Non-rationalizable Individuals, Stochastic Rationalizability, and Sampling
aa r X i v : . [ ec on . T H ] F e b Non-rationalizable Individuals, Stochastic Rationalizability, andSampling ∗ Changkuk ImDepartment of EconomicsThe Ohio State [email protected] John RehbeckDepartment of EconomicsThe Ohio State [email protected] 9, 2021
Abstract
Experimental work regularly finds that individual choices are not rational-ized by any preference. Nonetheless, recent work shows that data collectedfrom many individuals can be stochastically rationalized by a distributionof individuals with well-defined preferences. To examine this phenomenon,we study the relationship between deterministic and stochastic rationality.We show that a population can be stochastically rationalized even when halfof the individuals in the population cannot be rationalized by well-definedpreferences. We also find the ability to detect individuals in a populationwho are not stochastically rationalized can decrease as sample size increases.We discuss how these issues interact with cross-sectional and panel samplingschemes.
JEL Classification Numbers:
C00, D01, D11
Keywords:
Stochastic rationalizability, revealed preference, demand types, sam-pling ∗ We thank Roy Allen and P.J. Healy for helpful comments. Introduction
Experimental and empirical studies regularly find that individuals make choicesthat cannot be rationalized by any well-defined preference ordering. These find-ings suggest that any dataset with many individuals will contain some individualswho make choices that are not rationalized by any preference ordering. However,recent studies show that large cross-sectional datasets [Kitamura and Stoye, 2018,Deb et al., 2019] can be rationalized by a distribution of individuals with well-definedpreferences. When distributional choices from a population are rationalized in thisway, we say the population dataset is stochastically rationalized. This seems contra-dictory because even though there are likely individuals who make choices that arenot rationalized by a well-defined preference, the distributional data from the pop-ulation is rationalized by a distribution of individuals with well-defined preferences.This paper examines this discrepancy in a theoretical way.Throughout the paper, we focus on datasets with two consumption goods andobservations from two periods for simplicity. First, we show that even when thereare individuals who are not rationalized by any preference order in a population,the population can still be stochastically rationalized by a random utility model[McFadden and Richter, 1990]. This example holds with no sampling error. We saythat this is a false acceptance of stochastic rationalizability . We find that false ac-ceptances can occur even when half of the population make choices that cannot berationalized. Practically, this means that only populations with a majority of individ-uals are not rational is guaranteed to be detected as not stochastically rationalizable.Thus, when one finds a stochastic rationalization, one should be cautious when draw-ing conclusions from this data since a portion of the choices may be generated byindividuals who cannot be rationalized.Next, we take this insight to examine how cross-sectional sampling, panel sam-pling, and multinomial sampling interact with stochastic rationalizability. For cross- We take a preference ordering as a complete and transitive weak preference order. In-dividual choices are rationalizable by a preference ordering when their choices satisfy the re-vealed preference conditions of Richter [1966], Afriat [1967], and Varian [1982]. These conditionsare empirically violated in numerous domains such as household consumption [Echenique et al.,2011, Demuynck and Seel, 2018], risk and uncertainty [Choi et al., 2007, 2014, Carvalho et al.,2016, Carvalho and Silverman, 2019, Feldman and Rehbeck, 2020], altruistic allocation problems[Andreoni and Miller, 2002, Fisman et al., 2007], and so on. false rejection of stochastic rationalizability where a re-searcher erroneously rejects that the sample dataset is stochastically rationalizableeven when the population consists of only rational individuals. For panel sampling,we find that is a researcher ignores the panel structure (i.e. does not examine ra-tionality for each individual), then false acceptance persists but false rejections areimpossible. For multinomial sampling, we provide an analytical formula to computethe probability a population is stochastically rationalized for a given sample size.Here we show for a population with individuals who are not rationalizable that in-creasing the sample size reduces the ability to detect these people when looking for astochastic rationalization.Our findings in this paper contribute to the existing literature in several aspects.First, we clarify the relationship between individual