# Rationalization of indecisive choice behavior by majoritarian ballots111The
authors are grateful to Davide Carpentiere for several insightful comments.
José Carlos R. Alcantud thanks the audience of The Saarland Workshop in
Economic Theory for their suggestions. Alfio Giarlotta gratefully acknowledges
the financial support of “Ministero dell’Istruzione, dell’Università e della
Ricerca (MIUR) - PRIN 2017”, project Multiple Criteria Decision Analysis and
Multiple Criteria Decision Theory, grant 2017CY2NCA.

José Carlos R. Alcantud , Domenico Cantone , Alfio Giarlotta , Stephen Watson
BORDA Research Unit and IME, University of Salamanca, Spain.
<EMAIL_ADDRESS>of Mathematics and Computer Science, University of
Catania, Italy<EMAIL_ADDRESS>of Economics and Business,
University of Catania, Italy<EMAIL_ADDRESS>(corresponding
author)Department of Mathematics and Statistics, York University, Toronto,
Canada<EMAIL_ADDRESS>
###### Abstract

We describe a model that explains possibly indecisive choice behavior, that
is, quasi-choices (choice correspondences that may be empty on some menus).
The justification is here provided by a proportion of ballots, which are
quasi-choices rationalizable by an arbitrary binary relation. We call a quasi-
choice $s$-majoritarian if all options selected from a menu are endorsed by a
share of ballots larger than $s$. We prove that all forms of majoritarianism
are equivalent to a well-known behavioral property, namely Chernoff axiom.
Then we focus on two paradigms of majoritarianism, whereby either a simple
majority of ballots justifies a quasi-choice, or the endorsement by a single
ballot suffices – a liberal justification. These benchmark explanations
typically require a different minimum number of ballots. We determine the
asymptotic minimum size of a liberal justification.

JEL Classification: D71, D81.

Keywords: Bounded rationality; Chernoff property; ballot; voter;
rationalization by multiple rationales; majoritarian choice.

##  1 Introduction

Many models of multi-self decision making have been designed with the purpose
of explaining context-dependent behaviour: see, e.g., Ambrus and Rozen (2014),
Fudenberg and Levine (2006), Manzini and Mariotti (2007), May (1954), Silva-
Leander and Seth (2017). In a variety of manners, they accommodate choices
that violate the property of Independence of Irrelevant Alternatives (IIA).
Some authors (Ambrus and Rozen, 2014) have even measured the degree of
irrationality of a choice function by the number of violations of IIA that is
shows. In this paper we study a model that does the opposite job. Our
observable is a possibly indecisive choice correspondence, i.e., an utmost
general expression of selections. Then within a multi-selves formulation, the
model explains exactly all choices that satisfy IIA.

For comparison, consider the model of bounded rationality (rationalization by
multiple rationales, RMR) elaborated by Kalai, Rubinstein, and Spiegler
(2002). The authors argue that violations of principles of rational selection
can be explained if every choice set has an epistemic value, and the decision
maker (DM) applies a suitable rationale for the specific choice problem posed
by that menu.222Compare to Sen (1993), who provides arguments for ‘rational’
ways to contradict the property of IIA. The choice set triggers the
application of one of the feasible rationales (linear orderings), and the
selection may be dependent on the information that it embodies.

Since the RMR model is non-testable – that is, all choices do admit a multiple
rationalization – its focus is not on rationality per se, but instead on
‘degrees of rationality’: in fact, in Section 4 of the mentioned paper, the
authors say that “the appeal of the explanation depends on the number of
orderings involved”. Probably because choice functions are single-valued (a
unique item is selected from each menu), in the RMR model the epistemic value
of the chosen item is disregarded, or at least amalgamated with that of the
menu. As soon as the DM knows the choice problem, she selects the rationale
that produces the desired result, irrespective of the information that this
choice conveys.333 Remark 1 below refers to the application of the RMR spirit
to choice correspondences. For a structured version of the RMR model, where
the justifying rationale is linked to the most ‘salient’ items of the menu
itself, see Giarlotta, Petralia, and Watson (2022).

Suppose, on the other hand, that the DM is allowed to select more than one
item from a menu – that is, we deal with a multi-valued choice, also called a
choice correspondence. Then it may be the case that she finds different
reasons for testing the suitability of each of her selections. To illustrate
this point, consider the example used by Eliaz and Ok (2006) to argue how
incomplete preferences can explain violations of the Weak Axiom of Revealed
Preference (WARP).

######  Example 1.

(Mrs. Watson’s movie selection) (Eliaz and Ok, 2006, Example 1) Mrs. Watson
has to choose a movie to rent for her children, Alice and Tom. The set of
available movies is $X=\\{x,y,z\\}$. Alice’s and Tom’s preferences over $X$
are represented by the following rankings: $y\succ_{A}z\succ_{A}x$ (for
Alice), and $x\succ_{T}y\succ_{T}z$ (for Tom). Then Mrs. Watson’s choice $c$
is as follows (selected items are underlined):444This definition seems
redundant, because choices from menus with a single item are forced in case at
least an element must be selected from every nonempty menu. However, in this
paper we also consider indecisive DMs, whose selections are quasi-choices,
that is, selecting no item at all from a menu becomes a feasible option.

$\underline{x}\underline{y}z\,,\;\underline{x}\underline{y}\,,\;\underline{x}\underline{z}\,,\;\underline{y}z\,,\;\underline{x}\,,\;\underline{y}\,,\;\underline{z}\,,$

where a decision like $c(\\{x,z\\})=\\{x,z\\}$ (denoted by
$\underline{x}\underline{z}$ for brevity) means that she will flip a coin to
decide which movie to rent between $x$ and $z$. Although Mrs. Watson’s
approach contradicts WARP (because $z$ is chosen in both
$\underline{x}\underline{z}$ and $\underline{y}\underline{z}$ but not in
$\underline{x}\underline{y}z$), Eliaz and Ok (2006) illustrate how plausible
her way to settle conflicts may be considered.

Alternatively – and more in the flavor of this paper – Mrs. Watson’s
selections might be explained by the following mental process. In any choice
problem (say, the selection from the menu $\\{y,z\\}$), she flips a coin
between the eligible movies that are best for at least one of the children.
Then her apparently conflicting choices become a natural consequence of her
attitude. Indeed, she explains her choice from the menu $\\{x,z\\}$ by the
fact that both Tom’s favorite ($x$) and Alice’s favorite ($z$) cannot be
disregarded. For a similar reason, $z$ is rejected from $\\{x,y,z\\}$, because
it is never the best in any of the two available rankings. Mrs. Watson’s
selections are all justified by this mechanism: the flip of a coin between the
best alternatives for at least one of the two rationales, namely Alice’s and
Tom’s preferences.

The latter argument is subtly different from the spirit of the RMR model in a
critical feature: Mrs. Watson is (voluntarily) bound to the opinions of her
children. When the rationales are fixed, her opinion is determined by them.
The rationales are not a mere consultative tool, instead they enforce actions.
To emphasize the role of the rationales in the process of rationalization, we
shall refer to them as ‘voters’, and the selections they enforce will be
called ‘ballots’. Our approach departs from the RMR model of Kalai,
Rubinstein, and Spiegler (2002) in three main aspects.

(1)

The items selected from each menu may not be unique, and there might be empty
choices. The range of our analysis is much wider, because our observables are
quasi-choice correspondences rather than choice functions. In other words,
from any menu $A$ it is possible to select no item at all; furthermore, when
we observe the selection of $x$ from $A$, we are only allowed to infer $x\in
c(A)$, and not $c(A)=\\{x\\}$.

(2)

We do not ignore the epistemic value of the alternatives. The DM can find
different reasons for a selection: for each menu, she ponders whether a
justification exists for choosing any given item.

(3)

Justifications are not interpreted as in the RMR model, which permits the
existence of rationales that are never applied. On the contrary, we do not
allow irrelevant opinions: a justification ‘is’ a voter, who may select all,
some, or none of the options available in a menu via the ballot (that is, the
quasi-choice) derived from her opinion. In other words, justifications stem
from a natural voting procedure involving all justifications.

Another feature of our approach is that it makes simultaneous use of the
justifications. This characteristic distinguishes our approach from models
where rationales are sequentially applied, e.g., the Rational Shortlist
Methods (RSM) in Manzini and Mariotti (2007).

With these minor modifications, we can draw a neat demarcation line between
‘rational’ and ‘non-rational’ choice behavior. In fact, the explanation of a
choice correspondence by means of a voting process in the sense hinted at
above gives rise to a testable model, as the next example shows.

######  Example 2.

(A non-liberal choice) Suppose that a DM chooses from a grand set
$X=\\{x,y,z\\}$ as follows: $c(\\{x,z\\})=\\{x\\}$, and $c(A)=A$ for any other
menu. According to the most basic presentation of our model, a liberal
justification of this choice behavior is a finite collection $V$ of voters
over $X$ such that the options selected from each menu are exactly those
chosen by at least one voter in $V$. Here we represent a voter by an
unrestricted binary relation $\to_{i}$ on $X$, where ‘unrestricted’ means that
$\to_{i}$ is an arbitrary subset of $X\times X$, possibly displaying loops or
cycles. In other words, a liberal justification of $c$ consists of a finite
family $\\{\to_{i}\;\mathrel{\mathop{\mathchar 58\relax}}\>i\in I\\}$ of
binary relations over $X$ such that, for each menu $A\subseteq X$ and
candidate $p\in A$, we have

$p\in c(A)\quad\Longleftrightarrow\quad p\in\max(A,\to_{i})\mbox{ for some
}i\in I.\vspace{-0.1cm}\vspace{-0.1cm}$

(As usual, $\max(A,\to_{i})$ is the subset of $A$ composed of the items $p$
for which there is no $q\in A$ such that $q\to_{i}p$ holds.) Note that such a
binary relation $\to_{i}$ over $X$ naturally induces a quasi-choice $v_{i}$
over $X$, defined by $v_{i}(A)=\max(A,\to_{i})$ for all $A\subseteq X$: we
call such a quasi-choice the ballot over $X$ derived from $\to_{i}$. (In
technical terms, a ballot over $X$ is a rationalizable quasi-choice.)
Therefore, a liberal justification for $c$ is equivalently given by a family
$\\{v_{i}\mathrel{\mathop{\mathchar 58\relax}}i\in I\\}$ of ballots over $X$
such that for each $A\subseteq X$ and $p\in A$, we have

$p\in c(A)\quad\Longleftrightarrow\quad p\in v_{i}(A)\mbox{ for some }i\in
I.\vspace{-0.1cm}\vspace{-0.1cm}$

It turns out that such a family (of voters, or, equivalently, ballots) does
not exists, and so $c$ cannot be explained by the liberal paradigm. We
conclude that this choice behavior must be labeled non-rational by the
simplest instance of our approach.

A direct proof of the non-liberality of $c$ is not immediate, despite the tiny
number of alternatives. A simpler proof relies on the normative implications
of this model: in fact, liberality is equivalent to a classical condition of
choice consistency, namely Chernoff (1954) property, also called
Axiom$\>\alpha$ by Sen (1971).555This fact is known: see Aizerman and
Aleskerov (1995). Then $c$ cannot be explained by the liberal case of our
model, because Axiom$\>\alpha$ is violated: $z$ is chosen in the grand menu
$X$ but is not chosen in its submenu $\\{x,z\\}$.

