# Properties of Aggregation Operators Relevant for Ethical Decision Making in
Artificial Intelligence222This work began during Fioravanti’s research stay at
the MPI for Human Development. Thanks are due to the funding of the German
Academic Exchange Service (DAAD).

Federico<EMAIL_ADDRESS>(corresponding author)
Instituto de Matemática de Bahía Blanca, CONICET - UNS, Bahía Blanca,
Argentina  Departamento de Matemática, Universidad Nacional del Sur, Bahía
Blanca, Argentina  Iyad Rahwan  Center for Humans and Machines, Max Planck
Institute for Human Development, Berlin, Germany  Fernando Tohmé  Instituto de
Matemática de Bahía Blanca, CONICET - UNS, Bahía Blanca, Argentina
Departamento de Economía, Universidad Nacional del Sur, Bahía Blanca,
Argentina

###### Abstract

We present an axiomatic study of aggregation operators that could be applied
to ethical AI decision making. The information is given here by different
preferences over the decisions to be made by automated systems. We consider
two different but very intuitive notions of preference of an alternative over
another one, namely pairwise majority and position dominance. Preferences are
represented by permutation processes over alternatives and aggregation rules
are applied to obtain results that are socially considered to be ethically
correct. We address the problem of the stability of the aggregation process,
which is important when the information is variable. In this setting we find
many aggregation rules that satisfy desirable properties for an autonomous
system.
Keywords: Aggregation Operators; Permutation Process; Decision Analysis.

##  1 Introduction

Many relevant scenarios arising from the use of Artificial Intelligent (AI)
systems require making ethical decisions. So for instance, consider the case
of an autonomous vehicle suffering a brake failure in the midst of heavy urban
traffic. Should it swerve its course and hit a wall, killing its passengers?
Or stay its way and kill two pedestrians who are crossing the street? Of
course, a definite answer depends on the availability of several pieces of
information that are missing in this story, such as the age and gender of both
the passengers and pedestrians. But even with access to the complete
information about the situation, it is difficult from an ethical point of view
to make a decision. This kind of ethical decision-making problems, have
constituted, for a long time, a great challenge for Artificial Intelligence
[40]. Even when ground-truth ethical principles were available, the lack of
corresponding formal specifications make it difficult to solve these problems.

Some authors suggest that “an approximation as agreed upon society” must be
applied when ethical principles are not available [13]. Conitzer et al. [11]
discuss a game theoretic and a machine learning approach to the development of
a general framework, instead of appealing to the use of ad-hoc rules at each
scenario. Furthermore, it would also be convenient to automatize the decision-
making process, aggregating the diverse opinions held in the society on
ethical dilemmas [21, 11]. In this sense, Rossi et al. [36] discuss related
problems where preferences (not necessarily on moral issues) are aggregated by
morally-ranking all the alternatives.

Noothigattu et al. [30] propose and implement a concrete approach to ethical
decision-making based on tools drawn from Computational Social Choice [5, 6].
It follows four steps:

  * i.

Data Collection: human voters are asked to compare pairs of alternatives,

  * ii.

Learning: a model of each individual preference is generated,

  * iii.

Summarization: a single model incorporating those preferences is created, and

  * iv.

Aggregation: a voting rule aggregates the individual preferences into a
collective decision.

Notice that these steps involve the definition of an aggregation operator,
i.e. a mathematical procedure to combine information, in this case about the
preferences over decisions to be made over alternative decisions on ethical
settings. We seek to define aggregation processes yielding solutions as close
to the ground-truth ethical principles as possible. We intend to provide a
sound foundation for the automatizing of ethical decision-making in situations
like that of the case of the failure of the braking system of an autonomous
car [32], or the automatic assignation of priorities in kidney exchange
programs [17] or even in the assistance of juridical deliberation [1, 3].

This paper presents, essentially, a ‘technical’ contribution to the discipline
of Machine Ethics, the field concerned with the ‘ethical behavior’ of
autonomous intelligent systems operating in social environments [41, 12, 27].
Following Bjorgen et al. [4], who propose estimating the ethical performance
of an autonomous system on the basis of the responses to some ethical
dilemmas. The results presented in this work may help to set up these
‘benchmarks’.

To address our main research question, we need to establish a model of
preferences for each voter. There exists a vast literature that uses random
utility models for the elicitation of preferences [22, 31, 2]. That is, an
agent’s preferences are modeled by drawing, for each alternative, a real-value
from a parameterized distribution, yielding a ranking of the alternatives
according to their scores. In our setting, since an autonomous system may face
different scenarios, each of which with a finite set of alternatives that may
be obtained from an infinite set, we need a more general notion [9]. That is
why we consider general models known as permutation processes [30].

Once we have a model for each voter, as well as a summary of those models
(Step iii), we need a voting rule333More formally known as Social Choice
correspondence. to select the best alternatives. We consider two different
conceptions of what it means that an alternative is the ‘best’ one, adapting
to this setting the ideas of domination among alternatives of Caragiannis et
al. [8]. We present some properties that a good voting rule should verify in
terms of these dominance notions. The results obtained can be considered more
or less positive for each dominance notion, depending on how many voting rules
satisfy some desirable properties. We consider the different domains that may
arise depending on the definition of the models of the individuals.

This work intends to contribute to the theory of aggregation of permutation
processes [34, 30, 24]. In particular, we focus on the very highly relevant
property of stability, which ensures the consistency of decisions across
different scenarios of variable information. This property matters for the
characterization of information aggregation operators, since they are
susceptible to information losses [7, 19].

The plan of the paper is as follows. In Section 2, we introduce a general
model of the aggregation of permutation processes. Sections 3 and 4 introduce
different dominance definitions and important properties that voting rules
should verify. In Section 5 we consider the setting in which the aggregation
of permutations yields positive results. In Section 6 we introduce the concept
of stability and find some rules satisfying it. Finally, Section 7 concludes
with a brief discussion about some future lines of research.

