Quantitative Fairness -- A Framework For The Design Of Equitable Cybernetic Societies

Abstract

Advancements in computer science, artificial intelligence, and control systems of the recent have catalyzed the emergence of cybernetic societies, where algorithms play a significant role in decision-making processes affecting the daily life of humans in almost every aspect. Algorithmic decision-making expands into almost every industry, government processes critical infrastructure, and shapes the life-reality of people and the very fabric of social interactions and communication. Besides the great potentials to improve efficiency and reduce corruption, missspecified cybernetic systems harbor the threat to create societal inequities, systematic discrimination, and dystopic, totalitarian societies. Fairness is a crucial component in the design of cybernetic systems, to promote cooperation between selfish individuals, to achieve better outcomes at the system level, to confront public resistance, to gain trust and acceptance for rules and institutions, to perforate self-reinforcing cycles of poverty through social mobility, to incentivize motivation, contribution and satisfaction of people through inclusion, to increase social-cohesion in groups, and ultimately to improve life quality. Quantitative descriptions of fairness are crucial to reflect equity into algorithms, but only few works in the fairness literature offer such measures; the existing quantitative measures in the literature are either too application-specific, suffer from undesirable characteristics, or are not ideology-agnostic. Therefore, this work proposes a quantitative, transactional, distributive fairness framework, which enables systematic design of socially feasible decision-making systems. Moreover, it emphasizes the importance of fairness and transparency when designing algorithms for equitable, cybernetic societies.

Full text

Quantitative Fairness - A Framework For The Design Of
Equitable Cybernetic Societies
Kevin Riehl1*, Michail Makridis1and Anastasios Kouvelas1
1*Traffic Engineering Group, Institute for Transport Planning and Systems, ETH Zurich,
Stefano-Franscini-Platz 5, Z¨urich, 8093, Z¨urich, Switzerland.
*Corresponding author(s). E-mail(s): kriehl@ethz.ch;
Contributing authors: mmakridis@ethz.ch;kouvelas@ethz.ch;
Abstract
Advancements in computer science, artificial intelligence, and control systems of the recent have catalyzed
the emergence of cybernetic societies, where algorithms play a significant role in decision-making processes
affecting the daily life of humans in almost every aspect. Algorithmic decision-making expands into almost
every industry, government processes critical infrastructure, and shapes the life-reality of people and the
very fabric of social interactions and communication. Besides the great potentials to improve efficiency and
reduce corruption, missspecified cybernetic systems harbor the threat to create societal inequities, systematic
discrimination, and dystopic, totalitarian societies. Fairness is a crucial component in the design of cybernetic
systems, to promote cooperation between selfish individuals, to achieve better outcomes at the system level, to
confront public resistance, to gain trust and acceptance for rules and institutions, to perforate self-reinforcing
cycles of poverty through social mobility, to incentivize motivation, contribution and satisfaction of people
through inclusion, to increase social-cohesion in groups, and ultimately to improve life quality. Quantitative
descriptions of fairness are crucial to reflect equity into algorithms, but only few works in the fairness
literature offer such measures; the existing quantitative measures in the literature are either too application-
specific, suffer from undesirable characteristics, or are not ideology-agnostic. Therefore, this work proposes
a quantitative, transactional, distributive fairness framework, which enables systematic design of socially-
feasible decision-making systems. Moreover, it emphasizes the importance of fairness and transparency when
designing algorithms for equitable, cybernetic societies.
Keywords: Resource Allocation, Equitable Societies, Distributive Fairness, Procedural Fairness, Algorithmic Fairness
1 Introduction
Technology is increasingly employed for the automa-
tion of processes that were previously performed by
humans. This shift has expanded from early appli-
cations in agriculture, manufacturing, and mechani-
cal processes to now encompass planning, decision-
making, and control processes (Xu et al., 2018).
Especially the advancements in computer science,
artificial intelligence, and control systems of the recent
have catalyzed the emergence of cybernetic societies,
where algorithms play a significant role in decision-
making processes that affect the daily life of humans in
almost every aspect. This algorithmic decision-making
is becoming more prevalent across industries, from
finance and healthcare to media, retail and customer
service, in the life-reality of citizens of smart and mega
cities, and it also involves the design and operations
of energy and transportation networks. Algorithms
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arXiv:2411.13184v1 [eess.SY] 20 Nov 2024