Arguably, the liberal model described above appears to be excessively
permissive, as a single rationale (equivalently, a single ballot) suffices to
ensure the election of a candidate. In contrast, consider a different version
– here called democratic – of our approach, in which an option is selected
from a menu if and only if it is chosen by more than half of the
voters/ballots. More formally, a quasi-choice correspondence $c$ over $X$ is
democratic if there is a finite family $\\{\to_{i}\mathrel{\mathop{\mathchar
58\relax}}i\in I\\}$ of voters over $X$ such that, for each $A\subseteq X$ and
$p\in A$, we have

$p\in c(A)\quad\Longleftrightarrow\quad p\in\max(A,\to_{i})\mbox{ for more
than }\textstyle\frac{|I|}{2}\mbox{ voters}.\vspace{-0.1cm}\vspace{-0.1cm}$

Alternatively, $c$ is democratic, if there is a finite family
$\\{v_{i}\mathrel{\mathop{\mathchar 58\relax}}i\in I\\}$ of ballots over $X$
such that, for each $A\subseteq X$ and $p\in A$, we have

$p\in c(A)\quad\Longleftrightarrow\quad p\in v_{i}(A)\mbox{ for more than
}\textstyle\frac{|I|}{2}\mbox{ ballots}.\vspace{-0.1cm}\vspace{-0.1cm}$

Such a justification appears to have more solid grounds than a liberal one,
because a democratic support for a decision hints at a robust behavior.

However – and maybe surprisingly – the normative implication of a democratic
model is exactly the same as that of the liberal one: in fact, we shall show
that a quasi-choice is democratic if and only if Chernoff property holds. As a
consequence, outside observers who only monitor the choices made by a DM
cannot discard that she has taken the ‘democratic’ position, once they have
confirmed that the ‘liberal’ mechanism justifies her choices.

In fact, more is true: any majoritarian modification of the RMR model is
equivalent to Axiom$\;\alpha$, where by ‘any’ we mean that the share of voters
that witnesses the selection of an item may be any number between zero and
one. More formally, given a ‘share’ $s\in[0,1)$, we say that a quasi-choice
correspondence $c$ over $X$ is $s$-majoritarian if there exists a finite
family $\\{\to_{i}\;\mathrel{\mathop{\mathchar 58\relax}}\>i\in I\\}$ of
voters over $X$ such that, for each menu $A\subseteq X$ and alternative $p\in
A$, we have

$p\in c(A)\quad\Longleftrightarrow\quad p\in\max(A,\to_{i})\mbox{ for more
than }s|I|\mbox{ voters.}\vspace{-0.1cm}\vspace{-0.1cm}$

(Again, an equivalent version of this property can be given in terms of
ballots.) According to this general notion, liberal means $0$-majoritarian,
and democratic means $0.5$-majoritarian. In this paper, we prove that being
$s$-majoritarian for any share $s\in[0,1)$ is equivalent to the satisfaction
of Chernoff property: see Theorem 1 and Corollary 2. This result provides a
full generalization of the characterization given by Aizerman and Aleskerov
(1995), whereby liberalism is extended to all types of majoritarianism.

Now a natural question arises: How can we distinguish a specific majoritarian
paradigm from a different one? We shall partially answer this query by
focusing on liberalism and democracy. Similarly to the RMR model – where the
minimum number of linear rationales justifying a single-valued choice yields a
‘degree of rationality’ of that behavior – we distinguish liberal from
democratic behavior by the minimum number of voters that provide a
rationalization. Such proxies are respectively called the liberal and
democratic numbers of the observed choice correspondence. Specifically, we
examine the mutual relationship between the ‘liberal number’ and the
‘democratic number’ as the size of the grand set of candidates varies. Our
analysis yields a tight upper bound for the liberal number of a choice
satisfying Axiom$\;\alpha$ (Theorem 2). Then, using Stirling’s approximation,
we determine the asymptotic behavior of the minimal size of a liberal
justification (Corollary 3).

The paper is organized as follows. Section 2 collects preliminaries. In
Section 3 we give the notion of rationalization by ballots, and characterize
it by Chernoff property. In Section 4 we distinguish two specifications of our
model, liberalism and democracy, and determine a tight upper bound for the
minimum number of voters witnessing liberalism. Section 5 concludes the paper.
Minor proofs are presented in the main body of the paper, whereas main proofs
are in Appendix.

##  2 Rationalization, ballots, and voters

Let $X$ be a finite set of $n\geqslant 2$ alternatives (also called items or
candidates); we refer to $X$ as the grand set. As usual, $2^{X}$ denotes the
family of all subsets of $X$. A quasi-choice correspondence over $X$ is a map
$c\colon 2^{X}\to 2^{X}$ that satisfies

(contractiveness)

$c(A)\subseteq A$ for each $A\in 2^{X}$.

If, in addition, $c$ satisfies

(decisiveness)

$c(A)\neq\varnothing$ when $A\neq\varnothing$,

then it is a choice correspondence. Historically, only choice correspondences
have been an object of careful study, and quasi-choices correspondence have
been disregarded. However, very recently Costa-Gomes et al. (2022) have
reported on choice experiments that strongly suggest that nonforced-choice
models rejecting decisiveness may offer a powerful lens to study revealed
preferences. Following this stream of evidence, our main results avoid the
assumption of decisiveness, and concern quasi-choice correspondences in
general.

Hereafter, unless ambiguity may arise, we shall drop the term
‘correspondences’, and refer to choice correspondences and quasi-choice
correspondences as choices and quasi-choices, respectively. A very special
case of choice is given by a single-valued choice $c$, that is, $|c(A)|=1$ for
each nonempty $A\in 2^{X}$, which is classically referred to as a choice
function. In this case, $c$ is identified with the function $c\colon
2^{X}\setminus\\{\varnothing\\}\to X$ defined in the obvious way.

Each $A\in 2^{X}$ is a menu, and $c(A)$ is the choice set of $A$, which
collects all the items in $A$ that are deemed selectable by the DM. If $c$ is
a choice, then at least one item is selected from each nonempty menu. Instead,
for a quasi-choice the DM is allowed ‘not to decide’, selecting no item from
some (even all) menus. The two extreme types of quasi-choices over $X$ are
therefore the following:

  * •

the identity choice – a ‘neutral judgement’ – in which every candidate is
always selected from any menu, that is, $c(A)=A$ for all $A\in 2^{X}$;

  * •

the null quasi-choice – a ‘hypercritical judgement’ – in which no candidate is
selected from any menu, that is, $c(A)=\varnothing$ for all $A\in 2^{X}$.

In our setting, the identity choice can be thought of as representative of a
mild and socially responsible behavior, when there is no substantial argument
to reject any of the feasible candidates. On the contrary, a null quasi-choice
encodes a behavior that either abhors all available candidates or simply wants
no (even indirect) involvement in any political matter.

A binary relation $\succ$ over $X$ is an arbitrary subset of $X\times X$. In
particular, $\succ$ is asymmetric if $x\succ y$ implies $\neg(y\succ x)$ for
all $x,y\in X$; in this case, $\succ$ is also loopless (or irreflexive), that
is, $x\succ x$ holds for no $x\in X$. The classical notion of rationality in
revealed preference theory is encoded by the existence of an asymmetric
relation that explains choice behavior by maximization (Samuelson, 1938):

######  Definition 1.

A quasi-choice $c$ over $X$ is asymmetrically rationalizable if there is an
asymmetric relation $\succ$ over $X$ such that
$c(A)=\max(A,\succ\leavevmode\nobreak\ \\!\\!)$ for all $A\in 2^{X}$, where
$\max(A,\succ)$ is the set $\\{a\in A\mathrel{\mathop{\mathchar
58\relax}}b\succ a\hbox{ for no }b\in A\\}$.

Since we interpret $b\succ a$ as ‘candidate $b$ dominates candidate $a$’, the
set $\max(A,\succ)$ collects all non-dominated candidates of $A$. Note that if
$c$ is an asymmetrically rationalizable choice, then the asymmetric relation
$\succ$ that rationalizes $c$ is unique; moreover, $\succ$ must also be
acyclic, that is, there is no cycle $x_{1}\succ x_{2}\succ\ldots\succ
x_{k}\succ x_{1}$ of $k\geqslant 3$ items.

The notion of rationality given in Definition 1 does not fit well the very
concept encoded by a quasi-choice, because it may fail to consistently explain
indecisive choice behavior. Indeed, if $c$ is a quasi-choice that is
asymmetrically rationalizable, then all menus $A$ of size one or two must have
a nonempty choice set $c(A)$ by the very definition of asymmetric
rationalizability. In other words, a rational DM is allowed not to decide only
for menus with at least three items, whereas she is forced to select at least
one item from all menus of size two and one. The possibility of an empty
selection is banned for menus of the type $\\{x,y\\}$ or $\\{x\\}$. Is such a
constraint justifiable?

In this regard, Costa-Gomes et al. (2022) report on experimental findings in
psychology with a story from Shafir el al. (1993): Thomas Schelling “was
presented with two attractive encyclopedias and, finding it difficult to
choose between the two, ended up buying neither […] Had only one encyclopedia
been available he would have happily bought it.” This phenomenon has been
demonstrated in experiments with one and two alternatives (Tversky and Shafir,
1992, Section 4). To give a simple example, in a political ballot, if there
are only two candidates $x$ and $y$ who belong to extreme wings, and I have
very moderate political views, then I may decide not to vote at all (that is,
$c(\\{x,y\\})=\varnothing$). By the same token, if I am allergic to chocolate,
and a restaurant only offers a chocolate cake $x$ as dessert, then I will
simply avoid taking dessert (that is, $c(\\{x\\})=\varnothing$).

The awkward restriction on indecisive choices described above motivates the
next definition, where the notion of rationality is relaxed, in order to model
the very concept of an indecisive DM.

######  Definition 2.

A quasi-choice $c$ over $X$ is freely rationalizable (rationalizable, in case
no ambiguity may arise) if there is a binary relation $\to$ over $X$ such that

$c(A)=\max(A,\to)=\\{a\in A\,\mathrel{\mathop{\mathchar
58\relax}}\,b\not\rightarrow a\hbox{ for all }b\in
A\\}\vspace{-0.1cm}\vspace{-0.1cm}$

for each $A\in 2^{X}$, where $b\not\rightarrow a$ stands for the negation of
$b\to a$. Hereafter, any freely rationalizable quasi-choice on $X$ is called a
ballot over $X$, and is often denoted by $v,w,z$, etc. (instead of $c$). When
no confusion may arise, we shall abuse notation, and identify a ballot over
$X$ with any binary relation $\to$ over $X$ that rationalizes it: we call this
binary relation a voter.666Note that the binary relations (voters) that
rationalize a ballot need not be unique. However, this abuse of notation will
be harmless.

If a quasi-choice $c$ is asymmetrically rationalizable, then we use $\succ$
for the (unique) rationalizing preference to avoid confusion. Instead, for a
ballot $v$, we use the ‘arrow notation’ $\to$ for any rationalizing preference
(voter). Note that, in the latter case, the rationalizing relation $\to$ may

  * (a)

display loops, that is, $x\to x$ holds for some $x\in X$, in which case we
call the item $x$ repellent, and/or

  * (b)

fail to be asymmetric, hence $x\to y$ and $y\to x$ are allowed to coexist for
distinct $x,y\in X$, in which case we call the items $x,y$ pairwise exclusive,
and use the notation $x\leftrightarrows y$.