##  2 Model

Following the line of [34, 30] we consider a potentially infinite set of
alternatives $X$. Let $A\subseteq X$ be a finite subset of size $n$. A total
ordering over $A$ can be characterized by a bijection
$\sigma:A\rightarrow\\{1,\ldots,n\\}$ such that $\sigma(a)=j$ indicates that
the position of alternative $a$ in $\sigma$ is $j$. By a slight abuse of
language we call $\sigma$ a permutation of $A$.444We will also refer to
permutations over sets of alternatives as votes or rankings.. Let $S_{A}$ be
the set of all permutations of $A$. For each $\sigma\in S_{A}$,
$\sigma(a)<\sigma(b)$ is interpreted as indicating that $a$ is more preferred
than $b$ in the order characterized by $\sigma$. We denote this as
$a>_{\sigma}b$.

With $\sigma|_{B}$ we denote the restriction of $\sigma$ on $B\subseteq A$.
Given a probability distribution $P$ over $S_{A}$ and $B\subseteq A$, we have
that

$P_{B}(\sigma^{\prime})=\sum_{\sigma\in
S_{A}:\sigma|_{B}=\sigma^{\prime}}P(\sigma)$

where $\sigma^{\prime}\in S_{B}$.

A permutation process $\Pi$ is a collection of probability distributions over
sets of permutations $S_{B}$ such that $B\subseteq A$, with $\Pi(A)$ being a
distribution over $S_{A}$. A permutation process is consistent if
$\Pi(A)|_{B}=\Pi(B)$ for any finite subset $B\subseteq A$, that is, if the
distribution over $S_{B}$ is obtained by marginalizing the distribution over
$S_{A}$ on the extra alternatives in $A\setminus B$.

Consistent permutation processes can be naturally represented in terms of
utilities [24]. That is, given a stochastic process $U$, indexed by $X$, such
that for any $A=\\{x_{1},x_{2},\ldots,x_{n}\\}\subseteq X$, we can define the
probability of $\sigma\in S_{A}$ as the probability that
$sort(U_{x_{1}},U_{x_{2}},\ldots,U_{x_{n}})$ coincides with that of
$>_{\sigma}$, where $sort$ is an operation that yields a linear ordering of
$\\{U_{x_{j}}\\}_{j=1}^{n}$. Thus, $U:A\rightarrow\mathbb{R}$ can be conceived
as a stochastic utility function.

We follow the line of Parkes et al. [31] and focus our attention on
permutation processes in which the random utilities are independently drawn
from distributions in the exponential family, like the Normal, Poisson, Gamma,
Binomial and Negative Binomial distributions [28].

Consider a finite set of $N$ voters. The preferences of a voter $i$ over a
finite set of alternatives $A$, is denoted by $\sigma_{i}\in S_{A}$. A
preference profile is defined as a collection of $N$ individual rankings,
$\bar{\sigma}=(\sigma_{1},\ldots,\sigma_{N})$. In our setting, the identity of
the voters is not important. So we can consider an anonymous profile
$\pi\in[0,1]^{|A|!}$, where, for each $\sigma\in S_{A}$, $\pi(\sigma)\in[0,1]$
is the probability that a random voter has the ranking $\sigma$.

The relation between an anonymous preference profile and a permutation process
over a finite subset $A\subseteq X$, is that for $\sigma\in S_{A}$,
$\pi(\sigma)$ is the probability of $\sigma$ in $\Pi(A)$555Because of this, we
can refer to $\Pi(A)$ as an anonymous preference profile..

We want to analyze some axiomatic aspects of the aggregation of preferences
induced by a permutation process, defined by the selection of the winning
alternatives. We define a social choice correspondence (SCC) as a function $f$
that maps an anonymous preference profile defined over a finite subset
$A\subseteq X$ into a nonempty subset of $A$. Examples of SCC are the
plurality rule, that selects the alternatives that are on the top of the
largest number of individual preferences, $argmax_{a\in A}\sum_{\sigma\in
S_{A}:\sigma(a)=1}\pi(\sigma)$, or the antiplurality rule, that selects the
alternatives that are the least preferred by the smallest number of individual
preferences, $argmin_{a\in A}\sum_{\sigma\in S_{A}:\sigma(a)=N}\pi(\sigma)$.

##  3 Pairwise Majority Dominance

We can define what it means for an alternative to be better than another one
in aggregate terms, based on the concept of majoritarian power. That is, an
alternative $a$ is better than an alternative $b$, if $a$ is preferred to $b$
by a majority of the individuals, i.e., a group of cardinality larger or equal
than $|\frac{N}{2}|$. We have the following notion of dominance:

######  Definition 1.

An alternative $a\in A$ pairwise majority-dominates (PM-dominates) another
alternative $b\in A$ in an anonymous preference profile $\pi$ over $A$,
denoted $a\rhd^{pm}_{\pi}b$, if $|\\{\sigma\in
S_{A}:a>_{\sigma}b\\}|\geq|\\{\sigma\in S_{A}:b>_{\sigma}a\\}|$.

This binary relation is complete but not transitive, due to the possible
existence of Condorcet cycles.666A Condorcet cycle or ‘Condorcet Paradox’
occurs when $a\rhd^{pm}_{\pi}b$, $b\rhd^{pm}_{\pi}c$ and $c\rhd^{pm}_{\pi}a$.

The following is a desirable property to be satisfied by a social choice
correspondence. Intuitively, it means that the voting rule is ‘consistent’
with the dominance relation: if it selects an alternative $b$, then it must
also select the alternatives that are better than $b$.

######  Definition 2.

An anonymous SCC $f$ is said to be pairwise majority-dominance-efficient (PMD-
efficient) if for every anonymous preference profile $\pi$ and any two
alternatives $a,b\in A$, if $a\rhd^{pm}_{\pi}b$, then $b\in f(\pi)$ implies
that $a\in f(\pi)$.

