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Identifying Heterogeneous Decision Rules From Choices When Menus Are Unobserved
Consider the problem of explaining the distribution of choices in a heterogeneous population. Denote by $\lambda$ the probability distribution of chosen alternatives, the data. A common approach is to posit heterogeneity in decision rules (or underlying preferences) and possibly also in the menus from which alternatives are chosen. A decision rule $d$ specifies the alternative $d\left( A\right) $ chosen from each menu $A$; the set of all decision rules is $\mathcal{D}$. An individual with decision rule $d$ faces menu $A$ with probability $\pi_{d}\left( A\right) $. Decision rules are distributed according to a probability measure $Q$ that is to be inferred from the data, while the collection of probability measures $\{\pi_{d}\}_{d\in\mathcal{D}}$ is known to the analyst (possibly up to unknown parameters).\footnote{Filiz-Ozbay and Masatlioglu (2023) call this a random-choice model (RCM), defined by a probability distribution over a collection of choice functions (potentially irrational). They axiomatize a specific class of RCMs under the assumption of rich stochastic choice data.} Accordingly, she seeks $Q$ satisfying, for the given $\{\pi_{d} \}_{d\in\mathcal{D}}$,
for all alternatives $a$. Then empirical frequencies are rationalized by the heterogeneity in decision rules described by $Q$. Of particular interest is the set of all rationalizing $Q$s (the sharp identified region).
The above model is general in that it covers the bulk of the discrete choice literature where various special cases are adopted;\footnote{We are ignoring covariates that often appear in this literature, and that could be added below, because they are not germane to our contribution. We adopt a streamlined formulation in order to maximize transparency of the theoretical point of this paper.} for example, the traditional assumption (McFadden 1974) that the menu corresponding to each choice is observed corresponds to the special case where $\pi_{d}(A)=1$ for some $A$. However, data about menus that would support knowledge of the conditional probabilities $\pi_{d}$ are often unavailable (see Manski (1977) and the overviews and many references in Barseghyan et al (2021, pp. 2016-2017, 2041-2043) and Azrieli and Rehbeck (2023)). Notably, decision models based on consideration sets (Abaluck and Adams-Prassl 2021, Cattaneo et al 2020, Manzini and Mariotti 2014, Masatlioglu et al 2012) or rational inattention (Caplin et al 2019) view choices as made from subjective menus, thus arguing against their observability.\footnote{To be clear, we use "menu" to refer to the set from which an alternative is chosen by maximizing preference or by applying another decision rule. Consequently, it may be a strict subset of the objective feasible set, (for example, a consideration set), that is determined by the individual's cognitive deliberation process and is unobservable to the analyst.} One is led to the concern that conclusions about the identified set of measures $Q$ that are based on ((ref)) sometimes rely on ad hoc assumptions about menus.
An objective in this paper is to robustify the above model by incorporating the analyst's imperfect knowledge about menus. One alternative to the perfect information assumption is complete ignorance about menus - "anything goes" for specifications of $\pi_{d}$s. However, in general, one would expect there to be partial information about the menu process. Therefore, we admit a range of assumptions about the analyst's information that are intermediate between complete ignorance and perfect knowledge. In all cases, we show (Theorem (ref)) that the implied sharp identified set of distributions consists of all measures $Q$ satisfying a finite set of linear inequalities and hence forms a polytope (a convex set with finitely many extreme points); in particular, it is computationally tractable.
We adopt a novel formulation using convex (or supermodular) capacities and their cores. (The appendix collects the few basic definitions and facts regarding capacities that are used below; a very accessible and comprehensive reference is Grabisch (2016).) Capacities are set functions that generalize probability measures in order to permit a role in the representation of beliefs for limited information and the resulting limited confidence in any single probability measure - in other terms, uncertainty about probabilities. They arise in decision theory, notably in Schmeidler's (1989) Choquet expected utility theory, where convexity of the capacity is identified with aversion to such uncertainty and where convexity characterizes the Choquet models that conform also to multiple-priors utility (Gilboa and Schmeidler 1989).\footnote{Convex capacities, or equivalently their conjugates, known as 2-alternating, are important also in statistical theory (in proving an extension of the Neyman-Pearson Lemma (Huber and Strassen 1973) and in supporting a version of Bayes' theorem for capacities (Wasserman and Kadane 1990). They appear also in cooperative game theory as characteristic functions. However, the epistemic interpretation is a better fit here.} For our purposes, the key technical feature of convex capacities is that "the core of a mixture of capacities equals the mixture of their cores" (see ((ref)) for a formal statement). Given our formulation, this property leads to a short transparent (indeed elementary) proof of our theorem that applies to and unifies all of our specifications. We view this simplicity and the associated epistemic perspective as a strength and a contribution.
The scope of our results merits emphasis. Thus far we have interpreted the paper as addressing heterogeneity in choice assuming heterogeneity in decision rules and the unobservability of menus. However, with suitable reinterpretation of the symbols in the formal model, Theorem (ref) applies also to other contexts where one seeks the identification of heterogeneity that is robust to unobservables. A concrete setting where we describe such an application is to the rationalization of the distribution of beliefs in a population (see section (ref)).\footnote{Two other settings are outlined in the concluding section (ref).} Individuals often disagree about the likelihoods of future events. Two candidate reasons are differences in information and in updating, both of which are often unobservable to the analyst (outside the laboratory). Unobservability of information is a common assumption. However, it is typically accompanied by the assumption that all updating is Bayesian, contrary to the abundant experimental evidence of a number of systematic biases (see the seminal study by Tversky and Kahneman (1974), and the many references in Camerer (1995), Rabin (1998)); particularly relevant for the sequel is evidence of under/overreaction (see Benjamin (2019) and Ba, Bohren and Imas (2023) and the many references therein). Such evidence motivates our hypothesis that updating rules may differ between individuals, with universal Bayesian updating being a very special case. We admit a family of updating rules that deviate from Bayesian updating by either overreacting or underreacting to information and we (partially) identify the population distribution of updating rules, while respecting that little may be known about differences in information. We also provide conditions under which any rationalization of the distribution of beliefs exhibits underreaction (or overreaction, respectively) "on average."