The existence of a loop $x\to x$ means that the alternative $x\in X$ is
‘intrinsically bad’ in the voter’s eyes, and so will never be selected. The
item eliminates itself as in the case of a person allergic to chocolate: the
fact that $x$ contains chocolate deems $x$ ineligible, and this is independent
of the structure of $X$ and any possible binary comparison. On the other hand,
a double arrow $x\leftrightarrows y$ between distinct items means that there
are scenarios for which a voter considers $x$ strictly better than $y$ ($x\to
y$) and different scenarios in which the converse happens ($y\to x$): this
yields full indecisiveness, and so the voter will end up choosing none of the
two pairwise exclusive items (that is, the ballot associated to this voter
displays an empty choice on the menu $\\{x,y\\}$.)

The difference between Definitions 1 and 2 only materializes for DMs whose
selection is empty on at least a menu:

######  Lemma 1.

  * (i)

A choice is asymmetrically rationalizable if and only if it is freely
rationalizable.

  * (ii)

If a quasi-choice is asymmetrically rationalizable, then it is freely
rationalizable. The converse is false.

Proof. Asymmetric rationalizability obviously implies free rationalizability.

To prove the converse in (i), suppose $c\colon 2^{X}\to 2^{X}$ is freely
rationalizable by a voter $\to$ over $X$. Since $c(\\{x\\})=\\{x\\}$ for any
$x\in X$, the relation $\to$ has no loops. Further, we cannot have
$x\leftrightarrows y$ for any distinct $x,y\in X$, since otherwise
$c(\\{x,y\\})=\varnothing$, a contradiction. We conclude that the voter $\to$
representing $c$ is asymmetric and loopless, and so $c$ is asymmetrically
rationalizable.

To complete the proof of (ii), consider the quasi-choice $c$ over
$X=\\{x,y\\}$, defined by $c(A)=A$ if $|A|=1$, and $c(A)=\varnothing$
otherwise. Then $c$ is freely rationalizable (by the voter $\to$ defined by
$x\leftrightarrows y$) but fails to be asymmetrically rationalizable. ∎

The asymmetric rationalizability of a choice is characterized by the
satisfaction of two classical properties of choice consistency:

Axiom $\alpha\,$:

for all $A,B\subseteq X$ and $x\in X$, if $x\in A\subseteq B$ and $x\in c(B)$,
then $x\in c(A)$;

Axiom $\gamma\,$:

for all $A,B\subseteq X$ and $x\in X$, if $x\in c(A)$ and $x\in c(B)$, then
$x\in c(A\cup B)$.

Axiom$\>\alpha$ is due to Chernoff (1954). It says that any item selected from
a menu is also chosen from any smaller menu containing it: that is why this
property is also called Standard Contraction Consistency. Its role in the
abstract theories of rational individual choice and social choice is central,
sometimes in the form of Independence of Irrelevant Alternatives (Arrow,
1950). The normative appeal of Axiom$\>\alpha$ is rather strong, as discussed
in Eliaz and Ok (2006, Remark 1) and Heller (2012, Section 2.2). In fact,
Nehring (1997, p. 407) goes even further, calling Axiom$\>\alpha$ “the mother
of all choice consistency conditions’’. Axiom$\>\gamma$, often referred to as
Standard Expansion Consistency, is introduced by Sen (1971). It says that if
an item is chosen in two menus, then it is also selected from the larger menu
obtained as their union.

The connection between these properties and rational choice behavior is as
follows:

######  Theorem (Sen, 1971).

A choice is asymmetrically rationalizable if and only if both Axiom$\>\alpha$
and Axiom$\>\gamma$ hold.

This characterization extends to quasi-choices, provided that we employ the
relaxed notion of rationality described in Definition 2:

######  Proposition 1 (Aizerman and Aleskerov, 1995, Theorem 2.5).

A quasi-choice is freely rationalizable (equivalently, it is a ballot) if and
only if Axioms$\;\alpha$ and $\gamma$ hold.

For a proof of Proposition 1, see Aleskerov and Monjardet (2002, Theorem 2.8).
In connection to Proposition 1, observe that uniqueness of a free
rationalization is not guaranteed for quasi-choices: for instance, a null
quasi-choice is freely rationalizable by all (and only) the binary relations
in which all items are repellent.

##  3 Rationalization by ballots

Here we introduce the main notion of the paper, justify it, and characterize
it.

###  3.1 Preliminaries

######  Definition 3.

Let $s$ be any number in the interval $[0,1)$, here called a share. A quasi-
choice $c\colon 2^{X}\to 2^{X}$ is $s$-majoritarian if there exists a finite
family $\mathcal{V}=\\{v_{1},\ldots,v_{k}\\}$ of $k$ ballots over $X$, not
necessarily pairwise distinct, such that

$x\in
c(A)\quad\Longleftrightarrow\quad\frac{\big{|}\\{i\mathrel{\mathop{\mathchar
58\relax}}x\in v_{i}(A)\\}\big{|}}{k}>s\vspace{-0.1cm}$ (1)

for any $A\in 2^{X}$ and $x\in X$. We say that $c$ is rationalizable by
ballots if it is $s$-majoritarian for some share $s\in[0,1)$.

In words, $c$ is $s$-majoritarian provided that we can find a finite number of
voters such that a candidate is selected from a menu if and only if it is
endorsed by a share of voters – equivalently, a share of the ballots
associated with the voters – strictly larger than $s$. It will always be
understood that voters/ballots in a given family need not be pairwise
distinct.

The model of rationalization by ballots described in Definition 3 aims to
build a bridge between choice theory and voting theory. In fact, a selection
process is explained by means of a population $\mathcal{V}$ of voters, who
make a collective decision by appealing to a socially acceptable paradigm,
encoded by a threshold.

Next, we prove some simple properties of the model given in Definition 3.

######  Lemma 2.

Any quasi-choice that is rationalizable by ballots satisfies Axiom$\;\alpha$.

Proof. Suppose $c$ is a quasi-choice over $X$, which is rationalizable by
ballots. Thus, there is a family $\mathcal{V}=\\{v_{1},\ldots,v_{k}\\}$ of $k$
ballots satisfying (1) for some share $s\in[0,1)$. Let $A\subseteq B\subseteq
X$ and $x\in A\cap c(B)$. By hypothesis,
$\frac{|\\{i\,\mathrel{\mathop{\mathchar
58\relax}}\,x\,\in\,v_{i}(B)\\}|}{k}>s$. Since any ballot $v_{i}$ satisfies
Axiom$\;\alpha$ by Proposition 1, it follows that

$\frac{\big{|}\\{i\mathrel{\mathop{\mathchar 58\relax}}x\in
v_{i}(A)\\}\big{|}}{k}\geqslant\frac{\big{|}\\{i\mathrel{\mathop{\mathchar
58\relax}}x\in v_{i}(B)\\}\big{|}}{k}>s,\vspace{-0.1cm}$

and so $x\in c(A)$ by (1). This proves that $c$ satisfies Axiom$\;\alpha$. ∎

The case of a zero share is especially tractable.

######  Lemma 3.

A quasi-choice $c$ is $0$-majoritarian if and only if there exists a family
$\mathcal{V}=\\{v_{1},\ldots,v_{k}\\}$ of ballots such that
$c=\bigcup_{i=1}^{k}v_{i}$.777That is, $c(A)=\bigcup_{i=1}^{k}v_{i}(A)$ for
any menu $A$.

Proof. Let $c$ be a quasi-choice over $X$. By (1), $c$ is $0$-majoritarian if
and only if there is a family $\mathcal{V}=\\{v_{1},\ldots,v_{k}\\}$ of $k$
ballots over $X$ such that for any $A\in 2^{X}$,

$\displaystyle c(A)=\left\\{x\in A\mathrel{\mathop{\mathchar
58\relax}}\frac{\big{|}\\{i\mathrel{\mathop{\mathchar 58\relax}}x\in
v_{i}(A)\\}\big{|}}{k}>0\right\\}=\big{\\{}x\in A\mathrel{\mathop{\mathchar
58\relax}}x\in v_{i}(A)\text{ for some
}i\\}=\bigcup_{i=1}^{k}v_{i}(A).\vspace{-0.1cm}\vspace{-0.1cm}$

This proves the claim. ∎

Combining Lemma 3 with Theorems 2.5(a) and 5.4 in Aizerman and Aleskerov
(1995), we readily derive the following fact:

######  Corollary 1.

Any quasi-choice that satisfies Axiom$\;\alpha$ is $0$-majoritarian.

###  3.2 Interpretation and relation with literature

To start, let us point out that the standard maximization model is a basic
instance of rationalization by ballots. Specifically, if a quasi-choice $c$ is
freely rationalizable (hence $c$ is a ballot $v_{1}$ according to Definition
2), then for each $s\in[0,1)$ the family $\mathcal{V}=\\{v_{1}\\}=\\{c\\}$ of
ballots over $X$ is such that

$x\in
c(A)\quad\Longleftrightarrow\quad\frac{\big{|}\\{i\mathrel{\mathop{\mathchar
58\relax}}x\in v_{i}(A)\\}\big{|}}{1}>s$ (2)

for any $A\in 2^{X}$ and $x\in X$.

Remarkably, the case in Eliaz and Ok (2006) can be expressed in the language
of our model, too. Theorem 2 in the mentioned paper shows that a choice
correspondence satisfying the Weak Axiom of Revealed Non-Inferiority (WARNI)
can be explained by maximizing (in the sense of Definition 2) the asymmetric
part of a reflexive, transitive, and ‘regular’ binary relation. It follows
that all choice correspondences that satisfy WARNI are $s$-majoritarian for
each $s\in[0,1)$. In Section 3.3 we shall prove that Axiom$\;\alpha$, which is
a behavioral consequence of WARNI, is the exact property that justifies the
characterization of $s$-majoritarianism for each $s\in[0,1)$.

Next, consider the model of Kalai, Rubinstein, and Spiegler (2002), which
explains choice functions by appealing to a set of binary justifications.
Specifically, a rationalization by multiple rationales (RMR) of a choice
function $c\colon 2^{X}\setminus\\{\varnothing\\}\to X$ is a set $\L$ of
linear orders888A linear order is an asymmetric, transitive and complete
binary relation. (called rationales) over $X$ such that, for all nonempty
$A\subseteq X$, the item $c(A)$ is $\succ$-maximal for some $\succ$ in $\L$.

By definition, any choice function has an RMR: assign to each menu a linear
order having the selected item as its top element. As a consequence of the
non-testability of the RMR model, the authors study the ‘degree of
rationality’ of choice functions, intended as the minimum number of rationales
that are needed to justify selection on each menu. Then, the larger this
degree of rationality is, the less rational a choice behavior is labeled.
(Note that this degree is $1$ if and only if the choice is rationalizable.)
The authors show that a choice function on $n$ elements can be explained by
$n-1$ rationales; however, the proportion of choices that require the maximum
number of rationales tends to one as the number of items tends to infinity.
Thus, distinguishing choice functions by their degree of rationality fails to
be very effective, especially for large grand sets.

######  Remark 1.

One could also define rationalization by multiple rationales of a choice
correspondence $c$ as a set ${\mathcal{W}}$ of weak orders999A weak order is
an asymmetric and negatively transitive (hence transitive) binary relation.
over $X$ such that, for all nonempty $A\subseteq X$, $c(A)$ is the set of
$\succ$-maximal elements for some $\succ$ in ${\mathcal{W}}$. Then any choice
correspondence has a rationalization by multiple rationales, too. An analysis
of the minimum number of rationales that are needed to guarantee
rationalization appears to be interesting, but is beyond the scope of this
paper.

Apart from some drawbacks already put across by the authors (see, e.g., Kalai,
Rubinstein, and Spiegler, 2002, p. 2847), we detect an additional setback of
this model: the same RMR can simultaneously explain rather different choice
functions.