A stronger requirement states, roughly, that dominated alternatives should not
be selected, unless they also dominate the alternatives that dominate them.

######  Definition 3.

An anonymous SCC $f$ is said to be strongly PMD-efficient if for every
anonymous preference profile $\pi$ over $A$, and any two alternatives $a,b\in
A$ such that $a\rhd^{pm}_{\pi}b$, then:
-if $b\ntriangleright_{\pi}^{pm}a$, then $b\notin f(\pi)$,
-if $b\rhd_{\pi}^{pm}a$, then $b\in f(\pi)$ if and only if $a\in f(\pi)$.

It is clear that strong-efficiency implies efficiency. This notion sets the
bar too high on SCCs, since there are no rules satisfying it on unrestricted
domains. This problem arises because Condorcet cycles can arise in those
domains. But even efficiency is a very demanding condition:777The definitions
of the voting rules as well as the proof of Proposition 1 can be found in the
Appendix. The definitions may also be found in [16] and [14].

######  Proposition 1.

Black, Dodgson, Young, Kemeny, Nanson, Minimax and Fishburn rules are not PMD-
efficient voting rules.

This means that this notion of efficiency is not satisfied by many (important
and well studied) voting rules. An alternative intuition is that an
“efficient” rule should select the ‘most’ dominating alternatives, or at
least, the ones that are the ‘less’ dominated. The Schwartz set captures this
idea [38]:

######  Definition 4.

The Schwartz set of a profile of preferences $\pi$, denoted $Sc(\pi)$, is the
minimal set $Sc(\pi)\subseteq A$ such that $Sc(\pi)\neq\emptyset$ verifying
that for any $a\in Sc(\pi)$ there is no $b\in A\setminus Sc(\pi)$ such that
$b\rhd_{\pi}^{pm}a$.

This set is nonempty for any profile of preferences. The Schwartz voting rule
is defined as the SCC that selects as winning alternatives the entire Schwartz
set. The following result follows immediately from the definition of a
Schwartz set:

######  Proposition 2.

The Schwartz voting rule is PMD-efficient.

##  4 Position Dominance

In this section, we consider a different notion of dominance arising when an
alternative $a$ is positioned higher in more rankings than another one, $b$.
Formally:888We adapt the definition introduced by Caragiannis et al. [8],
allowing ties.

######  Definition 5.

Given an anonymous preference profile $\pi$ on $A$, an alternative $a\in A$,
$i\in N$ and $j\in\\{1,\ldots,|A|\\}$, let $s_{j}(a)=|\\{i:\sigma_{i}(a)\leq
j\\}|$. That is, $s_{j}(a)$ is the number of voters that rank alternative $a$
at a position $j$ or lower. For $a,b\in A$, we say that $a$ position-dominates
(Pos-dominates) $b$, denoted by $a\rhd^{pos}_{\pi}b$, if $s_{j}(a)\geq
s_{j}(b)$ for every $j\in\\{1,\ldots,|A|\\}$.

This binary relation is obviously not complete, since there may exist
alternatives that are not comparable under a given profile. But Pos-dominance
is transitive, ruling out the possibility of cycles.

The following example shows that neither one of the dominance relations
presented above (PM and Pos) implies the other.

######  Example 1.

In the preference profile $\\{a>_{1}b>_{1}c,\ a>_{2}b>_{2}c,\
b>_{3}c>_{3}a\\}$ we have that $a\rhd^{pm}_{\pi}b$ and
$a\ntriangleright_{\pi}^{pos}b$, while in the profile
$\\{a>_{1}c>_{1}d>_{1}b,\ b>_{2}a>_{2}c>_{2}d,\ c>_{3}d>_{3}b>_{3}a\\}$ we
have $a\rhd^{pos}_{\pi}b$ and $a\ntriangleright_{\pi}^{pm}b$.

The properties of Position Dominance-efficient (PosD-efficient) SCC and
strongly Position Dominance-efficient (strongly PosD-efficient) SCC are
analogous to those of PMD-efficient and strongly PMD-efficient SCC. We have
that:999The definitions of the voting rules as well as the proofs of the
claims in this section can be found in the Appendix.

######  Proposition 3.

The Bucklin SCC is PosD-efficient but not strongly PosD-efficient.

Moreover, there exists an entire family of PosD-efficient rules:

######  Proposition 4.

All positional scoring rules are PosD-efficient.101010If scores are
normalized, positional scoring rules can be seen as a family of weighted
arithmetic mean operators, one for each alternative [25, 20].

In turn, the strong PosD-efficiency of scoring rules depends on their
associated weights:

######  Proposition 5.

Scoring rules with associated decreasing weights vectors are strongly PosD-
efficient.

##  5 Aggregation of Permutation Processes

So far, we introduced two different notions of dominance, rather limited in
their efficiency. For richer results, we can turn our attention to the
aggregation of permutation processes.

######  Definition 6.

An alternative $a\in X$ PM-dominates (respectively Pos-dominates) an
alternative $b\in X$ in the permutation process $\Pi$, denoted
$a\rhd_{\Pi}^{pm}b$ $(a\rhd^{pos}_{\Pi}b)$, if for every finite set of
alternatives $A\subseteq X$ such that $a,b\in A$, we have that $a$ PM-
dominates (Pos-dominates) $b$ in the anonymous preference profile $\Pi(A)$.

######  Definition 7.

A permutation process $\Pi$ over $X$ is said to be PM-compatible (Pos-
compatible) if for every $A\subseteq X$, the binary relation
$\rhd_{\Pi}^{pm}|_{A}$ $(\rhd^{pos}_{\Pi}|_{A})$ is a total preorder over
$A$.111111A total preorder is a transitive and complete binary relation.