rationality from Afriat [1967]and Varian [1982] to stochastic rationality from McFadden and Richter [1990] andMcFadden [2005]. This example also clarifies the statement from Kitamura and Stoye[2018] that “a stochastic demand system is rationalizable if and only if it is a mixtureof rationalizable nonstochastic demand systems” (p.1889). In particular, we showa mixture of demand systems that include non-rational deterministic demand typescan be stochastically rationalized. In a broader sense, our findings are closely relatedto Becker [1962] since we show that aggregate stochastic data of many individualsmay be categorized as rational even when a fraction of the population cannot berationalized.Our results also give insight for furthering the research evaluating stochasticrationality. In this paper, we suggest how to account for the “power” of stochasticrationalizability by using a multinomial sampling scheme of estimated demand types.One interesting finding is that one may lose substantial “power” even when one hasa large dataset. This is an analogue to comparing deterministic rationalizability tothe “power” of a random sample to reject the model following Bronars [1987].The results are also important for other studies building on the random utilitymodel such as Aguiar et al. [2018] and Deb et al. [2019] when trying to make coun-3erfactual or welfare comparisons. For example, Deb et al. [2019] make welfare com-parisons based on the proportion of rationalizable demand types recovered from theaggregate stochastic choice dataset. However, since it is possible to falsely categorizea proportion of individuals as rational when looking for a stochastic rationalization, aresearcher may incorrectly estimate the proportion of rational individuals and obtainerroneous counterfactual and welfare estimates.The rest of this paper is organized as follows. Section 2 reviews the definitionsof deterministic rationalizability and stochastic rationalizability based on the strongaxiom of revealed preference. Since two budget sets and two consumption goods casesare sufficient to discuss our main research questions, we focus on this setting. Section 3provides an intuitive example and the main results without sampling error. Section4 extends the analysis by considering cross-sectional sampling, panel sampling, andmultinomial sampling. Section 5 provides our final remarks.
Here we define the standard consumer problem and the extension to randomutility models. We consider the random utility model [McFadden and Richter, 1990,McFadden, 2005] for the standard consumer problem following Hoderlein and Stoye[2015] and Kitamura and Stoye [2018]. It is enough to consider the standard con-sumer problem when there are two goods to discuss our main research question.Moreover, the insights on the relationship between deterministic and stochastic ratio-nalizability are most clearly seen when there are two budget sets as shown in Figure 1.Throughout the paper, we assume for normalized prices from observation one ( p ) andobservation two ( p ) that there exists ˆ x P R ` with p ¨ ˆ x “ p ¨ ˆ x so that budget linesoverlap and violations of rationality can be detected.4 x p p Figure 1: Two overlapping budget sets with two consumption goods
The deterministic dataset of interest is given by D D “ tp p , x q , p p , x qu whereprices in the t -th observation are given by p t P R `` and the observed consumptionbundle is given by x t P R ` . We consider the normalized budget set defined by B p p t q “ t x P R ` | p t ¨ x ď u . We consider the standard consumer problem whenthere is a locally non-satiated utility function u : R ` Ñ R that yields a uniquemaximizer. Thus, if the observed choices are rationalized by utility maximization,then x t “ argmax x P B p p t q t u p x qu . It is well known that choices are consistent with a non-satiated utility functionwith singleton demand when the dataset satisfies the strong axiom of revealed pref-erences. This is a strengthening of the general axiom of revealed preference [Afriat,1967, Varian, 1982]. The strong axiom of revealed preference for a dataset with twoobservations can be stated as two intuitive conditions. First, for distinct observations,if the s -th bundle chosen costs strictly less than expenditure from the t -th observa-tion at prices from the t -th observation, then the t -th consumption bundle must coststrictly more than than expenditures of the s -th consumption bundle at the s -th A utility function is defined as locally nonsatiated when for any x P R ` and any ε ą y P R ` with || y ´ x || ď ε such that u p y q ě u p x q . For a statement of the strong axiom of revealed preference, see Houthakker [1950] orChambers and Echenique [2016]. s -th ob-servation lies on the t -th budget line. We record the characterization of deterministicrationalizability with a unique maximizer below. Proposition 1.