######  Example 3.

Let $c_{1},c_{2}\colon 2^{X}\setminus\\{\varnothing\\}\to X$ be the choice
functions over $X=\\{x,y,z\\}$ respectively defined by

$(c_{1})\;\;\underline{x}{y}\,,\;{x}\underline{z}\,,\;y\underline{z}\,,\;\underline{x}yz\qquad\text{and}\qquad(c_{2})\;\;\underline{x}{y}\,,\;\underline{x}{z}\,,\;y\underline{z}\,,\;xy\underline{z}\,.\vspace{-0.1cm}\vspace{-0.1cm}$

Both $c_{1}$ and $c_{2}$ are non-rationalizable, and can be explained by the
same set of two rationales. Specifically, let $\succ_{1}$ and $\succ_{2}$ be
the linear orders over $X$ such that $x\succ_{1}y\succ_{1}z$ and
$z\succ_{2}x\succ_{2}y$. We claim that $\L=\\{\succ_{1},\succ_{2}\\}$ is an
RMR for both $c_{1}$ and $c_{2}$. Indeed, for $c_{1}$, the linear order
$\succ_{1}$ rationalizes the menus $X$ and $\\{x,y\\}$, whereas $\succ_{2}$
rationalizes $\\{x,z\\}$ and $\\{y,z\\}$. On the other hand, for $c_{2}$, the
linear order $\succ_{1}$ rationalizes the menus $\\{x,y\\}$ and $\\{x,z\\}$,
whereas $\succ_{2}$ rationalizes $\\{y,z\\}$ and $X$.

The fact that the same RMR may be employed to explain very different
selections is a shortcoming of the described model. However, this can be
easily fixed, rewriting the definition of an RMR according to a ‘functional
approach’. Specifically, let a rationalizer for $c$ be a pair
$\langle\L,f\rangle$ such that $\L$ is a family of linear orders over $X$, and
$f\colon 2^{X}\setminus\\{\varnothing\\}\to\L$ is a map such that
$c(A)=\max(A,f(A))$ for all $A\in 2^{X}\setminus\\{\varnothing\\}$. The
function $f$ assigns a rationale to each menu, and so the two choices in
Example 3 now do have distinct rationalizer. The functional approach suggested
for the RMR approach is, in fact, the one that we shall employ – mutatis
mutandis – in our model.

Alternative interpretations of a rationalization by ballots are related to
other bounded rationality approaches. Multi-self decision making – see, e.g.,
May (1954) and Manzini and Mariotti (2007) – provides interpersonal or
intrapersonal frameworks for aggregation of preferences. The essential idea of
this paradigm is that in order to derive choices over alternatives, a DM
resorts to a family $\mathcal{V}$ of fictitious selves (also called
‘motivations’ or ‘priorities’). In this respect, here we employ the suggestive
terms ‘ballots’ and ‘voters’ instead, and do not impose an interpersonal
interpretation. We also assume that each ballot $v\in\mathcal{V}$ can be
generally indecisive, including indecisiveness for singletons and doubletons.
Even more important, we consider the general case of quasi-choice
correspondences instead of the very restricted scenario of choice functions.
From this point of view, a quasi-choice correspondence is rationalizable by
ballots whenever there is a family $\mathcal{V}$ of (possibly fictitious or
unobservable) voters with the property that the items selected from each menu
are exactly those endorsed by more than a fixed share of voters.

Another interpretation of our model is related to multiple criteria decision
making. In such a scenario, alternatives are characterized by attributes, and
each attribute is explained by a rationale. When the DM only attends to
selected characteristics, she makes a choice by the attributes that stem from
the maximization of the corresponding rationales. Choices may be justified by
these ‘selections by attributes’ as follows: a DM decides to accept an
alternative from a menu when it is choosable for a fixed share of attributes.
Definition 3 considers choices that can be explained ‘as if’ the DM were
assigning attributes to the set of options, and in that frame of mind her
choices are justified by at least a certain number of attributes.

###  3.3 Characterization

Since a share $s$ in Definition 3 is, by its own nature, a proportion of a
finite society, we may assume that $s$ is a rational number. In such a setting
we have, as announced:

######  Theorem 1.

The following statements are equivalent for a quasi-choice $c\colon 2^{X}\to
2^{X}$:

  * (i)

$c$ satisfies Axiom$\;\alpha$;

  * (ii)

$c$ is $s$-majoritarian for some rational share $s\in[0,1)$;

  * (iii)

$c$ is $s$-majoritarian for each rational share $s\in[0,1)$.

Theorem 1 suffices for practical purposes. However, we wish to point out that
it also holds true for irrational shares:

######  Corollary 2.

The following statements are equivalent for a quasi-choice $c\colon 2^{X}\to
2^{X}$:

  * (i)

$c$ satisfies Axiom$\;\alpha$;

  * (ii)

$c$ is $s$-majoritarian for some share $s\in[0,1)$;

  * (iii)

$c$ is $s$-majoritarian for each share $s\in[0,1)$.

Theorem 1 and Corollary 2 are proved in the Appendix. They assert that all
$s$-majoritarian justifications are ‘equally sound’, because they are
witnessed by the same behavioral property: either all of them are valid
explanations of a given choice behavior, or none is.

However, this equivalence vanishes as soon as we consider the simplest feature
of a justification by ballots, namely the number of voters that is required in
the group. Section 4 investigates this issue by focussing on two benchmark
specifications of $s$-majoritarianism, where the share is either one half
(‘democracy’) or zero (‘liberalism’).

##  4 Group size matters: distinguishing majoritarian justifications

Rationalization by ballots is a parametric choice model, with the share
$s\in(0,1]$ being the parameter. For a fixed value of $s$, the complexity of
any $s$-majoritarian justification is measured by the minimum number of voters
needed for it. In this perspective, for any observed (possibly non-decisive)
choice behavior satisfying Standard Contracting Consistency, it may be
interesting to determine the share $s$ which yields the $s$-majoritarian
justification with the least minimum number of voters. This would entail the
‘least complex’ explanation within the majoritarian paradigm, also providing a
novel measure of the degree of rationality of quasi-choices.

Apparently, the general version of this problem appears to be rather
challenging; thus, we leave it open. However, we do have some partial results,
which allow one to gauge its difficulty. Specifically, in this section we
shall investigate the minimum size of populations witnessing two benchmark
types of majoritarian justifications: democratic and liberal.

######  Definition 4.

Let $c\colon 2^{X}\to 2^{X}$ be a quasi-choice over $X$ satisfying
Axiom$\>\alpha$. A democratic representation of $c$ of size $k$ is any family
$\mathcal{V}=\\{v_{1},\ldots,v_{k}\\}$ of $k$ ballots over $X$ such that for
any $A\in 2^{X}$ and $x\in X$,

$x\in c(A)\quad\Longleftrightarrow\quad\big{|}\\{i\mathrel{\mathop{\mathchar
58\relax}}x\in v_{i}(A)\\}\big{|}>\frac{k}{2}.\vspace{-0.1cm}\vspace{-0.1cm}$
(3)

Similarly, a liberal representation of $c$ of size $k$ is any family
$\mathcal{V}=\\{v_{1},\ldots,v_{k}\\}$ of $k$ ballots over $X$ such that for
any $A\in 2^{X}$ and $x\in X$,

$x\in c(A)\quad\Longleftrightarrow\quad x\in v(A)\text{ for some
}v_{i}\in\mathcal{V}.$ (4)

As a first evidence of the difficulty of the problem posed above, in Section
4.1 we shall prove a rather counterintuitive result, in relation to the
comparison of the complexity of democratic and liberal explanations.

###  4.1 Democratic vs liberal representations

The proof of Theorem 1 suggests that the minimum number of ballots/voters
needed to achieve a democratic representation is typically different from that
needed to obtain a liberal one. This observation yields a natural proxy that
differentiates these two ways of justifying choices by ballots:

######  Definition 5.

For any quasi-choice $c\colon 2^{X}\to 2^{X}$, we define the democratic number
$\textsf{dem}(c)$ as the size of a smallest democratic representation of $c$,
if there are any, and infinite otherwise, namely

$\textsf{dem}(c)\mathrel{\mathop{\mathchar
58\relax}}=\left\\{\begin{array}[]{llll}\hbox{size of smallest democratic
representation of $c$}&\hbox{ if }c\hbox{ satisfies Axiom$\>\alpha$,}\\\
\infty&\hbox{ otherwise.}\end{array}\right.\vspace{-0.1cm}$

Similarly, the liberal number $\textsf{lib}(c)$ is defined as the size of a
smallest liberal representation of $c$, if there are any, and infinite
otherwise, namely

$\textsf{lib}(c)\mathrel{\mathop{\mathchar
58\relax}}=\left\\{\begin{array}[]{llll}\hbox{size of smallest liberal
representation of $c$}&\hbox{ if }c\hbox{ satisfies Axiom$\>\alpha$,}\\\
\infty&\hbox{ otherwise.}\end{array}\right.$

Thus, we have:

$\textsf{dem}(c)<\infty\quad\Longleftrightarrow\quad\hbox{$c$ satisfies
Axiom$\>\alpha$}\quad\Longleftrightarrow\quad\textsf{lib}(c)<\infty.\vspace{-0.1cm}$

In a situation of full rationality, the two paradigms are indistinguishable:

######  Lemma 4.

A quasi-choice $c$ is freely rationalizable if and only if $\textsf{dem}(c)=1$
if and only if $\textsf{lib}(c)=1$.

Lemma 4 follows from the fact that a democratic representation of $c$ by a
single (arbitrary) binary relation is also a liberal representation of $c$.

However, for non-rationalizable quasi-choices, the liberal and the democratic
paradigms may display a different behavior concerning the minimal size of a
representation, even on small sets of alternatives. The next two examples
provide some simple instances of this kind, moreover in the case of decisive
choices.

######  Example 4.

Let $c$ be the choice over $X=\\{x,y,z\\}$ defined by
$\underline{x}\underline{y}\,,\;\underline{x}z\,,\;\underline{y}z\,,\;\underline{x}yz\,.$
Note that $c$ satisfies Axiom$\;\alpha$ but not Axiom$\;\gamma$, and so fails
to be rationalizable. In the Appendix we prove that $\textsf{lib}(c)=2$ and
$\textsf{dem}(c)=3$.

######  Example 5.

Let $c$ be the choice over $X=\\{0,1,2,3,4,5\\}$ defined by

$c(A)\;=\;\left\\{\begin{array}[]{lll}A\setminus\\{0\\}&\hbox{ if }0\in
A\hbox{ and }|A|\geq 4\\\ A&\hbox{
otherwise.}\end{array}\right.\vspace{-0.1cm}\vspace{-0.1cm}$

Again, $c$ satisfies Axiom$\;\alpha$ but not Axiom$\;\gamma$, which implies
$2\leqslant\textsf{dem}(c)<\infty$ and $2\leqslant\textsf{lib}(c)<\infty$. In
the Appendix we prove that $\textsf{dem}(c)\leq 5$ and $\textsf{lib}(c)\geq
10$. Note that the liberal and democratic paradigms are very far apart in this
case. This can be interpreted as saying that a situation in which a government
only disregards a single item when the latter appears in a large menu
(otherwise being fully neutral) is better suited for a democratic model rather
than a liberal one.