When a permutation process is PM-compatible, the total preorder is consistent
across all the different finite subsets of $X$:

######  Lemma 1.

Let $\Pi$ be a consistent permutation process that is PM-compatible. Then, for
any finite subset of alternatives $A\subseteq X$,
$(\rhd_{\Pi(A)}^{pm})=(\rhd_{\Pi}^{pm}|_{A})$

###### Proof.

First, by definition, we have that $a\rhd_{\Pi}^{pm}b$ implies
$a\rhd_{\Pi(A)}^{pm}b$ for every subset $A\subseteq X$ such that $a,b\in A$.
Then $(\rhd_{\Pi}|_{A})\subseteq(\rhd_{\Pi(A)})$.

For the other inclusion, we must prove that $a\rhd_{\Pi(A)}^{pm}b$ implies
that $a\rhd_{\Pi(C)}^{pm}b$ for any finite subset $C$ that contains $a$ and
$b$. Let $a\rhd_{\Pi(A)}^{pm}b$ and suppose that
$a\ntriangleright_{\Pi(C)}^{pm}b$ for $C$. Since $\rhd_{\Pi}^{pm}$ is PM-
compatible, it is a total preorder for every $C$. Then it must be that
$b\rhd_{\Pi(C)}^{pm}a$. Then

$|\\{\sigma\in S_{A}:a>_{\sigma}b\\}|\geq|\\{\sigma\in S_{A}:b>_{\sigma}a\\}|$
(1)

and

$|\\{\sigma\in S_{C}:a>_{\sigma}b\\}|<|\\{\sigma\in S_{C}:b>_{\sigma}a\\}|$
(2)

Since the permutation process is consistent, when we restrict the domain to
$\\{a,b\\}$, the relative positions of $a$ and $b$ do not change. So, (1)
leads to $|\\{\sigma\in S_{\\{a,b\\}}:a>_{\sigma}b\\}|\geq|\\{\sigma\in
S_{\\{a,b\\}}:b>_{\sigma}a\\}|$ and (2) leads to ${|\\{\sigma\in
S_{\\{a,b\\}}:a>_{\sigma}b\\}|<|\\{\sigma\in S_{\\{a,b\\}}:b>_{\sigma}a\\}|}$.
This is a contradiction, derived from the assumption that
$a\ntriangleright_{\Pi(C)}^{pm}b$ for any finite subset $C$ containing $a$ and
$b$. Then $a\rhd_{\Pi(C)}^{pm}b$ for any $C$, and thus $a\rhd_{\Pi}^{pm}b$
verifies that $(\rhd_{\Pi(A)})\subseteq(\rhd_{\Pi}|_{A})$. ∎

This result allows us to write $\Pi$ instead of $\Pi(A)$ whenever we are
referring to PM-dominance relations. The same is not true for a permutation
process that is Pos-compatible. The dominance order of the alternatives may
not be consistent between different subsets of $X$, as shown in the following
example.

######  Example 2.

Consider the set of alternatives $A=\\{a,b,c\\}$ and the profile of
preferences where one voter chooses $a>b>c$ and another one $b>c>a$. Then we
have that the only possible order is
$b\rhd_{\Pi(A)}^{pos}a\rhd_{\Pi(A)}^{pos}c$. If we now consider $B=\\{a,b\\}$,
we obtain that $b\rhd_{\Pi(A)}^{pos}a$ and $a\rhd_{\Pi(A)}^{pos}b$.

We can introduce a new definition of compatibility, according to which the
orders are consistent over all the subsets.

######  Definition 8.

A permutation process $\Pi$ over $X$ is said to be strongly compatible if it
is compatible and the preorder $\rhd_{\Pi}$ is consistent accross all subsets
$A\subseteq X$.

It is clear from Lemma 1 that if a permutation process is PM-compatible it is
also strongly PM-compatible.

The next result states that if the permutation process is compatible, then
under an efficient voting rule one of the winning alternatives is the one that
dominates the rest of the alternatives.

######  Theorem 1.

Let $f$ be an anonymous PMD-efficient SCC, and let $\Pi$ be a PM-compatible
permutation process. Then, for any finite subset of alternatives $A$, there
exists an $a\in A$ such that $a\rhd_{\Pi}^{pm}b$ for all $b\in A$. Moreover,
$a\in f(\Pi(A))$.

Let $f$ be an anonymous PosD-efficient SCC, and let $\Pi$ be a Pos-compatible
permutation process. Then, for any finite subset of alternatives $A$, there
exists an $a\in A$ such that $a\rhd_{\Pi(A)}^{pos}b$ for all $b\in A$.
Moreover, $a\in f(\Pi(A))$.

If $\Pi$ is strongly Pos-compatible, there exists an $a\in A$ such that
$a\rhd_{\Pi}^{pos}b$ for all $b\in A$. Moreover, $a\in f(\Pi(A))$

###### Proof.

We prove the theorem for pairwise majority-dominance, since the proof is
analogous for position-dominance.

Since $\Pi$ is PM-compatible, the relation $\rhd_{\Pi}^{pm}$ restricted to $A$
is a total preorder. Therefore, there exists an alternative $a\in A$ such that
$a\rhd_{\Pi(A)}^{pm}b$ for all $b\in A$.

The definition of a SCC states that the outcome in non-empty. Assume that
there is a $b\neq a$ such that $b\in f(\Pi(A))$. Since $a\rhd_{\Pi(A)}^{pm}b$
and $f$ is PM-efficient, it must be that $a\in f(\Pi(A))$. ∎

In the light of Proposition 1, it may seem that Theorem 1 is not very relevant
since there are not too many PM-efficient voting rules. But this is no longer
the case when we consider PM-compatible permutation processes:

######  Theorem 2.

Let $\Pi$ be a PM-compatible and consistent permutation process. Then the
Black, Nanson, Dodgson, Young, Minimax, Kemeny, Fishburn and Schwartz rules
select $f(\Pi(A))=\\{a\in A:a\rhd_{\Pi}^{pm}b\ \mbox{for all}\ b\in A\\}$.
Moreover, these rules are strongly PM-efficient.