even influence the very fabric of our social inter-
actions, personal relationships, and communication
using digital media and social networks. What’s more,
automated processes are increasingly employed even
in law-enforcement, budget allocation, and planning
at the governmental level (Friedman, 2019; Larsson,
2022; Ashby, 1956).
Automated processes offer numerous potential ben-
efits, such as increased efficiency and objectivity of
decisions, improved enforcement of legislation, reduced
corruption, acceleration of bureaucratic processes,
standardization of processes, and the automation of
repetitive tasks that negatively affect mental health
of human workers (Abbott et al., 2024). At the same
time, missspecified or purposefully misused technolo-
gies harbor the threat to create societal inequities
through systematic discrimination, and enable even
more corrupt systems through mass surveillance tech-
nology and extreme restriction of individual free-
dom (Hossain et al., 2020).
Together with efficiency, fairness plays a cru-
cial role when designing and implementing philan-
thropic, cybernetic systems, that serve and benefit
humans (Friedman, 2019).
•Only systems that promote equitable societies can
guarantee that these cybernetic systems serve peo-
ple, and not people serving these systems.
•The implementation of cybernetic systems in prac-
tice often fails due to public resistance and equity
concerns (Gu et al., 2018).
•Cybernetic systems often support self-coordination
of selfish, rational individuals, and to align egois-
tically optimal with societally optimal outcomes;
doing so, fairness is the foundation to achieve any
form of cooperation of individuals in large popula-
tions (Gurney et al., 2021).
Algorithm-driven, automated processes in some
form distribute resources to people. As these processes
are automated by technology, a discussion of the fair-
ness of cybernetic systems must be a discussion of
procedural and distributive fairness (Pereira et al.,
2017; Friedman, 2019) (Fig. 1). For example, credit
scoring algorithms determine whether a certain person
has access to a loan, transportation demand man-
agement and congestion pricing determine whether a
certain person has access to the road infrastructure
of a city, automatic screening algorithms in recruit-
ing determine whether a certain person is invited to a
job interview, and user-engagement-maximizing algo-
rithms determine which information is provided to the
consumer of a social media platform.
While most discussions in the fairness literature are
instrumental to discuss which resources can be consid-
ered as fairness-relevant (Rawls, 1971; Walzer, 1983;
Nussbaum, 2011; Sen, 2008), which groups shall be
compared and what fairness can be considered as con-
ceptually (Goppel et al., 2016), only few answers can
be found on how to quantitatively assess fairness in a
specific situation based on data, which is crucial for the
design and integration into algorithms, that in some
form act within their environment using data (Cor-
men et al., 2022). The quantitative measures proposed
in the literature are either too specific to a particular
application, suffer from some undesirable characteris-
tics, or are limited to specific ideologies (Jain et al.,
1984).
It is the mission of this article (i) to highlight the
importance of a quantitative discussion on the fairness
of cybernetic systems, (ii) to enable algorithm design
and evaluation based on a holistic, ideology-agnostic,
quantitative fairness framework for a distributive dis-
cussion, and (iii) to create a connection between
domain-specific literature and the fairness literature.
The remainder of this work is organized as follows.
Section 2 reviews the existing literature of fairness,
distributive justice, domain-specific discussions of fair-
ness in various fields, and elaborates in particular
on algorithmic fairness, biased training data, and
the challenges of algorithm transparency and explain-
able artificial intelligence (AI). Section 3 proposes the
quantitative fairness framework for distributive jus-
tice. Section 4 discusses the usefulness of the proposed
fairness framework and illustrates its application when
designing algorithms for automated decision-making.
Section 5 concludes this work.
2 Related Works
Exploring the concept of fairness presents significant
challenges, as it is an abstract philosophical notion
deeply rooted in specific social and cultural contexts.
Despite over two thousand years of philosophical dis-
course across diverse human civilizations, a universal
consensus on the definition and application of fair-
ness remains elusive. Fairness is widely regarded as a
cornerstone of human coexistence and is extensively
examined as a multidisciplinary concept across various
fields, including philosophy, ethics, biology, sociology,
political science, economics, and religion studies. The
2

Chance
Trans-
parency
What
HowWho
Decision-Making
System
Initial Situation
Resource
Allocation
What resources are
relevant to discuss?
How understandable are the
process and its component?
To whom / which groups
are resources allocated? How to make decisions?
Guiding principles to be used?
What chances do
individuals have to
influence the process?
Fig. 1 The Fairness of Decision-Making Systems.
Decision-Making is depicted as a resource allocating process, that takes an initial situation as an input, makes a decision, and
provides a resource allocation as an output. The fairness discussion of decision-making therefore covers procedural and distributive
fairness. For procedural fairness, aspects such as (i) the transparency of the process, and (ii) what chances individuals have to
influence the process are important questions to answer. For distributive fairness, aspects such as (i) what resources are fairness-
relevant and distributed to (ii) which groups and (iii) how decisions are made (based on which guiding principles) are important
questions to answer.
terms fairness, equity and justice are used interchange-
ably in the literature, where equity is often found in
the literature of economics and justice in the literature
of politics and law (Goppel et al., 2016).
In this section, we start to discourse distribu-
tive fairness and procedural fairness. Then, we review
domain-specific discussions of fairness to capture
its multidisciplinarity. As automated decision-making
impacts almost every aspect of our life, a review
of various domains enables us to gather a broader
perspective for the development of a quantitative fair-
ness framework for resource allocation in equitable,
cybernetic societies. At the end, we elaborate on the
algorithm aspects of cybernetic societies: algorithmic
fairness related to biased training data, algorithmic
transparency, and explainable AI.
2.1 Distributive Fairness
Distributive fairness (also distributive justice) tries to
find fair ways of allocating resources (goods, oppor-
tunities, etc.) across a population as a result of
transactions. This can include how benefits but also
Endowment
(Institution)
Interactions
(Market)
Transactions
Diorthotic Fairness
(iustitia commutativa)Dianemetic Fairness
(iustitia distributiva)
Legend: Fairness Theory
Ex-Ante
Perspective
Ex-Post
Ex-Ante
Resource
Allocation
Ex-Post
Fig. 2 A Transactional View on Distributive Fairness.
Aristotle distinguishes dianemetic and diorthotic, distributive
fairness. Dianemetic fairness is concerned with the fair distri-
bution of resources from top to bottom (endownment) from
usually one central decision maker (institution, government) to
the population. Diorthotic fairness is concerned with the fair
distribution of resources in a decentralized way as a result of
transactions between individuals (markets).
burdens can be distributed across individuals, reflect-
ing or neglecting individual properties such as wealth,
social status, contributions, needs, etc. Hume (1740)
3

