Justifications for Generalizations of Approval Voting

by

Hari Sarang Nathan

A dissertation submitted in partial fulfillment

of the requirements for the degree of

Master of Science (M.S.) in Mathematics

Courant Institute of Mathematical Sciences

New York University

May 2023

Professor Michael E. Orrison
---
Harvey Mudd College

###### Contents

  1. Acknowledgements
  2. 1 Introduction
  3. 2 Definitions and Notation
  4. 3 Axiomatic Justifications
    1. 3.1 Prior Results
    2. 3.2 New Axioms
    3. 3.3 Faithfulness, Consistency, Positive Scaling, and Cancellation
    4. 3.4 Faithfulness, Consistency, Positive Scaling, and Disjoint Equality
  5. 4 Utility Maximization and Maximum Likelihood Estimation Justifications
    1. 4.1 Preliminaries
    2. 4.2 k,p-Approval Voting as UM and MLE
  6. 5 Distance Rationalizability Justifications
    1. 5.1 Preliminaries
    2. 5.2 Distance Rationalizability of Approval Voting
    3. 5.3 Distance Rationalizability When k = 1
    4. 5.4 Distance Rationalizability for General k and p
  7. 6 Conclusion

## Acknowledgements

I am greatly indebted to my advisor, Professor Michael E. Orrison for helping
guide me on this thesis as well as Professor Katharine Shultis and Professor
Jessica Sorrells for their ideas and support throughout this work. I am also
grateful to Professor Vlad Vicol, Professor Sylvain Cappell, and Professor Ben
Blum-Smith for their encouragement and in helping me find my thesis advisor.
Finally, I could never have finished this without the support of my mother,
Ms. Pushpa Nathan.

## Abstract

Approval voting is a common method of preference aggregation where voters vote
by “approving” of a subset of candidates and the winner(s) are those who are
approved of by the largest number of voters. In approval voting, the degree to
which a vote impacts a candidate’s score depends only on if that voter
approved of the candidate or not, i.e., it is independent of which, or how
many, other candidates they approved of. Recently, there has been interest in
satisfaction approval voting and quadratic voting both of which include a
trade-off between approving of more candidates and how much support each
selected candidate gets. Approval voting, satisfaction approval voting, and
quadratic voting, can all be viewed as voting where a vote is viewed as
analogous to a vector with a different unit norm ($\mathcal{L}^{\infty}$,
$\mathcal{L}^{1}$, and $\mathcal{L}^{2}$ respectively). This suggests a
generalization where one can view a vote as analogous to a normalized unit
vector under an arbitrary $\mathcal{L}^{p}$-norm. In this paper, we look at
various general methods for justifying voting methods and investigate the
degree to which these serve as justifications for these generalizations of
approval voting.

##  1 Introduction

Approval voting is a common method of preference aggregation where voters vote
by “approving” of a subset of candidates and the winner(s) are those who are
approved of by the most voters. In approval voting, the degree to which a vote
impacts a candidate’s score depends only on if that voter approved of the
candidate or not, i.e., it is independent of which, or how many, other
candidates they approved of. However, other work has investigated the
advantages of alternatives where the contribution to a candidates depends on
the number of candidates approved.

In particular, [3] introduces “satisfaction approval voting” where a voter
contributes a total of a single point divided equally among all the candidates
they approved of. In addition, there has been recent interest in “quadratic
voting” (e.g., [13], [15]) where (in some forms) each voter has a fixed pool
of points and the contribution to a candidate is the square root of the number
of points spent on that candidate (e.g., [14]).

Approval voting, satisfaction approval voting, and quadratic voting, can all
be viewed as voting where a vote is viewed as a unit vector with a different
norm ($\mathcal{L}^{\infty}$, $\mathcal{L}^{1}$, and $\mathcal{L}^{2}$
respectively). This suggests a generalization where one can view a vote as a
unit vector under an arbitrary $\mathcal{L}^{p}$-norm. In this paper, we study
such generalizations which we call $k,p$-approval voting. In particular,
voters pick a subset of candidates to approve of and the vote is viewed as
analogous to a $0,1$-vector which is then normalized according to some
predetermined ($\mathcal{L}^{p}$) norm. We look at various general methods for
justifying voting methods and investigate the degree to which these serve as
justifications for $k,p$-approval voting.

The rest of this paper proceeds as follows. In section 2, we lay out the basic
definitions and notation. In section 3, we provide an axiomatic justification
of $k,p$-approval voting. In section 4, we look at justifications of
$k,p$-approval voting from the perspective of utility maximization and maximum
likelihood estimation. In section 5, we look at $k,p$-approval voting from the
perspective of distance rationalizability. In section 6, we conclude.

As a note, throughout this paper we use the vocabulary of “democratic”
elections, i.e., voters, candidates, etc. However, as [8] points out, there
are a number of other uses for approval voting. They list the following which
span numerous fields:111In the quote below, we omit the citations to various
references. The references can be found in [8].

[leftmargin=0pt, vskip=0pt]

  1. 1.

finding group recommendations where the possible recommendations can be
thought of as candidates and individual group members as voters,

  2. 2.

collaborative filtering where, for example, related movies are recommended
based on large data collections,

  3. 3.

diversifying search results where users sending a search query can be
interpreted as voters and the possible search results correspond to
candidates,

  4. 4.

locating public facilities where the candidates are possible locations in
which facilities can be built,

  5. 5.

the design of dynamic Q&A platforms, where participants propose and upvote
questions to be asked in a Q&A session,

  6. 6.

selecting validators in consensus protocols (blockchain), with the users of
the protocol corresponding to both voters and candidates, and

  7. 7.

genetic programming, a technique to solve global optimisation problems.

##  2 Definitions and Notation

Before we get to the specific definitions we need, we lay out some notation we
will use throughout the paper.

  * •

We use $\mathcal{P}(X)$ to refer to the power set of $X$, i.e.,
$\mathcal{P}(X)=\\{Y\subset X\\}$.

  * •

We use $\mathds{1}$ as the indicator function, i.e.,

$\mathds{1}_{g(x)}=\begin{cases}1&g(x)\text{ is true}\\\
0&\text{otherwise}.\end{cases}$

  * •

Given two sets, $X$ and $Y$, we use $X^{Y}$ to refer to the set of all
functions from $Y$ to $X$.

  * •

Given two sets, $X$ and $Y$, we use $X\setminus Y$ as the set difference i.e.
$X\setminus Y=\\{x\in X:x\notin Y\\}$.

  * •

Given a vector $v\in\mathbb{R}^{n}$ we use $\|v\|_{p}$ as the
$\mathcal{L}^{p}$ norm of the vector, i.e., for $p\in[1,\infty)$

$\|v\|_{p}=\left(\sum_{i=1}^{n}{|v_{i}|^{p}}\right)^{1/p}$

and, for $p=\infty$,

$\|v\|_{\infty}=\lim_{p\to\infty}{\|v\|_{p}}=\max_{i\in\\{1,...,n\\}}{|v_{i}|}.$

  * •

In this paper, we use both $\operatorname*{arg\,min}$ and
$\operatorname*{arg\,max}$ as per the standard definitions. Of note, however,
is that we treat these as returning the set of all arguments that
minimize/maximize the relevant quantity. For example, we have

$\operatorname*{arg\,min}_{x\in[0,10]}{\\{x-\lfloor
x\rfloor\\}}=\\{0,1,...,9,10\\},$

i.e., all the integers in $[0,10]$.

  * •

We write $A\sim B$ to mean “$A$ is proportional to $B$” i.e. there is some
$c>0$ such that $A=cB$.

We now turn to the abstract definition of a voting system. A voting system is
a way to take a finite collection of ballots and aggregate them into an
outcome. To specify all of these, we use the following ingredients ([16]).

######  Definition 1.

We define:

  * •

$\mathcal{O}$ a finite set, the set of outputs;

  * •

$\mathcal{I}$ a finite set, the set of inputs or ballots;

  * •

$\pi:\mathcal{I}\to\mathbb{Z}_{\geq 0}$ a profile; and

  * •

$\Pi=\left(\mathbb{Z}_{\geq 0}\right)^{\mathcal{I}}$, i.e., the set of all
functions from $\mathcal{I}$ to $\mathbb{Z}_{\geq 0}$, the set of all
profiles;

  * •

$\mathcal{F}:\Pi\to\mathcal{P}(\mathcal{O})\setminus\\{\emptyset\\}$ is a
voting rule.

######  Definition 2.

A tuple $(\mathcal{I},\mathcal{O},\mathcal{F})$ is an abstract anonymous
voting system (or just a voting system).222The definition in [16] also
includes the “set of possible outcomes” which is denoted as
$\mathcal{C}(\mathcal{O})$. However, we find this notation redundant, since
$\mathcal{F}(\Pi)$ plays this role and, below, we use $\mathcal{C}$ for the
set of candidates.

The “anonymous” part in the definition above refers to the fact that the
voting system doesn’t care which voters cast which vote, only how many voters
cast each vote. This comes from the fact that $\mathcal{F}$ takes profiles,
and not lists of votes, as inputs.333An example of a system that is not
anonymous is a dictatorship where multiple people vote but only the dictator’s
vote counts and every other vote is ignored. Note that $\mathcal{F}$ may not
return a single output. When $|\mathcal{F}(\pi)|>1$, we interpret this to mean
that the various outcomes in $\mathcal{F}(\pi)$ are “tied” (at least as far as
$\mathcal{F}$ and $\pi$ are concerned).

