Nonparametric Inference of Utilites

Dissertation / Doktorarbeit von Matthias Herfert

Abstract:

In Chapter 2, „Foundations”, we provide a description of selected parts of theories which we believe are helpful to better understand the contribution of this thesis. We start with the presentation of several behavioral hypotheses in preference and utility theory. Next, we describe the basics of inferential statistics and Conjoint Analysis. Then, we describe probabilistic entropy, in addition to that a later established version of it, and its axiomatization as a general inference principle. We conclude Chapter 2 by presenting La Mura's decision-theoretic entropy, a version of entropy as an inference technique for expected utilities. La Mura had developed this connection between probabilistic entropy and expected utilities in his Ph.D. thesis.

Based on his work, the initial research objective for this dissertation had been to make his approach applicable to the inference of unique consumer utilities given some observed evidence, having in mind the vast amounts of data that nowadays are available to analysts but still not used very effectively, in order to jointly overcome the limitations of Conjoint Analysis as mentioned above.

In the following five chapters you will see that our research has instead resulted in a new method, namely Entropy Analysis, which is not based on expected utility functions but on ordinary utility functions. We close Chapter 2 with a conclusion for the following chapters.

In Chapter 3, „Entropy Analysis”, we derive the new method combining probabilistic cross-entropy and ordinary utility functions. We start by imposing a set of conditions on the inference method. Then, we suggest a normalization of utility functions such that they become formally a probability measure. Finally, we present and prove our main result.

In Chapter 4, „Irrational Behavior”, we present a solution for the problem of how to treat observed „irrational” behavior (see Definition 4.1) with Entropy Analysis. This is motivated by two reasons. First, we are hardly able to observe „perfectly” rational data in any survey or for any given set of transaction data. Therefore, any utility inference method that cannot deal with irrational data will not be meaningful for research or commercial applications.

Second, our method is at first sight formally structured in a way in which its application to irrational data would return an inferred utility function that is trivial, i.e. uniform (to be further explained at the beginning of the chapter).

Our solution to this problem involves the principled use of a specific version of our method which we call relative Entropy Analysis, the cross-entropy version of Entropy Analysis. We start the chapter by presenting our general technique. Next, we substantiate our technique by suggesting one widely applicable heuristic.

In Chapter 5, „Consumer Choice Models”, we develop three consumer choice models to apply our method to marketing problems. We start by developing a basic model for consumer choices in which we consider preferences that relate product characteristics or bundles of goods with money.

Next, we constrain this basic model by imposing conditions on preference relations which imply utilities that are quasi-linear in money. We do this because such utilities reduce technical complexity for utility inference problems and because we believe that quasi-linear utilities (which imply the absence of income effects) are sufficiently representative for all items that have relatively low prices. Our third choice model uses von Neumann-Morgenstern expected utilities to apply our method to inference of utilities over risky alternatives.

In Chapter 6, „Applications”, we apply our method to synthetic, i.e. fictitious, data. We imagine hypothetical consumers whom we use to show the variety of possible applications of our method. We start with our quasi-linear choice model and infer estimates for additively separable utilities, inferior goods, perfect complements, and Cobb-Douglas preferences. Within these examples, we show that our method can compete favorably with Conjoint Analysis and that it is applicable to other structural forms of utilities besides additively separable ones as it is most often implied by Conjoint Analysis.

Then, we again use our quasi-linear consumer choice model and also our basic choice model to apply Entropy Analysis to inferences of willingness to pay functions for cases in which they are independent of the wealth of the consumer and for cases in which they are dependent on the wealth of the consumer.

One area of applications of inference of willingness to pay functions is the inference of price-demand curves. To illustrate this, we present an example with our quasi-linear consumer choice model, but in principle we are not constrained to that choice model. Next, we use our expected utility consumer choice model to infer estimates of utilities of lotteries over both money values and multiattributive alternatives. Finally, we show by the example of our quasi-linear consumer choice model how our method can return estimates from observed irrational behavior in the cases of both survey and transaction data.