Our main theorem, Theorem (ref), is, in fact, equivalent to Theorem 4 of Strassen (1965) when specialized as here so that all sets are finite. An important difference is our drastically simpler elementary proof. We view the simpler proof as significant not as a mathematical contribution, but rather because it enhances transparency and accessibility of the theorem which, we believe, may help to expose and promote it as a useful tool for economists. The other value-added over Strassen is our demonstration of the theorem's usefulness as outlined above.
First we relate our contribution to some recent papers in discrete choice (and related econometrics) that also weaken a priori assumptions about menus. Barseghyan et al (2021) study identification in a random utility model where the distribution of menus in the population is unknown. Two differences from our model are that: they assume preference maximization (particularly, Sen's $\alpha$ condition) rather than general decision rules, and they assume that all menus of size at least $\kappa$, ($\kappa\geq2$), a parameter specified by the analyst, are conceivable for any individual conditional on her preference order, while we allow the set of conceivable menus to be arbitrary. Moreover, they deal only with the case of complete ignorance of the menu process, for which their characterization of the sharp identified set corresponds (apart from their inclusion of covariates) to our complete-ignorance result in Theorem (ref).\footnote{Minor differences are described following the statement of our theorem in section (ref).} Their proof also differs from ours in that it applies the theory of random sets. The significance of this difference is that random sets induce a belief function which is a very special kind of convex capacity that precludes many of the richer information structures (those short of complete ignorance about menus) that are accommodated in our theorem. \ Lu (2022) assumes that all conceivable menus are bounded above and below in the sense of set inclusion, and that the bounding menus are known. He uses the latter, and the assumption that decision rules satisfy Sen's $\alpha$, to describe a superset of the identified region. In contrast, our conditions on $Q$ are both necessary and sufficient for $Q$ to rationalize the data, thus yielding the sharp identified region. Azrieli and Rehbeck (2023) also study what can be learned from aggregate choice frequencies, but with several differences from the present paper. A major difference is that they assume that the marginal empirical distributions of both menus and choices are known (constitute the data). In their study of random utility models, they assume that menus are homogeneous across decision makers, (that is, the distribution of menus does not depend on the decision rule), while we allow for correlation between menus and decision rules. An example in section (ref) shows that menu-homogeneity can strictly shrink the sharp identification region. Regarding proof arguments, they also highlight their use of \ "known properties of the core," though these do not not include the key property that we exploit here, and they borrow more from cooperative game theory than from decision theory and thus do not emphasize epistemics in their interpretations. Further, their proofs (specifically for their Proposition 9) use not only core properties but also network flow arguments (based on a version of Hall's marriage theorem), while we use only the single mixture property of the core noted above.\footnote{None of the above papers invoke, or cite, Strassen (1965).}
Dardanoni et al (2020) also explore what can be inferred from aggregate choice data, though their focus is on cognitive heterogeneity rather than on preference (or decision rule) heterogeneity. Individuals differ in cognitive "type" and, given an objective feasible set, they arrive at different consideration sets (menus in our terminology); further, they do so in a way that conforms to specific functional forms - the "consideration capacity model" (which limits the cardinality of the consideration set) or the "consideration probability model" (Manzini and Mariotti 2014). In the section most closely related to our paper, where preferences are unobservable and heterogeneous, they assume that choices are observed from multiple "occasions" across which both the feasible set and cognitive heterogeneity are stable. With this rich dataset and functional form restrictions they prove point identification of the distribution of cognitive types in the consideration capacity model. Roughly speaking, from the perspective of our formal framework, they severely restrict the distribution of decision rules and aim at identification of the menu formation process ($\pi_{d}$), which reflects the distribution of cognitive type. Unsurprisingly, their proof arguments are much different than ours.
Doval and Eilat (2023) study the setting where the analyst knows the marginal over an agent's actions and the prior over states of the world, but does not know the distribution of actions given realizations of the states of the world. They ask when two such marginals (over actions and states, similar to the dataset in Azrieli and Rehbeck 2023) can be rationalized (in the sense of a Bayes correlated equilibrium) as the outcome of the agent learning something about the state before taking an action. Their characterization result is two systems of linear inequalities that are necessary and sufficient for the dataset to be consistent with a Bayes correlated equilibrium. One of these can be established using our "mixture of cores" property. Their proof relies on both Strassen (1965, Theorem 3) and on network flow arguments.
The (finite) universal set of alternatives is $X$, and the set of probability distributions or measures on $X$ is denoted $\Delta\left( X\right) $. Each individual in a finite population faces a menu, a subset of $X$, from which she chooses one alternative. The collection of all "relevant" menus is denoted $\mathcal{A}$, with generic element $A$. The collection $\mathcal{A}$ is a primitive, determined by the analyst. Another primitive is a finite set $\mathcal{D}$ of decision rules, where, for each $d$ in $\mathcal{D}$, $d\left( A\right) $ denotes the alternative that $d$ chooses from the menu $A$ in $\mathcal{A}$. We do not impose any requirements on decision rules, for example, they need not be derived from preference maximization.