On the other hand, the democratic and liberal numbers are equal for some non-
rationalizable choices:

######  Example 6.

Let $c$ be the choice over $X=\\{x,y,z,w\\}$ defined by

$\underline{x}\underline{y}\,,\;\underline{x}\underline{z}\,,\;\underline{x}w\,,\;\underline{y}\underline{z}\,,\;\underline{y}w\,,\;\underline{z}\underline{w}\,,\quad\underline{x}\underline{y}z\,,\;\underline{x}\underline{y}w\,,\;\underline{x}zw\,,\;\underline{y}zw\,,\quad\underline{x}yzw\,.\vspace{-0.1cm}\vspace{-0.1cm}$

Again, Axiom $\alpha$ holds for $c$, but Axiom $\gamma$ does not. It follows
that $2\leqslant\textsf{dem}(c)<\infty$ and
$2\leqslant\textsf{lib}(c)<\infty$. In the Appendix we prove that
$\textsf{dem}(c)=\textsf{lib}(c)=3$.

We hinted above that some bounded rationality models define the complexity of
an observed choice behavior by representing it in terms of a parametric model,
and interpreting the minimum number of parameters needed for a representation
as a measure of this complexity: Kalai, Rubinstein, and Spiegler (2002) and
Ambrus and Rozen (2014) provide two notable examples of this kind. However,
this type of approach raises a challenging philosophical question: How
convincing is the underlying parametric representation as an actual model of
decision-making?

In this respect, our parametric model is not designed with the exclusive
purpose of measuring the complexity of choice behavior by appealing to the
minimum number of voters needed for an $s$-majority. Regrettably, Examples 4-6
yield a possibly counterintuitive conclusion, because even a comparison of the
complexity of the $0.5$-majority and $0$-majority explanations fails to hold
true. In view of this setback, our goal is first to produce concrete evidences
that tell the liberal paradigm apart from the democratic one, and only then
measuring how distant these two specifications of the $s$-majoritarian model
are from each other.

All in all, we can distinguish three types of quasi-choices for which Axiom
$\alpha$ holds: (i) those with a liberal number smaller than the democratic
number (see Example 4); (ii) those for which the reverse happens (see Example
5); (iii) those such that the two numbers are equal (see Example 6). A
systematic analysis of the democratic and liberal numbers of a quasi-choice
satisfying Axiom $\alpha$ appears to be non-trivial. We leave open the problem
of determining a testable characterization of the classes (i)–(iii). Finding
an $s$-majoritarian justification with a smallest minimum number of voters,
for each quasi-choice satisfying Axiom $\alpha$, is another open problem that
we mentioned above in this section.

The following result provides some ‘relative’ upper bounds, that is, bounds
for the democratic (resp. liberal) number in terms of the liberal (resp.
democratic) number:

######  Lemma 5.

For any quasi-choice $c$, we have:

  * (i)

$\textsf{dem}(c)\leqslant 2\,\textsf{lib}(c)$;

  * (ii)

$\textsf{lib}(c)\leqslant 2^{\,\textsf{dem}(c)-1}$.

Lemma 5 partially answers the following question: If we have a liberal (resp.
democratic) representation of a possible indecisive choice behavior, how large
should the population of voters be in order to obtain a democratic (resp.
liberal) justification for the same behavior? In this respect, the proof of
Lemma 5 is instructive, because it explicitly shows how to canonically
construct a liberal (resp. democratic) representation from a democratic (resp.
liberal) one.

###  4.2 Asymptotic behavior of liberalism

We conclude this section by establishing an upper bound for the liberal number
of any quasi-choice that satisfies Axiom$\>\alpha$ (in relation to the size of
the grand set). Since this bound is tight, we shall also get the asymptotic
behavior of $\textsf{lib}(c)$.

To that end, we need a combinatorial result (Lemma 6), in which we abstract
the main ingredients in the proof of Theorem 2. In it, we construct an
infinite collection $\\{c_{n,k}\mathrel{\mathop{\mathchar
58\relax}}n\in\mathbb{N},\,1\leqslant k\leqslant n+1\\}$ of choices over a
grand set $\\{0,1,\ldots,n\\}$ of size $n+1$ having the following features:

  * •

all of them satisfy Axiom$\>\alpha$ (hence they are $s$-majoritarian for any
$s\in[0,1)$);

  * •

they have a fixed (but quite ‘large’) liberal number;

  * •

they have a democratic number that is bounded above (by a ‘small’ integer).

These choices are very close to being neutral, because all elements, with the
possible exception of the item $0$, are selected in all menus.

######  Lemma 6.

For each $1\leqslant k\leqslant n+1$, let $c_{n,k}$ be a choice over the
nonempty grand set $X_{n}=\\{0,1,\ldots,n\\}$ satisfying the following
conditions for all nonempty $A\in 2^{X}$:

  1. (a)

$A\setminus c_{n,k}(A)\subseteq\\{0\\}$;

  2. (b)

$A\setminus c_{n,k}(A)=\\{0\\}$ if and only if $0\in A\hbox{ and }|A|>k$.

Then $c_{n,k}$ satisfies Axiom$\>\alpha$. Furthermore,

$\textsf{lib}(c_{n,k})={n\choose
k-1}\qquad\text{and}\qquad\begin{cases}\textsf{dem}(c_{n,k})\leqslant
2k-1&\text{if\leavevmode\nobreak\ }\frac{n}{2}<k\leqslant n+1\\\
\textsf{dem}(c_{n,k})\leqslant 2(n-k)&\text{if\leavevmode\nobreak\ }1\leqslant
k\leqslant\frac{n}{2}.\end{cases}$ (5)

From Lemma 6, we obtain what we were after:

######  Theorem 2.

For any quasi-choice $c$ over a grand set of size $n\geqslant 2$ satisfying
Axiom $\alpha$,

$\textsf{lib}(c)\,\leqslant\,{n-1\choose\lfloor\frac{n-1}{2}\rfloor}.\vspace{-0.1cm}\vspace{-0.1cm}$
(6)

Moreover, the upper bound (6) is tight.

Finally, we determine the asymptotic behavior of $\textsf{lib}(c)$. Since
Stirling’s approximation entails

${n-1\choose\lfloor\frac{n-1}{2}\rfloor}\,=\,\mathcal{O}\left(\frac{2^{n}}{\sqrt{n}}\right),$

the tightness of the upper bound in Theorem 2 yields the following result:

######  Corollary 3.

For any quasi-choice $c$ over a grand set of size $n\geqslant 2$ satisfying
Axiom$\;\alpha$,

$\textsf{lib}(c)\,=\,\mathcal{O}\left(\frac{2^{n}}{\sqrt{n}}\right).$

##  5 Final remarks

Several models show that effective theories of individual choice can be
founded on tenets that relax the classical Weak Axiom of Revealed Preference,
possibly in combination with some other common postulates. Our approach to
associate observed choice behavior with maximizing attitudes is ‘radical’,
because it addresses a root concern: we identify a liberal maximization
process that can be tested solely by Standard Contraction Consistency.
Moreover, in this liberal approach, we have explored the justifications that
require the minimal number of voters. A less radical justification of the same
class of choices places them in a majoritarian scenario with respect to an
arbitrarily large share. Here our numerical conclusions are less complete. Our
procedure is in continuation of the strand of literature on rationalizability
of choice with multi-self models that started with May (1954).

Alternatively, when choices are single-valued, Manzini and Mariotti (2007)
explain some empirically important ‘irrational’ patterns of choice by a
rational shortlist method (RSM). ‘Irrational’ means that basic rationality
tenets, such as Chernoff’s Axiom $\alpha$ and Sen’s Axiom $\gamma$, are
contradicted. In a RSM, a first rationale (an asymmetric binary relation)
gives a selection of alternatives (a shortlist); then a second rationale
(another asymmetric binary relation) determines the unique selection. Notice
that there are choice functions that are neither RSM nor $s$-majoritarian,
because they violate Axiom$\>\alpha$. A simple example of this kind is the
‘default route’ example in Manzini and Mariotti (2007, Section I.B). Indeed,
for $X=\\{A,B,C\\}$, the (single-valued) choice function defined by
$A\underline{B}$, $\underline{A}C$, $\underline{B}C$, and $\underline{A}BC$ is
not an RSM; moreover, since Axiom$\>\alpha$ fails to hold, this choice fails
to have an $s$-majoritarian representation.

Rationalization by multiple rationales (Kalai, Rubinstein, and Spiegler, 2002)
does not use rationales in a sequential order. Our model resorts to a
multiplicity of rationales but it does not use them sequentially either. As in
an RMR approach, one rationale justifies the choice of an item in the liberal
(i.e., $0$-majoritarian) model. Unlike that model, however, neither of the
rationales is irrelevant, and we allow for two framework effects, because both
the menu and the item may affect the mental argument – the particular self –
that acts for justification. Furthermore, the introduction of a new voter (a
bounding rationale) modifies the choice. Choices are not necessarily unique,
and we need not be explicit about the interpretation of multiple choices.

Eliaz and Ok (2006) show how a choice theory that allows ‘rational agents’ to
remain indecisive at times can be founded on WARNI. Since Axiom$\>\alpha$ is a
behavioral consequence of WARNI, our model places their ‘rational agents’ in a
(specific) multi-selves context (Eliaz and Ok, 2006, Footnote 29).

One issue remains unexplored: computational complexity. With respect to the
RMR model, Apesteguia and Ballester (2010) and afterwards Demuynck (2011) show
that the problems of rationalizing choices by a minimum number – or by a fixed
number – of rationales are NP-complete. The class of NP-complete problems
consists of the most difficult NP problems. In turn, any NP problem is deemed
‘easy to verify’ (i.e., there are algorithms that check in polynomial time
whether a purported solution is indeed a solution), although it might be
‘difficult’ to find a solution (e.g., because solving it might take
exponential time). Also the problem posed by sequential choice by $K$
rationales (Manzini and Mariotti, 2007) is NP-complete when $K\geq 3$
(Demuynck, 2011, Theorem 3). In the opposite direction, the latter result
proves that the problem of whether a choice correspondence is explained by the
status-quo bias model (Masatlioglu and Ok, 2005) is decidable in polynomial
time (i.e., it is in the class P of decision problems). By exploiting the
bounds in Theorem 2 and Lemma 5(i), it is not difficult to show that checking
whether $\textsf{lib}(c)\,\leq\,k$ is true, or $\textsf{dem}(c)\,\leq\,k$, is
an NP problem. This means that the answer can be verified in polynomial time.
However, we do not know whether the problems $\textsf{lib}(c)\,\leq\,k$ and
$\textsf{dem}(c)\,\leq\,k$ are also NP-hard (hence NP-complete) or not. The
bound established by Corollary 3 does not help for this purpose.

## Appendix: Main Proofs

Proof of Theorem 1. Since (iii)$\;\Longrightarrow\;$(ii) is obvious, and
(ii)$\;\Longrightarrow\;$(i) is Lemma 2 (note that $s$ is not required to be a
rational number), we only show (i)$\;\Longrightarrow\;$(iii).

(i)$\;\Longrightarrow\;$(iii): Suppose $c$ satisfies Axiom $\alpha$, and let
$s\in\mathbb{Q}\cap[0,1)$. We prove the claim in three steps:

  1. 1.

$c$ is $0$-majoritarian;

  2. 2.

$c$ is $t$-majoritarian for some $t\in(s,1)\cap\mathbb{Q}$;

  3. 3.

$c$ is $s$-majoritarian.