The above is somewhat abstract; a few examples will help clarify.

######  Example 1.

Perhaps the best-known voting system is plurality voting where each voter
votes for a single candidate and the candidate with the most votes wins. In
this system:

  * •

$\mathcal{O}=\\{c_{1},...,c_{n}\\}$ is the set of candidates;

  * •

$\mathcal{I}=\mathcal{O}$ since each vote is just for a single candidate; and

  * •

$\mathcal{F}$ returns the candidates with the most votes. This can be written
as

$\mathcal{F}(\pi)=\operatorname*{arg\,max}_{c\in\mathcal{O}}{\\{\pi(c)\\}}.$

For example, if there are three candidates, $\\{c_{1},c_{2},c_{3}\\}$, we can
look at two profiles.

Profile | $c_{1}$ | $c_{2}$ | $c_{3}$ | $\mathcal{F}(\pi)$
---|---|---|---|---
$\pi_{1}$ | 10 | 15 | 5 | $\\{c_{2}\\}$
$\pi_{2}$ | 20 | 15 | 20 | $\\{c_{1},c_{3}\\}$

Here $\mathcal{F}(\pi_{1})=\\{c_{2}\\}$ (the clear winner) but
$\mathcal{F}(\pi_{2})=\\{c_{1},c_{3}\\}$ since they are tied for winner. If
this were being used in a situation where a single clear winner was needed,
some tie-breaking methodology would be needed.

######  Example 2.

We now turn to an example of the type we will study in this paper. In approval
voting each candidate selects at least one candidate to approve of and the
candidate approved of by the largest number voters wins. In this system:

  * •

$\mathcal{O}=\\{c_{1},...,c_{n}\\}$ is the set of candidates;

  * •

$\mathcal{I}=\mathcal{P}(\mathcal{O})\setminus\\{\emptyset\\}$ since each
voter must approve of at least one candidate but can approve of as many as
they like; and

  * •

$\mathcal{F}$ is defined via a function $s:\mathcal{O}\times\Pi\to\mathbb{R}$
where

$s(c,\pi)=\sum_{b\in\mathcal{I}}{\pi(b)\cdot\mathds{1}_{c\in b}}$

and

$\mathcal{F}(\pi)=\operatorname*{arg\,max}_{c\in\mathcal{O}}{\\{s(c,\pi)\\}}.$

The function $s$ is an example of what we will call a “score function” (which
we will define formally below). Here, the score of each candidate is number of
voters who approved of them and $\mathcal{F}$ returns all the candidates with
maximum score.

Again, we use a three candidate example with $\\{c_{1},c_{2},c_{3}\\}$ and
look at two examples.

Profile | $\\{c_{1}\\}$ | $\\{c_{2}\\}$ | $\\{c_{3}\\}$ | $\\{c_{1},c_{2}\\}$ | $\\{c_{1},c_{3}\\}$ | $\\{c_{2},c_{3}\\}$ | $\\{c_{1},c_{2},c_{3}\\}$
---|---|---|---|---|---|---|---
$\pi_{1}$ | 2 | 0 | 0 | 2 | 0 | 3 | 0
$\pi_{2}$ | 2 | 0 | 0 | 0 | 0 | 3 | 0

This leads to the following results:

Profile | $s(c_{1},\pi)$ | $s(c_{2},\pi)$ | $s(c_{3},\pi)$ | $\mathcal{F}(\pi)$
---|---|---|---|---
$\pi_{1}$ | 4 | 5 | 3 | $\\{c_{2}\\}$
$\pi_{2}$ | 2 | 3 | 3 | $\\{c_{2},c_{3}\\}$

######  Example 3.

An alternative to approval voting is to “normalize” votes which approve of
more than one candidate. In satisfaction approval voting ([3]) $\mathcal{O}$
and $\mathcal{I}$ are the same as in example 2 above but we modify $s$ such
that

$s(c,\pi)=\sum_{b\in\mathcal{I}}{\frac{\pi(b)\cdot\mathds{1}_{c\in b}}{|b|}}.$

This can change the outcome of an election. For example, for the profiles in
the example 2:

Profile | $s(c_{1},\pi)$ | $s(c_{2},\pi)$ | $s(c_{3},\pi)$ | $\mathcal{F}(\pi)$
---|---|---|---|---
$\pi_{1}$ | 3 | 2 | 1.5 | $\\{c_{1}\\}$
$\pi_{2}$ | 2 | 1.5 | 1.5 | $\\{c_{1}\\}$

With these definitions and examples in hand, we now turn to defining the
specific objects of our study. In particular, we will study generalizations of
approval voting and satisfaction approval voting. We generalize these in two
ways. First, we allow for a broader range of score functions. Second, we allow
for the possibility of selecting a committee of multiple candidates, not just
a single candidate.

######  Definition 3.

For $k\in\mathbb{Z}_{>0}$ and $p\in[1,\infty]$, a $k,p$-approval voting
system, $(\mathcal{I},\mathcal{O},\mathcal{F}_{k,p})$, is an abstract
anonymous voting system where:

  * •

$k$ is the committee size;

  * •

there are $n$ candidates, i.e., $\mathcal{C}=\\{c_{1},...,c_{n}\\}$;

  * •

the possible outcomes are committees of size $k$, i.e.,
$\mathcal{O}=\\{K\subset\mathcal{C}:|K|=k\\}$;

  * •

the ballots are approval ballots, i.e.,
$\mathcal{I}=\mathcal{P}(\mathcal{C})\setminus\\{\emptyset\\}$;444Everything
here can be equally well formulated if we allow
$\mathcal{I}=\mathcal{P}(\mathcal{C})$, i.e., if we allow a voter to “approve
of nobody.” We only disallow it here because we will be dividing by
$|b|^{1/p}$ and we would need to create special cases to avoid diving by zero.

  * •

the score function is $s_{p}:\mathcal{C}\times\Pi\to\mathbb{R}$ defined by as
follows.555The inspiration for these score functions comes from thinking of
the ballots as $0-1$ vectors in $\mathbb{R}^{\mathcal{C}}\cong\mathbb{R}^{n}$.
For each $b\in\mathcal{I}$, we can consider the vector $\vec{b}$ where
$\vec{b}_{i}=1$ if $c_{i}\in b$ and $b_{i}=0$ otherwise. In this case, we can
see that $\|\vec{b}\|_{p}=|b|^{1/p}$ and we are, in effect, normalizing each
$\vec{b}$ by its $\mathcal{L}^{p}$ norm. For $p<\infty$,

$s_{p}(c,\pi)=\sum_{b\in\mathcal{I}}{\frac{\pi(b)\cdot\mathds{1}_{c\in
b}}{|b|^{1/p}}}$

and, for $p=\infty$,

$s_{\infty}(c,\pi)=\lim_{p\to\infty}{s_{p}(c,\pi)}=\sum_{b\in\mathcal{I}}{\pi(b)\cdot\mathds{1}_{c\in
b}};$

and

  * •

$\mathcal{F}_{k,p}$ is defined as

$\mathcal{F}_{k,p}(\pi)=\operatorname*{arg\,max}_{K\in\mathcal{O}}{\left\\{\sum_{c\in
K}{s_{p}(c,\pi)}\right\\}}.$

######  Example 4.

Using the definition above, $1,\infty$-approval voting is approval voting,
$1,1$-approval voting is satisfaction approval voting. In addition,
$1,2$-approval voting is quadratic voting.

######  Example 5.

Changing $p$ can dramatically change the outcome. We look at the following
profile, $\pi$:

$\\{c_{1}\\}$ | $\\{c_{2}\\}$ | $\\{c_{3}\\}$ | $\\{c_{1},c_{2}\\}$ | $\\{c_{1},c_{3}\\}$ | $\\{c_{2},c_{3}\\}$
---|---|---|---|---|---
800 | 600 | 122 | 100 | 622 | 966

With $k=1$ and for $p$ as $1$, $2$, and $\infty$ we get the following results:

$p$ | $s_{p}(c_{1},\pi)$ | $s_{p}(c_{2},\pi)$ | $s_{p}(c_{3},\pi)$ | $\mathcal{F}_{1,p}(\pi)$
---|---|---|---|---
$1$ | $1161$ | $1133$ | $916$ | $\\{\\{c_{1}\\}\\}$
$2$ | $1310.53$ | $1353.78$ | $1244.89$ | $\\{\\{c_{2}\\}\\}$
$\infty$ | $1522$ | $1666$ | $1710$ | $\\{\\{c_{3}\\}\\}$

##  3 Axiomatic Justifications

There is a long tradition, going back to at least Arrow’s Impossibility
Theorem in [2], of looking for axioms that various voting systems satisfy or
that characterize voting systems. Approval voting is no different. Indeed,
there are various axiom sets that uniquely characterize approval voting. In
this section, we look at some of these axioms and discuss how to generalize
them to uniquely characterize $k,p$-approval.

Throughout this section, and for the rest of the paper, we will assume both
$k$ and $p$ are fixed, $\mathcal{C}=\\{c_{1},...,c_{n}\\}$,
$\mathcal{O}=\\{K\subset\mathcal{C}:|K|=k\\}$, etc., as in definition 3 above.

