Rational Choice with Full-Comparability
Domains
Bruno A. Furtado
*Leandro Nascimento
Gil Riella
October, 2020
Abstract
We propose a new model of choice in the presence of incomplete preferences. Instead of simply choosing an element which is maximal according to her pref-erences, the decision maker divides the space of alternatives into subdomains inside which her preferences are complete. She then acts fully rationally and maximize her preferences inside these domains of full comparability. Repre-sentation theorems are given in which the decision maker always satis es a weaker form of the Weak Axiom of Revealed Preference and dierent postu-lates are imposed on a general notion of revealed preference. They identify a class of choice correspondences that is nested between choice correspondences represented by multiple rationales and the standard model of rational choice. In addition, we suggest that our results provide a useful tool to adapt mod-els that were otherwise restricted to deterministic choice to alternative setups, such as random choice and preferences over menus. In a way, they provide a bridge between dierent environments, with special focus on discussing incom-plete preferences.
JEL classi cation: D01
Keywords: Choice with categories, revealed preference relation, choice corre-spondences, incomplete preferences, linking deterministic and stochastic choice, consideration sets.
*Department of Economics, Columbia University. E-mail: [email protected]. Department of Economics, Universidade de Brasília. E-mail: [email protected].
1 Introduction
It is well-accepted among economists that the assumption that individuals are able to compare every pair of alternatives is inaccurate, both descriptively and normatively (see Aumann (1962), Bewley (2002) and Eliaz and Ok (2006), for example). That is, in general the preferences of economic agents are not complete. This may happen be-cause some decisions are simply too hard to make, or even bebe-cause some alternatives are too dierent to be compared to each other. Still, often individuals have to make a decision from choice problems that include alternatives she cannot compare. One possible model in such a situation is that the individual has an incomplete preference relation and considers `choosable' all options that are maximal with respect to this relation. This is the approach in Eliaz and Ok (2006), for example. Although the approach in Eliaz and Ok (2006) reconciles the incompleteness of preferences with maximization and optimality, as a description of behavior it is not clear what proce-dure the individual would follow in order to end up choosing one maximal option or another. This is particularly problematic if we not only observe the alternatives the individual may choose some day, but we also observe the frequency with which she makes each choice. That is, the model where the individual simply picks a maximal alternative in the presence of incomplete preferences has no natural counterpart in the setup of random choice. We have a similar di culty if the object of study is a preference over menus.
The goal of the present work is to characterize a model where the decision maker (henceforth DM) has a clear way of dealing with incomplete preferences and this way has a natural adaptation to contexts other than deterministic choice. In our model, the DM has a possibly incomplete preference relation, but divides the space of alternatives into a collection of (not necessarily disjoint) sets of fully comparable alternatives. The choosable alternatives in a given choice problem are then the options that maximize the individual's preferences inside any of these domains of full comparability. It is as if the impossibility of acting fully rationally due to the incompleteness of her preferences leads the DM to focus her attention on subsets of the available alternatives where she can be `rational' and simply pick the best option. We start from a basic rationality postulate and investigate the consequences of impos-ing dierent consistency properties on a suitable notion of revealed strict preference and the individual's choices from two-element sets. In words, we say that an alterna-tive x is revealed strictly preferred to another alternaalterna-tive y if there exists at least one
situation where y stops being a choosable option because x becomes available. We then show how by simply imposing that this notion of revealed preference is acyclic one obtain a model of choice under incomplete preferences that has a natural coun-terpart in other setups. In a way, the representation we derive gives us a language that connects dierent types of datait is a bridge between deterministic choice, stochastic choice and preferences over menus, specially useful to discuss incomplete preferences.
We label our model choice by full-comparability domains. The full-comparability domains model is nested among models that display dierent degrees of rationality. The basic rationality postulate we start from characterizes what is known as the pseudo-rational representation (see Aizerman and Malishevski (1981)). Therefore, the full-comparability domains model is less general than pseudo-rationalization. If we impose that the strict revealed preference we de ne agrees with the notion of revealed preferences derived from two-element sets, we obtain the standard model of choice under incomplete preferences, as in Eliaz and Ok (2006). This implies that maximization of an incomplete preference relation is a special case of our model. These relations are summarized in Figure 1.
pseudo-rationality full-comparability domains
incomplete preferences rational choice
Figure 1: Rational choice models.
This paper is organized as follows. In Section 2, we describe our framework and recall some de nitions. We characterize our model and show how it relates to the standard model of maximization of an incomplete preference in Section 3. In Section 4, we discuss how our model provides a language to talk about incomplete preferences in setups other than deterministic choice. Section 5 places the model within the context of rational choice, while we discuss the related literature in Section 6. Section 7
concludes with remarks on some open questions. In the main text, we restrict the analysis to nite choice spaces. We deal with the case of in nite choice spaces in Appendix A. Finally, we relegate the proofs of the results presented in the main text to Appendix B.
2 Setup and De nitions
For an arbitrary set X, a binary relation % on X is simply a subset of X × X. As usual, we write x % y to represent the fact that (x, y) ∈ %. We say that % is re exive if x % x for every x ∈ X, and say it is transitive if x % y and y % z imply x % z, for every x, y, z ∈ X. A re exive and transitive binary relation is called a preorder or a preference relation. Given a subset Y of X, we say that % is complete on Y if, for every x, y ∈ Y , we have x % y or y % x. The symmetric and asymmetric parts of % are denoted by ∼ and , respectively. Formally, for each x, y ∈ X, x ∼ y if x % y and y % x, and x y if x % y, but it is not true that y % x. We say that % is antisymmetric if ∼ ⊆ {(x, x) : x ∈ X}. We call an antisymmetric preorder a partial order and use the symbol < to denote it. If in addition the relation is complete, we say it is a linear order. Given a preorder % on X and a subset A of X, we de ne the set of maximizers of % on A by max(A, %) := {x ∈ A : x % y for all y ∈ A}. Similarly, we de ne the set of maximal elements with respect to % in A by MAX(A, %) := {x ∈ A : y x for no y ∈ A}.