The dataset D D is rationalized by a locally non-satiated utility func-tion with unique maximizers if and only if for all t, s P t , u with t ‰ sp t x s ă p t x t implies p s x t ą p s x s and p t x s “ p t x t implies p s x t “ p s x s . A stochastic demand system dataset is given by D S “ tp p , π q , p p , π qu whereprices in the t -th observation are given by p t P R `` and π t is a distribution ofconsumption bundles where supp p π t q Ď B p p t q “ t x P R ` | p t ¨ x ď u . For simplicity,we let each distribution π t be over a finite number of consumption bundles. Thesupport of the distribution π denoted supp p π q is the set of points assigned positiveprobability by π .We now define a random utility model following Kitamura and Stoye [2018].First, let π t p O q be the probability that a choice bundle is in the measurable set O Ď R ` . Let U be the space of strictly quasiconcave locally non-satiated utilityfunctions u : R ` Ñ R . A dataset D S is rationalized by a random utility model (RUM) when there is a probability measure ρ over the space of functions U suchthat, for all t P t , u : π t p O q “ ρ ´ u P U : argmax x P B p p t q u p x q P O (¯ , (1)for any measurable subset O Ď R ` . The argmax set is a singleton since U consists ofstrictly quasiconcave functions. In other words, the probability of choosing a bundlein the set O is equal to the probability of drawing a utility function that is maximizedover B p p t q at some point in the set O .First, we know that any locally non-satiated utility function generates choiceson the budget line with probability one. Looking at Figure 2, we see that demandcan fall in one of three regions for each budget line. For example, x | is the region6 x x | x | x | x | x | t p p Figure 2: Demand types for two budget sets and two goodsfrom the first observation that is above the second budget line. Moreover, x | t is theconsumption bundle where the budget lines intersect. In general, x r | t are choices inthe r -th region of the t -th budget set.As shown by Hoderlein and Stoye [2015], stochastic rationality of a stochasticdemand system only depends on the choice probabilities in each of these regions.Thus, we let π r | t denote the probability of choices from the r -th region in the t -thobservation. For instance, in the two budget set and two goods case, we denote theset of choice probabilities as π “ p π | , π | , π | , π | , π | , π | q .When there are only two budgets and two goods, we can easily relate the de-terministic consumer problem to random utility models. Rather than thinking ofutility functions or preferences, we look over a space of “demand types.” For exam-ple, an individual may choose x | from the first budget and x | from the secondbudget. We label this as demand type θ p , q where the first entry corresponds tothe demand region from the first budget and the second entry corresponds to thedemand region from the second budget. This generates nine different combinationsof demand represented in Table 1. However, only four of these are deterministicallyrationalizable. 7emand Type Budget 1 Budget 2 Deterministically RationalizedType 1 θ p , q x | x | YesType 2 θ p , q x | x | YesType 3 θ p , q x | x | YesType 4 θ p , q x | x | YesType 5 θ p , q x | x | NoType 6 θ p , q x | x | NoType 7 θ p , q x | x | NoType 8 θ p , q x | x | NoType 9 θ p , q x | x | NoTable 1: Demand TypesThe condition that characterizes a stochastic demand system dataset as stochas-tically rationalizable is studied in McFadden and Richter [1990], McFadden [2005],and Hoderlein and Stoye [2015]. Stochastic rationalizability requires one to finda probability distribution over rationalizable demand types that sums to the ob-served probabilities in each region. Let the set of rationalizable types be denoted RT “ t θ p , q , θ p , q , θ p , q , θ p , qu . We denote the set of all types as AT . For aset S , we let ∆ p S q be a probability distribution indexed over elements of the set. Letthe probability distribution over rationalizable demand types be given by µ P ∆ p RT q “ $’&’% µ P R | RT |` | ÿ p j,k q s.t. θ p j,k qP RT µ p j, k q “ ,/./- where µ p j, k q is the probability of type θ p j, k q . The linear programming characteri-zation of stochastic rationality is below. Proposition 2.
The dataset D S is stochastically rationalized if and only if there exists measure of rational demand types µ P ∆ p RT q such that »—————————– θ p , q θ p , q θ p , q θ p , q fiffiffiffiffiffiffiffiffiffifl »————– µ p , q µ p , q µ p , q µ p , q fiffiffiffiffifl “ »—————————– π | π | π | π | π | π | fiffiffiffiffiffiffiffiffiffifl . (2)Note that we can solve this system of linear equations by Gauss-Jordan elimi-nation. Using the elimination, we can find that the solution exists when π | “ π | , π | ´ π | “ π | ´ π | , π | ě π | and π | ě π | . Performing the elimination, wefind a solution of µ p , q “ π | µ p , q “ π | ´ π | “ π | ´ π | µ p , q “ π | µ p , q “ π | “ π | . (3)The above four existence conditions of (3) can be simplified by the two conditions, π | “ π | and π | ě π | . To see this, note that ř r π r | t “ t P t , u and theequality condition π | “ π | implies π | ` π | “ π | ` π | . By rearranging it, wehave the equality condition, π | ´ π | “ π | ´ π | . In addition, the inequality condi-tion, π | ě π | , and the previous equality imply that π | ě π | . This means that adataset is stochastically rationalizable if and only if the D S with π satisfies π | “ π | and π | ě π | . This result is closely related to the finding from Hoderlein and Stoye[2015] when supp p π t q is finite for all t . Corollary 1.