requires three conditions to discuss the question of fair-
ness: scarcity of resources, a conflict of interest, and
a relative balance of power between the negotiating
parties. Important aspects which need clarification in
the discussion of distributive fairness include: (i) which
resource is distributed (ii) across whom and (iii) how
(based on which guiding principle).
Three important concepts are useful to deter-
mine the fairness-relevance of a specific resource
include: primary goods, distributive spheres of jus-
tice, and capabilities. Rawls (1971) contends that
primary goods, which are universally valuable and
impact the well-being of all individuals regardless of
their personal preferences, are crucial considerations in
determining fairness. Walzer (1983) argues that goods
with a distinct social meaning need to be distributed
in dedicated, fair, distributive spheres, contrary to
normal goods. The capability approach advocates to
discuss resources that determine capabilities (freedom
of choice through many opportunities) rather than
functionings (actual outcomes) (Nussbaum, 2011; Sen,
2008).
Aristotle’s Nicomachean Ethics (Wolf, 2002)
presents a theory of transactional fairness that distin-
guishes between two types of justice: dianemetic and
diorthotic fairness (Fig. 2). Transactions can occur
between various parties under different circumstances,
including market exchanges, governmental resource
allocation, participation in political decision-making,
or legal proceedings. Dianemetic fairness focuses on
how resources are allocated from an authority to a pop-
ulation, typically involving governmental institutions
in cases such as subsidies or societal redistribution.
The initial distribution of resources can also be con-
sidered a dianemetic endowment. Diorthotic fairness,
on the other hand, deals with transactions between
individuals or groups within a population, often in the
context of market exchanges. Nozick (1974) bridges the
gap between these two concepts by arguing that fair,
diorthotic transactions can only occur if the initial,
dianemetic allocation was already fair.
Goppel et al. (2016) distinguish two types of per-
spectives on fairness: ex-post, and ex-ante fairness.
These perspectives describe what matters for the fair-
ness of a resource allocation; the first one (ex-post)
focuses only on the output of a transaction, the sec-
ond (ex-ante) one focuses only on the input of a
transaction. Teleological fairness focuses on outcomes
or consequences, while deontological fairness focuses
on the adherence to rules, duties, and principles
Difference Sufficiency Equality
Greater
Good Proportion
Equality of
Opportunity
Fig. 3 Philosophic Guiding Principles For Fairness.
Six guiding principles for distributive fairness that cover differ-
ent perspectives on fairness. The difference principle advocates a
situation in which the least-advantaged (poorest) are in the best
possible situation. The equality principle advocates an equal
distribution of outcomes. The equality-of-opportunity principle
advocates an equal distribution of the initial situation to pro-
vide every individual with the same chances to make their own
luck. The greater-good principle is the basis for Utilitarianism
and argues that the suffering of the few is acceptable if it serves
the greater good of the majority. The proportion principle advo-
cates distributions that stand in proportional relation to the
contributions and status of individuals. The sufficiency principle
advocates distributions in which everyone receives a certain suf-
ficient minimum to satisfy basic needs, inequalities above that
are not an issue.
regardless of the outcome. Moreover, a differentia-
tion between horizontal equity (fairness within groups
of similar individuals) and vertical equity (fairness
between different groups) is possible.
The literature of fairness provides various ideolo-
gies and moral guiding principles, that can be summa-
rized by following six principles: equality, proportion,
greater-good, difference, equality-of-opportunity, and
sufficiency (Fig. 3). The equality principle considers
an allocation fair, if it ensures equal allocation for
all. This principle is the foundation for Egalitarian
ideology (Scheffler, 2017). The proportion principle
considers an allocation fair, if allocated resources stand
in proportion to the status of individuals, where sta-
tus could refer to social status in a society, economic
power, contributions to the system, or needs. This
principle is the foundation for Aristocratic (also Aris-
totelian) ideology (Goppel et al., 2016; Wolf, 2002).
The greater-good principle considers an allocation fair,
4

if it maximizes the welfare of the many (society),
even if this means the suffering of few (individu-
als). This principle is the foundation for Utilitarian
ideology (Mill, 2016). The difference principle consid-
ers an allocation fair, if it achieves the best possible
outcome for the least-advantaged (Rawlsian ideol-
ogy) or the average (Harsanyian ideology) (Rawls,
1971; Harsanyi, 1975). The equality-of-opportunity
principle considers an allocation fair, if the initial
changes / opportunities of each participating individ-
ual to a resource-allocating process were equal upfront.
This principle is the foundation for Luck-Egalitarian
ideology (Dworkin, 2000). The sufficiency principle
considers an allocation fair, if it is guaranteed that
each individual receives at least a sufficient minimum.
This principle is the foundation for Sufficientarian
ideology (Shields, 2012).
2.2 Procedural Fairness
The concept of fairness has evolved throughout his-
tory. In ancient philosophy, it was viewed as a personal
virtue and a key attribute of divine beings. Dur-
ing the Renaissance, this divine association shifted
towards a universal, natural law. The Enlightenment
era saw philosophical discussions move away from reli-
gious contexts, emphasizing rational thought instead.
Contemporary debates on fairness primarily center
on political and economic aspects, expanding beyond
individual conduct to encompass institutional prac-
tices and societal processes (Goppel et al., 2016).
Procedural fairness discusses fairness in processes.
These processes usually include processes that resolve
disputes and distribute resources, particularly in
administrative and legal contexts, but are not limited
to those. Other examples could include promotions to
higher positions in organizations, hiring for new posi-
tions, admission to educational facilities, etc. Procedu-
ral fairness intersects with distributive fairness (when
distributing resources), and retributive fairness (when
punishing mistakes). Two important aspect to proce-
dural fairness are (i) the opportunities of users affect-
ing the process outcome, and (ii) the transparency of
the processes and underlying decision-making (Lind
and Tyler, 2013). Similarly to distributive fairness, ex-
ante and ex-post perspectives on procedural fairness
apply.
Three important schools of thought exist in the
equality-of-opportunities of processes: formal equal-
ity, substantive equality, and pure equality. Formal
equality (also meritocracy) often refers to merito-
cratic provision of opportunities based on performance
only, neglecting other discriminatory aspects such
as gender, race, or age. Substantive equality refers
to equal chances for different groups of people. For
example, substantive equality distributes opportuni-
ties according to gender and race, rather than merits,
to ensure the representation of certain groups. While
formal and substantive equality discriminate individ-
uals either based on performances or personal aspects
(e.g. gender, race), the pure equality (also known as
equality before the law) describes the equal distribu-
tion of opportunities to all individuals independent of
any personal features (Acemoglu and Wolitzky, 2021;
Barnard and Hepple, 2000).
2.3 Domain-specific discussions of
fairness
The discussions of fairness across different domains
have in common, that fairness - besides efficiency -
is a crucial component when designing systems. Not
only does fairness promote cooperation between selfish
individuals to achieve better outcomes at the system
(population) level, but also to confront public resis-
tance against policy, to gain trust and acceptance
for rules and institutions, to perforate self-reinforcing
cycles of poverty through social mobility, to incentivize
motivation, contribution and satisfaction of people
through inclusion, to increase social-cohesion in groups
and group performance, and ultimately to improve life
quality.
2.3.1 Natural Science & Engineering
Biology & Behavioural Psychology - Fairness
in biological systems often manifests as cooperative
behaviors. From an evolutionary biology perspective,
sensitivity to fairness is a behavioral trait that evolved
in social animals, whose survival relies on cooperation
and group dynamics. This perception of fairness can be
observed in various social animals, including Capuchin
monkeys, vampire bats, and humans (Brosnan and
de Waal, 2014). At the individual level, the subjective
perception of fairness plays a crucial role in both phys-
ical health and psychological well-being. The brain of
social animals, particularly the insula, reacts to per-
ceived unfairness with a sense of disgust. Interestingly,
this reaction is more pronounced when individuals feel
they are unfairly disadvantaged compared to when
they are unfairly advantaged (Jackson et al., 2006).
5

