e., subjects after seeing the same trial repeatedly could alternate their choice). It has been suggested by Bass, Pessemier, and Lehmann (1972) to explain the inconsistency they found. Finally, all of the stimuli in Tversky's counter example appear to be quite susceptible to variety-seeking (e.g., choosing repeatedly between a Beethoven and a Debussy recording). Thus a tested theory of variety-seeking may go far in explaining experimental inconsistency of choice.

The experiment we describe investigates how preferences for several fruit juices change over days when subjects are constrained to consume the same fruit juice day after day. (This constraint is imposed because it greatly simplifies model predictions and estimation, and because it highlights differences between alternative models.) Figure 1 plots the preference ratings for a pilot subject as a function of the day of week (the experiment began on Monday) with a separate curve for each fruit juice rated. The juices were Apple, Grape, Orange, Tomato, and V-8. This subject was given about 6 oz. of Orange Juice free of charge every morning and afternoon of the experiment. Ratings of preference on a 100-point scale were collected every afternoon before the Orange juice was given. To encourage careful ratings, the subject was actually awarded the juice which she ranked first on Thursday and Friday. Note that, consistent with a general satiation hypothesis, preference for Orange juice (the juice she consumed) decreases over time. The question of interest is how preferences change for the other juices.

Jeuland (1978) has suggested a model of variety-seeking. Hagerty (1979) has shown that this model reduces to a ratio model for brands not consumed. That is, each line in Figure 1 (except for Orange juice) is predicted to differ from the others only by a change in slope. However, under the constraints of this experiment, one cannot detect changes in slope. The reason is that the data are not completely ordered, as fundamental measurement techniques require. Instead, brands within each day are ordered (since all brands are rated on the same scale) but there appears to be no reasonable way for a subject to compare his preferences for a brand on two separate days. Thus a variety-seeking experiment seems to be limited to collecting partially ordered stimuli: the fiverated preferences for any day may be rescaled by any monotone function (since this would preserve the original orderings of the day's preferences), and this function may be different for each of the four days. This constraint on the data is even weaker than that assumed by MONANOVA, which allows rescaling the ratings by a __single__ monotone transformation. Nevertheless, the variety-seeking data are constrained enough that alternative models may be tested. For example, any crossovers in Figure 1 may not be scaled away even by a separate monotone function for each day. (Computing a main effect of Days, however, is meaningless, since monotone functions can always be found which render this effect zero.)

PREFERENCE RATINGS FOR 5 FRUIT JUICES ON EACH 4 DAYS WHEN ORANGE JUICE IS CONSUMED.

Jeuland's model, therefore, predicts in this experiment that all of the curves in Figure 1 (except Orange juice) can be rescaled to zero slope, so that unconsumed brands undergo no change in preference relative to each other. A reasonable alternative hypothesis is that brands which are similar to the consumed brand can decrease in preference almost as much as the consumed brand. For example, if brand A of Orange juice is consumed every day and decreases in preference enough to cause crossovers, then one might expect brand B of Orange juice to decrease almost as much and cause similar crossovers. Thus, this "similarity" hypothesis predicts a significant interaction (crossovers) which cannot be scaled away for the four unconsumed juices, and that the brands with lowest slope will be similar to the consumed brand. On the other hand, Jeuland's "null" hypothesis predicts no crossovers among the four unchosen brands. Since Figure 1 displays no crossovers (except for Orange juice which was consumed), this subject with these juices supports Jeuland's model. Work is progressing to run more subjects and to improve the measurement of preferences.

POSSIBLE APPLICATIONS

First, if SM is correct, then we have shown that there are certain conditions under which a consumer violates WST. Tversky (1969) points out that such consumers act as a "money pump." For example, if product A is preferred to B, which is preferred to C, but C is preferred to A, then the consumer would pay money to exchange C for B, would again pay to exchange B for A, and would pay once again to exchange A for C. Thus he would have paid money to finish with the same object he started with. Such a money pump would be of interest.

Second, SM reconciles the deterministic demand curves of classical economics with the stochastic character of individual choice. This can be seen in the demand curve for price by noting that SM predicts that in any given situation a given consumer has an indifference point. When the firm's price goes below this point, all of the consumer's purchases in that situation are devoted to the firm's product by the deterministic assumption of SM. Thus the aggregate demand function may be computed by (a) finding each subject's relevant situations, (b) finding each subject i's indifference points P_{ij} for each situation j, (c) finding how much product U_{ij} is used in each situation j (how heavy a user he is), and (d) cumulating all U_{ij}'s for which P_{ij} is greater than the price the firm is charging. Thus SM builds the demand curve up from individual consumers in individual situations, whereas economists tend to build it directly from individual consumers. [The demand curve arising from empirical applications of SM has a computational disadvantage in that the total profit curve is often discontinuous and not differentiable. Analyses assuming LCA, on the other hand, do yield differentiable profit functions under many conditions (see Hagerty, 1978), thus ensuring the existence of an expression for the firm's optimal price.] Noting that market researchers are already tending to analyze demand by situation (Shocker and Srinivasan 1979), this may be a case where marketing is ahead of theory in psychology.

Probabilistic-type models are often defended as parsimonious in that they are simple models which need not consider the many situations and external influences impinging on choice. However, considering the growing complexity needed by probabilistic models (e.g., EBA) to account for Tversky's experiments, the more deterministic models such as SM may now be simpler and more parsimonious theories.

REFERENCES

Bass, Frank (1974), "The Theory of Stochastic Preference and Brand Switching," __Journal of Marketing Research__, 11, 1-20.

Bass, Frank M., Pessemier, Edgar A, and Lehmann, Donald R. (1972), "An Experimental Study of Relationships between Attitudes, Brand Preference, and Choice," Behavioral Science,17, 532-541.

Becker, Gary M., De Groot, Morris H., and Marschak, Jacob (1963), "Probabilities of Choices among Very Similar Objects," __Behavioral Science__, 8, 306-311.

Best, Roger J. (1978), "Validity and Reliability of Criterion-Based Preferences," __Journal of Marketing Research__, 15, 154-160.

Block, H. D. and Marschak, Jacob (1960), "Random Orderings and Stochastic Theories of Responses," in __Contributions to Probability and Statistics__, eds. I. Olkin, S. Ghurye, W. Hoeffding, and H. Mann. Stanford: Stanford University Press.

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Hagerty, Michael R. (1978), "Model Testing Techniques and Price-Quality Tradeoffs," __Journal of Consumer Research__, 5, 194-205.

Jeuland, Abel P. (1978), "Brand Preference over Time," in __Research Frontiers in Marketing__, ed. S. C. Jain, 43, 33-37.

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Pessemier, Edgar A., Burger, Phillip C., Teach, Richard D., and Tigert, Douglas J. (1971), "Using Laboratory Brand Preference Scales to Predict Consumer Brand Purchases,'' __Management Science__, 6, 371-385.

Shocker, Allan D. and Srinivasan, V. (1979), "Multiattribute Approaches for Product Concept Evaluation and Generation," __Journal of Marketing Research__, 16, 159-180.

Silk, Alvin J. and Urban, Glen L. (1978), "Pre-Test-Market Evaluation of New Packaged Goods," __Journal of Marketing Research__, 15, 171-191.

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Tversky, Amos and Russo, J. E. (1969), "Similarity and Substitutability in Binary Choices," __Journal of Mathematical Psychology__, 6, 1-12.

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