###  3.1 Prior Results

There are several sets of axioms which uniquely characterize approval voting
with $k=1$. We will first present some of these and the relevant
characterizations before turning to generalizations of the axioms and new
characterizations.

######  Remark.

We note that since profiles are just functions, they can be added and scaled.
These are defined as $(\pi+\pi^{\prime})(b)=\pi(b)+\pi^{\prime}(b)$ and, for
$\ell\in\mathbb{R}$, $(\ell\cdot\pi)(b)=\ell\cdot\pi(b)$.

Before defining the axioms, we create a preliminary definition. We need to be
able to create a profile out of a single ballot.

######  Definition 4.

Given $b\in\mathcal{I}$, we define the single ballot profile as
$\pi^{b}:\mathcal{I}\to\mathbb{R}_{>0}$ where
$\pi^{b}(b^{\prime})=\mathds{1}_{b^{\prime}=b}$.

This means that profiles are just linear combinations of the $\pi^{b}$ with
non-negative coefficients.

We now define some axioms that are used to characterize approval voting. These
axioms assume that $\mathcal{F}$ is a voting rule applied to profiles on
approval ballots (such as those in definite 3). The axioms, which we find in
[9], are:

  * •

A voting rule, $\mathcal{F}$, satisfies faithfulness if when there is only one
voter, their vote specifies the winners, i.e., for all $b\in\mathcal{I}$,
$\mathcal{F}(\pi^{b})=\\{\\{c\\}:c\in b\\}$.

  * •

A voting rule, $\mathcal{F}$, satisfies consistency if the outcome of
combining two profiles is the intersection of their outcomes (when its not
empty), i.e., given two profiles, $\pi,\pi^{\prime}\in\Pi$,

$\mathcal{F}(\pi)\cap\mathcal{F}(\pi^{\prime})\neq\emptyset\Rightarrow\mathcal{F(\pi+\pi^{\prime})}=\mathcal{F}(\pi)\cap\mathcal{F}(\pi^{\prime}).$

  * •

A voting rule, $\mathcal{F}$, satisfies cancellation if every candidate wins
when they all have the same number of votes, i.e.,
$\mathcal{F}(\pi)=\mathcal{O}$ when for all $c,c^{\prime}\in\mathcal{C}$,
$s_{\infty}(c,\pi)=s_{\infty}(c^{\prime},\pi)$.

  * •

A voting rule, $\mathcal{F}$, satisfies disjoint equality if
$\mathcal{F}(\pi^{b}+\pi^{b^{\prime}})=\\{\\{c\\}:c\in b\cup b^{\prime}\\}$
when $b\cap b^{\prime}=\emptyset$.

With these axioms, for $k=1$, there are two well-known theorems.

######  Theorem 6.

([1], [7]) Let $\mathcal{F}$ be a voting rule.
$\mathcal{F}=\mathcal{F}_{1,\infty}$ if and only if it satisfies faithfulness,
consistency, and cancellation.

This was proved in [7] but that proof also required an additional assumption
called neutrality.666Neutrality refers to the idea that if the candidates were
renamed, the outcome of the election would just be the outcome original
outcome with the renaming applied. Later, [1] provided a simpler proof without
the assumption of neutrality.

######  Theorem 7.

([4]) Let $\mathcal{F}$ be a voting rule. $\mathcal{F}=\mathcal{F}_{1,\infty}$
if and only if it satisfies faithfulness, consistency, and disjoint equality.

###  3.2 New Axioms

When $k<1$ and/or $p<\infty$, we can see that $k,p$-approval voting does not
satisfy most of the axioms above. If $k>1$ then
$\mathcal{F}_{k,p}\left(\pi^{b}\right)$ consists of sets of size $k\neq 1$ and
so it fails faithfulness. Additionally, even with $k=1$, for $p<\infty$,
$\mathcal{F}_{k,p}\left(\pi^{\\{c_{1}\\}}+\pi^{\\{c_{2},c_{3}\\}}\right)=\\{\\{c_{1}\\}\\}$.
If $n=3$, this violates cancellation since all the candidates got the same
number of votes. For any $n\geq 3$, this violates disjoint equality since

$\mathcal{F}_{k,p}\left(\pi^{\\{c_{1}\\}}+\pi^{\\{c_{2},c_{3}\\}}\right)=\\{\\{c_{1}\\}\\}\neq\\{\\{c_{1}\\}\\}\cup\\{\\{c_{2}\\},\\{c_{3}\\}\\}=\mathcal{F}_{k,p}\left(\pi^{\\{c_{1}\\}}\right)\cup\mathcal{F}_{k,p}\left(\pi^{\\{c_{2},c_{3}\\}}\right).$

As such, we will need to generalize these axioms to characterize
$k,p$-approval voting.

######  Remark.

For the rest of this section, we will expand the definition of profiles so
that $\Pi=\mathbb{R}_{\geq 0}^{\mathcal{I}}$, i.e., a profile, $\pi$, is a
function from ballots to non-negative real numbers, not just non-negative
integers.

In order to generalize to $k>1$, we introduce the winners function which takes
a ballot as input and returns those committees with maximum intersection.

######  Definition 5.

We define the winners function
$\mathcal{W}_{k}\colon\mathcal{I}\to\mathcal{O}$ as

$\mathcal{W}_{k}(b)=\\{K\in\mathcal{O}:|K\cap b|=min(|b|,k)\\}.$

For convenience, we let $\mathcal{W}_{k}(\emptyset)=\mathcal{O}$.

In addition, to deal with $p<\infty$, we need to be able to transform profiles
into functions that return scores, not votes.

######  Definition 6.

For each $b\in\mathcal{I}$, we define the unit score ballot profile as

$\theta_{p}^{b}=|b|^{1/p}\cdot\pi^{b}$

If we take an arbitrary profile, $\pi$, we can write it as

$\pi=\sum_{b\in\mathcal{I}}{\beta_{b}\cdot\pi^{b}}\in
span\left\\{\pi^{b}:b\in\mathcal{I}\right\\}$

where the $\beta_{b}=\pi(b)\geq 0$. Similarly, for the same $\pi$ we can write
it as

$\pi=\sum_{b\in\mathcal{I}}{\gamma_{b}\cdot\theta_{p}^{b}}\in
span\left\\{\theta_{p}^{b}:b\in\mathcal{I}\right\\}$

where $\gamma_{b}=\beta_{b}/|b|^{1/p}\geq 0$. The main difference between
these two ways of writing $\pi$ is that, when written as an element of
$span\left\\{\theta_{p}^{b}:b\in\mathcal{I}\right\\}$ we can calculate a
candidates score as

$s_{p}(c,\pi)=\sum_{b\in\mathcal{I}}{\gamma_{b}\cdot\mathds{1}_{c\in b}}$

as opposed to how the score is calculated in definition 3.

Now that we have these definitions, we can generalize the axioms.

######  Definition 7.

A voting rule, $\mathcal{F}$, satisfies $k$-faithfulness if when there is only
one voter, their vote specifies the winners, i.e., for all $b\in\mathcal{I}$,
$\mathcal{F}(\pi^{b})=\mathcal{W}_{k}(b)$.

######  Definition 8.

A voting rule, $\mathcal{F}$, satisfies $p$-cancellation if every committee
wins when every candidate has the same score, i.e.,
$\mathcal{F}(\pi)=\mathcal{O}$ when for all $c,c^{\prime}\in\mathcal{C}$,
$s_{p}(c,\pi)=s_{p}(c^{\prime},\pi)$.

######  Definition 9.

A voting rule, $\mathcal{F}$, satisfies $p$-disjoint equality if
$\mathcal{F}\left(\theta_{p}^{b}+\theta_{p}^{b^{\prime}}\right)=\mathcal{W}_{k}(b\cup
b^{\prime})$ when $b\cap b^{\prime}=\emptyset$.

######  Definition 10.

A voting rule, $\mathcal{F}$, satisfies positive scaling if for
$\ell\in\mathbb{R}_{>0}$, $\mathcal{F}(\ell\pi)=\mathcal{F}(\pi)$ for all
$\pi\in\Pi$.

We can see that $k$-faithfulness generalizes faithfulness since
$\mathcal{W}_{1}(b)=\\{\\{c\\}:c\in b\\}$. In addition, $p$-cancellation and
$p$-disjoint equality generalize cancellation and disjoint equality
(respectively) since $\theta_{\infty}^{b}=\pi^{b}$. Positive scaling, however,
is an entirely new axiom. If $\mathcal{F}$ satisfies consistency, positive
scaling is unnecessary when $\ell=x/y$ for $x,y\in\mathbb{Z}_{>0}$ because

$\mathcal{F}\left(\frac{x}{y}\pi\right)=\mathcal{F}\left(\sum_{i=1}^{y}{\frac{x}{y}\pi}\right)=\mathcal{F}(x\pi)=\bigcap_{i=1}^{x}{\mathcal{F}(\pi)}=\mathcal{F}(\pi).$

However, for $\ell\in\mathbb{R}_{>0}\setminus\mathbb{Q}_{>0}$, the above
doesn’t work since we can’t write such an $\ell$ as the ratio of two positive
integers. The table below briefly compares the original versus the generalized
axioms.