Furthermore, perceptions of fairness significantly influ-
ence how individuals form relationships and interact
with one another (De Cremer et al., 2010). At the
societal level, fairness plays a crucial role in fostering
robust communities, encouraging adherence to social
norms, and promoting cooperation. The perception of
fairness is shaped by shared, group-specific norms and
cultural values. Moreover, fairness serves to enhance
social cohesion, mitigate conflicts, bolster group iden-
tity and trust, and ultimately leads to improved group
work outcomes (De Cremer et al., 2010; Hitti et al.,
2011). The evolution of fairness in biological systems
is closely tied to the concept of reciprocal altruism.
Organisms that engage in fair exchanges are more
likely to maintain beneficial relationships over time,
increasing their chances of survival and reproduction.
This has led to the development of various strategies
and mechanisms in nature to enforce fairness, such as
partner choice and reputation systems in some species.
Often, fairness reflects a balance between contribution
and reward that is sustainable and beneficial for the
species or ecosystem as a whole (Schino and Aureli,
2009).
Network Management - Human societies use
networks for the transportation and distribution of
resources, e.g. energy networks, telecommunication
networks, supply chain networks, rail- and road net-
works, computation networks, and social networks.
Fairness in network management is a critical aspect
of ensuring that resources are distributed equitably
among users and applications. This principle is essen-
tial for maintaining the efficiency, reliability, user
satisfaction, and motivation to contribute to a net-
work. Discussions of fairness in networks include
allocated resources, but also quality of service (e.g.
delays or bandwidth in the context of internet net-
works). Achieving fairness in network management
often involves trade-offs between efficiency and equity.
For instance, while it is desirable to allocate resources
fairly, it is also important to ensure that the network
operates efficiently and maximizes overall through-
put. This balance can be challenging, particularly
in heterogeneous networks where users have vary-
ing requirements and capabilities. Utility optimization
methods are often used to address this issue, where the
goal is to maximize the aggregate utility of all users
while ensuring fair resource allocation. Various fair-
ness models, such as max-min fairness, proportional
fairness, α-fairness, and the Jain metric have been
developed to address these allocation challenges (Jain
et al., 1984; Bonald et al., 2006).
2.3.2 Economics
Macro Economics - Economics studies the pro-
duction, consumption and distribution (allocation) of
resources in human societies. Fairness and economics
are deeply intertwined concepts that have signifi-
cant implications for market structures, competition,
policies and taxation. Governmental intervention into
markets is often justified with restoring competition
that is affected by monopolies, public goods, and exter-
nalities. Minimum wages, wealth redistribution and
income taxation play an important role in fairness-
promoting policies. Behavioral economics has provided
new insights into how people form judgments about
what is fair. Factors like framing, social norms, and
perceptions of intentions all influence whether a given
outcome is seen as fair or unfair, which has important
implications on public support for policies. Welfare
economics is a branch of economics that studies social
welfare, and to evaluate the overall well-being of a soci-
ety, where fairness is an important component to most
definitions of welfare. Especially welfare-economists
have shaped the fairness literature of the recent.
Micro Economics & fairness of market prices
-When can markets and market prices considered as
fair? The fairness of markets was closely linked with
the philosophical, political and economic discussions
of fairness. Aristotle argues, a diorthotic transaction
is fair, when the exchanged resources are of equal
value. Albertus Magnus and Thomas Aquinas intro-
duced the term of a fair price for transactions at
monetary markets. The fair price primarily reflects
the efforts for the generation of the resource, but can
also include marginal profits of traders. The school
of Salamanca puts the term of a fair price equal to
the market price, assuming efficient, ideal markets.
Smith (1776) developed the theory of the invisible
hand which claims, that any selfish, egoistic behavior
and any price in transactions is fair, as free markets
lead to societal optima as a result. Besides market fail-
ure and arbitrage, purposeful phenomena such as price
differentiation, dynamic pricing, price discrimination,
and personalized pricing are heavily discussed in the
literature. The fairness of markets is therefore closely
linked to the fairness of prices (Kahneman et al., 1986).
Education - Education systems are a main ingre-
dient for societal development and individual growth.
Education is a catalyst and one of the most powerful
6

