All things are subject to interpretation. Whichever interpretation prevails at a given time is a function of power and not truth.
Friedrich Nietzsche

## Abstract

An implementation of a non-compensatory modeling and simulation of product choice probability is presented. The procedure relies on the thresholds (cutoffs) of acceptability, under which the product is rejected regardless of other properties. Computed perceptances that respect the thresholds do not suffer from taking extremely large values, often observed with the standard compensatory model. Simulation appears to be more realistic especially for situations when an individual must decide whether to accept or reject an offered product when alternatives are unavailable.

## Problem

Analysis and simulation of choices from a survey is commonly based on assumption of additive contributions (part-worths) of individual attribute levels (the aspects of the product) to utility of the product. In this approach, a product is understood as a bundle of aspects. The formulation of utility of a product is equivalent to Cobb-Douglas utility function of the product aspects. While this approach has many good properties, it has also some less welcomed ones.

• Contributions of levels of different attributes can not be compared.
The standard model allows to specify only differences between the level part-worths of an attribute. Reference value, usually zero, is set as the average level part-worths of the attribute (effect coding), or one of the levels is set to zero (dummy coding).
• The attribute value under which the product is refused is unknown.
Additivity of contributions is the vehicle for outweighing part-worth of unacceptable levels to obtain acceptable utility of the product. In reality, subjects have a very clear idea of which characteristics an acceptable product must have, which are tolerable, and which are not. Such a view is in principle excluded from the standard model.
• Computed acceptance of a product has a value comparable only within the current analysis.
Acceptance is defined as the probability of choosing between the considered and a referential product with zero utility. In case of centered utilities, the referential product is a hypothetical product with all attributes set to the attribute centers. Its acceptance is exactly 50%.
• Estimated preferential shares include the subjects who might reject all profiles in the choice set.
This is because all profiles have principally positive odds that are re-scaled to give sum of 100% before being averaged.
As aside
• The above list is not exhaustive. These are only those that the suggested approach is addressing.

Marketing experts and psychologists have shown that a compensation of negative by positive aspects of a product does not correspond to the real behavior. A number of non-compensatory models have been proposed. Those cited most often are:

• Lexicographic model
Based on the level of satisfaction with aspects sorted by relevance: "I will take the best on the most."
• Disjunctive model
Based on the level of satisfaction with the most important aspect: "Okay I am flexible; must have either this or that."
• Conjunctive model
Based on sufficient satisfaction with all aspects or their subset: "Must have at least this much of these."
• EBA (Elimination By Aspects)
Exclusion on the basis of unfavorable or the least satisfactory aspect: "At least this much on the most."
• Combined models (e.g. disjunction of conjunctions), and other models (e.g. distance from the ideal combination of aspects)

In all models, a product is rejected when a certain acceptability threshold is not exceeded. An unsatisfactory aspect cannot be compensated by another, even highly satisfactory one. Linked to a little satisfying aspect is the fact that growth of satisfaction with other aspects may not necessarily lead to a higher satisfaction with the product. Other properties are a matter of opinion, and are interpreted differently by different authors.

Market research studies rely on a limited number of respondents who are supposed to represent the potential target. Non-compensatory models are qualitative in nature, and would provide at most ranking of products in a scenario. Since ranks cannot be averaged, namely in small samples, pure non-compensatory models are not suitable for a simulation.

Additive utility models give quite a good correlation between the observed and predicted results. However, such comparisons were done using the results from the study, rather than confronting market data. From the response of our clients we know that the trends derived from the additive model are correct, but the calculated characteristics (e.g. shares) often substantially differ from those found in the real market. The main causes can be seen in the method of interviewing, neglect of respondents' non-compensatory reasoning, and the estimation and simulation model.

In the traditional CBC - Choice Based Conjoint, a respondent chooses the product out of several product profiles offered. If the product is CPG/FMCG, fast food or other goods requiring little involvement, the situation induced in the test is very similar to a real situation. Thresholds are hardly significant because usually several products exceed the acceptability thresholds. The compensatory model is appropriate. The situation is different in case of durable products or repeated and/or long-term services that require high involvement of the decision maker. Acceptability thresholds play an important role when people decide about higher than common daily expenses, whether for household equipment, rent, ownership, lease agreement, financial services, etc.

## Solution

### Model

For the purpose of a quantitative preferential simulation, disjunctive model could have been excluded. If a subject were deciding by a single aspect, a sorting or sequential choice questions on the aspects would give the full answer. Any further analysis would be superfluous.