Now, let X be any set and let ΩX be a collection of nonempty subsets of X. We
interpret X as the grand set of alternatives and ΩX as the collection of all conceivable
choice problems the DM may face. For notation, we denote by F(X) the set of all nite and nonempty subsets of X.
We follow Eliaz and Ok (2006) and work with the following de nition: De nition 1. We say that (X, ΩX) is a choice space if
(i) {x} ∈ ΩX for every x ∈ X;
(ii) A ∪ B ∈ ΩX whenever A, B ∈ ΩX.
In words, the de nition above says that (X, ΩX) is a choice space whenever all
singletons belong to ΩX, and ΩX is closed under nite unions. Given a choice space
c(A) ⊆ A for every A ∈ ΩX. We can now formally introduce the type of
full-comparability decomposition our model will display.
De nition 2. Let X be an arbitrary set and % a preorder on X. A full-comparability decomposition of the set X relative to % is a family S of nonempty subsets of X such that X = ∪S and % is complete on every S ∈ S. Any S ∈ S is referred to as a full-comparability domain.
We may now introduce our main notion of representation.
De nition 3. Given a choice space (X, ΩX) and a choice correspondence c on
(X, ΩX), we say that c admits a representation by full-comparability domains if there
exist a partial order < on X and a full-comparability decomposition S of X relative to < such that, for every choice problem A ∈ ΩX,
c(A) = [
S∈S
max(A ∩ S, <).
3 Representation Theorems
In the main text, we will restrict our attention to nite choice spaces. That is, we will assume that X is a nite set and ΩX := 2X \ {∅}. We discuss arbitrary choice
spaces in Appendix A. Let c be a choice correspondence on (X, 2X\{∅}). As we have
discussed in the Introduction, the analysis here will be based on a general notion of strict revealed preference. Formally, consider the following de nition:
De nition 4. De ne the binary relation B ⊆ X × X by x B y if, and only if, there exists a choice problem A ∈ ΩX such that y ∈ c(A) but y /∈ c(A ∪ {x}), for any x
and y in X.
In words, the relation B declares x to be preferred to the alternative y if there exists at least one situation where y stops being chosen because x is added to the set of available alternatives.
Throughout the paper our axioms will always imply that the binary relation B is asymmetric (that is, x B y implies that y B x is not true), so that we can indeed interpret it as a form of strict revealed preference. As we have discussed in the Introduction, all models studied in this paper satisfy a basic rationality property
and dier from each other only with respect to the properties of the relation B. Formally, consider the following weakening of the standard Weak Axiom of Revealed Preference (WARP):
Axiom 1 (Pseudo-WARP). For all A, B ∈ ΩX: if x ∈ c(B) ∩ A and c(A) ⊆ B,
then x ∈ c(A).
As we have said above, Pseudo-WARP is a basic rationality property satis ed by sev-eral models. Recall that the original WARP1 postulate warranties the same
conclu-sion as Pseudo-WARP whenever c(A)∩B 6= ∅, so Pseudo-WARP is a clear weakening of WARP.
Additionally to Pseudo-WARP, the only other property we need in order to char-acterize the choice correspondences that have a representation by full-comparability domains is the acyclicity of the relation B. Formally:
Axiom 2 (Acyclicity). The binary relation B is acyclic. That is, for any nite collection {x1, . . . , xn} ⊆ X, if x1B · · · B xn, then it is not true that xnB x1.
Acyclicity can also be seen as some basic rationality postulate imposed on the re-vealed strict preference relation B. We note that if a choice correspondence c has a representation by full-comparability domains (<, S) and x B y for some pair of alter-natives x, y ∈ X, then it is clear that we must have x y. Therefore, the fact that the acyclicity of the relation B is a necessary condition for a choice correspondence cto have such a representation is an immediate implication of the transitivity of the partial order < that is part of that type of model. At the same time, it is easy to verify that any choice correspondence that has a representation by full-comparability domains satis es Pseudo-WARP. Our rst result shows that the converse is also true and these two postulates exactly characterize the choice correspondences that admit a representation by full-comparability domains.
Theorem 1. Let X be a nite set and c be a choice correspondence on (X, 2X\{∅}).
Then c has a representation by full-comparability domains if, and only if, it satis es Pseudo-WARP and Acyclicity.2
1WARP. For all A, B ∈ ΩX: if x ∈ c(B) ∩ A and c(A) ∩ B 6= ∅, then x ∈ c(A).
2Theorem 1 does not actually require the grand set of alternatives X to be nite. It is su cient
to assume that every choice problem is a nite set, that is, ΩX= F (X). Therefore, the axioms also
characterize a representation by full-comparability domains for any arbitrary choice set X as long as the DM is always presented with a nite number of alternatives to choose from at a time.
Theorem 1 is an immediate corollary of Theorem 7 below, so we omit its proof. Still, it is important to mention that two aspects of our axioms are key to the su ciency part of the proof of Theorem 1. First, they imply that if an alternative x is chosen from a choice problem A, then it is also chosen from all choice problems smaller than A in which x is still available. We refer to this property as Cherno's axiom or Sen's α postulate (see Section 5 below). Second, they imply the existence of an acyclic binary relation ∗ on X such that, for any pair of choice problems A and B
in ΩX with B ⊆ A, if x ∈ X is such that x ∈ c(B), but it is not true that x ∈ c(A),
then there exists y ∈ A \ B with y ∗ x and y ∈ c(B ∪ {y}). We can see this
property as a partial form of binariness (representability by a binary relation), since, for every A ∈ ΩX and x ∈ A, x ∈ max(A, ∗) implies x ∈ c(A). Moreover, it turns
out that these two properties characterize the choice correspondences that admit a representation by full-comparability domains even in arbitrary choice spaces. We have a detailed discussion of this fact in Appendix A.
We note that the full-comparability domains representation model (<, S) has a nat-ural counterpart in the random choices setup. All one needs for that is a probability measure over S. Now, given a choice problem A and an alternative x ∈ A, we may say that the probability of choosing x from A is simply the probability of drawing a full-comparability domain S such that x maximizes < in A ∩ S.