Consider a dataset D S “ tp p , π q , p p , π qu . Then the following state-ments are equivalent:(i) D S is stochastically rationalized.(ii) For data from D S , there exists a measure of rational demand types that satisfiesthe system of (2). iii) D S satisfies π | “ π | and π | ě π | . This section examines the relationship between deterministic and stochastic ra-tionalizability assuming no sampling error. In particular, we show that there existpopulations where individuals cannot be deterministically rationalized, but none-the-less are rationalized by a stochastic demand system. This means that even whenthere are individuals who cannot be rationalized by any preferences, the stochasticdemand system can be stochastically rationalized. We call this a false acceptanceof stochastic rationality . Following the example, we characterize properties of falseacceptance when there is no sampling error.
Example 1.
Let p “ p , q and p “ p , q be given normalized prices. Hence, wehave two normalized budgets B p p q and B p p q that overlap as in Figure 2.Suppose that a population consists of two “demand types:” of the popula-tion are type θ p , q and of the population are type θ p , q . Recall from Table 1that type θ p , q individuals are deterministically rationalizable whereas type θ p , q individuals are not deterministically rationalizable.Suppose that a researcher observes choices from the entire population with-out sampling error. This generates a distribution of observed choices π “p π | , π | , π | , π | , π | , π | q “ p , , , , , q . This dataset is stochastically ra-tionalized since it satisfies conditions π | “ π | and π | ě π | from Corollary 1. In-deed, a vector µ P ∆ p RT q with µ p , q “ , µ p , q “ , µ p , q “ , and µ p , q “ solves the corresponding system of linear equations (2) from Proposition 2.Note relative to the true demand types that generate the population, the stochasticrationalization under-estimates the proportion of type θ p , q , while over-estimatingthe proportion of types θ p , q and θ p , q in the population. Example 1 shows that the dataset can fail to refute stochastic rationalizabilityeven when the population contains individuals who are not deterministically rational-ized by any preference relation, i.e., type θ p , q . Moreover, there are demand typesin the stochastic rationalization that are given positive probability even though there10re no individuals of this type in the population.We now present relevant analytical results of false acceptance for the population.Let ν P ∆ p AT q “ t ν P R | AT |` | ř j,k Pt , , u ν p j, k q “ u be a probability distribution overall individual demand types in a population. For a population ν , a sample dataset isequivalent to choices made by the given population when there is no sampling error.Hence, for convenience, we say a population is stochastically rationalized wheneverits dataset is stochastically rationalized, and vice versa. In this special case, usingCorollary 1, we attain conditions of stochastically rationalizable populations expressedby the distribution of demand types. Proposition 3.
Consider a dataset D S as in Figure 2 and the researcher samples theentire population. A population distribution over demand types ν P ∆ p AT q satisfies ν p , q ` ν p , q “ ν p , q ` ν p , q and ν p , q ` ν p , q ě ν p , q ` ν p , q , (4) if and only if it is stochastically rationalized.Proof of Proposition 3. By Corollary 1, we know that the dataset is stochasticallyrationalized if and only if its observed choice probabilities satisfy π | “ π | and π | ě π | . Since the dataset contains the entire population, we have π r | “ ř k ν p r, k q and π r | “ ř j ν p j, r q for all r “ , ,
3. Hence, the condition π | “ π | is equivalentto ν p , q` ν p , q` ν p , q “ ν p , q` ν p , q` ν p , q and we obtain ν p , q` ν p , q “ ν p , q ` ν p , q . Similarly, the condition π | ě π | is equivalent to ν p , q ` ν p , q ` ν p , q ě ν p , q ` ν p , q ` ν p , q , and so we obtain ν p , q ` ν p , q ě ν p , q ` ν p , q .By Proposition 3, we can directly determine the stochastic rationalizability of apopulation by observing its distribution of demand types. For instance, in Example 1,we have ν p , q “ and ν p , q “ . Thus, we can see that the distribution ofdemand types satisfies conditions of (4).One surprising observation from Proposition 3 is that there are populationsthat consist entirely of individuals who are not deterministically rationalizable, butnonetheless the population data are stochastically rationalized. In contrast, if a pop-11lation consists entirely of deterministically rationalizable individuals, then it cannotbe rejected as stochastically rationalizable since ν p , q “ ν p , q “ ν p , q “ ν p , q “ ν p , q “
0. We record these results in the following proposition.
Proposition 4.