The "data" to be explained are represented by $\lambda\in\Delta\left( X\right) $, the empirical frequency distribution of chosen alternatives across the population. The analyst's view of the menu formation process determines what constitutes an "explanation." We assume that she is certain that only menus in $\mathcal{A}$ are relevant, but otherwise she has limited understanding of how menus are determined; in particular, she cannot be confident in any single conditional probability distribution over menus $\pi_{d}\in\Delta\left( \mathcal{A}\right) $, which suggests modeling via a set of conditional distributions. We proceed in this way, though with a slight twist as explained next.
Let
Thus $C_{d}$ denotes the set of all alternatives that can be chosen by $d$ for some menu.\ For a given $d$, the analyst can be sure that an element of $C_{d}$ will be selected, but since the choice depends on the menu, her limited knowledge of menus affects her view of which choice is associated with $d$. For any given distribution over menus $\pi_{d}\in\Delta\left( \mathcal{A}\right) $, induced beliefs over alternatives are given by $\rho_{d}\in\Delta\left( C_{d}\right) $, where
Using $\rho_{d}$, ((ref)) implies that
where menus have been eliminated and distributions over alternatives are described by both the empirical measure $\lambda$ and by the "explanatory" measures $\{\rho_{d}\}$. To proceed, we adopt as the benchmark notion of an explanation of $\lambda$ that "((ref)) is satisfied by the known $\{\rho_{d}\}$," thus replacing "((ref)) is satisfied by the known $\{\pi_{d}\}$." In fact, we describe later (section (ref)) why the two benchmarks lead to identical results in the present discrete choice setting. However, in other contexts, such as in the effort example in section (ref) there may be no obvious counterpart of ((ref)), while the model based on ((ref)) is applicable.
We define what it means for a measure $Q$ over decision rules to rationalize the empirical measure $\lambda$. In the extreme case where the analyst knows the distributions over menus this is expressed by ((ref)) using the known conditionals $\{\rho_{d}\}$. One can capture the other extreme of complete ignorance by requiring that ((ref)) is satisfied for some conditionals $\{\rho_{d}\}$, restricting them only to reflect certainty that $d$ chooses an element in $C_{d}$, that is, $\rho_{d}\in \Delta\left( C_{d}\right) $ for every $d$. The associated robustness may be desirable but comes with costs (that we formalize below). First, if "anything goes," then the identified region for any given data $\lambda$ is large. Second, with such weak maintained assumptions, (almost) every $\lambda$ can be rationalized by some $Q$. Consequently, and also because there are situations in which there exists partial information about the menu process, we propose a model that also accommodates such intermediate situations.
To model the presence of some information, we assume that, for each $d$, only distributions $\rho_{d}$ that lie in the set $\mathcal{R}_{d}\subset \Delta\left( C_{d}\right) $, determined by the analyst, are deemed relevant. This leads to the following definition: Say that $Q\in\Delta\left( \mathcal{D}\right) $ rationalizes $\lambda$ given $\{\mathcal{R}_{d}\}$ if there exists $\rho_{d}\in\mathcal{R}_{d}$ for all $d$, such that
Perfect information is the special case where each $\mathcal{R}_{d}$ is a singleton. More interesting specifications follow.\footnote{They are all well-known in both robust statistics and in decision theory. We have borrowed them and their properties described below from Wasserman and Kadane (1990). However, we have not seen the last three used previously in the present context.}
\noindentComplete ignorance: Let $\mathcal{R}_{d}=\Delta\left( C_{d}\right) $ for each $d$ as indicated above.
$\epsilon$-contamination: For each $d$, let $\widehat{\rho }_{d}\in\Delta\left( C_{d}\right) $ be a focal probability distribution over alternatives, perhaps the analyst's best "point estimate," but one in which she may not have complete confidence. As a reflection of her incomplete confidence she entertains as possible all contaminations of $\widehat{\rho }_{d}$ of the form \[ \rho_{d}=\left( 1-\epsilon\right) \widehat{\rho}_{d}+\epsilon\widetilde {\rho}_{d}\text{,} \] where $\widetilde{\rho}_{d}$ is any measure on $C_{d}$ and where $0\leq\epsilon\leq1$ is a parameter to be specified by the analyst. That is, let
The extremes $\epsilon=0,1$ correspond respectively to the complete confidence and complete ignorance models respectively. Further, it is easy to see that $\mathcal{R}_{d}$ grows larger in the sense of set inclusion as $\epsilon$ increases in $\left[ 0,1\right] $. This suggests the interpretation of decreasing confidence (or increasing ignorance) as $\epsilon$ increases.
The "$\epsilon$-contamination" model was proposed initially by Huber (1964) and has been used frequently in robust statistics (e.g. Huber and Ronchetti 2009, Wasserman and Kadane 1990), and also in decision theory and its many applications where it is a useful parametric specialization of the set of priors appearing in multiple-priors utility (Gilboa and Schmeidler 1989).