In fact, the random choices version of the full-comparability domains model has been axiomatized by Aguiar (2017), albeit with a dierent interpretation. Remarkably, the axiomatization of the model in the stochastic choice setup has a similar structure and also relies on the acyclicity of a suitable notion of revealed strict preference. That is, not only the full-comparability domains model has a natural counterpart in the setup of random choice, but also the behavioral foundations of the model in the deterministic and stochastic setups are very close to each other. We will talk more about how the full-comparability domains model allows one to talk about incomplete preferences in the setups of random choice and preferences over menus in Section 4. As we have discussed in the introduction, the standard model of choice in the pres-ence of incomplete preferpres-ences deems all maximal alternatives with respect to some incomplete preorder choosable. Formally:
De nition 5. Given a choice space (X, ΩX) and a choice correspondence c on
(X, ΩX), we say that c can be represented by the maximization of a possibly
in-complete preference relation if there exists a preorder %⊆ X × X, not necessarily complete, such that c(A) = max (A, %), for every A ∈ ΩX.
It turns out that the maximization of an incomplete preference relation is equivalent to the particular version of the full-comparability domains model where the collection of domains of full comparability is closed under intersections. Moreover, this model is characterized by a strengthening of the acyclicity postulate that requires that the relation B be entirely identi ed from binary choices.
Axiom 3 (Binary Dominance). For all x, y ∈ X: if x B y then {x} = c({x, y}). De ne a binary relation B∗ ⊆ X × X by x B∗ y if, and only if, y /∈ c({x, y}). It is
easy to check that Pseudo-WARP implies that B∗ is transitive, so Pseudo-WARP
plus Binary Dominance imply Acyclicity. Therefore, by Theorem 1, we know that a choice correspondence that satis es Pseudo-WARP and Binary Dominance has a representation by full-comparability domains. As we have mentioned above, Pseudo-WARP and Binary Dominance also characterize the choice correspondences that can be represented by the maximization of a possibly incomplete preference relation. Formally, we have the following result:
Theorem 2. Let X be a nite set and c be a choice correspondence on (X, 2X\{∅}).
The following statements are equivalent:
( i) The choice correspondence c satis es Pseudo-WARP and Binary Dominance; ( ii) The choice correspondence c has a representation by full-comparability domains
(<, S) such that S is closed under nonempty intersections. That is, for every S, T ∈ S with S ∩ T 6= ∅, S ∩ T ∈ S;
( iii) The choice correspondence c can be represented by the maximization of a pos-sibly incomplete preference relation.
Theorem 2 shows that maximization of a possibly incomplete preference relation is a special case of a full-comparability domains representation. This gives us a possible solution to the issue we raised in the introduction of the inexistence of a natural counterpart to the maximization of an incomplete preference model in a random choices setup. As long as one uses the language of full-comparability domains representations, the only thing necessary to write a random choice version of the model is a probability measure over the collection of full-comparability domains S.
4 Incomplete Preferences and Dierent Datasets
So far, we have worked in a deterministic choice setup. In terms of observed data, this corresponds to the implicit assumption that we can observe which alternatives the DM has ever chosen from each choice problem, but we do not have enough infor-mation to infer how often she has made each choice. In this section, we discuss how the full-comparability domains model gives us a language to talk about incomplete preferences when we have access to two other dierent types of data.
The rst type of data we will look at is the random choice setup. That is, we will assume that not only we observe which alternatives are chosen from each choice problem, but we also observe the frequency with which the DM chooses each option. In fact, we will even assume that we observe how often the DM does not make a choice. Formally, let X be a nite set and consider the choice space (X, 2X\ {∅}). A
stochastic choice function on (X, 2X\{∅})is a function P : X ×(2X\{∅}) → R + such
that, for each A ∈ 2X\ {∅}, P
x∈AP(x, A) ≤ 1 and P(y, A) = 0 for every y ∈ X \ A.3
The translation of the full-comparability domains model to the setup above is im-mediate:
De nition 6. Let X be a nite set and P be a stochastic choice function on (X, 2X\
{∅}). We say that (<, S, µ) is a full-comparability domains representation of P if < is a partial order on X, S is a full-comparability decomposition of X with respect to < and µ is a full-support probability measure on S such that, for each choice problem A ∈ 2X \ {∅} and alternative x ∈ A,
P(x, A) = µ({S ∈ S : {x} = max(A ∩ S, <)}).
Under a dierent interpretation, the representation above was axiomatized by Aguiar (2017). Curiously, it is again characterized by a basic rationality postulate plus the acyclicity of a notion of strict revealed preference that has a similar interpretation as the relation B de ned in Section 3. Precisely, the version of the B relation in the random choice setup says that x B y if there exists a choice problem A such that P(y, A) > P(y, A ∪ {x}), so, again, x B y if there exists at least one situation where
3The possibility that P
x∈AP(x, A) < 1 for some choice problem A captures the idea that the
DM may sometimes refrain from making a choice. Equivalently, one could impose the existence of an alternative that is available in every choice problem. This is the approach adopted by Manzini and Mariotti (2014), for example.
the presence of x makes y less desirable.4
Another type of data that is sometimes studied in the literature is the setup of preferences over menus (see Kreps (1979)). Formally, consider again the choice space (X, 2X \ {∅}) for a nite set X. Our primitive now will be a complete preorder % on 2X\ {∅}. We say that (<, S, µ, U) is a full-comparability domains representation
of % if < is a partial order on X, S is a full-comparability decomposition of X with respect to <, µ is a full-support probability measure on S and U : X → R++ is a
strictly <-increasing function5 such that the function V : 2X \ {∅} → R de ned by
V (A) := X
S∈S
π(S) max
x∈A∩SU (x), for each A ∈ 2
X \ {∅},
with the convention that maxx∈∅U (x) = 0, represents %.