Consider a dataset D S as in Figure 2 and the researcher samplesthe entire population.(i) False Acceptance:
There exist populations ν P ∆ p AT q with supp p ν q Ď AT z RT that satisfy stochastic rationality.(ii) No False Rejection:
All populations ν P ∆ p AT q with supp p ν q Ď RT satisfystochastic rationality.Proof of Proposition 4. [False Acceptance] Suppose that ν p , q “ and ν p , q “ so that no individual is deterministically rationalized by a utility function. Then thispopulation is stochastically rationalized since ν p , q ` ν p , q “ “ ν p , q ` ν p , q and ν p , q ` ν p , q “ ě “ ν p , q ` ν p , q .[No False Rejection] Suppose that ν P ∆ p AT q with supp p ν q Ď RT “t θ p , q , θ p , q , θ p , q , θ p , qu . Note that ν p j, k q ě p j, k q such that θ p j, k q P RT . The population is stochastically rationalized since ν p , q ` ν p , q “ “ ν p , q ` ν p , q and ν p , q ` ν p , q “ ν p , q ě “ ν p , q ` ν p , q .The first part of Proposition 4 heavily relies on the existence of demand typesthat choose the bundle in the third region of either of the budget. If we look forrationalizations of types that are not deterministically rationalized and never choosein region three, then we have ν p , q “ θ p , q which is not deterministically rationalized, and yet the dataset can bestochastically rationalizable. Recall that type θ p , q excludes the knife edge choicesin region three. 12 roposition 5. Consider a dataset D S as in Figure 2 and the researcher samples theentire population. Let the population ν P ∆ p AT q have ν p , q “ ε P R ` . There existdatasets that are stochastically rationalizable if and only if ε ď .Proof of Proposition 5. Example 1 can be modified to show stochastically rationaliz-able datasets exist when ε ď . When ε ď , let ν p , q “ ε , ν p , q “ ´ ε , andall other demand types occur with zero probability. Note that Proposition 3 holds.Next, we show that when ε ą there is no stochastic rationalization. When ε ą ,the second condition of (4) cannot be satisfied since ν p , q ` ν p , q ď ´ ε ă ă ε ď ν p , q ` ν p , q .In this section, we assumed that a researcher observes choices from the wholepopulation without sampling error and examined the relationship between determin-istic rationalizability and stochastic rationalizability. In the next section, we studythis relationship in the presence of sampling error. This section investigates the interaction of deterministic and stochastic ratio-nality for different sampling schemes. First, we analytically characterize propertiesconcerning cross-section sampling and panel sampling. Finally, we discuss a multino-mial sampling scheme of the true population and relate it to “power” from Bronars[1987]. Throughout this section, we assume the true distribution over all demandtypes in a population is given by ν P ∆ p AT q . We interpret a cross-section sample of the data for each period to be a randomsample of the population of individuals that is not necessarily related. We describethis in more detail below.A random sample in the t -th period describes individuals sampled in the t -thobservation. Let the random sample in period t be denoted by s t P S “ t s t P R | AT |` | s t p j, k q ď ν p j, k q @ j, k P t , , uu whose only restriction is the sample is lessthan or equal to the true proportion of individuals. If a researcher does not sample13ll individuals of a given type θ p j, k q , then s t p j, k q ă ν p j, k q . For example, s t p , q says that in the t -th period the researcher samples s t p , q ν p , q of all individuals who choosefrom region one when normalized prices are p and from region three when normalizedprices are p . Thus, cross-section sampling is defined by the samples in period one andtwo given respectively by s , s P S . Here the main feature of cross-section samplingis that the samples s and s do not need to be related in any particular way.We denote the stochastic datasets generated from a sample by ˆ π p s , s q . In par-ticular, for any r -th region, the probabilities are given by ˆ π r | p s , s q “ ř k Pt , , u s p r,k q ř j,k Pt , , u s p j,k q and ˆ π r | p s , s q “ ř j Pt , , u s p j,r q ř j,k Pt , , u s p j,k q . Here sampling in period one only affects the ob-served distribution of choices for observation one and sampling in period two onlyaffects the observed distribution of choices for observation two. To check the stochas-tic rationalizability of the sample dataset, we can straightforwardly apply the resultsfrom Corollary 1, i.e., ˆ π | “ ˆ π | and ˆ π | ě ˆ π | . Throughout the following results,we regularly drop dependence on the sample when discussing the sampled dataset ˆ π .The sampled dataset can have little relation to the true percentage of demandtypes. The following proposition shows that there are cross-sectional samples that arestochastically rationalized even when all individuals are not deterministically ratio-nalized. This is an example of a false acceptance of stochastic rationality generated bysampling error. Contrary to the case of perfect sampling, one can also reject stochas-tic rationality in the presence of cross-sectional sampling even when all individualsare deterministically rationalized. We call the rejection of stochastic rationality whenall individuals are deterministically rationalizable a false rejection of stochastic ratio-nality . Proposition 6.