\noindentVariation neighborhood: For any $p^{\prime}$ and $p$ in $\Delta\left( C_{d}\right) $, define \[ \delta_{d}\left( p^{\prime},p\right) =\sup_{K\subset C_{d}}\mid p^{\prime }\left( K\right) -p\left( K\right) \mid\text{,} \] Fix a reference/focal measure $P_{d}$ on $C_{d}$ and $\epsilon>0$, and let
\noindentInterval belief: Let $p_{d\ast}$ and $p_{d}^{\ast}$ be measures (not probability measures) on $C_{d}$, satisfying \[ p_{d\ast}\left( \cdot\right) \leq p_{d}^{\ast}\left( \cdot\right) \text{ and }0<p_{d\ast}\left( C_{d}\right) <1<p_{d}^{\ast}\left( C_{d}\right) \text{,} \] and define \[ \mathcal{R}_{d}=\{p_{d}\in\Delta\left( C_{d}\right) :~p_{d\ast}\left( \cdot\right) \leq p_{d}\left( \cdot\right) \leq p_{d}^{\ast}\left( \cdot\right) \text{ on }C_{d}\}\text{.} \] In the special case \[ p_{d\ast}=a_{d}P_{d}\text{ and }p_{d}^{\ast}=b_{d}P_{d}\text{,} \] where $a_{d}<1<b_{d}$ and $P_{d}~$is a probability measure on $C_{d}$, one obtains \[ \mathcal{R}_{d}=\{p_{d}\in\Delta\left( C_{d}\right) :~a_{d}P_{d}\leq p_{d}\leq b_{d}P_{d}\} \]
In all cases, the identified set is (weakly) smaller than the identified set under complete ignorance. More generally, it shrinks if confidence increases in the sense that each set $\mathcal{R}_{d}$ shrinks; this happens, for example, if $\epsilon$ is reduced in the $\epsilon$-contamination specification or in the variation neighborhood specification. (Similarly, if each set of alternatives $C_{d}$ shrinks.) It is easy to see also that an increase in confidence shrinks the set of empirical measures $\lambda$ that can be rationalized by some $Q$. For example, in the absence of any confidence (complete ignorance), every $\lambda$ with support in $\cup_{d}C_{d}$ can be rationalized by some $Q$, while in the $\epsilon$-contamination specification $\lambda$ can be rationalized only if it can be expressed as a mixture $\left( 1-\epsilon\right) \widehat{\lambda}+\epsilon\widetilde{\lambda}$ where $\widehat{\lambda}$ is rationalizable under complete confidence ($\epsilon=0$) and $\widetilde{\lambda}$ is rationalizable under zero confidence ($\epsilon=1$), which is equivalent to the statement \[ \lambda\in\left( 1-\epsilon\right) ch\left( \{\widehat{\rho}_{d}\}\right) +\epsilon\Delta\left( \cup_{d}C_{d}\right) \text{,} \] where $ch$ denotes `convex hull.'
The main question to be addressed is "which measures $Q$ can rationalize $\lambda$ given $\{\mathcal{R}_{d}\}$?" We provide a comprehensive answer under the assumption that each $\mathcal{R}_{d}$ is the core of a convex capacity, that is, for each $d$,\footnote{Since a convex capacity is uniquely determined by its core (see ((ref))), $\nu_{d}$ is necessarily unique. Another point (see the appendix), is that a capacity $\nu_{d}$ on $C_{d}$, hence satisfying $\nu_{d}(C_{d})=1$, can be uniquely extended to a capacity on all of $X$, just as a probability measure on $C_{d}$ can be so extended.}
Though limiting, ((ref)) is of interest in light of the role of convex capacities in decision theory and in statistics (as mentioned in the introduction); and we note also that it is satisfied by all of the preceding specifications (as shown shortly).
The main message is that the sharp identified set of measures $Q$ is the set of solutions $Q$ to the finite set of linear inequalities ((ref)), and constitutes a (convex) polytope. The proof is extremely simple.
Proof: Under the assumption ((ref)), rationalizability amounts to the statement that \[ \lambda\in\sum_{d}Q\left( d\right) ~core\left( \nu_{d}\right) , \] while condition ((ref)) is the statement that \[ \lambda\in core\left( \sum_{d}Q\left( d\right) \nu_{d}\right) \text{.} \] However, by ((ref)), the core of the mixture equals the mixture of the cores, which proves the required equivalence.
For the assertions regarding the special cases, one need only show that in each case the indicated capacity $\nu_{d}$ is convex and that it has core equal to the corresponding set $\mathcal{R}_{d}$. But these are well-known facts (Huber and Strassen 1973, Wasserman and Kadane 1990).{ $\blacksquare$}
\noindentRemark: It is noteworthy that the proof uses convexity of the $\nu_{d}$s only to justify applying the mixture-linearity property of their cores. That is, the characterization provided by ((ref)) is valid also if convexity is replaced by this mixture-linearity. In a sense, therefore, since Strassen's (1965) Theorem 4 assumes convexity, our result is (strictly) more general. Tijs and Branzei (2002) and Bloch and de Clippel (2010) give other assumptions, besides convexity, that imply mixture-linearity of cores.\footnote{They work in the context of cooperative games where capacities are typically not normalized to assign a common fixed value to the universal coalition, and hence they refer to additivity rather than mixture-linearity of the core.} Indeed, it follows from the latter paper that (generically) there exists a partition of the set of all capacities having nonempty cores such that the mixture-linearity property is satisfied if (and only if) all capacities lie in the same equivalence class. The set of convex capacities is one such equivalence class, but there are others and the theorem applies to each of them as well. It remains for future work to determine if any of the other equivalence classes provide alternatives to convexity that are interesting in our setting.
\
A brief discussion of the ignorance special case may be clarifying and provide perspective on the theorem. The associated capacities, written more fully, are given by \[ \nu_{d}\left( K\right) =\left\{
\right. \] The epistemic interpretation is that $C_{d}$ is certain but there is complete ignorance within $C_{d}$. The condition ((ref)) characterizing rationalizability specializes to the set of inequalities
Similar conditions have appeared previously in Barseghyan et al (2021, Theorem 3.1) and in Azrieli and Rehbeck (2023). As indicated in the introduction, the latter addresses different questions, and the former assumes a designated minimum size for menus. The ignorance special case admits alternative proofs. For one, the associated capacities $\nu_{d}$ are belief functions and hence the result for that case can be derived by using random sets, which is the approach taken by Barsheghyan et al (2021). Alternatively, it follows immediately from the well-known structure of the core of a belief function (Dempster (1967) or Wasserman (1990, Theorem 2.1)). Moreover, these alternatives apply also to the $\epsilon$-contamination specification since its $\nu_{d}$s are belief functions. However, they do not apply when $\nu_{d} $s are convex but not belief functions, such as in the other two special cases or at the level of generality in the theorem.