Nascimento et al. (2020) have shown that, as it was the case in the two previous setups, the representation above is characterized by standard basic rationality pos-tulates and the acyclicity of a suitable de nition of revealed preference.6 The precise
de nition of the relation B in the preferences over menus setup says that x B y if {x, y} ∼ {y}or there exists a menu A with A ∪ {y} A,7 but A ∪ {x, y} ∼ A ∪ {y}. That is, once more x B y if there exists at least one situation where y becomes less desirable because x is available.
5 Full-Comparability Domains and Rational Choice
The full-comparability domains model can be placed within the wider context of rational choice. Pseudo-WARP is a basic rationality postulate satis ed by several models, while the properties of the binary relation B help us distinguish between models with dierent degrees of rationality. We begin by recalling the de nition of a pseudo-rationalizable choice correspondence, rst axiomatized by Aizerman and Malishevski (1981).8
4Actually, Aguiar (2017) de nes the relation B by x B y if there exists a choice problem A such
that P(y, A) 6= P(y, A ∪ {x}), but the acyclicity of the relation B as de ned above is enough to deliver the representation.
5That is, for any pair of distinct alternatives x, y ∈ X, x < y implies that U(x) > U(y). 6The rationality postulates are exactly the ones that appear in Kreps (1979) and that characterize
Kreps' preference for exibility representation.
7The symbol refers to the strict part of the preorder % on 2X\ {∅}. 8See also Moulin (1985).
De nition 7. Given a choice space (X, ΩX) and a choice correspondence c on
(X, ΩX), we say that c has a pseudo-rational representation if there exists a
fam-ily P of linear orders on X such that c(A) = [
<∈P
max(A, <), for every A ∈ ΩX.
Two classical axioms introduced by Cherno (1954), Postulates 4 and 5∗, characterize
the choice correspondences that have a pseudo-rational representation. Together, they are equivalent to Pseudo-WARP, which shows that Pseudo-WARP contains two general principles of rational choice. One is Cherno's axiom, which was discussed in the previous section. The second principle asserts that when more options are added to the set of alternatives and none of the new options is chosen, the choices from the smaller set are also choices in the larger set of alternatives. It is sometimes referred to as Aizerman's axiom.
Cherno's Axiom. For all A, B ∈ ΩX with B ⊆ A: c(A) ∩ B ⊆ c(B).
Aizerman's Axiom. For all A, B ∈ ΩX with B ⊆ A: if c(A) ⊆ B then c(B) ⊆
c(A).
This discussion can be summarized by the following result:
Theorem 3. Let X be a nite set. A choice correspondence c on (X, 2X \ {∅})
has a pseudo-rational representation if, and only if, it satis es any of the following conditions.9
( i) Cherno and Aizerman's axioms; ( ii) Pseudo-WARP.
When we compare Theorem 3 with Theorem 1, we see that only the acyclicity of the binary relation B separates representation by full-comparability domains from pseudo-rationality. The next example con rms that pseudo-rationality is indeed a strictly more general concept than representation by full-comparability domains.
9It is well known that in nite settings Aizerman and Cherno are equivalent to Plott's (1973)
path independence axiom (see, e.g., Moulin, 1985). Therefore, path independence is also equivalent to Pseudo-WARP.
Example 1. Let X := {x, y, z, w} and consider a choice correspondence c pseudo-rationalized by the linear orders y 1 x 1 z 1 w and w 2 z 2 x 2 y.
Since x ∈ c({x, y}) but x /∈ c({x, y, z}), we have z B x. On the other hand, since z ∈ c({z, w}), but z /∈ c({x, z, w}), we also have x B z, thus violating Acyclicity. Therefore, c does not have a representation by full-comparability domains. We have seen above that an important special case of representation by full-comparability domains is the maximization of a possibly incomplete preference relation, which is obtained when Acyclicity is replaced by Binary Dominance. If you strengthen Binary Dominance, we obtain the rational choice model. Consider the following postulate. Axiom 4 (Transitive Binary Dominance). For all x, y ∈ X: if x B y, then {x} = c({x, z}) for every z ∈ X with y ∈ c({y, z}).
Pseudo-WARP and Transitive Binary Dominance characterize (in nite choice spaces) the choice correspondences that maximize a complete preference relation.
Theorem 4. Let X be a nite set and c be a choice correspondence on (X, 2X\{∅}).
Then, c satis es Pseudo-WARP and Transitive Binary Dominance if, and only if, there exists a complete preorder %⊆ X × X such that c(A) = max(A, %) for every A ∈ 2X \ {∅}.
This discussion is summarized by Figure 1, in the Introduction. The analysis shows that the concept of representation by full-comparability domains connects several models and all these connections rely on a single rationality property, satis ed by all the models, with their dierences characterized by properties of the revealed preference relation B. We note also that, although we have so far restricted the analysis to nite choice spaces, in Appendix A.3 we show that all inclusions in Figure 1 are valid in arbitrary choice spaces.
6 Related Literature
6.1 Rational Choice with Incomplete Preferences
Several papers have results related to the rationalization of choices with an incom-plete preference relation. In particular, versions of part (iii) of Theorem 2 have
appeared in Sen (1971), Schwartz (1976) and Bandyopadhyay and Sengupta (1993). The three papers mentioned above stated the result in terms of a complete and quasi-transitive relationthat is, a complete relation whose strict part is quasi-transitive. Eliaz and Ok (2006) worked with incomplete preorders and, on top of having axiomatized choice correspondences that can be rationalized by such relations, also worried about the issue of dierentiating indierence from incomparability. The analysis in Eliaz and Ok (2006) is based on a behavioral notion of incomparability. Ribeiro and Riella (2017) argued that indierence is a more fundamental and more applicable concept than incomparability and introduced the concept of behavioral indierence.