Suppose normalized prices give the demand regions in Figure 2.(i)
False Acceptance:
There exist populations ν P ∆ p AT q with supp p ν q Ď AT z RT and cross-section random samples s , s P S such that the dataset ofprices and observed choices ˆ π is stochastically rationalized.(ii) False Rejection:
There exist populations ν P ∆ p AT q with supp p ν q Ď RT and cross-section random samples s , s P S such that the dataset of prices andobserved choices ˆ π is not stochastically rationalized. roof of Proposition 6. [False Acceptance] Suppose that ν p , q “ and ν p , q “ so that no individual is deterministically rationalized. The sample s p , q “ and s p , q “ s p , q “ s p , q “ is stochastically rationalized sinceˆ π | “ ˆ π | “ π | “ ě “ ˆ π | . In fact, the resulting stochastic demandsystem is deterministically rationalized.[False Rejection] Suppose that ν p , q “ and ν p , q “ so that all individualsare deterministically rationalized. The sample s p , q “ s p , q “ and s p , q “ and s p , q “ π | “ ă “ ˆ π | . In fact, the resulting stochastic demand system is not deterministicallyrationalized.The above proposition shows that rejecting or failing to reject stochastic ratio-nalizability can greatly depend on the sampling scheme applied to the population. Inparticular, a population of individuals who are not deterministically rationalized cangenerate stochastically rationalizable datasets. Similarly, individuals who are deter-ministically rational can fail to produce stochastically rationalizable datasets from across-section sample.The proof of the first part of Proposition 6 uses demand types that have purchasesin the third region of each budget. As discussed in Section 3, this is a knife edge casesince it places probability mass on a single consumption bundle. However, one canshow that there exists cross-section sampling with an arbitrarily large proportion ofindividuals who are not deterministically rationalized and do not choose in the thirdregion that can still be stochastically rationalized. Proposition 7.
Suppose normalized prices give the demand regions in Figure 2.For every ε P p , s there exist populations ν P ∆ p AT q with ν p , q “ ´ ε and ν p , q “ ν p , q “ ν p , q “ ν p , q “ and cross-section random samples s , s P S such that the dataset of prices and observed choices ˆ π is stochastically rationalized.Proof of Proposition 7. Suppose ν p , q “ ´ ε and let ν p , q “ ε . The sample s p , q “ s p , q “ ε and s p , q “ s p , q “ ε is stochasticallyrationalized since ˆ π | “ “ ˆ π | and ˆ π | “ εε “ ě “ εε “ ˆ π | . In fact, theresulting dataset is deterministically rationalized.15he issue with cross-section sampling as shown through Proposition 7 is that onecannot guarantee that those who are not deterministically rationalized were accountedfor in the sample. This is an empirically relevant observation since some individualsare hard to reach which can result in sampling error.We later discuss how deterministic rationalizability, stochastic rationalizability,and multinomial sampling interact since one might assume demand types are selectedinto the sample independently. However, we show through simulation that even forlarge multinomial samples false acceptance of stochastic rationalizability can stillregularly occur. To intuitively understand why this can occur, consider Example 1.Here if the population is sampled multinomially, then a researcher will converge tothe true proportion of individuals in the population. Nonetheless, the true proportionof the population still leads to a false acceptance. Let s t be the random sample from the t -th observation as defined above. Panelsampling has the same individuals present in observation one and two. Thus, panelsampling is represented by s “ s . Note that when s “ s , the resulting stochasticdataset ˆ π results from a convex combination of types in the support of the popu-lation. This section examines the dangers of not using the full structure of panelsampling. In particular, when a researcher has panel data they could look directlyat deterministic rationality conditions for each individual which will lead to correctresults. Alternatively, a researcher could look for a stochastic rationalization whichthrows away information on individual choices. Here, we show that not using thepanel structure when looking for a stochastic rationalization can lead to false accep-tances of stochastic rationality. However, panel sampling prevents false rejections ofstochastic rationality. Proposition 8.
