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## Predictive Power of SUCR and TC

The completeness of SUCR and TC gives us a sense for the precision of their welfare guidance. However, it does not tell us exactly the precision of their welfare guidance for the observed choice sets, so we also calculate the predictive power of SUCR and TC “within sample” for both data sets.

To determine the predictions made by a relation for the observed choice sets, we follow Schwartz (1976) and Efe A. Ok (2002) in saying that the choice correspondence C induced by a (possibly incomplete) strict relation ≻ is C≻(Z ) = {x ∈ Z |y ≻ x for no y ∈ Z }.29 The tightness of the predictions given by C is useful for studying the precision of welfare guidance because what is predicted to be selected from a choice set based on C≻ is what is welfare optimal for that choice set. In the language of Bernheim and Rangel (2009), the elements of C≻ are the “weak individual welfare optimum” of choice set Z .

A natural measure of the predictive power of C≻ for an individual is the average size of C≻ for that individual. For this measure, the highest possible predictive power corresponds to a value of 1 (a single alternative predicted from all choice sets), and larger values represent less predictive power (more alternatives predicted).

In the experimental data, SUCR predicts that an average of 1.32 alternatives could be chosen for

29Bernheim and Rangel (2009) propose the same correspondence, which they denote as m≻(Z ).

Proportion of Individuals individuals with cyclic RP, whereas TC predicts that on average, 1.38 alternatives could be chosen.30 In the consumption data, the average number of predicted alternatives is 1.33 for SUCR, and 1.65 for TC.31 Figures 2.4 and 2.5 shows that SUCR makes tighter predictions than TC at the individual level. For a KS-test of equality of the distributions of SUCR and TC for the average number of predicted alternatives at the individual level is 0.0095 on the experimental data and <0.001 on the consumption data.

As our primary measure of predictive power, we use the average value of Selten’s index (Selten (1991)) instead because it has a theoretical grounding, has been used for related questions in the literature, and accounts for the number of available alternatives. With Selten’s index, the proportion of choices that a theory predicts successfully within-sample is reduced by the “area”, which is the relative size of the predicted subset compared with the set of all possible outcomes. In the notation of Selten (1991)}, it is written as m = r − a, where r is the relative frequency of correct predictions and a is the area. Because SUCR and TC always successfully predict the chosen option, r is always equal to 1, but a can vary by relation and choice set. For a relation ≻ and choice set Z , we deine a as the proportion of alternatives that are predicted to be chosen, so that a = |C≻(Z )| .

The choice set and its size are very straightforward to determine in the experimental data. As mentioned previously, in the consumption data we take the grand set of alternatives X to be the

30This diference is signiicant, as the two-sided paired t-test p-value is 0.0025.

set of all bundles that an individual has chosen at some point (in order to provide a stricter test of completeness). Analogously, we take the choice set to be the set of all bundles that an individual has chosen at some point (all bundles in X ) that are afordable at a given price and expenditure level.

We could consider the choice set to be every possible bundle on the budget line (as in Beatty and Crawford (2011)), but we consider this restricted space for three main reasons. First, it is far more computationally feasible to determine the set of predicted options given the large number of choices in our data set. Second, only bundles chosen elsewhere can generate inconsistencies. Third, this allows us to use the same metric across data sets. Fortunately, this approach provides a wide variety of choice set sizes, as shown in Appendix A.1.

One reason that we use Selten’s index is that it has an axiomatic foundation, and another is that it has been used elsewhere in the literature on empirical revealed preference analysis (Manzini and Mariotti (2010), Beatty and Crawford (2011), Dean and Martin (2016)). For example, Beatty and Crawford (2011) determine the fraction of demands that would pass a revealed preference test and then subtract this from an indicator for whether or not the observed choices passed the test. Their goal is to determine whether or not it is diicult for a set of choices to pass the revealed preference test for a given data set. Alternatively, Dean and Martin (2016) determine the average “distance” from rationality for all possible demands and then subtract this from the “distance” for observed choices.

### Average obtained by weighting by the num-ber of sets.

In the experimental data, the average value of Selten’s index for SUCR is 0.46 for individuals with cyclic RP, and for TC it is 0.44. The diference between the average Selten’s index for TC and SUCR is signiicant, as the two-sided paired t-test p-value is equal to 0.0022. For the experimental data, the average theoretical maximum of Selten’s index is just 0.58, as shown in Table 2.7. For a set size of 2, the maximum value of Selten’s index is 0.5; for a set size of 3, it is 0.66; and for a set size of 4, it is 0.75.

In the consumption data, the average Selten’s index is 0.95 for SUCR and 0.94 for TC. The diference here is signiicant, as the two-sided paired t-test p-value is <0.001. For the consumption data, the average theoretical maximum is 0.96.32 Selten’s index is higher in the consumption data compared to the experimental data because the size of the choice set appears in the denominator, and the choice set sizes are on average much higher on consumption data than they are on the experimental data.