Application of the theorem requires that when considering whether to adopt a specification of interest for the $\mathcal{R}_{d}$s one is able to check whether it satisfies ((ref)). For the particular specifications addressed in the theorem, the literature has confirmed ((ref)). More generally, an important observation is that, given $\{\mathcal{R}_{d}\}$, then, for each $d$, there is only one candidate for a suitable capacity $\nu_{d}$, namely the lower probability corresponding to $\mathcal{R} _{d}$ and defined by \[ \nu_{d}\left( K\right) =\inf\{\rho\left( K\right) :\rho\in\mathcal{R} _{d}\}\text{, \ for all }K\subset X\text{.} \] In other words, ((ref)) is equivalent to the assumption that the lower probability capacity is convex and has $\mathcal{R}_{d}$ as its core. (This follows directly from ((ref)).) Convexity of $\nu_{d}$ can be checked, in principle, by using its definition ((ref)) or any of its equivalent characterizations (Grabisch (2016, Theorem 3.15), for example). Since $\mathcal{R}_{d}\subset core\left( \nu_{d}\right) $ follows from the definition of lower probability, equality amounts to the requirement that $\mathcal{R}_{d}$ be sufficiently large in the sense that, for every $\rho \in\Delta\left( C_{d}\right) $, \[ \rho\left( K\right) \geq\nu_{d}\left( K\right) \,\ \text{for all }K\subset C_{d}\text{ implies }\rho\in\mathcal{R}_{d}\text{.}\footnote{Alternatively, given convexity, one can compute the cores, for example, by using the greedy algorithm Ichiishi (1981), or the algorithm in Chambers and Melkonyan (2005) that uses information about willingness to buy or sell and thus may help the analyst to calibrate parameters like $\epsilon$.} \]
Related is the question what can be done if one drops the assumption ((ref)) entirely.\footnote{If the sets $\{C_{d}\}$ are disjoint, then, for any $\{\mathcal{R}_{d}\}$, there is point identification - $\lambda$ is rationalized by the unique measure $Q$ given by $Q\left( d\right) =\lambda\left( C_{d}\right) $ for all $d$.} In fact, it is straightforward to show that the counterpart of ((ref)) given below is necessary for rationalizability given $\{\mathcal{R}_{d}\}$: If $Q\in\Delta\left( \mathcal{D}\right) $ rationalizes $\lambda$, then, for the lower probability capacity $\nu_{d}$, (using $\mathcal{R}_{d}\subset core\left( \nu_{d}\right) $ and ((ref))),
The theorem is relevant also to a rationalizability notion such as ((ref)), suitably modified, where menus appear explicitly. Modify ((ref)) by allowing $\pi_{d}$, for each $d$, to vary over a set $\Pi _{d}\subset\Delta\left( \mathcal{A}\right) $, where $\Pi_{d}$ is determined by the analyst. (For example, $\Pi_{d}=\Delta\left( \mathcal{A}\right) $ would model complete ignorance about menus.) Say that $Q\in\Delta\left( \mathcal{D}\right) $ menu-rationalizes $\lambda$ given $\{\Pi_{d}\}$ if there exists $\pi_{d}\in\Pi_{d}$ for all $d$, such that
Each $\pi_{d}$ induces a distribution $\rho_{d}$ on $C_{d}$ as in ((ref)); let $\mathcal{R}_{d}$ be the set of all such distributions $\rho_{d}$ as $\pi_{d}$ varies over $\Pi_{d}$. Then it is immediate that menu-rationalization of $\lambda$ given $\{\Pi_{d}\}$ implies rationalization ((ref)) given $\{\mathcal{R}_{d}\}$. Moreover, if $\Pi_{d}$ is the core of a convex capacity on $\mathcal{A}$, then $\mathcal{R}_{d}$ is the core of a convex capacity $\nu_{d}$ on $C_{d}$ (see appendix). Hence ((ref)) is necessary for menu-rationalizability by $Q$. To prove sufficiency, suppose that $Q$ rationalizes $\lambda$ given $\{\mathcal{R}_{d}=core\left( \nu _{d}\right) \}$ and define $\{\Pi_{d}\}\subset\Delta\left( \mathcal{A} \right) $ as follows: for each $d$ and $\rho_{d}\in\mathcal{R}_{d}$, and for each $a\in C_{d}$, select one menu $A_{a,d}$ satisfying $d\left( A_{a,d}\right) =a$, and define \[ \pi_{d}\left( A\right) =\left\{
\right. \] Then $\pi_{d}$ is a probability measure because \[ \sum_{A\in\mathcal{A}}\pi_{d}\left( A\right) =\sum_{a\in C_{d}}\rho _{d}\left( a\right) =\rho_{d}\left( C_{d}\right) =1\text{.} \] Let $\Pi_{d}$ be the set of all such $\pi_{d}$s as $\rho_{d}$ varies over $\mathcal{R}_{d}$. Then $Q$ menu-rationalizes $\lambda$ given $\{\Pi_{d}\}$ because
(Note that under complete ignorance ($\mathcal{R}_{d}=\Delta\left( C_{d}\right) $ or $\Pi_{d}=\Delta\left( \mathcal{A}\right) $), the above proves the equivalence of the two notions of rationalizability.)