Suppose normalized prices give the demand regions in Figure 2.(i)
False Acceptance:
There exist populations ν P ∆ p AT q with supp p ν q Ď AT z RT and panel random samples s “ s P S such that the dataset of pricesand observed choices ˆ π is stochastically rationalized. There are dedicated statistical methods to handle panel sampling studied inAguiar and Kashaev [2018]. ii) No False Rejection:
For all populations ν P ∆ p AT q with supp p ν q Ď RT andpanel random samples s “ s P S , the dataset of prices and observed choices ˆ π is stochastically rationalized.Proof of Proposition 8. [False Acceptance] Suppose that ν p , q “ and ν p , q “ so that no individual is deterministically rationalized. The sample with s p , q “ and s p , q “ and s “ s is stochastically rationalized since ˆ π | “ “ ˆ π | andˆ π | “ ě “ ˆ π | . In fact, it is stochastically rationalized by the random utilitymodel with µ p , q “ and µ p , q “ . This is a case with no sampling error.[No False Rejection] Suppose that supp p ν q Ď RT and s “ s is a randomsample. For p j, k q such that θ p j, k q P RT , let the probability over rational types begiven by µ p j, k q “ s p j,k q ř p ˜ j, ˜ k q s.t. θ p ˜ j, ˜ k qP RT s p ˜ j, ˜ k q . This is a random utility model by definitionand rationalizes the data that results from the random sample s “ s .The proof of Proposition 8 for the false acceptance uses demand types thatchoose in region three which is a point. However, as with the cross-sectional randomsampling, we show there are stochastically rationalizable datasets with an arbitrarilylarge proportion of individuals who are not deterministically rationalized and do notchoose in region three. Proposition 9.
Suppose normalized prices give the demand regions in Figure 2.For every ε P p , s there exist populations ν P ∆ p AT q with ν p , q “ ´ ε and ν p , q “ ν p , q “ ν p , q “ ν p , q “ and panel samples s “ s P S such that thedataset of prices and observed choices ˆ π is stochastically rationalized.Proof of Proposition 9. Suppose that ν p , q “ ´ ε and let ν p , q “ ε . If ε ď ,then consider the sample s p , q “ ε ď ´ ε “ ν p , q and s p , q “ ε “ ν p , q . Thesample is stochastically rationalized since ˆ π | “ εε ` ε “ ě “ εε ` ε “ ˆ π | . If ε ą ,then consider the sample s p , q “ ´ ε “ ν p , q and s p , q “ ε “ ν p , q . Thesample is stochastically rationalized since ˆ π | “ εε ` ´ ε “ ε ą ą ´ ε “ ´ εε ` ´ ε “ ˆ π | . 17 .3 Multinomial Sampling and Bronars Power So far, we have not placed many assumptions on the sampling process whenexamining the properties of false acceptance and false rejection of stochastic rational-izability. Another way to consider sampling is to examine random samples generatedby a multinomial distribution. This sampling process has the convenient propertythat the sample average of observed types almost surely converges to the true popu-lation probabilities.Let ν P ∆ p AT q be the true distribution of all demand types in a population. Formultinomial sampling with replacement, any demand type θ p j, k q with j, k P t , , u is sampled with probability ν p j, k q . In this subsection, we assume that the samplesize of each observation is the same, denoted by n P N , and that samples for eachobservation are independent. We also only consider demand types that never choosethe third region for any observation. Recall that by the third condition from Corollary1, a sampled dataset is stochastically rationalized if and only if ˆ π | ě ˆ π | . Giventhese parameters and information, we can calculate the probability that a sampleddataset is stochastically rationalized.The computation is straight forward, but tedious, so we provide details. Sincethe sample size for each observation are the same size, effectively we can turn thecondition ˆ π | ě ˆ π | into one that checks whether there are more choices from thesample in region one of the first budget than region one of the second budget. Thisrealization produces a tractable formula to compute the probability of a stochasticrationalization.To see how this works, suppose that for the second observation we see no samplechoices in region one. Using the multinomial theorem for n observations this oc-curs with probability ` n ˘ p ν p , q ` ν p , qq p ν p , q ` ν p , qq n . Conditional on thissample, any sample choices for the first observation are stochastically