###### Proof.

Since $\Pi$ is PM-compatible, $\rhd_{\Pi(A)}$ is a total preorder. Therefore
there are no ‘cycles’ and by Theorem 1 we have that there exists at least one
alternative $a\in A$ that PM-dominates the other alternatives. According to
the results in Fishburn [16] and Felsenthal and Tideman [14], we know that
Black, Dodgson, Young, Minimax and Fishburn under the absence of cycles are
Strict Condorcet Consistent. That is, they select the set of candidates that
beat or tie with all other alternatives.121212The undefeated alternatives are
called in the literature as Weak Condorcet winners or Quasi-Condorcet
candidates. It is also clear that the Schwartz rule is Weak Condorcet
Consistent. Therefore we have that $f(\Pi(A))=\\{a\in A:a\rhd_{\Pi}^{pm}b\
\mbox{for all}\ b\in A\\}$ for these rules. It is straightforward from the
definition of Weak Condorcet winners that these rules are strongly PM-
efficient. ∎

For position dominance, we obtain an analogous result, but under the proviso
that the order of alternatives may not be consistent among subsets of $X$:

######  Theorem 3.

Let $\Pi$ be a Pos-compatible and consistent permutation process, $f$ a
scoring rule with an associated decreasing weights vector and $A$ a finite set
of alternatives. Then $f(\Pi(A))=\\{a\in A:a\rhd_{\Pi(A)}^{pos}b\ \mbox{for
all}\ b\in A\\}$.

###### Proof.

According to Theorem 1 there exists an alternative $a\in A$ such that
$a\rhd_{\Pi(A)}^{pos}b$ for all $b\in A$ and $a\in f(\Pi(A))$. Since $\Pi$ is
Pos-compatible, any other alternative $b\in A$ is comparable with $a$. By
Proposition 5, all the scoring rules with associated decreasing weights are
strongly PD-efficient. If $b\rhd_{\Pi(A)}^{pos}a$, it must be that $b\in
f(\Pi(A))$, and $b\rhd_{\Pi(A)}^{pos}c$ for all $c\in A$ (because of
transitivity). If $b\ntriangleright_{\Pi(A)}^{pos}a$, then $b\notin
f(\Pi(A))$. ∎

As a consequence of Theorem 3, we get a positive result with strong
compatibility, since the orders are consistent across different subsets
containing the same alternatives.

######  Corollary 1.

Let $\Pi$ be a strongly Pos-compatible and consistent permutation process and
$f$ a scoring rule with an associated decreasing weighted vector. Then for any
finite set of alternatives $A\subseteq X$, $f(\Pi(A))=\\{a\in
A:a\rhd_{\Pi}^{pos}b\ \mbox{for all}\ b\in A\\}$.

Theorems 2, 3 and Corollary 1 are particular cases of a stronger result:

######  Theorem 4.

Let $\Pi$ be a strongly compatible and consistent permutation process and $f$
a strongly efficient SCC. Then for any finite set of alternatives $A\subseteq
X$, $f(\Pi(A))=\\{a\in A:a\rhd_{\Pi}b\ \mbox{for all}\ b\in
A\\}$.131313Whenever the type of dominance relation is not explicitly
specified, the result must be valid for both PM and Pos dominance.

###### Proof.

Let $A$ be a finite set of alternatives. By Theorem 1 and the fact that strong
efficiency implies efficiency, we have that $\\{a\in A:a\rhd_{\Pi}b\ \mbox{for
all}\ b\in A\\}\subseteq f(\Pi(A))$.

Now let $a\in f(\Pi(A))$ and assume that there exists $b\in A$ such that
$a\ntriangleright_{\Pi}b$. Since $\Pi$ is strongly compatible, $\Pi(A)$ has
the same total preorder, for every $A\subseteq X$. Then $b\rhd_{\Pi(A)}a$ and
$a\ntriangleright_{\Pi(A)}b$. Since $f$ is strongly efficient, it must be that
$a\notin f(\Pi(A))$, a contradiction. Then $f(\Pi(A))\subseteq\\{a\in
A:a\rhd_{\Pi}b\ \mbox{for all}\ b\in A\\}$. ∎

Then, if the permutation process is compatible, an efficient social choice
correspondence should select the alternatives that beat or tie up with every
other alternative. The following example, due to Caragiannis et al. [8] shows
that even when the permutation process is compatible, the ‘best’ alternative
may not coincide under the two dominance relations.

######  Example 3.

Consider $\Pi$, the consistent permutation process which given the
alternatives $a,b$ and $c$, gives the following profile: $(a>_{1}b>_{1}c)$,
with weight $\frac{4}{11}$; $(b>_{2}a>_{2}c)$, with weight $\frac{2}{11}$;
$(b>_{3}c>_{3}a)$ with weight $\frac{3}{11}$ and $(c>_{4}a>_{4}b)$ weighted
$\frac{2}{11}$. Then we have that $a\rhd_{\Pi}^{pm}b\rhd_{\Pi}^{pm}c$ and
$b\rhd_{\Pi}^{pos}a\rhd_{\Pi}^{pos}c$.

While the two dominance relations may not yield the same outcome, we seek to
find out when a permutation process is compatible. First, let us recall that a
permutation process can be interpreted in terms of utilities, which allows us
to introduce the following definition.

######  Definition 9.

Alternative $a\in X$ dominates $b\in X$ in the utility process $U$ if for
every finite subset of alternatives containing $x_{1}=a$ and $x_{2}=b$,
$\\{x_{1},x_{2},\ldots,x_{m}\\}\subseteq X$, and every vector of utilities
$(u_{1},u_{2},\ldots,u_{m})$ with $u_{1}\geq u_{2}$ we have that

$p(u_{1},u_{2},\ldots,u_{m})\geq p(u_{2},u_{1},\ldots,u_{m})$

where $p$ is the probability function.

The following two lemmas state that ‘natural’ utility processes are indeed
compatible.

######  Lemma 2.

Let $\Pi$ be a consistent permutation process and $U$ be its corresponding
utility process. If alternative $a$ dominates $b$ in $U$, then $a\rhd_{\Pi}b$.