tools for promoting social mobility. It provides indi-
viduals with knowledge, skills, and opportunities that
can lead to better job prospects and higher earning
potential, potentially allowing them to improve their
socioeconomic status (Brown, 2017). Especially early
childhood education has been identified as a key strat-
egy for promoting social mobility. Fairness plays an
important role in this context to make sure, that each
child has the chance to grow up to its full poten-
tial. Education offers resources and opportunities. As
children have different socio-economic backgrounds,
abilities, development stages, and abilities when enter-
ing the school, access to the same quality of teachers,
textbooks, and learning environments are important.
However, in addition to that, more resources and
opportunities might be necessary for students with
special needs, such as inclusion of children with dis-
abilities. Governments play a crucial role in providing
public education systems, to overcome the signifi-
cant disparities in society that hinder children from
disadvantaged contexts (Tharp, 2018).
Housing & gentrification - Gentrification is
a complex urban phenomenon that sits at the inter-
section of housing, economic development, and social
equity. While it can bring positive changes to neighbor-
hoods, such as improved infrastructure and increased
economic activity, it also raises significant concerns
about fairness and displacement of long-time resi-
dents. At its core, gentrification involves the influx of
wealthier residents into previously lower-income neigh-
borhoods, often accompanied by rising property values
and rents, and changes in the local culture and ameni-
ties. This process can lead to the displacement of
long-term residents who can no longer afford to live in
their communities, which affects social cohesion, and
raises important questions about housing fairness and
social justice (Krings and Schusler, 2020). In many
societies, the housing market and rental prices are
therefore highly regulated, and challenges to account
for discrimination, racial segregation, economic dispar-
ities, and marginalised communities bring complexity
to this issue (Von Hoffman, 2000).
Healthcare - Access to medical services is vital
for sustaining healthy and worth-living lives, and for
keeping up productivity of the workforce. In many
societies, access to healthcare and quality of treat-
ment outcomes depends upon socio-economic contexts.
At the same time, disease and sickness have been
identified as one of the most important reasons for
people turning into poverty (and even homelessness
in extreme cases) (Jamison, 2018). Fairness-relevant
discussions in this context include health insurance
systems on the societal level, and various ethical ques-
tions on the individual level, such as prioritization of
patients in the context of emergency rooms and organ
transplants (Daniels et al., 1996; Ding et al., 2019).
Transportation planning & policy - Trans-
portation infrastructure design determines how effec-
tively people and goods can move around. Fairness
arises as an important theme in the field of trans-
portation. Accessible, affordable, safe, inclusive, and
barrier-free transportation are the key aspects of dis-
cussion. The planning of transportation infrastructure,
such as roads, railways, and public transport are
important to enable an equitable access for as many as
possible, with positive outcomes for the economy and
life-quality. Often, transportation infrastructure faces
a demand which is higher than its supply, resulting in
congestion, and long waiting queues; policies for traf-
fic demand management such as congestion pricing are
part of highly controversial debates. Especially fair-
ness and equity-concerns are the major impediments
for the real-world implementation of transportation
policies (Martens, 2016; Gu et al., 2018).
2.3.3 Business administration
Management - Fairness in management is a critical
component of effective leadership and organizational
success. It encompasses the equitable treatment of
employees, transparent decision-making processes, and
the creation of inclusive work environments. When
managers prioritize fairness, they foster trust, boost
morale, and enhance overall productivity within their
teams. Consistent and unbiased treatment of employ-
ees, transparency in decision-making processes, equal
opportunities for growth and development, conflict
resolution, and compensation and recognition prac-
tices count amongst the most important fairness-
relevant aspects of management. By prioritizing equi-
table treatment, transparency, equal opportunities,
and impartial conflict resolution, managers can create
a work environment where all employees feel valued
and respected. This approach not only benefits indi-
vidual team members, but also contributes to the long-
term success and sustainability of the organization as
a whole (Simons and Roberson, 2003).
Recruiting & hiring processes - Ensuring fair-
ness in recruitment and selection not only promotes
diversity and inclusion, but also enhances the repu-
tation of the organization as an equitable employer,
7