Original Axiom | Generalized Axiom | Purpose of Generalization
---|---|---
faithfulness |  $k$-faithfulness | handle $k>1$
consistency | N/A | N/A
cancellation |  $p$-cancellation | handle $p<\infty$
disjoint equality |  $p$-disjoint equality | handle $p<\infty$
N/A | positive scaling | handle profiles with irrational images

In the following sub-sections, we turn to generalizing theorems 6 and 7.

###  3.3 Faithfulness, Consistency, Positive Scaling, and Cancellation

Here, we generalize theorem 6 in two ways: for $k>1$ and for all
$p\in[1,\infty]$.

######  Theorem 8.

Let $\mathcal{F}$ be a voting rule. $\mathcal{F}=\mathcal{F}_{k,p}$ if and
only if it satisfies $k$-faithfulness, consistency, $p$-cancellation, and
positive scaling.

First we prove a preliminary lemma.

######  Lemma 9.

Let $\mathcal{F}$ be a voting rule, $\mathcal{\pi}\in\Pi$ a profile,
$b\in\mathcal{I}$ a ballot, $\ell>0$ a positive real number. If $\mathcal{F}$
satisfies consistency, $p$-cancellation, and positive scaling, then

$\mathcal{F}\left(\pi+\ell\cdot\theta_{p}^{b}\right)=\mathcal{F}\left(\pi+\sum_{c\in
b}{\ell\cdot\theta_{p}^{\\{c\\}}}\right).$

###### Proof.

We denote the complement of $b$ as $a=\mathcal{I}\setminus b$. By
$p$-cancellation and positive scaling,

$\mathcal{F}\left(\ell\cdot\theta_{p}^{b}+\ell\cdot\theta_{p}^{a}\right)=\mathcal{O}=\mathcal{F}\left(\sum_{c\in
b}{\ell\cdot\theta_{p}^{\\{c\\}}}+\ell\cdot\theta_{p}^{a}\right),$

since, for each profile, all the candidates have a score of $\ell$. Since
$\mathcal{O}$ is the set of all possible committees of size $k$, we use
consistency to get

$\displaystyle\mathcal{F}\left(\pi+\ell\cdot\theta_{p}^{b}\right)$
$\displaystyle=\mathcal{F}\left(\pi+\ell\cdot\theta_{p}^{b}\right)\cap\mathcal{O}$
$\displaystyle=\mathcal{F}\left(\pi+\ell\cdot\theta_{p}^{b}+\sum_{c\in
b}{\ell\cdot\theta_{p}^{\\{c\\}}}+\ell\cdot\theta_{p}^{a}\right)$
$\displaystyle=\mathcal{F}\left(\pi+\sum_{c\in
b}{\ell\cdot\theta_{p}^{\\{c\\}}}+\ell\cdot\theta_{p}^{b}+\ell\cdot\theta_{p}^{a}\right)$
$\displaystyle=\mathcal{F}\left(\pi+\sum_{c\in
b}{\ell\cdot\theta_{p}^{\\{c\\}}}\right)\cap\mathcal{O}$
$\displaystyle=\mathcal{F}\left(\pi+\sum_{c\in
b}{\ell\cdot\theta_{p}^{\\{c\\}}}\right).$

∎

######  Corollary 10.

The outcome of $\mathcal{F}(\pi)$ only depends on the scores $s_{p}(c,\pi)$
for $c\in\mathcal{C}$.

###### Proof.

Given a profile, $\pi$, we can create a new profile, $\pi^{\prime}$, by
starting with $\pi^{\prime}\equiv 0$ and iteratively applying Lemma 9 to get
$\pi^{\prime}$ such that $\pi^{\prime}(\\{c\\})=s_{p}(c,\pi)$ and
$\pi^{\prime}(b)=0$ for $|b|>1$. In doing this,
$\mathcal{F}(\pi)=\mathcal{F}(\pi^{\prime})$. ∎

We now proceed to prove theorem 8.

###### Proof.

(Theorem 8) ($\Rightarrow$) First, we show that $\mathcal{F}_{k,p}$ satisfies
the axioms. $\mathcal{F}_{k,p}$ satisfies $k$-faithfulness since
$s_{p}(c,\pi^{b})=\mathds{1}_{c\in b}/|b|^{1/p}$ and so
$\mathcal{F}_{k,p}(\pi^{b})=\mathcal{W}_{k}(b)$.

For consistency, by basic properties of $\operatorname*{arg\,max}$, if
$\mathcal{F}_{k,p}(\pi)\cap\mathcal{F}_{k,p}(\pi^{\prime})\neq\emptyset$, we
get that

$\displaystyle\mathcal{F}_{k,p}(\pi+\pi^{\prime})$
$\displaystyle=\operatorname*{arg\,max}_{K\in\mathcal{O}}{\left\\{\sum_{c\in
K}{s_{p}(c,\pi+\pi^{\prime})}\right\\}}$
$\displaystyle=\operatorname*{arg\,max}_{K\in\mathcal{O}}{\left\\{\sum_{c\in
K}{s_{p}(c,\pi)}\right\\}}\cap\operatorname*{arg\,max}_{K\in\mathcal{O}}{\left\\{\sum_{c\in
K}{s_{p}(c,\pi^{\prime})}\right\\}}$
$\displaystyle=\mathcal{F}_{k,p}(\pi)\cap\mathcal{F}_{k,p}(\pi^{\prime}).$

For $p$-cancellation, if $s_{p}(c,\pi)=s_{p}(c^{\prime},\pi)$ for all
$c,c^{\prime}\in\mathcal{C}$, we get that $\sum_{c\in
K}{s_{p}(c,\pi)}=\sum_{c\in K^{\prime}}{s_{p}(c^{\prime},\pi)}$ for all
$K,K^{\prime}\in\mathcal{O}$ meaning that

$\mathcal{F}_{p}(\pi)=\operatorname*{arg\,max}_{K\in\mathcal{O}}{\left\\{\sum_{c\in
K}{s_{p}(c,\pi)}\right\\}}=\mathcal{O}.$

Finally, for positive scaling, let $\ell>0$. So,
$s_{p}(c,\ell\cdot\pi)=\ell\cdot s_{p}(c,\pi)$ and so we get

$\displaystyle\mathcal{F}_{k,p}(\ell\cdot\pi)$
$\displaystyle=\operatorname*{arg\,max}_{K\in\mathcal{O}}{\left\\{\sum_{c\in
K}{s_{p}(c,\ell\cdot\pi)}\right\\}}$
$\displaystyle=\operatorname*{arg\,max}_{K\in\mathcal{O}}{\left\\{\sum_{c\in
K}{\ell\cdot s_{p}(c,\pi)}\right\\}}$
$\displaystyle=\operatorname*{arg\,max}_{K\in\mathcal{O}}{\left\\{\sum_{c\in
K}{s_{p}(c,\pi)}\right\\}}$ $\displaystyle=\mathcal{F}_{k,p}(\pi).$

($\Leftarrow$) Now, assume $\mathcal{F}$ satisfies the axioms. Given a profile
$\pi$, we construct a new profile, $\pi^{*}$. To do this, we let
$t_{1}>t_{2}>\cdots>t_{z}=0$ be the sorted sequence of scores in
$s_{p}(\mathcal{O},\pi)$, i.e., all the scores actually obtained by some
committee. (Note, we include $t_{z}=0$ even if no committee has a score of
zero). We also let $\kappa_{i}=\\{c\in\mathcal{C}:s_{p}(c,\pi)=t_{i}\\}$. So,
we construct $\pi^{*}$ as

$\displaystyle\pi^{*}=$
$\displaystyle(t_{1}-t_{2})\cdot\theta_{p}^{\kappa_{1}}$
$\displaystyle+(t_{2}-t_{3})\cdot(\theta_{p}^{\kappa_{1}}\cup\theta_{p}^{\kappa_{2}})$
$\displaystyle\cdots$
$\displaystyle+(t_{z-1}-t_{z})\cdot\left(\theta_{p}^{\kappa_{1}}\cup\cdots\cup\theta_{p}^{\kappa_{z-1}}\right).$

First, we note that the scores in $\pi$ and $\pi^{*}$ are the same so, by
corollary 10, they have the same outcomes. Also, by $k$-faithfulness and
positive scaling, the outcome of each row is

$\mathcal{F}((t_{y}-t_{y+1})\cdot(\theta_{p}^{\kappa_{1}}\cup\cdots\cup\theta_{p}^{\kappa_{y}}))=\mathcal{W}_{k}(\kappa_{1}\cup\cdots\cup\kappa_{y}).$

Now, we note that for ballots $b_{1},b_{2}\in\mathcal{I}$, if $b_{1}\subset
b_{2}$ and $|b_{2}|\leq k$ then $\mathcal{W}(b_{1})\supset\mathcal{W}(b_{2})$
but if $b_{1}\supset b_{2}$ and $|b_{2}|\geq k$ then
$\mathcal{W}(b_{1})\subset\mathcal{W}(b_{2})$.