6.2 The Structure of Incomplete Preferences
Recently, Gorno (2018) has shown that every incomplete preference relation can be decomposed into maximal domains of comparability in a way that satis es some nice properties. A representation by full-comparability domains has a similar inter-pretation. Although the DM's preferences are incomplete, she divides the space of alternatives into sets such that all alternatives in the same set are comparable to each other. The alternatives she considers choosable in a given choice problem are the ones which are optimal in these domains of full comparability. Let X be any set and suppose that % is a preorder on X. For notation, by %|D we mean the restriction of
the relation % to the set D. That is, %|D :=% ∩(D × D). Gorno (2018) shows that
there exists a collection DX(%) of subsets of X such that
(i) % is complete on D for every D ∈ DX(%);
(ii) every set D ∈ DX(%) is maximal with respect to (i);
(iii) %= SD∈DX(%)%|D;
(iv) max(X, %) = SD∈DX(%)max(D, %).
The main dierence between what we call a full-comparability decomposition and the de nition above is that we do not demand that the domains of full-comparability be maximal. Not having this requirement allows us to obtain a decomposition that explains the individual's choices in every choice problem, while, in general, the re-quirement of maximality limits this fact to the big set X. Gorno (2018) has a detailed discussion about the subsets of X whose choices can be explained by his decomposition.
6.3 Choices with Limited Consideration
Several authors have worked with decision makers who focus only on a subset of the available options when making a choice. In some works, this happens because of cognitive limitations that make it impossible for the DM to pay attention to all al-ternatives. See Masatlioglu et al. (2012), Dean et al. (2017) and Lleras et al. (2017), for example. Another possibility is that the decision making procedure occurs in two stages and some available alternatives are eliminated in the rst stage. See Manzini and Mariotti (2007), Cherepanov et al. (2013) and Ok et al. (2015), for example. Here we oer a third possibility for this to happen. Because of the impossibility to compare all options, the DM concentrates her choices on subsets of fully compa-rable alternatives inside which she can be completely rational and simply pick the alternative that maximizes her preferences.
As we have metioned above, Aguiar (2017) has axiomatized a version of the ratio-nal choice with full-comparability domains model in a random choice setup. The interpretation given in that paper was dierent, though. Aguiar (2017) interpreted the domains of full comparability as categories that are randomly selected when the individual has to make a choice from a given choice problem. Under this interpreta-tion, the probability of choosing an alternative x from a choice problem A is simply the probability of drawing a category from which x is the best option in A.
6.4 Bridging Deterministic and Stochastic Choice
The link between deterministic and stochastic choice has only recently attracted some attention, so the literature about it is still sparse. One work that stands out is Ok and Tserenjigmid (2020). Ok and Tserenjigmid (2020) investigate which stochastic choice functions may be deemed rational. They start from a stochastic choice function P and study two notions of choice correspondences induced by P. Formally, let X be a nite set and P be a stochastic choice function on (X, 2X \ {∅}). They de ne the
following choice correspondences from P:
cP(A) := {x ∈ A : P(x, A) > 0},
cP(A) := {x ∈ A : P(x, A) ≥ P(y, A) for all y ∈ A},
for every A ∈ 2X \ {∅}. They then investigate, for several well-known stochastic
choice models, when the choice correspondences induced by those models can be represented by the maximization of a complete or an incomplete preorder.
The most important stochastic choice model is the random utility model (RUM). In the RUM, the DM has a collection L of linear orders over X and a probability measure µ on L. Given a choice problem A and some alternative x ∈ A, the probability of choosing x from A is given by
P(x, A) = µ({<∈ L : x < y for every y ∈ A}).
The RUM was axiomatized by Falmagne (1978) (see also Barberá and Pattanaik (1986)) and several stochastic choice models are special cases of it. We note that, for every RUM (µ, L), L is a pseudo-rational representation for cP. The RUM is, thus,
the natural counterpart of the pseudo-rational representation in the setup of random choice.
Ok and Tserenjigmid (2020) study the conditions for the choice correspondences induced by a RUM to be rationalizable by the maximization of a complete or an incomplete preorder. We shall focus on the problem of identifying when the choice correspondence cP induced by a RUM can be represented by the maximization of
a possibly incomplete preorder. Ok and Tserenjigmid (2020) show that the choice correspondence cP induced by a RUM (µ, L) maximizes a possibly incomplete
pref-erence relation if, and only if, for every choice problem A and x ∈ A, whenever x /∈ max(A, <) for every <∈ L, there exists some z ∈ A with z x for every <∈ L.10
We may ask the same question as Ok and Tserenjigmid (2020) for the full-comparability domains model. Recall that a stochastic choice function P on (X, 2X \ {∅}) has a
representation by full-comparability domains if there exist a partial order < on X, a full-comparability decomposition S of X with respect to < and a full-support probability measure µ on S such that, for each choice problem A ∈ 2X \ {∅} and
10Ok and Tserenjigmid (2020) write the RUM in terms of a collection of utility functions that
represent the relations in L, so they write their condition in a slightly dierent format. It is easy to see that the statement above is equivalent to their condition, though.
x ∈ A,
P(x, A) = µ({S ∈ S : x ∈ max(A ∩ S, <)}).
We now have the following characterization of when cP may be represented by the
maximization of a possibly incomplete preorder:
Theorem 5. Consider a stochastic choice function P on (X, 2X \ {∅}) that has a
representation by full-comparability domains (<, µ, S). The choice correspondence cP
induced by P can be represented by the maximization of a possibly incomplete preorder if, and only if, for every pair S, T ∈ S and every x ∈ S ∩ T , there exists Sx∈ S with
x ∈ Sx and {y ∈ Sx: y < x} ⊆ S ∩ T .
Theorem 5 characterizes, among the stochastic choice functions which have a repre-sentation by full-comparability domains, the ones whose support is compatible with the maximization of a possibly incomplete preference relation. Dierently from the condition in Ok and Tserenjigmid (2020), the characterization relies only on the el-ements of the representation and not on what happens in speci c choice problems. This result illustrates how the language of the full-comparability domains represen-tation gives us a uni ed way to discuss incomplete preferences in deterministic and stochastic setups.