increases the chance to find the most skilled workers
for a position, and improves organizational efficiency.
Procedural fairness, transparency on the recruiting
process, and bias awareness play an important role. To
account for diversity and inclusion, both the applicant
pool diversity and hiring outcome diversity are com-
mon measures to quantify fairness in this context. The
increased use of automated decision-making allow for
the consideration of larger number of applications, but
also harbor the threat for technological biases, that
need to be considered carefully (van den Broek et al.,
2020).
Banking & loan approval - Access to finan-
cial services, regardless of race, gender, residence, and
other factors, is important to achieve inclusion and
participation of customers. Historically, certain demo-
graphic groups, particularly those from marginalized
communities, have faced systemic barriers, especially
in obtaining loans. These barriers can include discrimi-
natory lending practices, biased credit scoring models,
and a lack of transparency in loan approval processes.
Addressing these issues is crucial for creating a more
equitable financial landscape and a greater trust of
the public in the industry. With the increasing use of
machine learning and algorithms in credit scoring and
loan approval, the concept of algorithmic fairness has
gained prominence. Algorithms can inadvertently per-
petuate biases present in historical data, leading to
unfair outcomes for certain groups. Therefore, various
fairness metrics and bias mitigation strategies have
been proposed to ensure that algorithms do not dis-
criminate against marginalized groups. While fairness
is crucial, financial institutions must also consider risk
management in their lending practices. Banks need
to assess the creditworthiness of applicants to mini-
mize default risks. However, this assessment should be
conducted in a way that does not disproportionately
impact certain groups. Striking a balance between
ensuring fair access to loans and managing financial
risk is a complex challenge that requires innovative
approaches and ongoing evaluation (Lee and Floridi,
2021).
2.3.4 Social Sciences
Social justice - Social justice is a concept that
encompasses the fair and equitable distribution of
resources, opportunities, and privileges within a soci-
ety. It is rooted in the principles of equality, human
rights, and collective responsibility. The pursuit of
social justice aims to address systemic inequalities
and promote a more inclusive and just society for
all individuals, regardless of their background or cir-
cumstances. At its core, social justice seeks to rectify
historical, ongoing, and self-reinforcing disparities in
areas such as economic inequality (e.g. rich vs. poor),
racial, and ethnic inequalities (e.g. white vs. black),
gender equality (e.g. men vs. women), or disability
rights. The pursuit of social justice often involves chal-
lenging existing power structures and advocating for
systemic changes. This can include policy reforms,
grassroots activism, education, and awareness cam-
paigns. Social justice movements have played a crucial
role in advancing civil rights, workers’ rights, and other
progressive causes throughout history (Harvey, 2010).
The digital age has brought new dimensions to social
justice efforts, with social media and online platforms
serving as powerful tools for organizing, raising aware-
ness, and amplifying marginalized voices. However, it
has also highlighted new challenges, such as digital
divides and online harassment (Eubanks, 2012).
Environmental & intergenerational justice
-A particular form of social justice discusses the
interplay between humans and their ecologic environ-
ment, as well as how social justice between generations
in the presence of demographic transformation and
climate change can be addressed. Environmental jus-
tice focuses on ensuring that all people, regardless
of race, color, national origin, or income, have equal
protection from environmental and health hazards. It
emerged as a social movement in the United States
in the 1980s, highlighting how marginalized commu-
nities often bear a disproportionate share of envi-
ronmental risks. This movement has since expanded
globally, addressing issues such as hazardous waste
disposal, resource extraction, and land use that neg-
atively impacts vulnerable populations (Mohai et al.,
2009). Intergenerational justice extends this concept
across time, emphasizing the responsibility of current
generations to preserve the environment for future
ones. It recognizes that today’s actions have long-term
consequences that will affect the quality of life and
opportunities available to future generations (Barry,
1997).
Sports & competitions - Fairness is a funda-
mental principle in sports and competitions, serving as
the foundation for meaningful and equitable contests.
Equal opportunity in the sense of skill, effort, and
merit rather than external advantages, is at the core
of fairness in sports. Sport organizations strive to cre-
ate rules and structures that give competitors an equal
8

chance to succeed based on their abilities and prepa-
ration. Fairness also involves adherence to established
rules and regulations. These rules define the bound-
aries of acceptable behavior and performance, ensuring
that all participants compete under the same con-
ditions. Enforcement of rules, including anti-doping
measures, helps maintain the integrity of competi-
tions and prevents unfair advantages. The principle
of fair play extends beyond just following rules. It
values sportsmanship, respect for opponents, and eth-
ical behavior both on and off the field. This broader
concept of fairness contributes to the overall positive
culture and values associated with sports. Fairness in
sports also has implications beyond individual com-
petitions. It plays a crucial role in maintaining public
trust and interest in sports. When spectators believe
competitions are fair, it enhances the excitement and
credibility of the events, contributing to the overall
appeal and sustainability of sports (Loland, 2010).
2.3.5 Government & policy
Policy making - Fairness is a critical considera-
tion in policy development and implementation across
various domains of governance and public administra-
tion. Procedural fairness is essential in policy creation
and enforcement. Policies should be developed through
transparent processes that allow for public input and
stakeholder consultation. When implementing poli-
cies, authorities must apply rules consistently and
provide clear explanations for decisions. Policymak-
ers must consider how different groups are impacted
and strive for outcomes that are perceived as just.
This may involve targeted interventions to address
historical inequities or support vulnerable popula-
tions. Policies should aim to promote equality-of-
opportunity while recognizing that strict equality of
outcomes is not always achievable or desirable. Anti-
discrimination policies, for example, seek to ensure
fair access and treatment across gender, racial, and
other lines. Evidence-based policy-making can help
promote fairness by relying on objective data rather
than subjective biases. Perceptions of fairness can vary
across cultural contexts and change over time. Pol-
icymakers must remain attuned to evolving societal
values and engage in ongoing dialogue with diverse
communities (Gilley, 2017).
Legal system & criminal justice - Fairness
in the legal system is a principle that underpins the
rule of law and ensures that justice is administered
equitably and impartially. This principle is crucial for
maintaining public trust and confidence in the legal
system. Procedural fairness in this context includes the
right to a fair hearing, and the right to an impartial
decision-maker. In criminal cases, procedural fairness
is essential to protect the rights of the accused, includ-
ing the presumption of innocence until proven guilty,
the right to legal representation, and the right to a
fair trial. Ensuring that these procedures are followed
helps to prevent miscarriages of justice and upholds
the integrity of the legal system. Equality before the
law is another aspect of fairness in the legal system.
This principle asserts that all individuals, regardless of
their background, are subject to the same laws and are
entitled to equal protection under the law. This means
that the legal system must be free from discrimination
based on race, gender, ethnicity, socioeconomic status,
or any other characteristic. Achieving true equality
before the law requires ongoing efforts to address sys-
temic biases and ensure that legal processes do not
disproportionately disadvantage certain groups (Berk
et al., 2021; Hurwitz and Peffley, 2005).
Police operations - Fairness plays an impor-
tant role for policing. Procedural fairness influences
both the public’s perception and acceptance of law
enforcement, and the internal dynamics, efficiency and
cohesion within police organizations. When police offi-
cers engage in fair procedures, such as treating people
with dignity, providing explanations for their actions,
and being neutral in decision-making, they are more
likely to be perceived as legitimate authorities. This
perception of legitimacy is crucial because it fosters
public trust and cooperation with law enforcement
efforts, even when the outcomes are not favorable
to the individuals involved. A study involving New
York City residents found that procedural fairness
was a key antecedent of police legitimacy, which in
turn influenced people’s willingness to comply with
the law (Sunshine and Tyler, 2003). Racial profil-
ing in law enforcement is a complex and contentious
issue that highlights the tension between fairness and
perceived effectiveness in policing. At its core, racial
profiling uses race or ethnicity as a key factor in
deciding whether to stop, search, or investigate indi-
viduals. While proponents argue it can be an efficient
crime prevention tool, racial profiling raises serious
concerns about discrimination, civil rights violations,
and the erosion of public trust in law enforcement.
Racial profiling is controversial because it judges indi-
viduals based on group characteristics rather than
individual behavior or evidence. This can lead to
9