So, there are three possibilities. If $|\kappa_{1}|>k$ then, by consistency,
$\mathcal{F}(\pi)=\mathcal{F}(\pi^{*})=\mathcal{W}_{k}(\kappa_{1})=\mathcal{F}_{k,p}(\pi)$
as desired. Similarly, if $|\kappa_{1}\cup\cdots\cup\kappa_{z-1}|<k$, then
$\mathcal{F}(\pi)=\mathcal{F}(\pi^{*})=\mathcal{W}_{k}(\kappa_{1}\cup\cdots\cup\kappa_{z-1})=\mathcal{F}_{k,p}(\pi)$
as desired. Otherwise, there is a $y$ such that
$|\kappa_{1}\cup\cdots\cup\kappa_{y}|\leq k$ and
$|\kappa_{1}\cup\cdots\cup\kappa_{y}\cup\kappa_{y+1}|>k$. We split the above
profile into the two profiles

$L=\sum_{i=1}^{y}{(t_{i}-t_{i+1})\cdot(\theta_{p}^{\kappa_{1}}\cup\cdots\cup\theta_{p}^{\kappa_{i}}))}$

and

$R=\sum_{i=y+1}^{z-1}{(t_{i}-t_{i+1})\cdot(\theta_{p}^{\kappa_{1}}\cup\cdots\cup\theta_{p}^{\kappa_{i}}))}.$

By consistency, we know that

$\mathcal{F}(L)=\mathcal{W}_{k}(\kappa_{1}\cup\cdots\cup\kappa_{y})=\left\\{K\in\mathcal{O}:\kappa_{1}\cup\cdots\cup\kappa_{y}\subset
K\right\\}$

and

$\mathcal{F}(R)=\mathcal{W}_{k}(\kappa_{1}\cup\cdots\cup\kappa_{y}\cup\kappa_{y+1})=\left\\{K\in\mathcal{O}:K\subset\kappa_{1}\cup\cdots\cup\kappa_{y}\cup\kappa_{y+1}\right\\}.$

So, we get that

$\displaystyle\mathcal{F}(L)\cap\mathcal{F}(R)$
$\displaystyle=\left\\{K\in\mathcal{O}:\kappa_{1}\cup\cdots\cup\kappa_{y}\subset
K\subset\kappa_{1}\cup\cdots\cup\kappa_{y}\cup\kappa_{y+1}\right\\}$
$\displaystyle=\\{\kappa_{1}\cup\cdots\cup\kappa_{y}\cup
S:S\subset\kappa_{y+1}:|S|=k-|\kappa_{1}\cup\cdots\cup\kappa_{y}|\\}$
$\displaystyle=\mathcal{F}_{k,p}(\pi),$

where the last line is true because: (a) the intersection is non-empty; and
(b) this is all the committees with the candidates who “clearly” win and then
any selection of the next tier candidates needed to get to $k$ candidates. In
all cases, we get $\mathcal{F}(\pi)=\mathcal{F}_{k,p}(\pi)$ as desired. ∎

This proof mirrors that in [1]. Lemma 9 and Corollary 10 mirror step 1 and
step 2 in [1]. The construction of $\pi^{*}$ and the analysis of
$\mathcal{F}(\pi^{*})$ mirrors step 3 in [1]. The key changes are the
introduction of $\mathcal{W}_{k}$ to allow for committees and $\theta_{p}^{b}$
to allow ballots of unit score instead of unit vote.

###  3.4 Faithfulness, Consistency, Positive Scaling, and Disjoint Equality

Here, we generalize Theorem 7. Unfortunately, unlike with Theorem 8, although
we are able to generalize to $p<\infty$, we are unable to generalize to $k>1$.

######  Theorem 11.

Let $\mathcal{F}$ be a voting rule. $\mathcal{F}=\mathcal{F}_{1,p}$ if and
only if it satisfies faithfulness, consistency, $p$-disjoint equality, and
positive scaling.

###### Proof.

($\Rightarrow$) We showed that $\mathcal{F}_{1,p}$ satisfies faithfulness,
consistency, and positive scaling the proof of theorem 8. It remains to show
that $\mathcal{F}_{1,p}$ satisfies $p$-disjoint equality. Let
$b,b^{\prime}\in\mathcal{I}$ such that $b\cap b^{\prime}=\emptyset$. So, for
$c\in\mathcal{C}$, we get that

$s_{p}(c,\theta_{p}^{b}+\theta_{p}^{b^{\prime}})=\mathds{1}_{c\in
b}+\mathds{1}_{c\in b^{\prime}}.$

Since $b\cap b^{\prime}=\emptyset$ at most one of the indicator functions
above can return a $1$ and so we get

$s_{p}(c,\theta_{p}^{b}+\theta_{p}^{b^{\prime}})=\mathds{1}_{c\in b\cup
b^{\prime}}.$

Thus, we maximize the total score of the members of a committee by selecting
as many candidates as possible from $b\cup b^{\prime}$, i.e.,

$\mathcal{F}_{1,p}(\theta_{p}^{b}+\theta_{p}^{b^{\prime}})=\operatorname*{arg\,max}_{K\in\mathcal{O}}{\left\\{\sum_{c\in
K}{s_{p}(c,\theta_{p}^{b}+\theta_{p}^{b^{\prime}})}\right\\}}=\mathcal{W}_{k}(b\cup
b^{\prime}).$

($\Leftarrow$) Let $\mathcal{F}$ be a voting rule that satisfies the axioms
and $\pi\in\Pi$ an arbitrary profile where

$\pi=\sum_{b\in\mathcal{I}}{\gamma_{b}\cdot\theta_{p}^{b}}$

with $\\{c\\}\in\mathcal{F}(\pi)$ and
$\\{c^{\prime}\\}\in\mathcal{F}_{1,p}(\pi)$. We’ll show that
$\\{c^{\prime}\\}\in\mathcal{F}(\pi)$ and $\\{c\\}\in\mathcal{F}_{1,p}(\pi)$.
To do this, we first let

$\pi[c^{\prime}/c]=\sum_{b\in\mathcal{I},c\notin b,c^{\prime}\in
b}{\gamma_{b}},\,\,\,\,\,\,\,\,{}\,\,\,\,\,\,\,\,{}\pi[c/c^{\prime}]=\sum_{b\in\mathcal{I},c^{\prime}\notin
b,c\in b}{\gamma_{b}},\text{
and}\,\,\,\,\,\,\,\,{}\,\,\,\,\,\,\,\,{}\pi[\cdot/cc^{\prime}]=\sum_{b\in\mathcal{I},c^{\prime},c\notin
b}{\gamma_{b}}.$

We create a new score profile, $\pi^{\prime}$ via

$\pi^{\prime}=\pi[c^{\prime}/c]\cdot\theta_{p}^{\\{c\\}}+\pi[c/c^{\prime}]\cdot\theta_{p}^{\\{c^{\prime}\\}}+\pi[\cdot/cc^{\prime}]\cdot\theta_{p}^{\\{c,c^{\prime}\\}}.$

So we can write $\pi+\pi^{\prime}$ in two different ways. In the first way:

$\displaystyle\pi+\pi^{\prime}=$
$\displaystyle\sum_{\begin{subarray}{c}b\in\mathcal{I}\\\ c\notin
b,c^{\prime}\in
b\end{subarray}}{\pi(b)(\theta_{p}^{b}+\theta_{p}^{\\{c\\}})}+\sum_{\begin{subarray}{c}b\in\mathcal{I}\\\
c^{\prime}\notin b,c\in
b\end{subarray}}{\pi(b)(\theta_{p}^{b}+\theta_{p}^{\\{c^{\prime}\\}})}+$
$\displaystyle\sum_{\begin{subarray}{c}b\in\mathcal{I}\\\ c,c^{\prime}\notin
b\end{subarray}}{\pi(b)(\theta_{p}^{b}+\theta_{p}^{\\{c,c^{\prime}\\}})}+\sum_{\begin{subarray}{c}b\in\mathcal{I}\\\
c,c^{\prime}\in b\end{subarray}}{\pi(b)(\theta_{p}^{b})}.$

We can see that $\\{c\\},\\{c^{\prime}\\}\in\mathcal{F}(\pi+\pi^{\prime})$ by
seeing they are returned by $\mathcal{F}$ for each term via:

  * •

for each term in the first three summations, $\\{c\\}$ and $\\{c^{\prime}\\}$
are returned due to positive scaling, $p$-disjoint equality, and faithfulness;
and

  * •

for each term in the last summation, $\\{c\\}$ and $\\{c^{\prime}\\}$ are
returned due to positive scaling, cancellation, and faithfulness.

Alternatively, we can write $\pi+\pi^{\prime}$ as:

$\pi+\pi^{\prime}=\pi+\pi[c/c^{\prime}]\left(\theta_{p}^{\\{c\\}}+\theta_{p}^{\\{c^{\prime}\\}}\right)+\left(\pi[c^{\prime}/c]-\pi[c/c^{\prime}]\right)\left(\theta_{p}^{\\{c\\}}\right)+\pi[\cdot/cc^{\prime}]\left(\theta_{p}^{\\{c,c^{\prime}\\}}\right).$

Note that since $\\{c^{\prime}\\}\in\mathcal{F}_{1,p}(\pi)$,
$\pi[c^{\prime}/c]\geq\pi[c/c^{\prime}]$. First, we show that $\\{c\\}$ is
returned for each term by:

  * •

$\\{c\\}\in\mathcal{F}(\pi)$ by assumption;

  * •

$\\{c\\}\in\mathcal{F}\left(\pi[c/c^{\prime}]\left(\theta_{p}^{\\{c\\}}+\theta_{p}^{\\{c^{\prime}\\}}\right)\right)$
by disjoint equality;

  * •

$\\{c\\}\in\mathcal{F}\left(\left(\pi[c^{\prime}/c]-\pi[c/c^{\prime}]\right)\left(\theta_{p}^{\\{c\\}}\right)\right)$
by faithfulness; and

  * •

$\\{c\\}\in\mathcal{F}\left(\pi[\cdot/cc^{\prime}]\left(\theta_{p}^{\\{c,c^{\prime}\\}}\right)\right)$
by faithfulness.