Conjunctive model can represent "satisficing", i.e. a way when the chosen product is the first product that exceeds thresholds in all important attributes. Such a product is therefore both sufficient and satisfying. The model is suggested for situations when searching for the optimal product would require additional costs (time, money, effort, etc.) for the decision maker. Because conjunctive model contains a large percentage of randomness, its pure form is not suitable for products being selected with a certain amount of caution. In this respect, lexicographical and EBA models suit better.

In lexicographical model, subject accepts and replaces products in their consideration set gradually, as evaluating products one by one. In EBA model, subject gradually eliminates the products until the final consideration set is left. Both models lead to the same consideration set of some size from which the final choice is made.

Non-compensatory models allow for ranking of products in a set. In a quantitative simulation, product characteristics must be continuous numeric and capable of averaging. Therefore, a hybrid model had to be developed.

The conjunctive model has been taken as the default non-compensatory model. A product can be accepted only if all its aspects are acceptable. When a product is understood as a bundle of aspects valued independently, the product acceptability is assumed as (not necessarily linearly) proportional to a weighted product of the aspect perceptances. In the simplest case, the weights are constant exponents. This choice has some important consequences in the space of positively perceived aspects.

• The model is an application of Weber-Fechner law for cognitive processes completed with Michaelis-Menten quasi-steady-state approximation of accepting and refusing an aspect. A resemblance with the neuron-based explanation of mental scaling in psycho-physics may not be accidental.
• The sum of part-worths in the traditional logit formula in the standard compensatory model has been replaced with weighted logarithms of perceptances. As the model has some traits common to prospective theory we name it prospective model
• The model is additive in logarithms of subjective aspect perceptances. From this point of view, the model is weighted compensatory. However, as perceptances have a saturation limit (100% satisfaction), neither profile perceptances nor profile utilities are a linear, or linearizable, function of (isolated) aspect odds.
• Perceptance of a product cannot exceed perceptance of its least acceptable aspect. From this point of view, the model is fully non-compensatory.
• The model is descriptive rather than normative. It is aimed to model real-life choices, rather than rationally optimal decisions, as would normative models do.

When at least one of the aspects is negative, only the negative aspects are used for product perceptance estimation. This reflects the fact people mostly ignore positive aspects if some other aspects are the cause of the product refusal.

The basic questionnaire for obtaining data for analysis and prospective simulation is composed of sections described on page CSDCA - Common Scale Discrete Choice Analysis.

As aside
• The model, interviewing method and estimation of parameters are designed so that it is possible to use commercially available tools for analyzing discrete choice problems. There is an implicit contradiction in the solution. While analytical software assumes additive contributions to the utility of the product, the simulation model is based on non-compensatory behavioral model. The contradiction is mitigated, but not fully resolved, with the following assumptions and practices.
• The questionnaire contains a block of questions on acceptability of levels in each of the attributes. Additional choice tasks are created that allow for estimation of thresholds. In the literature, such tasks are referred to as "soft constraints".
• All attribute levels in an acceptable profile, e.g. chosen in CBC, are considered as acceptable, even if they were indicated as unacceptable in the separate questions. The stated threshold values are corrected appropriately.
• The values from the DCM analysis, namely part-worths, are not used in a simulation directly, but are corrected according to the probabilistic model of non-compensatory behavior.
• The entire process is to be regarded as a merger of several non-compensatory and weighted compensatory rules comprised into in a practical solution.
• The proposed model does not address the IIA - Independence from Irrelevant Alternatives problem. The ratio of computed choice probabilities of any two profiles is constant regardless of the presence or absence of other profiles.

### Constant alternative "None"

Choice sets in a CBC questionnaire are usually completed with the constant alternative "None of the offered" enabling refusal of any of the offered product profiles. It often seems to be an escape for respondents in place of giving an answer. We use the estimated value only for test and comparison reasons, but do not rely on it.

### Product perceptance

Perceptance of a product is a complex function of perception of product attributes. In our model it is estimated as a weighted product of attribute perceptances. It is positive if perceptances of all aspects are positive for the subject. If perceptance of any of the important attributes is zero or negative, the product is rejected and its perceptance is set zero. With this approach the problem of counterbalancing unacceptable levels with acceptable levels is avoided.

A product may have some relatively unimportant or insignificant attributes. The threshold for such an attribute is identified below all the attribute level part-worths, and the attribute cannot be cause of rejection of the product in a simulation. The model gives the answer about attribute importance in a way completely different from the standard compensatory model.