Another paper that discusses the connection between deterministic and stochastic choice models is Costa et al. (2020). In that paper, the authors axiomatize pseudo-rational choice correspondences that satisfy the single-crossing and single-peak prop-erties, repeating, in a deterministic setup, the exercise in Apesteguia et al. (2017), which was performed in a random choice setup. Costa et al. (2020) then use the exercise to make three conceptual points, the last one being particularly relevant to the analysis here. As the exercise in Costa et al. (2020) points out, some models with poor uniqueness properties in deterministic setups are fully unique if one observes the exact frequencies of choice. Moreover, sometimes if one understands the exact connection between the two setups, one may realize that the full power of the exact choice frequencies is not necessary for one to recover the uniqueness of the repre-sentation. For example, in the case of the single-crossing pseudo-rational model of Costa et al. (2020), all one needs to uniquely identify the single-crossing collection of linear orders is knowledge of the comparative probabilities among two-element sets. Knowledge of the exact choice probabilities is not necessary.
The situation with the full-comparability domains model is similar, in the sense that its deterministic version is not unique, while the stochastic one is. At this
point, understanding how much knowledge of the choice probabilities is necessary to uniquely identify the elements of the full-comparability domains representation remains an open question.
7 Conclusion
We have presented a model that suggests a dierent way to deal with incomplete preferences. Instead of simply choosing the maximal elements according to her incom-plete preference relation, the DM divides the space of alternatives into subdomains where her preferences are complete. She then restricts her choices to these domains, where she can simply maximize her preferences. We have shown how a basic ra-tionality postulate plus variations on the properties of a suitably de ned revealed preference relation characterize not only the model introduced here, but also a full range of models related to the rationality paradigm.
As we have discussed in the introduction, our results give us a language to build a bridge between deterministic choice theory and other setups, with a special focus on discussing incomplete preferences. In fact, as we have mentioned above, Aguiar (2017) has axiomatized exactly a version of the full-comparability domains represen-tation in the setup of random choice, while Nascimento et al. (2020) have done that in the setup of preferences over menus. Interestingly, the property that characterizes the versions of the full-comparability domains model in these other setups is again the acyclicity of a suitably de ned notion of revealed preference.
The empiric interpretation of a choice correspondence is that it represents the choices which were made at least once from each choice problem, after repeated observations. In theory, if one has enough observations, one can even infer the choice frequencies and work instead with a stochastic choice function. In practice, this would demand an impossibly large set of observations, so in many situations researchers may have access only to the support of a given stochastic choice functionthat is, a choice correspondence. Because of this fact, working with models that `make sense' in both setups seems to be a desirable practice. Our work gives a possible solution to this issue in the case of incomplete preferences.
This work leaves some open questions. First, it fails to provide a general characteri-zation of representation by full-comparability domains when the choice space is not nite. Theorems 6 and 7 in Appendix A provide partial characterizations, but, while
Theorem 7 deals with a special case, the result in Theorem 6 exogenously assumes the existence of a binary relation that interacts with the choice correspondence in a certain way, which makes it unacceptable as a behavioral characterization of this type of representation. To the best of our knowledge, the same type of question remains open for pseudo-rational representations, as we are not aware of any axiomatization of that model in arbitrary choice spaces.
Finally, we note that at this point we also lack a better understanding of the unique-ness properties of representations by full-comparability domains. It is easy to see that such representations are not unique in general, but it is not clear which prop-erties characterize when two distinct representations by full-comparability domains generate the same choices.
A In nity Choice Spaces
We now discuss the representation by full-comparability domains when the choice space is not necessarily nite.
A.1 A Partial Characterization
Let (X, ΩX) be an arbitrary choice space. That is, let X be any set and ΩX be a
collection of nonempty subsets of X that includes all singletons and is closed under nite unions. Let c be a choice correspondence on (X, ΩX). Consider the following
postulate:
Axiom 5 (Weak Binariness). There exists an acyclic binary relation ∗⊆ X ×X
such that, for any A, B ∈ ΩX with B ⊆ A, if x ∈ c(B), but x /∈ c(A), then there
exists y ∈ A \ B with y ∗ x and y ∈ c(B ∪ {y}).
When put together with Cherno's Axiom the postulate above characterizes the choice correspondences that have a representation by full-comparability domains in arbitrary choice spaces. Formally:
Theorem 6. Let (X, ΩX) be an arbitrary choice space and c be a choice
correspon-dence on (X, ΩX). Then c has a representation by full-comparability domains if, and
Proof. It is easy to see that if c has a representation by full-comparability domains, then c satis es Cherno's Axiom and Weak Binariness, with the acyclic relation in the statement of Weak Binariness being the strict part of the relation < that appears in the statement of the full-comparability domains representation.
Conversely, suppose c satis es Cherno's Axiom and Weak Binariness, for an acyclic binary relation ∗⊆ X × X. Let < be any linear order that extends ∗.11
Fix any A ∈ ΩX and x ∈ c(A). De ne SA,x := {x} ∪ {y ∈ X \ A : y < x and y ∈
c(A ∪ {y})}. Notice that A ∩ SA,x = {x}and, consequently, {x} = max(A ∩ SA,x, <).
Now x any choice problem B ∈ ΩX and suppose that y ∈ max(B ∩ SA,x, <). If
y /∈ c(B), then, by Cherno's Axiom, y /∈ c(A ∪ B), and, by Weak Binariness, there must exist z ∈ B \ A such that z ∗ y and z ∈ c(A ∪ {y, z}). Applying
Cherno's Axiom once more, we get that z ∈ c(A ∪ {z}). Moreover, since < extends ∗, we have that z y < x. But then z ∈ S
A,x, which contradicts the fact that
y ∈ max(B ∩ SA,x, <). We conclude that y ∈ c(B). We have just shown that,
for every choice problem A ∈ ΩX and x ∈ c(A), there exists SA,x ⊆ X such that
{x} = max(A∩SA,x, <) and, for every choice problem B ∈ ΩX, max(B∩SA,x) ⊆ c(B).
Consequently, if we de ne S := {SA,x : A ∈ ΩX and x ∈ c(A)}, we obtain the desired
representation.