The following two examples relate to questions that arise from Barseghyan et al (2021) and Azrieli and Rehbeck (2023) respectively.
\noindentExample 1 (singleton menus): If $\mathcal{A}$ includes all singletons and there is complete ignorance about menus, then $C_{d}=X$ for all $d$ and any $\lambda$ is rationalized by any $Q$. Barseghyan et al (2021) assume that the minimum menu size is at least two to avoid this scenario in their setup. However, in our setup we can allow a subset of all singleton menus, in which case they can affect (strictly expand) the identified region.
We illustrate here for the case where every decision rule $d$ is derived from maximization of a preference order, and where complete ignorance is assumed. Then each $Q$ describes a probability distribution over preferences. Let $X=\{a,b,c\}$. The six possible preference orders are:
Finally, let $\mathcal{A}=\{\{a\},\{a,b\},\{a,b,c\}\}$.\footnote{Thus menus are nested. Think, for example, of expanding budget sets.} Then, after deleting redundant inequalities, ((ref)) reduces to:
The first inequality provides an upper bound on the probability of the set of preferences that rank alternative $a$ highest, the second provides an upper bound on the probability of preferences that rank $a$ or $b$ highest, and the third gives an upper bound on the probability of preferences that rank $a$ above $b$.
Consider now what happens if the singleton $\{a\}$ is deleted from $\mathcal{A}$. Let $\mathcal{A^{\prime}}=\{\{a,b\},\{a,b,c\}\}$ be the new set of menus. Then the following additional inequalities are implied by ((ref)):
As a result, the sharp identification region shrinks strictly. (The intuition is that when $\{a\}$ is removed, then the sets $C_{3}$ and $C_{4}$ shrink, leading to the lower bound for $\lambda\left( b\right) $, and $C_{6}$ shrinks, which leads to the lower bound for $\lambda\left( \{b,c\}\right) $.)
For a numerical example, take $\lambda\left( a\right) =1/2,\lambda\left( b\right) =1/4,$ and $\lambda\left( c\right) =1/4$. The preference distribution given by $Q\left( \succ_{1}\right) =Q\left( \succ_{3}\right) =Q\left( \succ_{4}\right) =Q\left( \succ_{6}\right) =1/4$ rationalizes $\lambda$ when $\{a\}$ is included, but not if it is removed. The presence of a singleton menu does not preclude meaningful inference, but it does weaken inference by expanding the sharp identification region.
\noindentExample 2 (menu homogeneity): Menu-rationalizability as defined in ((ref)) permits heterogeneity in both decision rules and in the menu formation processes, the latter because $\pi_{d}$ and $\pi_{d^{\prime}}$ are allowed to differ. Refer to menu-homogeneity if $\pi_{d}=\pi$ for all $d$. This hypothesis has been adopted in several applied works where one can interpret the different menus as arising from feasibility rather than consideration (Tenn and Yun 2008, Tenn 2009, Conlon and Mortimer 2013, Lu 2022), and in the theoretical contribution by Azrieli and Rehbeck (2023, section 4), while its limitations have been noted by Barsheghyan et al (2021). Where menus are based on consideration one would expect them to depend on preference (or decision rule), as in the applied papers by Goeree (2008), and Abaluck and Adams-Prassl (2021). Here we demonstrate that imposing menu-homogeneity\ can lead to different conclusions about the sharp identified set.
Let $X=\{a,b,c\}$, and assume preference maximization, with only two possible preference orders \[ a\succ_{1}b\succ_{1}c\text{, \ }a\succ_{2}c\succ_{2}b\text{.} \] Finally, let $\mathcal{A}=\{\{a,b\},\{a,c\},\{b,c\},\{a,b,c\}\}$, and let the empirical measure be given by \[ \lambda\left( a\right) =\lambda\left( b\right) =\lambda\left( c\right) =1/3. \]
Then $\lambda$ can be rationalized by $Q$, where $Q\left( \succ _{1}\right) =2/3$ and $Q\left( \succ_{2}\right) =1/3$, because the inequalities ((ref)) can be verified. A corresponding version of ((ref)) uses the (heterogeneous) distributions over menus given by \[ \pi_{1}\left( \{b,c\}\right) =1/2\;\text{and }\pi_{2}\left( \{b,c\}\right) =1\text{.} \] However, $Q$ cannot rationalize $\lambda$ if one insists on menu-homogeneity: Under the latter condition, ((ref)) implies
which would force $Q$ to assign equal probabilities to both preferences, a contradiction.
In general, any restriction on the distributions over menus admitted in ((ref)) makes menu-rationalizability more difficult and thus shrinks the sharp identified set. The example confirms that in the case of menu-homogeneity the shrinkage can be strict. (Finally, note that in the example the sharp identification set with menu-homogeneity is not empty. For the two preference orders given, the unique rationalizing measure $Q$ assigns equal probability to the two preference orders.)
Given the motivation described in the introduction, proceed as follows. Let $\Omega=\{\omega_{a},\omega_{b}\}$ be the set of possible future states of the world. (The binary specification is enough to capture uncertainty about whether an event will or will not happen.) There is a common prior $\mu=\left( \mu_{a},\mu_{b}\right) >0$. Every individual $i$ receives some information, (or observes a signal), and updates beliefs to the posterior $p_{i}=\left( p_{ia},p_{ib}\right) >0$. Beliefs are represented equivalently by their implied odds ratios (in logarithmic form) \[ x^{0}=\log\left( \mu_{a}/\mu_{b}\right) \text{ and }x_{i}=\log(p_{ia} /p_{ib})\newline\text{.} \] These ratios lie in a finite set $X$.\footnote{One can specify primitives of the model so that sets below are finite wherever needed by the formalism. We will not elaborate.} The data to be rationalized is $\lambda\in\Delta\left( X\right) $, the empirical cross-sectional distribution of posterior odds ratios. Neither the updating rules used by individuals nor their information (viewed as being generated by an `experiment') is observable to the analyst.