###### Proof.

Noothigattu et al [30] prove in Lemma 4.9 that if $a$ dominates $b$ in $U$,
then $a$ swap-dominates $b$. Swap dominance is a stronger notion that implies
PM-dominance and Pos-dominance. Then both conditions are satisfied. ∎

We consider consistent permutation processes $U$ such that given a set of
alternatives $\\{x_{1},\ldots,x_{m}\\}$, the utilities
$(U_{x_{1}},\ldots,U_{x_{m}})$ are independent and have a distribution drawn
from the Exponential Family [28]:

$p_{U_{a}}(u_{1})=\exp({\eta(\mu_{a})T(u_{1})-A(\mu_{a})+B(u_{1})})$

Examples of this are the Gaussian or Normal (Thurstone-Mosteller Process [39,
29]), the Gumbel (Placket-Luce Process [33, 23]), Poisson, Gamma, Binomial and
Negative Binomial distributions. Then:

######  Lemma 3.

Under an utility process with a distribution belonging to the Exponential
Family, the alternative a dominates alternative b if
$\eta(\mu_{a})>\eta(\mu_{b})$ and $T(u_{1})>T(u_{2})$.141414For the Normal
$(N(\mu_{x_{i}},\frac{1}{2}))$ and the Gumbel $(G(\mu_{x_{i}},\gamma))$
processes this means that $\mu_{a}\geq\mu_{b}$. For a Poisson process
$(P(\lambda_{x_{i}}))$, it means that $\lambda_{a}\geq\lambda_{b}$. For a
Gamma process with fixed shape $(\Gamma(r,\lambda_{x_{i}}))$, the implication
is that $\lambda_{b}\geq\lambda_{a}$. For a Binomial process
$(B(n_{x_{i}},p_{x_{i}}))$ with fixed $n$, it means that $p_{a}\geq p_{b}$. If
$p$ is fixed and $n$ is variable, it corresponds to $n_{a}\geq n_{b}$. For a
Negative Binomial process $(NB(r_{x_{i}},p_{x_{i}}))$ with fixed $r$, it means
that $p_{b}\geq p_{a}$ while if $p$ is fixed and $r$ is variable, it is
$r_{a}\geq r_{b}$.

###### Proof.

We have to find the conditions according to which
$p(u_{1},u_{2},\ldots,u_{m})\geq p(u_{2},u_{1},\ldots,u_{m})$ when $u_{1}\geq
u_{2}$. Since utilities are sampled independently, this implies checking when
$p_{U_{a}}(u_{1})p_{U_{b}}(u_{2})\geq p_{U_{a}}(u_{2})p_{U_{b}}(u_{1})$. That
is,

$\exp({\eta(\mu_{a})T(u_{1})-A(\mu_{a})+B(u_{1})})\exp({\eta(\mu_{b})T(u_{2})-A(\mu_{b})+B(u_{2}))}\geq$
$\exp({\eta(\mu_{a})T(u_{2})-A(\mu_{a})+B(u_{2})})\exp({\eta(\mu_{b})T(u_{1})-A(\mu_{b})+B(u_{1})})$

and thus

$\eta(\mu_{a})T(u_{1})-A(\mu_{a})+B(u_{1}))+\eta(\mu_{b})T(u_{2})-A(\mu_{b})+B(u_{2})\geq$
$\eta(\mu_{a})T(u_{2})-A(\mu_{a})+B(u_{2})+\eta(\mu_{b})T(u_{1})-A(\mu_{b})+B(u_{1})$

Then

$(\eta(\mu_{a})-\eta(\mu_{b}))T(u_{1})+(\eta(\mu_{b})-\eta(\mu_{a}))T(u_{2})\geq
0$

and

$(\eta(\mu_{a})-\eta(\mu_{b}))(T(u_{1})-T(u_{2}))\geq 0$

Since $u_{1}\geq u_{2}$, it follows that $T(u_{1})\geq T(u_{2})$ and thus
$\eta(\mu_{a})-\eta(\mu_{b})\geq 0$ and $\eta(\mu_{a})\geq\eta(\mu_{b})$.

The proof for each particular probability distribution is analogous to the
proof of the general case. ∎

Lemma 3 implies that a consistent permutation process with an utility process
drawn from the Exponential Family (which includes some of the best known
distributions), is compatible under both notions of dominance. This ensues
from the fact that dominance only depends on the parameters of the
distributions, all of which have real values (and thus are element of a
complete preorder). Moreover, Lemma 2 states that the preorders are the same
under both dominance notions satisfying strong compatibility. This allows us
to get rid of a fixed $A$ even for Pos-dominance. The downside of this, is
that we may rule out permutation processes that could be compatible under one
dominance notion but not under the other.

According to Theorem 2, Corollary 1 and Lemma 3, in many cases of interest we
only need to find the most dominant alternatives, which, depending on the
permutation process, are the alternatives with a maximum defined parameter.

##  6 Stability

In this section we introduce another property that we expect that a SCC should
verify.

######  Definition 10.

Let $f$ be an anonymous SCC and $\Pi$ a permutation process over $X$. We say
that $f$ is stable if for any non empty and finite subset of alternatives $A$
and $B$ such that $B\subseteq A\subseteq X$, $f(\Pi(A))\cap B=f(\Pi(B))$
whenever $f(\Pi(A))\cap B\neq\emptyset$.

Stability is a highly relevant property of aggregation operators meaning that,
in rough terms, ‘small input errors’ do not lead to ‘big output errors’ [42,
7, 26, 35]. In our setting this means that voting rules must yield consistent
choices on restricted sets of alternatives. For example, we do not want the
decision of an autonomous vehicle with three passengers to be to go straight
in the case that no pedestrians are in front of it, but swerve and crush a
wall in the same situation when it transports only two passengers.

The stability of a SCC $f$ depends on how both $f$ and the permutation process
are defined.