So, since $\\{c\\}$ is returned by $\mathcal{F}$ for each term, we know that
$\mathcal{F}(\pi+\pi^{\prime})$ is the intersection of what $\mathcal{F}$
returns for each term and that $\\{c\\}\in\mathcal{F}(\pi+\pi^{\prime})$.
Given this, if $\\{c^{\prime}\\}\notin\mathcal{F}(\pi)$ then
$\\{c^{\prime}\\}\notin\mathcal{F}(\pi+\pi^{\prime})$ which contradicts what
we proved above. Conversely, if $\\{c\\}\notin\mathcal{F}_{1,p}(\pi)$, then
$\pi[c^{\prime}/c]-\pi[c/c^{\prime}]>0$ which means that
$c^{\prime}\notin\mathcal{F}(\pi+\pi^{\prime})$ which also contradicts what we
proved above. Thus, $\mathcal{F}(\pi)=\mathcal{F}_{1,p}(\pi)$. ∎

The proof above is almost the same as that in [4] unit score ballots instead
of (regular) ballots.

##  4 Utility Maximization and Maximum Likelihood Estimation Justifications

In this section, we look at justifications of $k,p$-approval voting in two
different, but related, ways. The first is relates to utility maximization,
i.e., investigating when a voting system maximizes the total utility of the
voters. The second relates to likelihood maximization. Here, we assume there
is a “correct” answer (i.e., “correct” committee) and that the votes are
“noisy” signals that can be aggregated to help find the correct committee. A
voting system is a maximum likelihood estimator if it finds the “best answer”
to a given noise model.

###  4.1 Preliminaries

Before we discuss this, we present some definitions specific to this topic.

######  Definition 11.

A utility function is a function
$U:\mathcal{I}\times\mathcal{O}\to\mathbb{R}$. The interpretation is once we
know a voter’s ballot, we know their utility for each possible outcome. A
voting rule, $\mathcal{F}$, is utility maximizing (UM) with respect to $U$ if,
for all $\pi\in\Pi$,

$\mathcal{F}(\pi)=\operatorname*{arg\,max}_{K\in\mathcal{O}}{\left\\{\sum_{b\in\mathcal{I}}{\pi(b)\cdot
U(b,K)}\right\\}}.$

######  Definition 12.

A noise model is a function $P:\mathcal{O}\times\mathcal{I}\to\mathbb{R}_{\geq
0}$ such that for any fixed $K\in\mathcal{O}$,

$\sum_{b\in\mathcal{I}}{P(K,b)}=1.$

That is, when we hold $K\in\mathcal{O}$ constant, $P$ forms a probability
distribution over $\mathcal{I}$. A voting rule is a maximum likelihood
estimator (MLE) with respect to $P$ if, for all $\pi\in\Pi$,

$\mathcal{F}(\pi)=\operatorname*{arg\,max}_{K\in\mathcal{O}}{\left\\{\prod_{b\in\mathcal{I}}{P(K,b)^{\pi(b)}}\right\\}}.$

A voting rule as UM is a simple concept. The idea of a voting rule as an MLE
is explained in detail in [5]. Briefly, the idea is that there is a “correct”
outcome, $K^{*}$, and each voter is trying to “guess” the correct outcome in a
noisy environment. In this context, a voting rule is a method for taking the
noisy guesses and trying to determine the correct outcome.

######  Example 12.

We look back at plurality voting as in example 1. If it is the case that each
voter gets a utility of $1$ if their chosen candidate wins and $0$ otherwise,
then plurality voting returns the candidate that maximizes the total utility.

Similarly, assume if the “correct” candidate is some $c^{*}$ then each voter
has some $p_{1}$ chance of voting for that candidate and $p_{2}<p_{1}$ chance
of voting for any other candidate. In this case, plurality voting returns the
maximum likelihood estimator for the correct candidate.

We can link the two concepts via the following theorems.

######  Theorem 13.

([12]) A voting rule is UM for some $U$ if and only if it is MLE for some $P$.

The proof is in [12] but the key transformation between $U$ and $P$ is

$U(b,K)\sim\ln{P(K,b)}.$

This allows us to go from maximizing a sum to maximizing a product and vice
versa.

###  4.2 k,p-Approval Voting as UM and MLE

We now look at the specific forms of $U$ and $P$ for which $k,p$-approval
voting is UM and MLE.

######  Theorem 14.

$\mathcal{F}_{k,p}$ is UM for

$U(b,K)=U_{0}+\frac{\alpha|b\cap K|+\beta|b\setminus K|}{|b|^{1/p}}$

with $U_{0}\in\mathbb{R}$ and $\alpha>\beta$.

###### Proof.

Let $m=\sum_{b\in\mathcal{I}}{\pi(b)}$. For any $K\in\mathcal{O}$,

$\displaystyle\sum_{b\in\mathcal{I}}{\pi(b)\cdot U(b,K)}$
$\displaystyle=\sum_{b\in\mathcal{I}}{U_{0}+\frac{\alpha|b\cap
K|+\beta|b\setminus K|}{|b|^{1/p}}}$ $\displaystyle=mU_{0}+\alpha\sum_{c\in
K}{\sum_{b\in\mathcal{I}}{\frac{\pi(b)}{|b|^{1/p}}\cdot\mathds{1}_{c\in
b}}}+\beta\sum_{c\notin
K}{\sum_{b\in\mathcal{I}}{\frac{\pi(b)}{|b|^{1/p}}\cdot\mathds{1}_{c\in b}}}$
$\displaystyle=mU_{0}+\alpha\sum_{c\in K}{s(c)}+\beta\sum_{c\notin K}{s(c)}.$

Since $\alpha>\beta$ this is maximized by having $K$ be the candidates with
the highest scores. ∎

For satisfaction approval voting, or $k,1$-approval voting in general, when
$\beta=0$, this utility function is used in [3] to justify satisfaction
approval voting.

One might wonder if we can generalize this further by adding terms for the
candidates we did not consider, i.e., $|K\setminus b|$ and
$|\mathcal{C}\setminus(K\cup b)|$. To try this, for some fixed ballot $b$, we
let $w=|b\cap K|$ and $\ell=|b\setminus K|$. Writing out a (seemingly) more
general form gets us

$\displaystyle U(b,K)$ $\displaystyle=U_{0}+\frac{\alpha
w+\beta\ell+\gamma(k-w)+\delta(m-w-\ell-(k-w))}{|b|^{1/p}}$
$\displaystyle=U_{0}+\frac{\gamma
k+\delta(m-k)}{|b|^{1/p}}+\frac{(\alpha-\gamma)w+(\beta-\delta)\ell)}{|b|^{1/p}}.$

Since $k$ and $m$ are constants, this is of the same form as the equation of
above with

$U^{\prime}_{0}=U_{0}+\frac{\gamma
k+\delta(m-k)}{|b|^{1/p}},\,\,\,\,\,\,\,\,{}\,\,\,\,\,\,\,\,{}\alpha^{\prime}=\alpha-\gamma,\text{
and }\,\,\,\,\,\,\,\,{}\,\,\,\,\,\,\,\,{}\beta^{\prime}=\beta-\delta.$

As such, nothing is gained by adding in these additional terms. That is to
say, we do not gain any additional mathematical expressive power. However,
adding these terms could be useful when trying to determine utility functions
in a practical context.

Theorems 13 and 14 also gives us the following corollary.

######  Corollary 15.

$\mathcal{F}_{k,p}$ is MLE for

$P(K,b)\sim\alpha^{|b\cap K|/|b|^{1/p}}\cdot\beta^{|b\setminus K|/|b|^{1/p}}$

when $\alpha>\beta$.