Step 1:

This is Corollary 1, which owes to Aizerman and Aleskerov (1995), Theorems
2.5(a) and 5.4. Let $\mathcal{V}=\\{v_{1},\ldots,v_{n}\\}$ be a finite family
of ballots witnessing the fact that $c$ is $0$-majoritarian.101010Recall that
we allow ballots in $\mathcal{V}$ to be equal.

Step 2:

We build on the family $\mathcal{V}$ of ballots given at Step 1. Since the
sequence $\left\\{\frac{k^{2}+k-1}{k^{2}+nk}\right\\}_{k=1}^{\infty}$
converges to 1, and $\frac{k^{2}+k-1}{k^{2}+nk}<1$ for any $k\geqslant 1$,
there exists $m\in\mathbb{N}$ such that $s<\frac{m^{2}+m-1}{m^{2}+nm}<1$. Set
$t\mathrel{\mathop{\mathchar
58\relax}}=\frac{m^{2}+m-1}{m^{2}+nm}\in\mathbb{Q}\in(0,1)$. We claim that $c$
is $t$-majoritarian.

To prove the claim, we construct a finite family $\mathcal{W}$ of ballots
witnessing that $c$ is $t$-majoritarian. Let $\mathcal{V}^{\prime}$ be a
family of ‘neutral’ ballots having size $|\mathcal{V}^{\prime}|=m^{2}$, that
is, for each $v^{\prime}\in\mathcal{V}^{\prime}$, the equality
$v^{\prime}(A)=A$ holds for all menus $A\subseteq X$.111111Since
$\mathcal{V}^{\prime}$ is a family (and not a set, because elements can be
repeated), the size $|\mathcal{V}^{\prime}|$ of $\mathcal{V}^{\prime}$ is the
cardinality of the index set. Furthermore, let $\mathcal{V}^{\prime\prime}$ be
the family of ballots obtained by replicating $m$ times each ballot in the
original family $\mathcal{V}$. Set $\mathcal{W}\mathrel{\mathop{\mathchar
58\relax}}=\mathcal{V}^{\prime}\cup\mathcal{V}^{\prime\prime}$, hence
$\mathcal{W}$ is made of $m^{2}+mn$ ballots. To check that $\mathcal{W}$ is a
$t$-majoritarian representation of $c$, let $a\in A\subseteq X$. The forward
implication in (1) holds, because if $a\in c(A)$, then

$\displaystyle\frac{\big{|}\\{w\in\mathcal{W}\mathrel{\mathop{\mathchar
58\relax}}a\in w(A)\\}\big{|}}{|W|}\;$
$\displaystyle=\;\frac{\big{|}\\{v^{\prime}\in\mathcal{V}^{\prime}\mathrel{\mathop{\mathchar
58\relax}}a\in
v^{\prime}(A)\\}\big{|}+m\cdot\big{|}\\{v\in\mathcal{V}\mathrel{\mathop{\mathchar
58\relax}}a\in v(A)\\}\big{|}}{|\mathcal{W}|}$
$\displaystyle\geqslant\;\frac{m^{2}+m}{m^{2}+mn}\;>\;t.$

For the converse, suppose
$\frac{|\\{w\,\in\,\mathcal{W}\,\mathrel{\mathop{\mathchar
58\relax}}\,a\,\in\,w(A)\\}|}{|\mathcal{W}|}>t$. Then there is
$v\in\mathcal{V}$ such that $a\in v(A)$, since otherwise
$\frac{|\\{w\,\in\,\mathcal{W}\,\mathrel{\mathop{\mathchar
58\relax}}\,a\,\in\,w(A)\\}|}{|\mathcal{W}|}=\frac{m^{2}}{m^{2}+mn}<t$. Since
$\mathcal{V}$ witnesses that $c$ is $0$-majoritarian, we obtain $a\in c(A)$.
This proves that (1) holds.

Step 3:

We suitably modify the family $\mathcal{W}$ defined at Step 2. For brevity,
denote by $p\mathrel{\mathop{\mathchar 58\relax}}=m^{2}+mn$ the size of the
family $\mathcal{W}$. First we argue that we can assume that the number
$p^{\prime}\mathrel{\mathop{\mathchar 58\relax}}=p(\frac{t}{s}-1)$ is a
positive integer. Indeed, $\frac{t}{s}-1$ is a positive rational number, hence
a sufficiently large replication of $\mathcal{W}$ satisfies
$p^{\prime}\in\mathbb{N}$. (Observe that all replications of $\mathcal{W}$ are
obviously $t$-majoritarian justifications of $c$.) Now consider a family
$\mathcal{W}^{\prime}$ of $p^{\prime}$ ‘hypercritical’ ballots (that is,
ballots such that the selection from each menu is always empty). Finally, set
$\mathcal{Z}\mathrel{\mathop{\mathchar
58\relax}}=\mathcal{W}\cup\mathcal{W}^{\prime}$. Note that
$|\mathcal{Z}|=p+p^{\prime}=p\cdot\frac{t}{s}$.

We check that $\mathcal{Z}$ is an $s$-majoritarian representation of $c$.
Indeed, for each $a\in X$, we have

$\displaystyle\frac{\big{|}\\{z\in\mathcal{Z}\mathrel{\mathop{\mathchar
58\relax}}a\in z(A)\\}\big{|}}{\big{|}\mathcal{Z}\big{|}}$
$\displaystyle=\;\frac{\big{|}\\{w\in\mathcal{W}\mathrel{\mathop{\mathchar
58\relax}}a\in w(A)\\}\big{|}}{p\cdot\tfrac{t}{s}}$
$\displaystyle=\;\frac{\big{|}\\{w\in\mathcal{W}\mathrel{\mathop{\mathchar
58\relax}}a\in w(A)\\}\big{|}}{\big{|}\mathcal{W}\big{|}}\cdot\frac{s}{t}.$

It follows that

$\displaystyle\frac{\big{|}\\{z\in\mathcal{Z}\mathrel{\mathop{\mathchar
58\relax}}a\in z(A)\\}\big{|}}{\big{|}\mathcal{Z}\big{|}}>s$
$\displaystyle\Longleftrightarrow\quad\frac{\big{|}\\{w\in\mathcal{W}\mathrel{\mathop{\mathchar
58\relax}}a\in w(A)\\}\big{|}}{\big{|}\mathcal{W}\big{|}}>t.$

This completes the proof of Theorem 1. ∎

Proof of Corollary 2. We have already observed that the proof of
(ii)$\;\Longrightarrow\;$(i) in Theorem 1 goes through without requiring the
share $s$ be rational. Thus, it suffices to show that
(i)$\;\Longrightarrow\;$(iii) holds.

To that end, suppose $c$ satisfies Axiom$\>\alpha$, and let $s\in(0,1)$.
Select $t\in\mathbb{Q}$ such that $s<t<1$. By Theorem 1, $c$ is
$t$-majoritarian; let $\mathcal{V}$ be a family of $p$ ballots over $X$
witnessing that $c$ is $t$-majoritarian. By replicating the society as many
times as needed, we can assume without loss of generality that $p\geqslant 2$,
$pt\in\mathbb{N}$, and $t<\frac{p-1}{p}$.

Consider the sequence $\left\\{t-\frac{tk}{p+k}\right\\}_{k=0}^{\infty}$,
which converges to $0$ as $k$ diverges. We claim that the inequality

$t-\frac{t(k-1)}{p+(k-1)}\;<\;t-\frac{tk-1}{p+k}\vspace{-0.1cm}$ (7)

holds for all $k\in\mathbb{N}$. Since $t<\frac{p-1}{p}$, the claim holds for
$k=0$. Routine computations show that (7) is equivalent to $p(t-1)+1-k<0$,
which is true for $k\geqslant 1$. This proves (7). Let $q$ be the minimum
positive integer such that $t-\frac{tq}{p+q}\leqslant s$. Since

$t-\frac{tq}{p+q}\;\leqslant\;s\;<\;t-\frac{t(q-1)}{p+q-1}\;<\;t-\frac{tq-1}{p+q},\vspace{-0.1cm}\vspace{-0.1cm}$

there is $\delta\in[0,1)$ such that
$s=t-\frac{tq-\delta}{p+q}=\frac{\left(t+\frac{\delta}{p}\right)p}{p+q}$. To
complete the proof, we show that $c$ is $s$-majoritarian in two steps.

Step 1:

$c$ is $\left(t+\frac{\delta}{p}\right)$-majoritarian. By assumption, for any
$A\in 2^{X}$, we have

$a\in
c(A)\quad\Longleftrightarrow\quad\big{|}\\{v\in\mathcal{V}\mathrel{\mathop{\mathchar
58\relax}}a\in v(A)\\}\big{|}>t|\mathcal{V}|.\vspace{-0.1cm}\vspace{-0.1cm}$

Furthermore,

$\big{|}\\{v\in\mathcal{V}\mathrel{\mathop{\mathchar 58\relax}}a\in
v(A)\\}\big{|}\;\geqslant\;t|\mathcal{V}|+1\;>\;t|\mathcal{V}|+\delta\;\geqslant\;t|\mathcal{V}|,\vspace{-0.1cm}\vspace{-0.1cm}$

since $\big{|}\\{v\in\mathcal{V}\mathrel{\mathop{\mathchar 58\relax}}a\in
v(A)\\}\big{|}$ and $t|\mathcal{V}|$ are integers. Thus the claim follows from

$a\in
c(A)\quad\Longleftrightarrow\quad\big{|}\\{v\in\mathcal{V}\mathrel{\mathop{\mathchar
58\relax}}a\in v(A)\\}\big{|}>t|\mathcal{V}|+\delta=tp+\delta.$

Step 2:

$c$ is $s$-majoritarian. Let $\mathcal{V}^{\prime}$ be a family of $q$
hypercritical ballots (empty quasi-choices) over $X$. Set
${\mathcal{W}}\mathrel{\mathop{\mathchar
58\relax}}=\mathcal{V}\cup\mathcal{V}^{\prime}$, hence $|{\mathcal{W}}|=p+q$.
For each $A\in 2^{X}$, we have

$\displaystyle a\in c(A)\quad$
$\displaystyle\Longleftrightarrow\quad\frac{\big{|}\\{v\in\mathcal{V}\mathrel{\mathop{\mathchar
58\relax}}a\in v(A)\\}\big{|}}{p}>t+\frac{\delta}{p}$
$\displaystyle\Longleftrightarrow\quad\frac{\big{|}\\{w\in{\mathcal{W}}\mathrel{\mathop{\mathchar
58\relax}}a\in w(A)\\}\big{|}}{p}>t+\frac{\delta}{p}$
$\displaystyle\Longleftrightarrow\quad\frac{\big{|}\\{w\in{\mathcal{W}}\mathrel{\mathop{\mathchar
58\relax}}a\in
w(A)\\}\big{|}}{|{\mathcal{W}}|}>\frac{(t+\frac{\delta}{p})p}{p+q}=s.\vspace{-0.1cm}\vspace{-0.1cm}$

Thus the family ${\mathcal{W}}$ provides a $s$-majoritarian justification of
$c$, as claimed. ∎

Proof of Example 4. We know that the choice $c$ fails to be asymmetrically
rationalizable. By Lemma 1(i), $c$ is not freely rationalizable either. Lemma
4 and Theorem 2 yield $2\leqslant\textsf{dem}(c)<\infty$ and
$2\leqslant\textsf{lib}(c)<\infty$. We show that $\textsf{lib}(c)=2$ and
$\textsf{dem}(c)=3$ by explicitly exhibiting the binary relations (voters)
that rationalize $c$.