A.2 Full-comparability Domains under Finitariness
As we have discussed above, we were unable to provide a complete characterization of the choice correspondences that admit a representation by full-comparability domains in an arbitrary choice space. Instead, we will now concentrate the analysis on choice correspondences that satisfy an additional property due to Nehring (1996, 1997). Axiom 6 (Finitariness). For all A ∈ ΩX: if x ∈ c(D) for every nite subset D of
A then x ∈ c(A).
Finitariness is trivially satis ed when X is a nite set or ΩX is the collection of
nonempty nite subsets of X. It is also implied by a speci c continuity property of c when X is a separable metric space and ΩX is the space of nonempty compact
subsets of X (see Nehring (1996)).
11Since ∗ is acyclic, its transitive closure is a strict partial order that extends ∗. Now the
Szpilrajn Theorem (see Szpilrajn (1930)) guarantees that there exists a linear order < that extends ∗.
We will also need to adapt the Pseudo-WARP postulate in order to make it com-patible with the possibility that some domains of full comparability may have no maximum alternatives in choice problems with an in nity number of elements. Axiom 7 (Pseudo-WARP∗). For all A, B ∈ Ω
X: if x ∈ c(B)∩A and c(B∪{y}) ⊆
B for all y ∈ A, then x ∈ c(A).
Suppose that c satis es Pseudo-WARP and A, B ∈ ΩX and x ∈ X are such that
x ∈ c(B) ∩ A and c(B ∪ {y}) ⊆ B for all y ∈ A. By Pseudo-WARP, for every y ∈ c(A ∪ B)we have y ∈ c(B ∪{y}) ⊆ B. That is, c(A∪B) ⊆ B. Applying Pseudo-WARP twice we get rst that x ∈ c(A ∪ B) and then that x ∈ c(A). This shows that Pseudo-WARP implies Pseudo-WARP∗. The converse implication is true in nite
choice spaces, but, in general, Pseudo-WARP is stronger than Pseudo-WARP∗.
It turns out that Pseudo-WARP∗ and Acyclicity are not enough to characterize the
choice correspondences that admit a representation by full-comparability domains.12
They do so, however, if we restrict ourselves to the class of choice correspondences that satisfy Finitariness.
Theorem 7. Let (X, ΩX) be an arbitrary choice space and c be a choice
correspon-dence on (X, ΩX) that satis es Finitariness. Then, c satis es Pseudo-WARP∗ and
Acyclicity if, and only if, it has a representation by full-comparability domains. Proof. Necessity. Suppose c is a choice correspondence that satis es Finitariness and has a representation by full-comparability domains (<, S). It is clear that, for every x, y ∈ X, x B y implies x y, so that acyclicity of B is an immediate consequence of the transitivity of <. Now suppose that x ∈ c(B) ∩ A and y /∈ c(B ∪ {y}) for every y ∈ A \ B. By the representation of c, there must exist S ∈ S such that x ∈ max(B ∩ S, <), and x y for every y ∈ (A \ B) ∩ S. Therefore, x < y for every y ∈ A∩S, which implies that x ∈ c(A). We conclude that c satis es Pseudo-WARP∗. Su ciency. Suppose that c is a choice correspondence that satis es Pseudo-WARP∗
and Acyclicity, in addition to Finitariness. Let A, B ∈ ΩX and x ∈ X be such
12Let X := N ∪ {−1, 0} and ΩX := F (X) ∪ {X, X \ {−1}}. Consider the choice correspondence
c such that c(A) := {−1, 0} ∪ max(A, ≥) if A ∈ F(X) and {−1, 0} ⊆ A, c(A) := A if A ∈ F(X) and it is not true that {−1, 0} ⊆ A, c(X) := {−1} and c(X \ {−1}) := N. It is easy to check that c satis es Pseudo-WARP∗ and Acyclicity. However, suppose that there exists a representation by full-comparability domains, (S, <), of c. Since, for every x ∈ N, we have that x ∈ c({−1, x, x + 1}), but x /∈ c({−1, 0, x, x + 1}), we must have that 0 x for every x ∈ N. But then we would have 0 ∈ c(X \ {−1}), which is not the case. We conclude that c does not admit a representation by full-comparability domains.
that A ⊆ B, x ∈ c(A), but x /∈ c(B). Since c satis es Finitariness, there must exist a nite subset F of B such that x ∈ F \ c(F ). By Pseudo-WARP∗, we must
have x /∈ c(A ∪ F ). Since x ∈ c(A), a simple inductive argument guarantees that there exists D ⊆ F and y ∈ F such that x ∈ c(A ∪ D), but x /∈ c(A ∪ D ∪ {y}). This implies that y B x. Moreover, by Pseudo-WARP∗, this can happen only if
y ∈ c(A ∪ D ∪ {y}), which, in turn, implies that y ∈ c(A ∪ {y}). This shows that c satis es Weak Binariness with B playing the role of the acyclic binary relation that appears in the statement of that postulate. Since Pseudo-WARP∗ implies Cherno's
Axiom, we can now invoke Theorem 6 to guarantee that c has a representation by full-comparability domains.
A.3 Relationship with Other Models in Arbitrary Choice Spaces
In this section, we show that all inclusions portrayed in Figure 1 are valid in arbitrary choice spaces. For that, x an arbitrary set X and let ΩX be any collection of
nonempty subsets of X such that (X, ΩX) is a choice space.
Although we are not aware of any axiomatization of the pseudo-rational choice model in arbitrary choice spaces, we note that De nition 7 makes perfect sense in any choice space.13 This is also the case for the de nition of a representation by
full-comparability domains. Suppose now that c is a choice correspondence on (X, ΩX)
that has a representation by full-comparability domains (<, S). Pick any complete extension <? of <.14 For each S ∈ S, de ne a linear order <
S by x <S y if, and only
if, x ∈ S and y /∈ S, or x <? y and either {x, y} ⊆ S or {x, y} ⊆ X \ S. It is easy
to check that {<S: S ∈ S} is a pseudo-rational representation of c. This discussion
can be summarized by the following proposition:
13In addition, we note that, while Pseudo-WARP∗ is still a necessary postulate for a
pseudo-rational representation in any choice space, as the example below shows, the same is not true for Pseudo-WARP nor the Aizerman's axiom.