######  Theorem 5.

Let $f$ be a strongly efficient SCC and let $\Pi$ be a strongly compatible
permutation process. Then $f$ is stable.

###### Proof.

Let $B\subseteq A\subseteq X$ and assume that $f(\Pi(A))\cap B\neq\emptyset$.
Let $a\in f(\Pi(A))\cap B$. By Theorem 4 we have that $a\rhd_{\Pi}b$ for all
$b\in A$, and thus, for all $b\in B$. Then $a\in f(\Pi(B))$.

Now consider $a\in f(\Pi(B))$. Then $a\in B$ and $a\rhd_{\Pi}b$ for all $b\in
B$. Suppose that $a\notin f(\Pi(A))$. Then there is a $c\in A$ such that
$a\ntriangleright_{\Pi}c$. Since $\Pi$ is strongly compatible, we have that
$c\rhd_{\Pi(A)}a$. Since we assume that $f(\Pi(A))\cap B\neq\emptyset$, there
is an alternative $d\in f(\Pi(A))\cap B$. Then $d\rhd_{\Pi(A)}c$ and by
transitivity we have that $a\ntriangleright_{\Pi(A)}d$. Since $d\in B$ and
$\Pi$ is strongly compatible, we have that $a\ntriangleright_{\Pi(B)}d$. But
$f$ is strongly efficient, so we have that $a\notin f(\Pi(B))$, which is a
contradiction. ∎

This as a very encouraging result, since according to Proposition 5 and
Theorem 2 there exist many well known voting rules that are strongly
efficient. Thus, these rules must also verify stability.

##  7 Conclusions

We have presented an axiomatic study of the method developed by Noothigattu et
al. [30] for automating ethical AI decision making under different notions of
dominance. According to one of those notions, an alternative is better than
another if a majority prefers it. According to the other notion, an
alternative is dominant if it is better positioned than the other alternatives
in most rankings. At first sight, the dominance relations here introduced,
seem more intuitive and natural than the dominance relation used in [30],
namely swap-dominance. It seems that PM and Pos dominance have clearer
meanings and are better suited for the design of an autonomous system.

We showed that, depending on how we learn the preferences of the voters, there
are many well known voting rules that behave well, in the sense that they
select the most dominating alternatives. Moreover, it is only necessary to
find the alternative with a maximum defined parameter (depending on the
distribution used to learn the model). When we use ‘natural’ distributions for
the permutation process we do not need to distinguish between dominance
notions. Another important property is stability, which amounts to the
consistence of choices across different sets of alternatives. For example, an
autonomous vehicle is likely to face potentially infinite scenarios, and we
want decisions to be similar in similar scenarios. We found again that are
many voting rules verifying this property.

Our work yields stronger results than those in [30]. While the alternatives
selected are the same using theirs and our dominance notions, we obtain them
under voting rules satisfying different sets of properties. This is useful
since this unifies the arguments that can be used to justify the choice of a
given alternative.

We consider this paper as a contribution to the axiomatization of theory of
the aggregation of permutations. Its full development in this light may help
to overcome classical problems of aggregation theory.

A possible future line of research consists in analyzing the strategic aspects
of the aggregation of permutation processes. It is important to guarantee the
non-manipulability of this decision making process. For example, there should
not exist advantages from lying about preferences (this property is known in
the literature as strategy-proofness [18, 37]) or from voting several times
(known in the literature as false name-proofness [10, 15]). The results on
manipulation avoidance are rather negative, since there exist very few rules
satisfying non-manipulability, when considering unrestricted domains. We
believe that in the case of the aggregation of permutation processes, new
rules can be found overcoming these problems.

## Acknowledgements

The authors are especially grateful to Ariel Procaccia and Felix Brandt for
comments and suggestions that helped to improve the paper.

## Conflict of interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

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##  Appendix A Voting rules

Black’s rule:151515The Black and Dodgson rules described are called Revised
Black and Simplified Dodgson rules in [14]. this rule is applied in two
stages. In the first stage, the candidates that beat or tie with every other
candidate are selected. If there are not such winners, each candidate receives
a score depending on the positions on every ranking (the better the position
the higher the score). The alternative with the largest overall score is
elected.

Dodgson’s rule: a Condorcet winner is a candidate that beats every other
alternative. The rule computes the minimum number of times that it is
necessary to swap two adjacent alternatives on some rankings in order to make
each candidate a Condorcet winner. The alternative that requires the minimum
steps is considered the winner.

Young’s rule: this rule deletes candidates in order to make an alternative the
Condorcet winner. The candidate that requires less deletions is declared the
winner.

Kemeny’s rule: this rule selects the most preferred alternative of the
rankings that minimize the number of pairs of candidates that are ranked
opposite by all the voters.

Nanson’s rule: as in Black’s rule, a score is given to the candidates
according to their position. Every candidate with a score below the average
score is deleted. The process is repeated with the candidates left, an so on.
The rule selects the undeleted candidates.

Minimax rule: the candidate whose maximum losses in the paired comparisons are
the least is declared winner.

Fishburn’s rule: this rule checks out whether everything that beats x also
beats y under simple majority, and if x beats or ties something that beats y,
then x is ‘better than’ y under simple majority. The rule selects the best
alternatives under this notion.

Bucklin’s rule: the Bucklin score of an alternative $a$, $B(a)$, is the
minimum $k$ such that $a$ is among the first $k$ positions in the majority of
input votes. The Bucklin rule selects the alternatives with the least Bucklin
scores.

Scoring rules: these rules have an associated weights vector
$(\alpha_{1},\alpha_{2},\ldots,\alpha_{|A|})$, where
$\alpha_{i}\geq\alpha_{i+1}$ for $i=1,\ldots,|A|-1$. An alternative $a$ in a
profile $\sigma_{j}$ gets $\alpha_{i}$ points if $\sigma_{j}(a)=i$. The score
of an alternative is the sum of all the points across all voters. The
candidate with the most overall points is declared the winner. The Plurality
and the Antiplurality rules are examples of scoring rules with associated
weights vector $(1,0,\ldots,0)$ and $(1,\ldots,1,0)$ respectively. We say that
a scoring rule has an associated decreasing weights vector if
$\alpha_{i}>\alpha_{i+1}$ for $i=1,\ldots,|A|-1$.