Figures 2.6 and 2.7 provide the CDF of Selten’s index for both SUCR and TC for both data sets. The KS-test for equality of distributions has a p-value of <0.001 between SUCR and TC for both data sets.

As these results show, SUCR and TC have high predictive power on average for both the experi-mental and consumption data, which means that they provide precise welfare guidance. It qualiies our results on completeness shown in Section 2.3.3: the relations induced by SUCR and TC are complete enough to provide precise welfare guidance for observed choice sets.

#### Predictive Power and Revealed Preference Properties

In this section, we indicate when and why SUCR and TC have high predictive power and provide empirical evidence of these relationships. Speciically, we show that there are two properties of revealed preferences that are especially important for the predictive power of SUCR and TC: the of direct RP cycles and the fraction of RP cycles that are direct.

32In order to compute the theoretical maximum of Selten’s index, we assume that in all choice sets exactly one alternative is predicted to be chosen, and then take the average over all choice sets.

**Table of contents :**

**1 Introduction **

1.1 Revealed Preferences

1.2 Beyond Classical Preferences

1.2.1 Relaxing Transitivity and Menu-Independence

1.2.1.1 Partial Order

1.2.1.2 Interval Order

1.2.1.3 Semi-Order

1.2.1.4 Monotone Threshold

1.2.1.5 Menu-Dependent Threshold

1.2.1.6 Context-Dependent Threshold

1.2.2 Robust Revealed Preferences

1.2.2.1 The Strict Unambiguous Choice Relation

1.2.2.2 The Transitive Core

1.2.2.3 Other Welfare Relations

1.3 Organization of the Dissertation

**2 Predictive Power in Behavioral Welfare Economics **

2.1 Introduction

2.1.1 Empirical Findings in Behavioral Welfare Economics

2.2 Data

2.2.1 Experimental Data

2.2.2 Consumption Data

2.2.2.1 Analysis Sample: Panelists

2.2.2.2 Analysis Sample: Demographic Characteristics

2.2.2.3 Analysis Sample: Bundles

2.2.2.4 Analysis Sample: Prices

2.2.2.5 Additional Considerations

2.3 Results

2.3.1 Inconsistencies in Revealed Preferences

2.3.2 Inconsistencies in SUCR and TC

2.3.3 Completeness of SUCR and TC

2.3.4 Predictive Power of SUCR and TC

2.3.5 Predictive Power and Revealed Preference Properties

2.3.5.1 Number of Direct RP Cycles

2.3.5.2 Fraction of RP Cycles that are Direct

2.3.5.3 Regression Analysis

2.4 Discussion and Conclusion

**3 Identifying Choice Correspondences **

3.1 Introduction

3.2 Deinition of the Pay-For-Certainty Method

3.2.1 General Setup

3.2.2 Deinition of the Pay-For-Certainty Method

3.2.2.1 Incentive to Choose Several Alternatives

3.2.2.2 Selection Mechanisms

3.2.2.3 Linear versus Proportional Bonus Payment

3.2.3 Pay-For-Certainty with a Uniform Selection Mechanism

3.3 Identiication of the Choice Correspondence

3.3.1 Assumptions

3.3.2 Identiication of the Choice Correspondence

3.3.2.1 Partial Identiication of the Choice Correspondence

3.3.2.2 Full Identiication of the Choice Correspondence

3.3.3 Consistency when Full Identiication Fails

3.3.3.1 Compatibility with Classical Preferences

3.3.3.2 Compatibility with a Partial Order

3.3.3.3 Compatibility with Menu-Dependent Threshold

3.3.3.4 Compatibility with Fixed Point

3.4 Conclusion

**4 Pay-for-certainty in an Experiment **

4.1 Introduction

4.1.1 Menu Choice in the Literature

4.2 Design of the Experiment

4.2.1 Tasks

4.2.2 Timing

4.2.3 Information Treatments

4.2.4 Choices

4.2.5 Questionnaire

4.2.6 Data

4.2.7 Demographics Characteristics

4.3 Comparing Three Revelations Method

4.3.1 The Benchmark: Forced Single Choice

4.3.2 0-Correspondences

4.3.3 1-Correspondences

4.3.4 Identiied Choice Correspondence

4.3.4.1 Fully Identiied Choice Correspondences

4.3.4.2 Partially Identiied Choice Correspondences

4.3.5 Related Literature and Discussion

4.4 Going Further with Correspondences

6 CONTENTS

4.4.1 Intransitive Indiference

4.4.2 Menu-Dependent Choices

4.4.3 Discussion

4.5 Inluence of the Information Provided

4.6 Conclusion

**5 Conclusion**