$\textsf{lib}(c)=2\,$:

Let $\to_{1}$ and $\to_{2}$ be the two binary relations over $X$ defined by

  * (1)

$x\to_{1}y$, $x\to_{1}z$, and $y\to_{1}z$ (that is, $\to_{1}$ a strict linear
order over $X$),

  * (2)

$x\to_{2}z$ and $y\leftrightarrows_{2}z$.

One can readily check that the set $\\{\to_{1},\to_{2}\\}$ is a liberal
representation of $c$.

$\textsf{dem}(c)=3\,$:

Let $\to_{3}$ be the empty relation, which implies that the associated ballot
$c_{3}$ is the neutral choice (i.e., $c_{3}(A)=A$ for any menu $A\subseteq
X$). One can readily check that $\\{\to_{1},\to_{2},\to_{3}\\}$ is a
democratic representation of $c$. This proves $\textsf{dem}(c)\leqslant 3$. To
complete the proof, we show that no democratic representation of $c$ may have
size two. Toward a contradiction, suppose
$\\{\to^{\prime}_{1},\to_{2}^{\prime}\\}$ is a democratic representation of
$c$. Since $c(\\{x,y\\})=\\{x,y\\}$, the democratic paradigm implies that
$x\not\rightarrow_{i}^{\prime}x,y$, and $y\not\rightarrow^{\prime}_{i}x,y$ for
all $i\in\\{1,2\\}$. Moreover, from $c(\\{x,z\\})=\\{x\\}$ and
$c(\\{y,z\\})=\\{y\\}$ we deduce that $z\not\rightarrow_{i}^{\prime}x,y$ for
all $i\in\\{1,2\\}$. We conclude
$x,y\in\max(X,\to_{1}^{\prime})\cap\max(X,\to_{2}^{\prime})=c(X)=\\{x\\}$, a
contradiction. ∎

Proof of Example 5. To prove $\textsf{dem}(c)\leq 5$, let
$S=\\{\to_{i}\;\mathrel{\mathop{\mathchar 58\relax}}1\leqslant i\leqslant
5\\}$, where, for each $i$, the voter $\to_{i}$ only displays only the single
preference $i\to_{i}0$, and nothing else. It is immediate to check that $S$
democratically represents $c$. The claim follows.

Next, we prove $\textsf{lib}(c)\geq 10$. Let $\mathcal{F}$ be the family of
all menus of size $2$ not containing the item $0$; therefore,
$|\mathcal{F}|={5\choose 2}=10$. We claim that each menu in $\mathcal{F}$
requires a different voter over $X$ in order to obtain a liberal
representation of $c$: this will prove the claim. Toward a contradiction,
suppose that $\mathcal{V}$ is a liberal representation of $c$, and let $A$ and
$B$ be distinct menus in $\mathcal{F}$ such that $\max(A,\to)=c(A)=A$ and
$\max(B,\to)=c(B)=B$ for some voter $\to$ in $\mathcal{V}$. The definition of
$c$ yields $0\in\max(A\cup\\{0\\})=c(A\cup\\{0\\})=A\cup\\{0\\}$ and
$0\in\max(B\cup\\{0\\})=c(B\cup\\{0\\})=B\cup\\{0\\}$. It follows that
$i\not\rightarrow 0$ for all $i\in A\cup B$, and so $0\in\max(A\cup
B\cup\\{0\\})=c(A\cup B\cup\\{0\\})$. However, $|A\cup B\cup\\{0\\}|\geq 4$
implies $0\notin A\cup B=(A\cup B\cup\\{0\\})$, which is impossible. ∎

Proof of Example 6. First, we prove $\textsf{dem}(c)=3$. It is easy to show
$\textsf{dem}(c)>2$. Indeed, if $\\{\to_{1},\to_{2}\\}$ were to be a
democratic representation of $c$, then we would have
$x\perp_{i}y\perp_{i}z\perp_{i}x$ for $i=1,2$, because
$\underline{x}\underline{y}$, $\underline{x}\underline{z}$, and
$\underline{y}\underline{z}$.121212By $x\perp_{i}y$ we mean
$x\not\rightarrow_{i}y$ and $y\not\rightarrow_{i}x$. However, this would imply
that $z$ is chosen in $\\{x,y,z\\}$, a contradiction. To conclude the proof,
we exhibit a democratic representation of $c$ having size three. Let
$\to_{1}$, $\to_{2}$, and $\to_{3}$ be the binary relations on $X$ defined as
follows:

$x\to_{1}y,z,w\;\;\hbox{ and }\;\;y\to_{1}w\;,\qquad x\to_{2}w\;\;\hbox{ and
}\;\;y\to_{2}z,w\;,\qquad w\to_{3}y,z\;.\vspace{-0.1cm}\vspace{-0.1cm}$

Then $\\{\to_{1},\to_{2},\to_{3}\\}$ is a democratic representation of $c$,
hence $\textsf{dem}(c)=3$.

To prove that $\textsf{lib}(c)=3$, we first show $\textsf{lib}(c)>2$. Toward a
contradiction, suppose $\\{\to_{1},\to_{2}\\}$ is a liberal representation of
$c$. Since $\underline{x}w$ and $\underline{x}w$, we have $x\to_{i}w$ and
$y\to_{i}w$ for $i=1,2$. Since $\underline{x}\underline{z}$,
$\underline{y}\underline{z}$, and $\underline{x}\underline{y}z$, we have
(without loss of generality) $x\to_{1}z$ and $y\to_{2}z$. Since
$\underline{x}\underline{z}$ and $\underline{x}zw$, we get $w\to_{2}z$.
Similarly, since $\underline{y}\underline{z}$ and $\underline{y}zw$, we get
$w\to_{1}z$. However, the join of $w\to_{1}z$ and $w\to_{2}z$ contradicts
$\underline{z}\underline{w}$. We conclude that $\textsf{lib}(c)>2$. (Notice
that so far we have not used the selection $\underline{x}yzw$.) To exhibit a
liberal representation of $c$ of size three, let $\to_{1}$ be the strict
linear order such that $x\to_{1}y\to_{1}z\to_{1}w$, whereas $\to_{2}$ and
$\to_{3}$ are the voters over $X$ defined by

$x\to_{2}z,w\hbox{ and }y\to_{2}w\to_{2}z\to_{2}y\;,\quad
x\to_{3}w\to_{3}z\,,\;y\to_{3}z\,,\hbox{ and
}y\leftrightarrows_{3}w\;.\vspace{-0.1cm}\vspace{-0.1cm}$

(Note that $\to_{2}$ has a strict cycle, and $\to_{3}$ fails to be
asymmetric.) One can readily check that $\\{\to_{1},\to_{2},\to_{3}\\}$ is a
liberal representation of $c$, and so $\textsf{lib}(c)=3$. ∎

Proof of Lemma 5. Let $c$ be a quasi-choice over $X$. If Axiom$\;\alpha$ fails
for $c$, then $\textsf{dem}(c)=\textsf{lib}(c)=\infty$, hence (i) and (ii) are
verified. Next, suppose $c$ satisfies Axiom$\;\alpha$. By Theorem 2, $c$ is
both democratic and liberal, and so $\textsf{dem}(c)$ and $\textsf{lib}(c)$
are finite. In what follows, we prove (i) and (ii) by deriving a
representation of a certain type (resp. democratic, liberal) from one of the
other type (resp. liberal, democratic).

(i):

This inequality follows from the fact that for any liberal representation of
$c$, we can always create a democratic representation of $c$ by doubling the
number of voters, where all new voters are neutral (i.e., they choose
everything).

(ii):

Let $\mathcal{V}$ be a democratic representation of $c$ such that
$|\mathcal{V}|=\textsf{dem}(c)$. Below we construct a family

$\mathcal{W}_{\mathcal{V}}=\left\\{w_{\mathcal{U}}\mathrel{\mathop{\mathchar
58\relax}}\mathcal{U}\subseteq\mathcal{V}\hbox{ and
}|\mathcal{U}|>\textstyle\frac{|\mathcal{V}|}{2}\right\\}\vspace{-0.1cm}\vspace{-0.1cm}$

of ballots over $X$, which liberally represents $c$. The ballot
$w_{\mathcal{U}}\in\mathcal{W}_{\mathcal{V}}$ is such that, for each $A\in
2^{X}$ and $a\in X$,

$a\in w_{\mathcal{U}}(A)\quad\Longleftrightarrow\quad a\in u(A)\hbox{ for all
}u\in{\mathcal{U}}\,.\vspace{-0.1cm}\vspace{-0.1cm}$ (8)

To prove that ${\mathcal{W}}_{\mathcal{V}}$ liberally represents $c$, we show
that, for each $A\in 2^{X}$ and $a\in X$, the equivalence

$a\in c(A)\quad\Longleftrightarrow\quad a\in w_{\mathcal{U}}(A)\hbox{ for some
}w_{\mathcal{U}}\in{\mathcal{W}}_{\mathcal{V}}\vspace{-0.1cm}\vspace{-0.1cm}$
(9)

holds. Let $A\in 2^{X}$ and $a\in X$. If $a\in c(A)$, then, since
$\mathcal{V}$ is a democratic representation of $c$, there is a subfamily
${\mathcal{U}}\subseteq\mathcal{V}$ such that
$|{\mathcal{U}}|>|\mathcal{V}|/2$ and $a\in u(A)$ for all $u\in{\mathcal{U}}$.
Now (8) yields $a\in w_{\mathcal{U}}(A)$, which proves the forward implication
in (9). We prove the reverse implication in (9) by contrapositive. Suppose
$a\notin c(A)$. If $a\in w_{\mathcal{U}}(A)$ for some
${\mathcal{U}}\subseteq\mathcal{V}$ such that
$|{\mathcal{U}}|>|\mathcal{V}|/2$, then (8) gives $a\in u(A)$ for all
$u\in{\mathcal{U}}$, which contradicts the fact that $\mathcal{V}$ is a
democratic representation of $c$. Thus, there is no
${\mathcal{U}}\subseteq\mathcal{V}$, with $|{\mathcal{U}}|>|\mathcal{V}|/2$,
such that $a\in w_{\mathcal{U}}(A)$. Thus ${\mathcal{W}}_{\mathcal{V}}$
liberally represents $c$.

Now the bound (ii) follows easily. Indeed, letting
$n\mathrel{\mathop{\mathchar 58\relax}}=|\mathcal{V}|=\textsf{dem}(c)$, we
have:

$|{\mathcal{W}}_{\mathcal{V}}|\;=\;\displaystyle\sum_{k=\lfloor\frac{n}{2}\rfloor+1}^{n}{n\choose
k}\;\leqslant\;\frac{2^{n}}{2}\;=\;2^{\textsf{dem}(c)-1}\,,\vspace{-0.1cm}\vspace{-0.1cm}$

and so $\textsf{lib}(c)\leqslant|{\mathcal{W}}_{\mathcal{V}}|\leqslant
2^{\,\textsf{dem}(c)-1}$, as claimed. ∎

Proof of Lemma 6. Let $n\geqslant 0$ and $1\leqslant k\leqslant n+1$. One can
readily check that, for $n=0$, the choice $c_{0,1}$ is the unique choice over
$X_{0}=\\{0\\}$. Clearly, this choice satisfies Axiom$\>\alpha$, has liberal
number $\textsf{lib}(c_{0,1})={0\choose 0}=1$, and has democratic number
$\textsf{dem}(c_{0,1})=1$. Thus the claim holds for $n=0$.