For the most reliable indicator of a product to be successful on the market is usually an estimate of preferential share not accounting for the volatile constant alternative. To obtain such a value, a reasonable choice set must be created using competing products. A good indicator of acceptability independent from other items in a choice set should be some form of a stated acceptability. The perceptance of the product is a suitable candidate.

As aside
• Please note that the formula for perceptance does not contain any value related to the constant alternative. The stated and estimated thresholds are essential.

## Comparison of compensatory and non-compensatory approaches

Property Compensatory model Non-compensatory model
Attribute thresholds Unknown Initial values for thresholds are estimated in separate PRIORS tasks used as soft constraints in level part-worth estimation. They are is rectified using answers from a conjoint-like exercise
Level acceptability Undefined Level perceptance is based on estimated attribute threshold and is defined in the interval (-100%, 100%) to reveal either positive or negative attitudes to the level. The zero value corresponds to disinterest in the level.
Product acceptability Acceptance is a relative measure related to the average product seen by respondents in the conjoint exercise. Comparison between exercises is mostly impossible Product perceptance reflects thresholds of all important attributes. Under constant methodology conditions, it can be compared between different conjoint exercises or even studies. Unlike the level perceptance, it is not defined for negative values.

Additive nature of the model allows any attribute to outweigh all the other attributes Non-additive nature of the model excludes outweighing of all the other important attributes

Acceptance is controlled by all attributes indiscriminately Perceptance is mainly controlled by perception of an important attribute with the lowest acceptability. This is a fundamental difference
Questionnaire
A standard CBC exercise consist of repeating tasks of a similar feel. The tasks are often characterized by respondents as complicated an annoying. While the whole questionnaire block is a bit longer than a conjoint exercise, it is composed of several short exercises, each of a different type and character as described on page CSDCA - Common Scale Discrete Choice Analysis. The fatigue from interviewing can be lower.

A calibration exercise may be used to transform relative CBC-based utilities to reflect the stated intentions. A calibration is inherent to non-compensatory modeling.

## Example

### Questionnaire

The model was tested on a financial product – a bond – with three attributes:

Attribute Levels Attribute type
1. Bond issuer A, B, C, D nominal
2. Annual interest rate 1.5, 2, 2.5, 3, 3.5% p.a. ordinal
3. Bond maturity 3, 4, 5 years nominal

The questions were asked in two blocks.

• The purpose of the block PRIORS was to determine the preferences of attribute levels, and initial estimates of acceptability thresholds. For bond issuer and bond maturity attributes, respondents were presented with all randomized levels to sequentially choose the levels one by one. Each chosen level was asked for its acceptability. The questioning was stopped when the threshold was reached, i.e. all remaining levels were refused.
Annual interest rates were presented one by one (the Gabor-Granger "price wheel" method) starting with the worst 1.5% p.a., and each value was queried for acceptability. The questioning was stopped when the answer was "definitely yes".
As aside
At present, this method has been replaced with the algorithm that merges both randomized and systematic "price wheel" querying [Lipovetsky, S., Magnan, S., and Polzi, A.Z. (2011). Pricing Models in Marketing Research. Intelligent Information Management, 3, 167-174].
• The second block was the standard CBC with 8 choice tasks of 3 profiles completed with a constant alternative "None of them". More tasks were unnecessary because part of the required information was obtained from the first block.
As aside

Of the 1231 recruited respondents, 927 expressed potential interest in the product and answered questions from both blocks of the questionnaire. The average interviewing time was 4.5 min.

The web questionnaire was created in Sawtooth SSI ver. 8.4.8 system. Data was processed using Sawtooth CBC/HB 5.5.4 and IBM SPSS Rel. 23 software. The simulator was written in MS Excel 2007.

### Results and discussion

As aside
• Presentation of attribute thresholds, perceptances of the levels and their distribution in the sample is omitted in order to preserve the rights of the client.
• Results from choice-based studies using the standard compensatory model provide only relative values. This must be accounted in comparison of the two approaches.

Below is a picture of a preferential simulator output. It contains 3 groups of results for the standard compensatory (additive) simulation model, each group having 4 lines, and 3 analogous groups for the perceptional non-compensatory model. The first 3 lines in a group are for 3 segments of respondents categorized by their savings. The 4-th line shows the results for the whole sample.

The simulated profiles make two groups. The first four profiles, tinged with red, are created for the four bond issuers. All have the least favorable interest rate of 1.5% p.a., and repayment period of 5 years. The second five-member group of profiles, tinged with green, is made up of profiles for each of the five levels of interest. All have the preferred issuer C, and the most popular maturity of 3 years. The selection of profiles defined this way guarantees covering the whole acceptability range of profiles.