rationalizable.Thus, at least ` n ˘ p ν p , q ` ν p , qq p ν p , q ` ν p , qq n proportion of samples arestochastically rationalized.Next, suppose that the sample of observation two has one choice in the firstregion. The probability this occurs is ` n ˘ p ν p , q ` ν p , qq p ν p , q ` ν p , qq n ´ .For the sampled choices to be stochastically rationalizable, at least one choicefrom the first observation must be in region one. The probability this occurs is18 nℓ “ ` nℓ ˘ p ν p , q ` ν p , qq ℓ p ν p , q ` ν p , qq n ´ ℓ . Thus, multiplying these probabili-ties gives the probability a sampled dataset is stochastically rationalized when onechoice is in region one of the second observation.We can iterate and sum the above procedure to find the probability a multinomialsample of size n in both periods is stochastically rationalized. In particular, theprobability of a size n multinomial sample being stochastically rationalized is n ÿ i “ ` p | ˘ i ` p | ˘ n ´ i ˆ ni ˙ « n ÿ ℓ “ i ` p | ˘ ℓ ` p | ˘ n ´ ℓ ˆ nℓ ˙ff (5)where p j | “ ν p j, q ` ν p j, q and p j | “ ν p , j q ` ν p , j q for all j “ ,
2. Here the termin the brackets is the probability the sample from observation one has more choicesin region one than the sample from observation two. If ν p , q ą
0, then (5) indicatesthe probability of a false acceptance of stochastic rationalizability. If ν p , q “
0, thenone less (5) is the probability of false rejections of stochastic rationalizability.The above calculations will allow us to generate information related to the“power” of stochastic rationalizability in a sense closely related to Bronars [1987].Here we interpret the “power” as the probability a dataset is not stochastically ra-tionalized when there are some individuals in the population who are not determin-istically rationalizable, i.e., ν p , q ą p “ p , q and p “ p , q . The populations we considerare1. Uniform Distribution: We assume a uniform distribution over demand types,i.e., ν p , q “ ν p , q “ ν p , q “ ν p , q “ .2. Proportional Choices: We assume ν p , q “ ν p , q “ , ν p , q “ , and ν p , q “ . This is related to random behavior discussed by Becker [1962]since the distribution is proportional to size of the budget regions. These two methods closely follow the intuition of Bronars [1987] and Becker To see this, note that for prices p “ p , q and p “ p , q the intersection of budget linesgives regions where the proportional size of regions are and . If people choose uniformly over thebudget line, then we have a benchmark of ν p , q “ . We present the results of simulations for different sample sizes in Table 2.Sample Size10 50 100 500 1,000Uniform Sampling 0.5881 0.5398 0.5282 0.5126 0.5089Proportional Sampling 0.9624 0.9998 1 1 1Table 2: Probability a multinomial sample is stochastically rationalized according totwo benchmark populationsThe tables show that the ability to detect when there are individuals who arenot deterministically rational from stochastic choice data is low. In particular fromthe uniform sample simulations, we see that even when one fourth of the populationis not rational, the population dataset is rationalized over 50% of the time and thisdoes not improve much with large samples. That this is around 50% likely resultsfrom a uniform sample being on the boundary of the condition from Proposition 3.The results are worse for the proportional sampling. Even though there is asubstantial fraction of individuals who are not deterministically rationalizable ` ˘ , itis almost impossible to detect this group of people. Moreover, the ability to detectthis group of individuals worsens as the sample size increases. The reason this occursis exactly because Proposition 3 holds on the population. Thus, as the sample growslarger, it becomes harder to detect individuals who are not rational. This paper shows that it is difficult to detect violations of stochastic rationaliz-ability even when there are large fractions of the population who are not determin-istically rational. Thus, while stochastic choice models and non-parametric methodshave risen in popularity, the old problems of aggregate behavior not representingindividual behavior as mentioned in Becker [1962] still re-appear for these methods See Kitamura and Stoye [2018] and De Rock et al. [2019] for details.
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