##  Appendix B Proofs

###### Proof.

Proposition 1

Consider the profile
$\sigma=(a>_{1}b>_{1}c>_{1}d>_{1}e,e>_{2}d>_{2}a>_{2}c>_{2}b,b>_{3}c>_{3}d>_{3}e>_{3}a)$.
We have that $a\rhd^{pm}_{\pi}b$.

Black’s rule: there is no alternative that beats or tie every other
alternative. Then we use a scoring rule with an associated decreasing weighted
vector, for example $(5,4,3,2,1)$. We obtain that $score(a)=9$ and
$score(b)=10$, with $b$ getting the largest score. Thus we have that $b\in
Black(\pi)$ and $a\notin Black(\pi)$.

Dodgson’s rule: alternative $b$ needs $1$ swap in the first vote to become the
Condorcet winner, while all the other alternatives require $2$ or more. Then
$b\in Dodgson(\pi)$ and $a\notin Dodgson(\pi)$.

Young’s rule: alternative $b$ needs alternative $a$ to be eliminated in order
to become the Condorcet winner, while all the other alternatives need $2$ or
more alternatives to be eliminated. Then $b\in Young(\pi)$ and $a\notin
Young(\pi)$.

Kemeny’s rule: the ranking that minimizes the number of candidates that are
ranked opposite by all the voters is $b>c>d>a$. Then $b\in Kemeny(\pi)$ and
$a\notin Kemeny(\pi)$.

Consider the profile
$\sigma=(e>_{1}a>_{1}b>_{1}c>_{1}d,b>_{2}c>_{2}d>_{2}e>_{2}a)$. We have that
$a\rhd^{pm}_{\pi}b$.

Nanson’s rule: we use a scoring rule with an associated decreasing weighted
vector, for example $(5,4,3,2,1)$. During the first round, $a$ and $d$ are
eliminated. In the second round $c$ is eliminated. Finally we have that $b$
and $e$ are selected. Then $b\in Nanson(\pi)$ and $a\notin Nanson(\pi)$.

Consider the profile
$\sigma=(d>_{1}c>_{1}a>_{1}b,b>_{2}c>_{2}d>_{2}a,d>_{3}c>_{3}a>_{3}b,a>_{4}b>_{4}c>_{4}d,b>_{5}c>_{5}d>_{5}a)$.
We have that $a\rhd^{pm}_{\pi}b$.

Minimax rule: alternative $b$, $c$ and $d$ have $2$ as a maximum loss while
$a$ has $3$. Then $b\in Minimax(\pi)$ and $a\notin Minimax(\pi)$.

Consider the profile $\sigma=(a>_{1}b>_{1}c>_{1}d,b>_{2}c>_{2}d>_{2}a)$. We
have that $a\rhd^{pm}_{\pi}b$.

Fishburn’s rule: in this profile $b$ beats $c$ and $d$, ties with $a$ and beat
all the alternative that beat or tie with $a$. Then $b\in Fishburn(\pi)$ and
$a\notin Fishburn(\pi)$. ∎

###### Proof.

Proposition 3

Let $a\rhd^{pd}_{\pi}b$ and $b\in Bucklin(\pi)$. Let $B(b)=t$, that is,
$s_{t}(b)=\alpha\geq[\frac{n+1}{2}]$ Because of position dominance, we have
that $s_{t}(a)\geq s_{t}(b)=\alpha$. Then, $B(a)\leq t$. So $a\in
Bucklin(\pi)$.

The following example shows that it is not strong-pd-eff. Let $\pi$ be the
preference profile such that
$\sigma=(a>_{1}b>_{1}c,a>_{2}b>_{2}c,b>_{3}a>_{3}c,c>_{4}a>_{4}b)$. Then
$B(a)=B(b)=2$, $B(c)=3$, $s_{1}(a)=2$, $s_{1}(b)=$, $s_{2}(a)=4$ and
$s_{2}(b)=3$. So, we have that $a\rhd^{pd}_{\pi}b$ and
$b\ntriangleright^{pd}_{\pi}a$ but $Bucklin(\pi)=\\{a,b\\}$. ∎

###### Proof.

Proposition 4

For ease of demonstration we assume, without loss of generality, that
$A=\\{a,b,c\\}$. The score of alternative $a$ is given by
$Score(a)=(s_{1}(a),s_{2}(a)-s_{1}(b),3-s_{2}(a))\cdot(\alpha_{1},\alpha_{2},\alpha_{3})=(\alpha_{1}-\alpha_{2})s_{1}(a)+(\alpha_{2}-\alpha_{3})s_{2}(a)+3\alpha_{3}$
with $\alpha_{i}-\alpha_{i+1}\geq 0$ for $i=1,2$. From the fact that
$a\rhd^{pd}_{\pi}b$ we have that $s_{j}(a)\geq s_{j}(b)$ for $j=1,2$. Then
$Score(a)\geq Score(b)$. If $b$ is such that $b\in f(\pi)$, then $a\in f(\pi)$
as it has more or equal points than $b$. ∎

###### Proof.

Proposition 5

We assume again that $A=\\{a,b,c\\}$. And consider the same argument as in the
proof of Proposition 4. The main difference is that
$\alpha_{i}-\alpha_{i+1}>0$ for $i=1,2$. Then if $a\rhd^{pd}_{\pi}b$ and
$b\ntriangleright_{\pi}^{pd}a$, we have that $Score(a)>Score(b)$ and then
$b\notin f(\pi)$. If instead $b\rhd^{pd}_{\pi}a$, then $Score(a)=Score(b)$ and
both alternatives are selected if they have the most overall points. ∎