Comparison of compensatory and non-compensatory models

#### Acceptance vs. perceptance, choice "None" ignored

Acceptance is a conventional characteristic in the interval (0%, 100%) spanning a range from definitely unacceptable to definitely acceptable profile. As specification of the reference profile is unknown, so is meaning of its value. The actual value is influenced by the span of attributes and, through it, by the environment created by the other profiles shown in the interview. In contrast, perceptance is a measure based on acceptability of levels stated by respondents. It is positive for profiles, which have all the attributes above the threshold value. In all other cases, it is zero.

Both compensatory and non-compensatory models give values of similar magnitudes for the four-member group of profiles. This is probably only coincidence. However, striking differences exist when the five-member group of profiles is compared. For example, changing maturity from 5 years (profile no. 3) to 3 years (profile no. 5) results in a change of the standard acceptance from 10.7% to 52.2%, almost five times. Change in maturity is clearly outweighing the small interest rate. This hardly believable change is due to additivity. In contrast, perceptance is changed from 6.0% to 10.6% only, which is plausible. Increase of interest rate from 2.0% to higher values is showing only small increase in standard acceptance, as if the saturation were reached. Perceptance, in contrast, shows steady growth along the tested values, which seems to better reflect the general understanding of interest rate.

#### Acceptance vs. perceptance, choice "None" respected

The shown values are the probabilities of choice from two alternatives - the profile and the constant alternative "None", i.e. probabilities of accepting the profile when availability of some other profiles is known to the subject. The term acceptance in this case is fully justified for both models because it corresponds to the original definition. Both probabilities are computed in a similar way. As thresholds are respected in the perceptances, the respective values are generally lower than those of standard acceptances.

For the first group of profiles, the standard acceptances not reflecting and reflecting the alternative "None" do not differ much. This result is surprising. It was expected that the standard acceptances would drop substantially when accounted for presence of the alternative "None". In contrast, this is observed with the perception-based values. It is believed that perceptance without considering the None value is more likely to reflect real behavior. In a more detailed inspection of the source data we have noticed respondents answered "None" even when profiles with all levels above the thresholds stated by them in the preceding questions were present.

As aside
• Simultaneous consideration of both attribute thresholds and the refusals indicated by the alternative None is not without pitfalls. When inspecting individual data from later studies, we have observed that inclusion of the alternative None introduced (for some respondents a very heavy) bias in the computed perceptances and decreased reproducibility of choice data. The reason is most probably in insufficient reliability of the None choices. Use of the characteristic "perceptance with 'None' respected" has been abandoned.

#### Standard vs. perceptional shares

Shares are results of applying the multinomial logit using the estimated utilities of profiles. The constant alternative "None" does not enter this formula. The total share is 100%.

The share values computed from both models for the first four-member group of profiles are very similar. Differences are observed for the profiles in the second five-member group of profiles.

In the unrealistic situation when all profiles of the five-member group were offered at a time, the subject would choose the profile with the highest interest rate. Bot models give the same result of orders, nevertheless closer to the truth is the standard compensation model with the dominating preference of the highest interest rate. However, the differences between profiles are unrealistically high. If profiles from this group were offered independently, as is usual in reality, or randomly one by one, the result of simulation based on non-compensatory model is more likely.

The non-compensatory model allowed to include in preference shares 813 respondents for whom at least one of the 9 profiles was acceptable. The other 114 respondents with no willingness to accept any of the profiles could have been excluded. The conventional compensatory model lacks this property.

## Conclusions

Our experience suggests the compensatory model is suitable for a situation when choice decision takes place in presence of products that are more or less mutually replaceable. This is typical for FMCG/CPG products in stores with a rich selection, for goods in catalogs of online stores, and the like. However, it cannot be used for prediction of satisficing, i.e. choice of a just both sufficient and satisfying product.

There are situations where the offered products are not interchangeable, are interchangeable only with difficulty or some other cost, or essentially a single product is offered. When only adoption or rejection of a product is possible, and perception prevails in making the decision, a non-compensatory model of behavior might be preferred. To build the model, the knowledge of acceptability thresholds is essential. The model may be also useful in cases where prediction of satisficing is desirable.

The consideration of thresholds provides an alternative to calibration. A calibration is basically a transformation of the raw utilities obtained from the standard compensatory model. Sensitivity to all attributes is adjusted with the same multiplier thus leaving levels of different attributes incomparable. In contrast, perceptional thresholds of individual attributes are mutually independent and can indicate critical attributes and their levels.