Now suppose $n\geqslant 1$. Let $A\subseteq B\subseteq X_{n}$ and $x\in A\cap
c_{n,k}(B)$. To prove that Axiom$\>\alpha$ holds for $c_{n,k}$, we show that
$x\in c_{n,k}(A)$. If $x\neq 0$, then the claim readily follows from condition
(a). Therefore, let $x=0$. Since $0\in c_{n,k}(B)$ by hypothesis, condition
(b) yields $|B|\leqslant k$, which in turn implies $|A|\leqslant k$ because
$A\subseteq B$. Another application of (b) delivers $0\in c_{n,k}(A)$. This
shows that Axiom$\>\alpha$ holds for all $c_{n,k}$.

Next, we prove the equality in (5). Set

$\mathscr{F}_{n,k}\;\coloneqq\;\big{\\{}A\in
2^{X_{n}}\mathrel{\mathop{\mathchar 58\relax}}0\in A\text{ and
}|A|=k\big{\\}}\,.\vspace{-0.1cm}\vspace{-0.1cm}$

Plainly, $|\mathscr{F}_{n,k}|={n\choose k-1}$. Let
$\mathcal{V}_{n,k}=\\{v_{A}\colon A\in\mathscr{F}_{n,k}\\}$ be the family of
quasi-choices on $X_{n}$ defined by

$v_{A}(B)=\begin{cases}B&B\subseteq A;\\\ B\setminus\\{0\\}&B\not\subseteq
A.\\\ \end{cases}$

We show that each $v_{A}$ is rationalizable, and thus it is a ballot over
$X_{n}$. To that end, suppose $B\subseteq D$ and $x\in B\cap v_{A}(D)$. If
$x\neq 0$, we are done. If $x=0$, we obtain $B\subseteq D\subseteq A$, whence
$x\in v_{A}(B)$. Next, suppose $x\in v_{A}(B)\cap v_{A}(D)$. If $x\neq 0$, we
are done. Suppose $x=0$ and note that $(B\cup D)\subseteq A$. It follows that
$0\in v_{A}(B\cup D)$.

Next, we prove that the family of ballots $\mathcal{V}_{n,k}$ liberally
represents $c_{n,k}$. Suppose $x\in c_{n,k}(B)$ and $x\neq 0$. In this case,
every $v_{A}$ is defined in a way such that $x\in v_{A}(B)$. If $x=0$ and
$0\in c_{n,k}(B)$, then we obtain $|B|\leq k$. Therefore we can choose a menu
$A$ such that $|A|=k$ and $B\subseteq A$. For such a set, $0\in v_{A}(B)$.

Suppose there exists $A\in\mathscr{F}_{n,k}$ such that $x\in v_{A}(B)$. If
$x\neq 0$, then we are readily done, because $B\setminus
c_{n,k}(B)\subseteq\\{0\\}$. If $x=0$, then we have $B\subseteq A$. We
conclude $|B|\leq|A|$, and so $0\in c_{n,k}(B)$. This completes the proof that
$\mathcal{V}_{n,k}$ liberally represents $c_{n,k}$. Since
$|\mathcal{V}_{n,k}|={n\choose k-1}$, we conclude
$\textsf{lib}(v_{n,k})\leqslant{n\choose k-1}$.

To complete the proof of the equality $\textsf{lib}(c_{n,k})={n\choose k-1}$,
we show that no family $\mathcal{V}$ of ballots over $X_{n}$ having size
strictly less that ${n\choose k-1}$ can liberally represent $c_{n,k}$. Toward
a contradiction, suppose such a family $\mathcal{V}$ exists. Recall that
$|\mathscr{F}_{n,k}|={n\choose k-1}$, and for all $A\in\mathscr{F}_{n,k}$ we
have $0\in c_{n,k}(A)$. By the pigeonhole principle, there are two distinct
menus $A_{1},A_{2}\in\mathscr{F}_{n,k}$ and a ballot $v\in\mathcal{V}$ such
that $0\in v(A_{1})\cap v(A_{2})$. Apply Axiom$\>\gamma$ to obtain $0\in
v(A_{1}\cup A_{2})$. By the liberal rationalizability of $c_{n,k}$ by
$\mathcal{V}$, it follows that $0\in c_{n,k}(A_{1}\cup A_{2})$. By the
definition of $c_{n,k}$, we also obtain $|A_{1}\cup A_{2}|\leqslant k$.
However, the latter fact is impossible, since $|A_{1}|=|A_{2}|=k$, and
$A_{1}\neq A_{2}$ implies $|A_{1}\cup A_{2}|>k$.

To complete the proof of Lemma 6, we show that the two inequalities in (5)
hold. Define $n$ distinct ballots $\widetilde{v}_{1},\ldots,\widetilde{v}_{n}$
over $X_{n}$ by letting, for each $i=1,\ldots,n$ and $A\in 2^{X_{n}}$,

$\widetilde{v}_{i}(A)\coloneqq\begin{cases}A\setminus\\{0\\}&\text{if }0,i\in
A\\\ A&\text{otherwise.}\end{cases}\vspace{-0.1cm}\vspace{-0.1cm}$

Next, let $v^{N}$ be the neutral ballot over $X_{n}$ (the identity choice),
and, for $n\geqslant 3$, let $v^{H}$ be the hypercritical ballot over $X_{n}$
(the empty quasi-choice).131313For our purposes, at most $n+1$ replications of
the neutral ballot will suffice. Similarly, at most $n-2$ replications of the
hypercritical ballot will suffice, as long as $n\geq 3$.

The next two claims complete the proof of the two inequalities in (5):

Claim 1: For $\frac{n}{2}<k\leqslant n+1$, $c_{n,k}$ is democratically
represented by the family

$\mathcal{V}_{k}\coloneqq\\{\widetilde{v}_{1},\ldots,\widetilde{v}_{n}\\}\cup\big{\\{}v^{N}_{1},\ldots,v^{N}_{2k-n-1}\big{\\}},\vspace{-0.1cm}\vspace{-0.1cm}$

where $v^{N}_{j}=v^{N}$ for all $j$’s.

Claim 2: For $1\leqslant k\leqslant\frac{n}{2}$, $c_{n,k}$ is democratically
represented by the family

$\mathcal{V}_{k}\coloneqq\\{\widetilde{v}_{1},\ldots,\widetilde{v}_{n}\\}\cup\big{\\{}v^{H}_{1},\ldots,v^{H}_{n-2k}\big{\\}},\vspace{-0.1cm}\vspace{-0.1cm}$

where $v^{H}_{j}=v^{H}$ for all $j$’s.

In what follows, we prove Claim 1 (Claim 2 can be proved in a similar way):
this will complete the proof of Lemma 6. It suffices to show that
$\mathcal{V}_{k}$ is such that

$p\in
c_{n,k}(A)\quad\Longleftrightarrow\quad\big{|}\\{v\in\mathcal{V}_{k}\mathrel{\mathop{\mathchar
58\relax}}p\in
v(A)\\}\big{|}>\textstyle\displaystyle\frac{\left|\mathcal{V}_{k}\right|}{2}$
(10)

for each $A\in 2^{X_{n}}$ and $p\in X_{n}$. For $p\neq 0$, the equivalence
(10) holds. Suppose $p=0$. Then

$\big{\\{}i\in\\{1,\ldots n\\}\mathrel{\mathop{\mathchar
58\relax}}0\in\widetilde{v}_{i}(A)\big{\\}}=\big{\\{}i\in\\{1,\ldots,n\\}\mathrel{\mathop{\mathchar
58\relax}}i\notin A\big{\\}}=X_{n}\setminus
A\,,\vspace{-0.1cm}\vspace{-0.1cm}$

hence

$\big{|}\big{\\{}i\in\\{1,\ldots,n\\}\mathrel{\mathop{\mathchar
58\relax}}0\in\widetilde{v}_{i}(A)\big{\\}}\big{|}=n+1-|A|\,.\vspace{-0.1cm}$

Furthermore, we have

$\big{|}\big{\\{}i\in\\{1,\ldots,2k-n-1\\}\mathrel{\mathop{\mathchar
58\relax}}0\in
v_{i}^{N}(A)\big{\\}}\big{|}=2k-n-1.\vspace{-0.1cm}\vspace{-0.1cm}$

Since

$\big{(}n+1-|A|\big{)}+\big{(}2k-n-1\big{)}>\frac{2k-1}{2}\quad\Longleftrightarrow\quad|A|\leqslant
k\,,\vspace{-0.1cm}$

we conclude that the family $\mathcal{V}_{k}$ satisfies (10) also for $p=0$. ∎

Proof of Theorem 2. Let $c$ satisfy Axiom$\>\alpha$ over a grand set $X$ of
size $n\geqslant 2$. For simplicity, in the following we identify $X$ with
$\\{1,\ldots,n\\}$.

(Upper bound)

For each $p\in X$, denote by $\mathscr{A}_{p}$ the collection of the
$\subseteq$-maximal sets $A\subseteq X$ such that $p\in c(A)$. Let
$\mathcal{V}^{*}$ be any collection of ballots over $X$ satisfying the
following properties:

  1. (i)

for every ballot $v\in\mathcal{V}^{*}$, there exist
$C_{1}\in\mathscr{A}_{1},\ldots,C_{n}\in\mathscr{A}_{n}$ such that

$v(B)=\\{p\in X\mathrel{\mathop{\mathchar 58\relax}}B\subseteq
C_{p}\\},\quad\hbox{for $B\in 2^{X}$};\vspace{-0.1cm}\vspace{-0.1cm}$

  2. (ii)

for every item $p\in X$,

$\displaystyle\Big{\\{}\bigcup_{\overset{B\in 2^{X}}{p\in
v(B)}}B\mathrel{\mathop{\mathchar
58\relax}}v\in\mathcal{V}^{*}\Big{\\}}=\mathscr{A}_{p};\vspace{-0.1cm}\vspace{-0.1cm}$

  3. (iii)

$|\mathcal{V}^{*}|\leqslant\max_{p\in X}|\mathscr{A}_{p}|$.

It is not difficult to check that $\mathcal{V}^{*}$ liberally represents $c$.
To complete the proof, we show
$|\mathcal{V}^{*}|\leqslant{n-1\choose\lfloor\frac{n-1}{2}\rfloor}$. For $p\in
X$, let

$\mathscr{A}_{p}^{\prime}\coloneqq\\{A\setminus\\{p\\}\mathrel{\mathop{\mathchar
58\relax}}A\in\mathscr{A}_{p}\\},\vspace{-0.1cm}$

so that $|\mathscr{A}_{p}^{\prime}|=|\mathscr{A}_{p}|$ holds. Plainly, each
$\mathscr{A}_{p}^{\prime}$ is a Sperner family over $X\setminus\\{p\\}$, as
its members are mutually $\subseteq$-incomparable., Therefore, by Sperner’s
theorem (Sperner, 1928; Lubell, 1966), we have
$|\mathscr{A}_{p}^{\prime}|\leqslant{n-1\choose\lfloor\frac{n-1}{2}\rfloor}$,
and so by (iii) we get
$|\mathcal{V}^{*}|\leqslant{n-1\choose\lfloor\frac{n-1}{2}\rfloor}$. We
conclude that
$\textsf{lib}(c)\,\leqslant\,{n-1\choose\lfloor\frac{n-1}{2}\rfloor}$ holds.

(Tightness)

Take the liberal choice $c=c_{n-1,\lfloor\frac{n-1}{2}\rfloor+1}$ over
$X_{n-1}$ defined in Lemma 6, and observe that $|X_{n-1}|=n$ and
$\textsf{lib}(c)={n-1\choose\lfloor\frac{n-1}{2}\rfloor}$. ∎

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