Our formal model begins with the representation of experiments. Since a signal leads to a posterior odds ratio $x$ in $X$ and hence to the change from the prior view equal to $x-x^{0}$, we define an $\emph{experiment}$ to be a probability measure $E$ in $\Delta\left( X-\{x^{0}\}\right) $ .\footnote{$X-\{x^{0}\}=\{x-x^{0}:x\in X\}$.} To motivate this definition further, note that in the Bayesian framework standard application of Bayes rule implies
where $LLR$ denotes the log- likelihood ratio. Thus every experiment amounts to a distribution over log-likelihood ratios. (Equivalently, it is a distribution over posteriors odds ratio $x$, or over the posterior probability measure which is how experiments are frequently described.) Though the preceding presumes Bayesian updating, we show shortly that ((ref)) is compatible also with other updating rules. Accordingly, we continue to refer to signals as LLRs and as points in $X-\{x^{0}\}$.
The set of all experiments deemed relevant by the analyst, and satisfying $E\left( 0\right) <1$, is $\mathcal{E}$. An updating rule $\psi$ is a mapping from an experiment into a distribution over posterior odds: \[ \psi\left( E\right) \in\Delta\left( X\right) \text{.} \] Based on ((ref)), the focal Bayesian updating rule $\psi^{BR}$ is given by
for all $E\in\mathcal{E}$, that is, under Bayesian updating, the change in odds $x-x^{0}$ has the same distribution as $LLR$.
We also admit non-Bayesian updating rules defined as follows: For each parameter $\kappa$, $\kappa$$\leq\kappa\leq1$, $\psi^{\kappa}$ is given by\footnote{$\delta_{x^{0}}$ is the measure on $X$ that assigns probability 1 to the prior odds ratio $x^{0}$.}
where the lower bound $\kappa$ satisfies \[ \underline{\kappa}\geq-E\left( 0\right) /\left( 1-E\left( 0\right) \right) \text{ for all }E\text{.} \] (The latter condition implies that, for all $E$, $\psi^{\kappa}\left( E\right) \left( x\right) \geq0$ for all $x$, and hence that $\psi^{\kappa }\left( E\right) \left( \cdot\right) $ is a probability measure on $X$.) For instance, if we take Bayesian updating as the "correct" balancing of prior beliefs and responsiveness to signals, then, because $\delta_{x^{0}}$ gives no weight to signals, $\kappa>0$ implies "too much" weight to prior beliefs and thus underreaction to information. In contrast, $\kappa<0$ is interpretable as implying overreaction. The negative weight given to the prior view represented by $\delta_{x^{0}}$ upsets the proper balance captured by $\psi^{BR}\left( E\right) $ and implies "too much" sensitivity to the signal.\footnote{Epstein (2006) axiomatizes a closely related model of updating, (formulated for absolute probabilities rather than odds ratios), and we adapt his interpretations to ((ref)). Augenblick, Lazarus and Thaler (2023) postulate updating rules in terms of odds ratios and study how under/over-reaction varies with the strength of the signal as measured by LLR. They consider data on individual updating in response to a known experiment.}
Let $\Psi$ be a (finite) subset of $\{\psi^{\kappa}:\underline{\kappa} \leq\kappa\leq1\}$ that includes $\psi^{0}=\psi^{BR}$, and, for each $\psi \in\Psi$, let \[ \mathcal{P}_{\psi}=\{\psi\left( E\right) :E\in\mathcal{E}\}\text{,} \] the set of all distributions over odds ratios induced by an experiment from $\mathcal{E}$ and an update rule from $\Psi$. The analyst assumes only that every individual employs an experiment $E$ from $\mathcal{E}$ and an updating rule $\psi$ from $\Psi$, (their distribution across individuals is unknown), in which case she faces the set $\mathcal{P}_{\psi}$ of posterior distributions over odds ratios. Roughly, the foursome $\left( X,\Psi ,\mathcal{E},\{\mathcal{P}_{\psi}\}\right) $ is the counterpart of the foursome $\left( X,\mathcal{D},\mathcal{A},\{\mathcal{R}_{d}\}\right) $ in the main choice model.
As the counterpart of ((ref)) for this setting, say that $Q\in \Delta\left( \Psi\right) $ rationalizes $\lambda$ given $\{\mathcal{P}_{\psi}\}_{\psi\in\Psi}$ if, for every $\psi$, there exists $\rho_{\psi}\in\mathcal{P}_{\psi}$ such that
The sum is over distinct update rules and thus $Q$ describes their distribution explicitly.