In Chapter 7, „Summary and Outlook”, we provide a summary of our results and a research outlook for future studies.

Table of Contents:

1.	Introduction	1
1.1	Overview	1
1.2	Context, Motivation and Objective	2
1.3	Outline	8
2.	Foundations	12
2.1	Introduction	12
2.2	Preferences and Utility Functions	13
2.3	Methods of Inference	19
2.4	Conjoint Analysis	24
2.5	Probabilistic Entropy	33
2.5.1	Measure	33
2.5.2	The Principle of Entropy Maximization	38
2.5.3	The Principle of Cross-Entropy Minimization	43
2.5.4	General Inference Technique	44
2.6	Decision-Theoretic Entropy	52
2.7	Conclusion	58
3.	Entropy Analysis	60
3.1	Introduction	60
3.2	Setup	61
3.3	The Method	62
3.4	Conclusion	79
4.	Irrational Behavior	80
4.1	Introduction	80
4.2	The General Principle	82
4.3	A Heuristic	86
4.4	Conclusion	89
5.	Consumer Choice Models	91
5.1	Introduction	91
5.2	The Basic Utility Setup	92
5.3	Quasi-Linear Utilities	94
5.4	Expected Utilities	96
5.5	Conclusion	98
6.	Applications	100
6.1	Introduction	100
6.2	Additively Separable Utilities	104
6.2.1	Setup	104
6.2.2	Results with Small Amounts of Data	105
6.2.3	Implementation of Entropy Analysis	106
6.2.4	Implementation of Conjoint Analysis	112
6.2.5	Results with Larger Amounts of Data	114
6.3	Inferior Goods	116
6.3.1	Setup	121
6.3.2	Data and Result	123
6.4	Perfect Complements	125
6.4.1	Setup	127
6.4.2	Data and Result	127
6.4.3	Implementation of Entropy Analysis	130
6.4.4	Implementation of Conjoint Analysis	132
6.5	Cobb-Douglas Preferences	134
6.5.1	Setup	134
6.5.2	Small Amounts of Data	135
6.5.3	Implementation of Entropy Analysis	137
6.5.4	Larger Amounts of and Partially Lost Data	138
6.6	Wealth-Independent Willingness to Pay	139
6.6.1	Definitions	140
6.6.2	Setup	142
6.6.3	Data and Results	142
6.7	Wealth-Dependent Willingness to Pay	145
6.7.1	Setup	146
6.7.2	Data and Results	148
6.8	Price-Demand Curves	152
6.8.1	Setup	153
6.8.2	Data and Result	154
6.9	Money Lotteries	156
6.9.1	Setup	157
6.9.2	Small and Larger Amounts of Data	157
6.9.3	Implementation of Entropy Analysis	161
6.10	Multiattributive Lotteries	162
6.10.1	Setup	163
6.10.2	Small and Larger Amounts of Data	163
6.10.3	Implementation	166
6.11	Irrational Survey Data	167
6.11.1	Setup	167
6.11.2	Small Amounts of Irrationality	169
6.11.3	Implementation of Entropy Analysis	174
6.11.4	Larger Amounts of Irrationality	175
6.12	Irrational Transaction Data	175
6.12.1	Setup	176
6.12.2	Small Amounts of Irrationality	176
6.12.3	Larger Amounts of Irrationality	178
6.13	Conclusion	179
7.	Summary and Outlook	180
8.	List of References	182

Table of Contents:

1.	Introduction	1
1.1	Overview	1
1.2	Context, Motivation and Objective	2
1.3	Outline	8
2.	Foundations	12
2.1	Introduction	12
2.2	Preferences and Utility Functions	13
2.3	Methods of Inference	19
2.4	Conjoint Analysis	24
2.5	Probabilistic Entropy	33
2.5.1	Measure	33
2.5.2	The Principle of Entropy Maximization	38
2.5.3	The Principle of Cross-Entropy Minimization	43
2.5.4	General Inference Technique	44
2.6	Decision-Theoretic Entropy	52
2.7	Conclusion	58
3.	Entropy Analysis	60
3.1	Introduction	60
3.2	Setup	61
3.3	The Method	62
3.4	Conclusion	79
4.	Irrational Behavior	80
4.1	Introduction	80
4.2	The General Principle	82
4.3	A Heuristic	86
4.4	Conclusion	89
5.	Consumer Choice Models	91
5.1	Introduction	91
5.2	The Basic Utility Setup	92
5.3	Quasi-Linear Utilities	94
5.4	Expected Utilities	96
5.5	Conclusion	98
6.	Applications	100
6.1	Introduction	100
6.2	Additively Separable Utilities	104
6.2.1	Setup	104
6.2.2	Results with Small Amounts of Data	105
6.2.3	Implementation of Entropy Analysis	106
6.2.4	Implementation of Conjoint Analysis	112
6.2.5	Results with Larger Amounts of Data	114
6.3	Inferior Goods	116
6.3.1	Setup	121
6.3.2	Data and Result	123
6.4	Perfect Complements	125
6.4.1	Setup	127
6.4.2	Data and Result	127
6.4.3	Implementation of Entropy Analysis	130
6.4.4	Implementation of Conjoint Analysis	132
6.5	Cobb-Douglas Preferences	134
6.5.1	Setup	134
6.5.2	Small Amounts of Data	135
6.5.3	Implementation of Entropy Analysis	137
6.5.4	Larger Amounts of and Partially Lost Data	138
6.6	Wealth-Independent Willingness to Pay	139
6.6.1	Definitions	140
6.6.2	Setup	142
6.6.3	Data and Results	142
6.7	Wealth-Dependent Willingness to Pay	145
6.7.1	Setup	146
6.7.2	Data and Results	148
6.8	Price-Demand Curves	152
6.8.1	Setup	153
6.8.2	Data and Result	154
6.9	Money Lotteries	156
6.9.1	Setup	157
6.9.2	Small and Larger Amounts of Data	157
6.9.3	Implementation of Entropy Analysis	161
6.10	Multiattributive Lotteries	162
6.10.1	Setup	163
6.10.2	Small and Larger Amounts of Data	163
6.10.3	Implementation	166
6.11	Irrational Survey Data	167
6.11.1	Setup	167
6.11.2	Small Amounts of Irrationality	169
6.11.3	Implementation of Entropy Analysis	174
6.11.4	Larger Amounts of Irrationality	175
6.12	Irrational Transaction Data	175
6.12.1	Setup	176
6.12.2	Small Amounts of Irrationality	176
6.12.3	Larger Amounts of Irrationality	178
6.13	Conclusion	179
7.	Summary and Outlook	180
8.	List of References	182

Text Sample:

Chapter 2.4, Conjoint Analysis:

As mentioned in Chapter 1, Conjoint Analysis is the most popular utility inference method available today. The name Conjoint Analysis does not represent one single, in some sense well-defined, formula or technique to infer a utility function in a given context. Instead, it is a collection of approaches that has been extended by many researchers with a plentitude of refinements and improvements.

A common core that all approaches under the umbrella Conjoint Analysis share is a link to the initial and seminal contribution that introduced conjoint measurement and the usage of conjoint measurement in marketing for utility inference. The differences between most approaches can be found in how data are collected and parameters for the inferred utility function are estimated.

In contrast to the so-called expectancy-value models, a compositional approach in which the utility for some object is determined by the weighted sum of the object's perceived attribute levels and associated value ratings separately judged by the respondent, Conjoint Analysis is a decompositional approach. Respondents judge a set of product descriptions, and then the analyst finds so-called part-worths for the individual attributes that are most consistent with the respondents' overall preferences.

Since its start in the early 1970s, a plethora of new Conjoint Analysis models has been introduced to improve various aspects of the method. Nevertheless, the basic framework has not changed. Therefore, we would like to follow the lines of an overview and procedural description of the Conjoint Analysis methodology given by Green and Srinivasan.