a self-fulfilling prophecy, where increased investiga-
tion of certain groups results in higher findings and
arrest rates, which are then used to justify further
profiling (Hurwitz and Peffley, 2005).
2.4 Algorithmic Fairness
Algorithms can be used to formalize and automate
decision-making processes. An algorithm is a finite
sequence of well-defined steps to process inputs and
produce outputs. There are two sources that can be
used to define algorithms: First, algorithms can be for-
malized by humans, that specify inputs, outputs, and
process-steps manually to mimic their own decision-
making. Second, the advances in computer science,
especially in supervised machine learning, allowed
computers to derive algorithms directly from data
(input-output pairs). Cybernetic systems use algo-
rithms to control systems, and they cannot only be
used by computers, but also by human-driven pro-
cesses, such as governments that follow bureaucratic
and legal processes, and by physical machines.
The fairness of algorithms massively depends on
the choice of inputs, outputs, process-steps, and the
source of the algorithm. Algorithmic biases can lead
to systematic and repeatable errors and discrimination
of certain individuals or groups, and cause algorithmic
unfairness. Algorithmic unfairness in cybernetic sys-
tems can lead to self-reinforcing cycles of inequality,
take racial profiling as an example:
Algorithmic unfairness in cybernetic system, such
as group discrimination that is even justified by data,
can be harmful, as the Hirshleifer-effect (Hirshleifer,
1978) in the context of credit scoring illustrates:
Assuming that discriminating a certain group when
deciding on loan approvals is justified by data and
increases efficiency of decisions (improved default-risk
management), and as a consequence increases profits
for the banks and enables lower loan fees for the other
customers. While using more data and develop better
algorithms seems like a good idea on the individual
firm level for banks and insurances, it might be harm-
ful on a societal level. The Hirshleifer-effect describes
the paradoxical situation, that the release of more and
better information to the public (market) mutually
prevents beneficial risk-sharing trades, leading to a sit-
uation where all agents are worse off compared to a
situation where fewer or incomplete information was
available.
Algorithms with biases often stem from imbal-
anced data, where certain groups may be underrepre-
sented or misrepresented, leading to unfair outcomes.
Addressing these issues involves various strategies,
including data preprocessing to balance datasets, in-
process adjustments to algorithms during training,
and post-process evaluations to ensure equitable out-
comes (Pessach and Shmueli, 2023). The challenge of
imbalanced data is particularly pronounced in fields
like healthcare, where disparities in data collection can
exacerbate existing inequalities in treatment and diag-
nosis. For example, the design and development of
medical treatments have historically exhibited a gen-
der bias, often optimizing for male physiology at the
expense of female patients (Agyemang et al., 2023).
2.5 Transparency and Explainable AI
Increasingly, automated decision-making processes
employ deep-learning with neural network models
for machine learning, when developing automated
decision-making algorithms. These models have an
ever-growing number of parameters and thus com-
plexity, which allows them to achieve super-human
performance (for example in image recognition and
classification). At the same time, these models are
highly problematic, as due to their complexity it is
challenging to understand how and based on which
specific properties and features from the data, decision
are made. Transparency and traceability of decision-
making processes are a crucial component to procedu-
ral fairness. As a consequence, the application of even
better performing models is impeded in industry- and
governmental applications due to explainability con-
cerns, the often unclear origin of training data and
process, the inability to describe guarantees in terms
of safety, and equity issues (Xu et al., 2019).
3 Quantitative Fairness
Framework
The quantitative fairness framework aims to pro-
vide a holistic tool-set to quantitatively assess the
fairness of situations and hence make fairness accessi-
ble to algorithmic design in general. This framework
distinguishes itself from previous approaches, by its
general applicability, integrative and ideology-agnostic
approach, and its multi-perspective view including
teleological and deontological considerations. Rather
than advocating a specific ideology, the framework
enables an integrative analysis that combines different
10