The conditions characterizing rationalizability follow from Theorem (ref) if the sets $\mathcal{P}_{\psi}$ satisfy the appropriate form of the core condition ((ref)).\footnote{The theorem is applicable also given any other specifications of $\mathcal{E}$ and $\Psi$ for which the implied sets $\mathcal{P}_{\psi}$ satisfy the noted condition. In principle, therefore, alternative non-Bayesian updating rules could be accommodated.} Rather than simply assuming the latter, we derive it from the following assumption about the set of experiments $\mathcal{E}\subset\Delta\left( X-\{x^{0}\}\right) $: There exists a convex capacity $\nu$ on $X-\{x^{0}\}$ such that
Section (ref) describes and motivates examples of convex capacities and their cores that could be used to specify $\mathcal{E}$ and $\nu$ satisfying ((ref)). In addition, ((ref)) implies that, for each $\psi^{\kappa}\in\Psi$,
for some convex capacity on $X$ (in particular, $\mathcal{P}_{\psi^{\kappa}}$ is a convex set). In fact, for every $K\subset X$,\footnote{When $K=\varnothing$, take $K-\{x^{0}\}=\varnothing$ and\ hence $\nu\left( K-\{x^{0}\}\right) =0$.} \
Conclude from Theorem (ref) that $Q$ rationalizes $\lambda$ in the sense of ((ref)) if and only if, for all $K\subset X$,
where $\kappa_{Q}\equiv\Sigma_{\psi\in\Psi}Q\left( \psi^{\kappa}\right) \kappa$ is the average updating bias implied by $Q$. Notably, whether or not $Q$ rationalizes $\lambda$ depends only on the average bias that it implies. In particular, any rationalization of $\lambda$ can be mimiced by a population where everyone exhibits this average updating bias. This suggests the reframing whereby we ask if the average bias $\kappa_{av}$, $\min _{i}\kappa_{i}\leq\kappa_{av}\leq\max_{i}\kappa_{i}$, can rationalize $\lambda$.
The following additional implications are straightforward (use ((ref) )). There exists an Bayesian average rationalization if and only if, for all $K\subset X$,
where we use ((ref)), that is, $\nu$ is the lower envelope of the set $\mathcal{E}$ of experiments. \ A non-zero average updating bias is required to rationalize $\lambda$ under the following conditions. If $\kappa_{av}$ is any average bias that rationalizes $\lambda$ and if there exists $K^{\ast }\subset X$ such that
then \[
\] that is, there is underreaction on average if $x^{0}\not \in K^{\ast}$ and overreaction on average otherwise.
Finally, it follows that $\lambda$ cannot be rationalized by any average bias (or measure $Q$) if there exist subsets $K^{1}$ and $K^{2}$ of $X$ such that ((ref)) is satisfied by both $K^{\ast}=K^{i}$, $i=1,2$, and if $x^{0}\not \in K^{1}$, $x^{0}\in K^{2}$.
We conclude by describing two other settings where our theorem can be applied to characterize the robust identification of heterogeneity. The first concerns the identification of (unobservable) heterogeneous effort; notably, the story is not directly connected to choice or beliefs. The second is an instance of our choice model where decision rules are not based on preference maximization; rather, all choices are from a single known objective feasible set and are due to satisficing with heterogeneous unobserved reservation values which distribution is identified.
Consider a population of workers with common observable characteristics (e.g. education and experience), and working independently. Each produces a homogeneous output in quantity represented by an element of $X$.\footnote{To make clear the connection to the main choice model, we use the same symbols, though with different interpretations.} The empirical frequency distribution of outputs is given by $\lambda\in\Delta\left( X\right) $. Heterogeneity in output is attributed to differences in unobservable characteristics. The first unobservable is effort - there are finitely many effort levels $d\in \mathcal{D}$. The other unobservable is "everything else." The analyst may not be able to describe these other factors precisely, or even at all. However, she takes a stand on the set of their possible output consequences. Formally, for each effort $d$, denote by $C_{d}\subset X$ the set of outputs possible given the effort level and given what may ensue from "everything else." The analyst specifies the sets $C_{d}$, but is ignorant about likelihoods $\emph{within}$ these sets.\footnote{The assumption that $C_{d}$ can be specified even though "everything else" is poorly understood brings to mind Maskin and Tirole (1999) who argue that optimal contracts survive even with unforeseen contingencies when agents can forecast future payoffs .}
With this reinterpretation, rationalizability of $\lambda$ is well-defined, and Theorem (ref) can be applied to yield the (computationally tractable) sharp identified set of measures $Q$ over effort levels.
The rationalizability notion ((ref)) could also be accommodated by introducing a parameter $\theta\in\Theta$ to represent "everything else," and, for each $d$, a production function $f$ such that the pair $\left( d,\theta\right) $ yields output $f\left( d,\theta\right) $, and $C_{d}=\{f\left( d,\theta\right) :\theta\in\Theta\}$. Ignorance about $\Theta$ would be captured by admitting any distribution $\pi_{d}$ over $\Theta$ in the counterpart of definition ((ref)). (Roughly, $\Theta$ would play the role of the set of menus $\mathcal{A}$ above.) However, such a formulation involving production functions $f$ and probability distributions over $\Theta$ is arguably problematic in situations where the analyst cannot even conceive of what is included in "everything else."
There is a population of satisficing decision makers whose aspiration thresholds may differ. They each choose an alternative from the set $X$ and they agree that the value of alternatives is described by $v:X\mapsto\mathbb{R}$. However, individuals differ in two respects. First, aspiration thresholds differ; the set of distinct thresholds is $\mathcal{V}$. Second, individuals differ in the order in which they consider alternatives (this may be a subjective choice of procedure or exogenously imposed). Each sequential procedure follows a strict total order $>$ on $X$: the individual chooses the $>$-first alternative with a value at least as large as her threshold $v\in\mathcal{V}$, and if there are no such "satisfactory" alternatives then she chooses the $>$-last element in $X$. The empirical frequency of choices $\lambda$ is observed, but both aspiration levels and orders $>$ are unobserved. Theorem (ref), suitably reinterpreted, can be used to partially identify the distribution of aspiration levels while respecting limited knowledge (or complete ignorance) of the distribution of orders $>$.
Similar applications can be made to other problems of choice with frames (Salant and Rubinstein 2008) where frames vary across individuals and are unobserved by the analyst.