For $p=\infty$ the formula in corollary 15 has an interesting interpretation.
$P(K,b)$ reduces to

$\mathbb{P}(K,b)\sim\alpha^{|b\cap K|}\cdot\beta^{|b\setminus K|}.$

Given a “correct” committee $K$ we posit a voter who looks at the candidates
one at a time and either approves of each of them or not independently. If she
approves of a candidate $c\in K$ with probability $p_{1}$ and a candidate
$c\in K^{c}=\mathcal{C}\setminus K$ with probability $p_{2}$ then we get

$\displaystyle P(K,b)$ $\displaystyle=\left(\prod_{c\in K\cap
b}{p_{1}}\right)\cdot\left(\prod_{c\in K\setminus
b}{1-p_{1}}\right)\cdot\left(\prod_{c\in K^{c}\cap
b}{p_{2}}\right)\cdot\left(\prod_{c\in K^{c}\setminus b}{1-p_{2}}\right)$
$\displaystyle=p_{1}^{|K\cap b|}\cdot(1-p_{1})^{|K\setminus b|}\cdot
p_{2}^{|K^{c}\cap b|}\cdot(1-p_{2})^{|K^{c}\setminus b|}$
$\displaystyle=p_{1}^{w}\cdot(1-p_{1})^{(k-w)}\cdot
p_{2}^{\ell}\cdot(1-p_{2})^{(n-k-\ell)}$
$\displaystyle=(1-p_{1})^{k}\cdot\left(\frac{p_{1}}{1-p_{1}}\right)^{w}\cdot(1-p_{2})^{(n-k)}\cdot\left(\frac{p_{2}}{1-p_{2}}\right)^{\ell}.$

Because $(1-p_{1})^{k}$ and $(1-p_{2})^{(n-k)}$ are constants we get

$P(K,b)\sim\left(\frac{p_{1}}{1-p_{1}}\right)^{w}\left(\frac{p_{2}}{1-p_{2}}\right)^{\ell}.$

When $p_{1}>p_{2}$, this is in the same form as the expression in corollary
15. Thus, at least for approval voting, this has a reasonable interpretation
in terms of a voter selecting candidates one-by-one. For other norms, however,
no such interpretation seems readily available.

##  5 Distance Rationalizability Justifications

Finally, we turn to distance rationalizability. Distance rationalizability is
a concept used to justify various voting rules including approval voting [6].
In general, a voting rule is distance rationalizable if there is a class of
“consensus elections” (where the voters all agree on the correct outcome) and
the voting rule is equivalent to finding the nearest consensus election and
returning the consensus of that election.

###  5.1 Preliminaries

Until now, we’ve considered profiles which can be thought of as counting the
number of voters who submitted each ballot. However, an election can also be
thought of as a “list” showing which ballot each voter submitted. Based on
this, we define an election as follows.

######  Definition 13.

An election is a finite ordered sequence of ballots, i.e., something of the
form

$E=\langle v_{1},...,v_{m}\rangle$

where each $v_{i}\in\mathcal{I}$.

######  Remark.

Throughout this section, we will assume all elections have some fixed number,
$m$, of votes.

Note that every election induces a profile such that
$\pi(b)=|\\{i\in\\{1,...,m\\}:v_{i}=b\\}|$.777This also means that the
considerations in the previous section could be written in terms of elections.
If one wished to do so, one would sum over the votes in the election instead
of the ballots in $\mathcal{I}$. As such, it makes sense to apply
$\mathcal{F}$ to elections. Thus, we will, with reckless abandon, abuse
notation and write $\mathcal{F}(E)$. Before jumping into the details, we give
an example.

######  Example 16.

We return to plurality voting in example 1. We let $\mathcal{S}$ be the set of
“unanimous” elections, i.e., the $n$ elections, $E_{c}$, where each voter
votes for candidate $c$. Given two elections, $E=\langle
v_{1},...,v_{m}\rangle$ and $E^{\prime}=\langle
v^{\prime}_{1},...,v^{\prime}_{m}\rangle$, let the distance between them be

$d(E,E^{\prime})=|\\{i:v_{i}\neq v^{\prime}_{i}\\}|.$

With this metric, we can see that for a given $c\in\mathcal{C}$,

$d(E,E_{c})=|\\{i:v_{i}\neq c\\}|=m-s(c).$

So the closet unanimous elections are the $\\{E_{c}:c\in\mathcal{F}(E)\\}$. In
other words, $\mathcal{F}$ returns exactly those candidates whose unanimous
election is closest. As we will see with the definitions below, this makes
plurality voting distance rationalizable.

We now lay out some definition from [6] that we will need for our discussion.

######  Definition 14.

For $K\in\mathcal{O}$ we define

$\mathcal{U}_{K}=\left\\{E:K\subset\bigcap_{i=1}^{m}{v_{i}}\right\\}$

and

$\mathcal{U}=\bigcup_{K\in\mathcal{O}}{\mathcal{U}_{K}}.$

The set $\mathcal{U}$ is the set of unanimous elections.

There are other consensus classes in the literature with respect to approval
ballots ([6]) as well as relevant consensus classes and for other types of
ballots ([6], [11]). Here, however, we focus on $\mathcal{U}$ as defined in
definition 14.

######  Definition 15.

Given a metric, $d$, on the space of elections, the $(\mathcal{U},d)$-score of
a committee $K\in\mathcal{O}$ is the distance from $E$ to the nearest
unanimous election where $K$ is a winner, i.e.,

$d_{\mathcal{U}}(K;E)=\min_{E^{\prime}\in\mathcal{U}_{K}}{d(E,E^{\prime})}.$

The set of $(\mathcal{U},d)$-winners are those committees with minimum scores.

There are only a finite number of elections with $m$ votes so the minimum
always exists.

######  Definition 16.

Let $d$ be a metric between elections. A voting rule, $\mathcal{F}$, is
$(\mathcal{U},d)$-distance rationalizable if for all elections, the winners
$\mathcal{F}$ returns is exactly the set of $(\mathcal{U},d)$-winners.

At least in cases where a voting rule always returns the unanimous decision
for unanimous elections, i.e., where
$\mathcal{F}\left(\mathcal{U}_{K}\right)=K$, one could use the Jaccard
distance ([10]) on the outcomes, i.e.,

$d(E,E^{\prime})=1-\frac{|\mathcal{F}(E)\cap\mathcal{F}(E^{\prime})|}{|\mathcal{F}(E)\cup\mathcal{F}(E^{\prime})|}.$

Under this metric, almost any voting rule can be made distance rationalizable
with respect to the unanimity class. However, this somewhat trivializes the
concept of distance rationalizability since it directly measures how different
the outcomes, not the elections, are. More interesting is when a we have a
metric, $d$, on $\mathcal{I}$ and we use this to induce a metric, $\hat{d}$,
on elections via

$\hat{d}(E,E^{\prime})=\sum_{i=1}^{m}{d(v_{i},v^{\prime}_{i})}.$

######  Definition 17.

A voting rule is vote distance rationalizable if there is a metric, $d$, on
$\mathcal{I}$ such that the rule is distance rationalizable with respect to
the induced metric $\hat{d}$.

If such a metric can be found, one can look at how “reasonable” or “natural”
it is as a way of justifying a voting rule. In the remainder of this section,
we look into the existence of such metrics.

###  5.2 Distance Rationalizability of Approval Voting

It was shown in [6] that, with $k=1$, approval voting (i.e.
$1,\infty$-approval voting) is vote distance rationalizable. Here, we
generalize that to $k,\infty$-approval voting for any $k\geq 1$.

######  Theorem 17.

For any $k\geq 1$, $k,\infty$-approval voting is vote distance rationalizable
with respect to $\mathcal{U}$ with the metric $d(v,v^{\prime})=|v\triangle
v^{\prime}|$ where $\triangle$ represents the symmetric difference between two
sets.

###### Proof.

For $E^{\prime}=\langle
v^{\prime}_{1},...,v^{\prime}_{m}\rangle\in\mathcal{U}_{K}$,

$\displaystyle\hat{d}(E,E^{\prime})$
$\displaystyle=\sum_{i=1}^{m}{d(v_{i},v^{\prime}_{i})}$
$\displaystyle=\sum_{i=1}^{m}{|v_{i}\triangle v^{\prime}_{i}|}$
$\displaystyle=\left(\sum_{i=1}^{m}{\left|\left(v_{i}\cap
K\right)\triangle\left(v^{\prime}_{i}\cap
K\right)\right|}\right)+\left(\sum_{i=1}^{m}{\left|\left(v_{i}\cap
K^{C}\right)\triangle\left(v^{\prime}_{i}\cap K^{C}\right)\right|}\right)$
$\displaystyle=\left(\sum_{c\in K}{\sum_{i=1}^{m}{\left(1-\mathds{1}_{c\in
v_{i}}\right)}}\right)+\left(\sum_{i=1}^{m}{\left|\left(v_{i}\cap
K^{C}\right)\triangle\left(v^{\prime}_{i}\cap K^{C}\right)\right|}\right)$
$\displaystyle=\left(\sum_{c\in
K}{m-s_{\infty}(c)}\right)+\left(\sum_{i=1}^{m}{\left|\left(v_{i}\cap
K^{C}\right)\triangle\left(v^{\prime}_{i}\cap K^{C}\right)\right|}\right)$
$\displaystyle=mk-\sum_{c\in
K}{s_{\infty}(c)}+\left(\sum_{i=1}^{m}{\left|\left(v_{i}\cap
K^{C}\right)\triangle\left(v^{\prime}_{i}\cap K^{C}\right)\right|}\right).$

For fixed $K$, this is minimized by picking the consensus election
$E^{\prime}=\langle
v^{\prime}_{1},...,v^{\prime}_{m}\rangle\in\mathcal{U}_{K}$ such that for all
$i\in\\{1,...,m\\}$, $v^{\prime}_{i}=K\cup(v_{i}\setminus K)$. Doing this, we
get

$\hat{d}(E,E^{\prime})=mk-\sum_{c\in K}{s_{\infty}(c)}.$

This is minimized by picking $K$ with the highest scores, i.e.,
$K\in\mathcal{F}_{k,\infty}(E)$ as desired. ∎

###  5.3 Distance Rationalizability When k = 1

We now turn to general $1,p$-approval voting for all $p\in[1,\infty]$. To do
this, we construct a graph, $G(\mathcal{I},Z)$, where the nodes are the
ballots and the edges are

$Z=\\{\\{b,b^{\prime}\\}\in\mathcal{I}^{2}:|b\triangle b^{\prime}|=1\\}.$

We also have a cost function on the edges,

$c(\\{b,b\cup\\{c\\}\\})=\frac{1}{|b|^{1/p}}.$

With this, the distance, $d$, between two ballots is the sum of the costs on
the minimum cost path between the two ballots. We can think of the graph as
$n$ layers, each corresponding to votes of the same cardinality and layers
corresponding to larger cardinalities as “above” those with lower
cardinalities. We can see an example with $n=4$ in Figure 1.