Example 2. Assume that X := {x1, x2, . . . } is a countably in nite grand set of alternatives,
ΩX := 2X \ {∅}, and c(A) := max(A, <1) ∪ max(A, <2), where each <i is a linear order with
xk+1 1 xk for all k ≥ 1, x1 2 xk for all k ≥ 2, and xk 2 xl whenever k > l > 1. Then
c({x1, x2}) = {x1, x2} but c(X) = {x1} and thus c({x1, x2}) * c(X). Therefore, c does not
satisfy Pseudo-WARP nor Aizerman's axiom, while it has a pseudo-rational representation and, consequently, satis es Pseudo-WARP∗.
14That is, pick any linear order <?⊆ X × X such that x < y implies x <?y, for every x, y ∈ X.
Proposition 1. Let (X, ΩX) be an arbitrary choice space and c be a choice
corre-spondence on (X, ΩX). If c has a representation by full-comparability domains, then
it also has an alternative expression as a pseudo-rational choice correspondence. As we have seen in the main text, in nite choice spaces every choice correspondence that maximizes a possibly incomplete preorder also has a representation by full-comparability domains. Again, this is true in arbitrary choice spaces. That is, we can prove the following result:
Proposition 2. Let (X, ΩX) be an arbitrary choice space and c be a choice
corre-spondence on (X, ΩX). If c has a representation by the maximization of a possibly
incomplete preference relation, then it also has an alternative expression as a repre-sentation by full-comparability domains.
Proof. Let (X, ΩX)be an arbitrary choice space and consider a choice correspondence
c on ΩX. Suppose the choice correspondence c has a representation by the
maxi-mization of a possibly incomplete preorder, %. It is clear that c satis es Cherno's Axiom. Moreover, it is also clear that c satis es Weak Binariness with playing the role of the relation B∗ that appears in the statement of that postulate. Now
Theorem 6 guarantees that c has a representation by full-comparability domains. Since the rational choice model is a special case of maximization of an incomplete preference, we obtain that all inclusions in Figure 1 are valid in arbitrary choice spaces.
B Proofs
B.1 Proof of Theorem 2
Suppose rst that c is a choice correspondence that satis es Pseudo-WARP and Binary Dominance. We have already argued in the main text that this implies that chas a representation by full-comparability domains (<, S). Now let S1, . . . , Sn∈ S
be such that Tn
i=1Si 6= ∅ and x x ∈ T n
i=1Si. Now suppose A ⊆ X is such that
x ∈ A and x < y for every y ∈ A ∩ Tni=1Si. This implies that x ∈ c({x, y}) for
every y ∈ A. By Binary Dominance, we obtain that for no y ∈ A we have y B x. Now a simple inductive argument gives us that x ∈ c(A). This shows that if we add
Tn
i=1Si to S, we obtain a new full-comparability domains representation of c. More
generally, the collection of all nonempty intersections of elements of S together with < is a full-comparability domains representation of c, which shows that (i) implies (ii). Now suppose statement (ii) is true. That is, suppose c has a representation by full-comparability domains (<, S) such that S, T ∈ S and S ∩ T 6= ∅ implies S ∩ T ∈ S. De ne the binary relation <∗∈ X × X by x <∗
y if, and only if, {x} = c({x, y}). Suppose x, y, z ∈ X are distinct elements such that x <∗ y and
y <∗ z. This implies that x y, y z, x ∈ S for every S ∈ S with y ∈ S, and y ∈ S for every S ∈ S with z ∈ S. This now implies that x z and x ∈ S for every S ∈ S with z ∈ S, which, in turn, implies that x <∗ z. That is, <∗ is a partial order. Now x A ∈ ΩX and suppose that x ∈ c(A). It is immediate that we cannot have
that y ∗ x for some y ∈ A. Conversely, suppose that x ∈ MAX (A, <∗). Let S x be
the minimum S ∈ S with x ∈ S. We must have that x < y for every y ∈ A ∩ Sx,
which implies that x ∈ c(A). This shows that c maximizes the partial order <∗. We
conclude that (ii) implies (iii). The argument that (iii) implies (i) is standard.
B.2 Proof of Theorem 5
Consider a stochastic choice function P that admits a full-comparability domains representation (<, µ, S). Suppose rst that cP can be represented by the
maximiza-tion of a possibly incomplete preference relamaximiza-tion. By Theorem 2, cP satis es Binary
Dominance. It is also clear that (<, S) is a full-comparability domains representation of cP. Fix S, T ∈ S and x ∈ S ∩ T . Suppose that for every ˆS ∈ S with x ∈ ˆS there
exists y ∈ ˆS \(S ∩T )with y < x. Pick one such y for each ˆS ∈ Swith x ∈ ˆSand label the set consisting of all such elements A. Note that, by construction, x /∈ cP(A∪{x}).
However, for each y ∈ A, we have y /∈ S or y /∈ T , which implies that x ∈ cP({x, y})
for every y ∈ A. This contradicts Binary Dominance, as x /∈ cP(A ∪ {x}) implies
that there exists y ∈ A with y B x.
Conversely, suppose (<, µ, S) satis es the conditions in the statement of the theorem. Pick x, y ∈ X such that x ∈ cP(A), but x /∈ cP(A ∪ {y}) for some choice problem A.
This implies that there exists S ∈ S with {x, y} ⊆ S, y x and x < z for every z ∈ A ∩ S. Now x any T ∈ S with x ∈ T . By assumption, there exists Sx ∈ S
with x ∈ Sx and {z ∈ Sx : z < x} ⊆ S ∩ T . This implies that x < z for every
z ∈ A ∩ Sx. Since x /∈ cP(A ∪ {y}), we must have y ∈ Sx and, consequently, y ∈ T .
which implies that x /∈ cP({x, y}). That is, cP satis es Binary Dominance. Since
(<, S) is a pseudo-rational representation for cP, it also satis es Pseudo-WARP and,
by Theorem 2, admits a representation by a possibly incomplete preorder.
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