Figure 1: a “layer” graph with $n=4$. Following the path indicated by the two
red dashed lines, we can see
$d(\\{c_{1}\\},\\{c_{1},c_{2},c_{3}\\})=1+2^{-1/p}$.

######  Theorem 18.

For $p\in[1,\infty]$, any $1,p$-approval voting rule is vote distance
rationalizable with respect to $\mathcal{U}$ and $d$ as defined above.

###### Proof.

Fix an election $E$ and pick a candidate $c$. For each $v\in E$,
$v^{\prime}=v\cup\\{c\\}$ is the closest ballot containing $c$. If $c\in v$
then $v^{\prime}=v$ so this is obvious. Otherwise, traversing from $v$ to
$v^{\prime}$ on $G$ requires just traversing up just one layer with a distance
of $1/|v|^{1/p}$. For any other $v^{\prime\prime}\ni c$, traversing from $v$
to $v^{\prime\prime}$ will require either traversing either down a layer (at a
cost greater than $1/|v|^{1/p}$ since $z^{p}$ is increasing in $z$) or going
up more than one layer (which includes a cost of $1/|v|^{1/p}$ and then more).

So, to find the nearest election $E^{\prime}\in{\mathcal{U}}_{\\{c\\}}$, we
need only add $c$ to each vote in $E$. The distance is

$d(E,E^{\prime})=\sum_{i=1}^{m}{\frac{1}{|v_{i}|^{1/p}}\mathds{1}_{c\notin
v_{i}}}=\sum_{i=1}^{m}{\frac{1}{|v_{i}|^{1/p}}-\frac{1}{|v_{i}|^{1/p}}\mathds{1}_{c\in
v_{i}}}=\sum_{i=1}^{m}{\frac{1}{|v_{i}|^{1/p}}}-s(c).$

This is minimized by taking $c$ to be the candidate (or one of the candidates)
with the highest score. ∎

Of course, one can question if the metric above is “natural” enough. As a
rough justification, one can see this as a “cost” to adding a candidate to a
vote. Given a vote, the cost of adding a candidate decreases as the number of
candidates already in the vote increases. This makes sense if we consider that
adding a candidate to a “big” vote is not changing it as much as adding a
candidate to a “small” vote.

###  5.4 Distance Rationalizability for General k and p

We now turn to the question of distance rationalizability for $k,p$-approval
voting for $k>1$ and $p<\infty$. However, as we will show, for $p<\infty$ and
$k>1$, $k,p$-approval voting is not vote distance rationalizable. Before we
can do this, however, we need a lemma.

######  Lemma 19.

Let $d$ be a metric on ballots, $E=\langle v_{1},...,v_{m}\rangle$ an
election, and $U=\langle u_{1},...,u_{n}\rangle$ one of the nearest unanimous
elections (with respect to $\hat{d}$). If we let $\tilde{E}$ be $E$ with one
vote changed to match $U$ i.e. for some $\ell\in\\{1,...,m\\}$,
$\tilde{E}=\\{v_{1},...,v_{\ell-1},u_{l},v_{\ell+1},...,v_{m}\\}$ then $U$ is
still among the closest unanimous elections to $\tilde{E}$.

###### Proof.

Let $U^{\prime}=\langle u^{\prime}_{1},...,u^{\prime}_{m}\rangle$ be another
unanimous election. We show that
$\hat{d}(\tilde{E},U)\leq\hat{d}(\tilde{E},U^{\prime})$. We note that

$\hat{d}(\tilde{E},U)=\sum_{i=1}^{\ell-1}{d(v_{1},u_{i})}+\sum_{i=\ell+1}^{m}{d(v_{i},u_{i})}=\hat{d}(E,U)-d(v_{\ell},u_{\ell}),$

and

$\hat{d}(\tilde{E},U^{\prime})=\sum_{i=1}^{\ell-1}{d(v_{1},u^{\prime}_{i})}+d(u_{\ell},u^{\prime}_{\ell})+\sum_{i=\ell+1}^{m}{d(v_{i},u_{i})}=\hat{d}(E,U)-d(v_{\ell},u^{\prime}_{\ell})+d(u_{\ell},u^{\prime}_{\ell})$

Since $d$ is a metric it satisfies the triangle inequality we get

$d(u_{\ell},u^{\prime}_{\ell})+d(v_{\ell},u_{\ell})\geq
d(v_{\ell},u^{\prime}_{\ell})\Rightarrow-d(v_{\ell},u_{\ell})\leq-d(v_{\ell},u^{\prime}_{\ell})+d(u_{\ell},u^{\prime}_{\ell})$

This means that

$\hat{d}(\tilde{E},U)=\hat{d}(E,U)-d(v_{\ell},u_{\ell})\leq\hat{d}(E,U)-d(v_{\ell},u^{\prime}_{\ell})+d(u_{\ell},u^{\prime}_{\ell})=\hat{d}(\tilde{E},U^{\prime}).$

∎

Now, we can prove our negative result.

######  Theorem 20.

For $p\neq\infty$ and $k>1$, $k,p$-approval voting is not vote distance
rationalizable.

###### Proof.

We assume otherwise for the sake of a contradiction. Since there are at least
$k+1\geq 3$ candidates, we name them

$\mathcal{C}=\\{a_{1},a_{2},a_{3},b_{1},...,b_{k-2},c_{1},...,c_{n-k-1}\\}.$

We setup an election, $E$, with the following votes:

  * •

$m_{1}$ votes for $\\{a_{1},a_{3},b_{1},...,b_{k-2}\\}$;

  * •

$m_{2}$ votes for $\\{a_{1},b_{1},...,b_{k-2}\\}$; and

  * •

$m_{3}$ votes for $\\{a_{2},b_{1},...,b_{k-2}\\}$

where we set $m_{1},m_{2},m_{3}\in\mathbb{Z}_{\geq 0}$ such that $m_{2}$ is
big enough that $a_{1}$ beats both $a_{2}$ and $a_{3}$ and

$\frac{m_{3}}{k^{1/p}}<\frac{m_{1}}{k^{1/p}}<\frac{m_{3}}{(k-1)^{1/p}}.$

This is always possible since $z^{p}$ is strictly increasing in $z$. This
means that in $E$, $a_{1}$ beats $a_{2}$ and $a_{2}$ beats $a_{3}$ and so
$\mathcal{F}_{k,p}(E)=\\{\\{a_{1},a_{2},b_{1},...,b_{k-2}\\}\\}$. Letting $U$
be the nearest election in $\mathcal{U}$ we know that its consensus is
$\mathcal{F}_{k,p}(E)$.

Let $\tilde{E}$ be the election resulting from changing the last $m_{3}$ votes
to match $U$. This means that these votes each have at least $k$ candidates
which drops $a_{2}$’s score below $a_{3}$’s in $\tilde{E}$. By Lemma 19, the
closest election should still be $U$. But, in $\tilde{E}$, $a_{3}$ beats
$a_{2}$ so $U$ can’t be the closest consensus election. This gives a
contradiction. ∎

More succinctly, because adding a candidate to a vote decreases the score of
all the candidates in that vote, this can change the outcomes for all the
other candidates.

##  6 Conclusion

In this paper, we have looked at generalizations of approval voting where an
approval ballot is viewed as analogous to a $0,1$-vector normalized by
different $p$-norms. In many cases, any norm could, in fact, be used in place
of a $p$-norm.

The idea that one cannot just add a candidate to an approval “for free” is a
natural one - a person who approves of one candidate can be considered to want
that candidate more than a person who votes for ten candidates wants any of
those candidates. Alternatively, in a machine learning context, an agent that
reports one possibility might be considered to have more confidence in that
possibility that an agent that returns ten possibilities. The degree to which
this desire, or confidence, decreases can be calibrated by picking the correct
$p$-norm (or other norm). As such, it is helpful to see that there are several
justifications for most of these methods.

The main exception to this in the paper is with respect to distance
rationalizability for $k,p$-approval voting for $k>1$ and $p<\infty$. The
degree to which this is a problem depends on the context where a voting method
is being used.

Looking into $k,p$-approval voting is a new area. Concurrent work in this area
is looking into, among other things, the degree to which the outcome can
change as $p$ changes, the Condorcet efficiency of $k,p$-approval voting for
different $p$, the montonicity of $k,p$-approval voting as defined here as
well as in an instant runoff context, variations where the vote is not a
$0,1$-vector but rather an $a,b$-vector for some $0<a<b$, etc.

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