Principle Perspective Metric Optimization
Difference Ex-Post average or minimum of yimaximize
Equality Ex-Post dispersion of yiminimize
Equality-of-opportunity Ex-Ante dispersion of ximinimize
Greater-good Ex-Post sum of uimaximize
Proportion Both dispersion of ratio yi/ximinimize
Sufficiency Ex-Post threshold share of yimaximize
Table 1 Dianemetic Fairness Measures.
Statistical dispersion and economic concentration metrics are used to assess the equality of distributions. The formulas cover the
most important measures and are denoted to determine the equality of a distribution of nvalues xi, where ¯xrepresents the
average over all xi.
Principle Perspective Welfare Function
Difference Ex-Post Rawlsian welfare function
Equality Ex-Post Sen and Foster welfare function
Equality-of-opportunity Ex-Ante dispersion as Leontief-Lerner function
Greater-good Ex-Post Benthamite welfare function
Proportion Both none or dispersion of ratios yi/xi
Sufficiency Ex-Post threshold-share of yi
Table 2 Diorthotic Fairness Measures.
Statistical dispersion and economic concentration metrics are used to assess the equality of distributions. The formulas cover the
most important measures and are denoted to determine the equality of a distribution of nvalues xi, where ¯xrepresents the
average over all xi.
guiding principles and allows for systematic compar-
ison. In this section, we model decision-making pro-
cesses as resource allocation mechanisms, and develop
quantitative fairness metrics based on a transactional,
distributive fairness discussion.
3.1 Modelling as resource allocating
process
We model any decision-making system as a pro-
cess with an input x(initial situation), an output y
(resource allocation), and differentiate the discussion
of distributive fairness in the Nicomachean, trans-
actional view as diorthotic and dianemetic fairness.
Arguably, any decision-making system in some form
uses information about an initial situation to make an
informed decision, and the outcome of any decision is
related in some form to the distribution of resources,
be it useful goods, opportunities to be selected, or
burdens in retributive contexts.
While we value the importance of the transparency
and chance aspect of procedural fairness in this con-
text, we advocate that a distributive fairness discus-
sion is more instrumental when designing algorithms
for decision-making. We acknowledge the challenge of
a unifying definition and guiding principle for fairness,
and therefore advocate the use of an integrative frame-
work, that has the capacity to reflect multiple ideology,
rather than to focus on dedicated ideologies.
As a result, we provide a framework that is able to
reflect following guiding principles: equality, propor-
tion, greater-good, difference, equality-of-opportunity,
and sufficiency. As not all of these ideologies combine
the ex-post and ex-ante perspective on fairness, some
will require input and / or output information. Addi-
tionally, following the Utilitarian ideology based on
the greater-good principle, we denote the utility uas
a function fof y:u=f(y). We consider a population
with nindividuals, that each experience a treatment of
the decision-making system, based on their individual
input xi, output yi, and utility ui.
In the following parts of this section we propose the
use of statistic and dispersion measures for the quan-
tification of dianemetic fairness, and the use of dedi-
cated social welfare functions for the quantification of
diorthotic fairness.
11

Metric Reference Formula
Atkinson-Index Atkinson et al. (1970) Aϵ=



























1−1
x(1
nPn
i=1 x1−ϵ
i)1/(1−ϵ)
for 0≤ϵ < 1
1−1
x(Πn
i=1xi)1/N
for 0≤ϵ < 1
1−1
xmini(xi)
for ϵ = +∞
Gini-Coefficient Dorfman (1979) G=Pn
i=1 Pn
j=1 ∥xi−xj∥
2nPn
i=1 xi
Herfindahl-Index Herfindahl (1950) HHn=HH −1
n
1−1
n
, HH =Pn
i=1(xi
Pn
j=1 xj)2
Hirschmann-Index
Hoover-Index Hoover (1936) H=1
2
Pn
i∥xi−x∥
Pn
ixi
Palma-Index Palma (2011) P=R40%
0% L(x)dx
R100%
90% L(x)dx
Standard Deviation σ=1
nPn
i=1(xi−x)2
Theil-Index T Theil (1965) TT=1
nPn
i=1
xi
xln(xi
x)
Theil-Index L Theil (1965) TL=1
nPn
i=1 ln(x
xi)
Table 3 Dispersion metrics.
Statistical dispersion and economic concentration metrics are used to assess the equality and thus dispersion of distributions. The
formulas cover the most commonly used measures and are denoted to determine the equality of a distribution of nvalues xi,
where ¯xrepresents the average over all xi.
3.2 Metrics for dianemetic fairness
The suggested quantitative metrics for dianemetic fair-
ness for different guiding-principles are summarized in
Table 1. Following notes shall be mentioned in addition
to Table 1:
•As a note for the difference principle, the goal is
the maximization of the average (Harsanyian) or
minimum of any y(Rawlsian).
•As a note for the sufficiency principle, the goal is
the maximization of the share of all individuals for
which the outcome exceeds a certain threshold T
(yi> T).
•An Utilitarian interpretation of the difference,
equality, proportion, and sufficiency principle could
involve uinstead of yas well. Similarly, the greater-
good principle could also involve yinstead of u.
For equality, equality-of-opportunity, and propor-
tion principle, the use of dispersion metrics is sug-
gested. We provide a compilation of common disper-
sion metrics in Table 3. Some of them originate from
statistics, where dispersion is a common term, while
others originate from economics, where concentration
is a common term.
3.3 Metrics for diorthotic fairness
We advocate the use of Welfare functions as quan-
titative metric for diorthotic fairness, as summarized
in Table 2 for different guiding-principles. Welfare
12

Title Formalization
Leontief-Lerner W=F(x1, ..., xn)
Bergson-Samuelson W=F(u1, ..., un)
Isoelastic W=1
1−ρPn
i=1 αiu1−ρ
i
Benthamite W=Pn
i=1 ui
Rawlsian W= minn
i=1 ui
Bernoulli-Nash W= Πn
i=1αiui
Capability-approach W=F(y1, ..., yn)
Sen W=y(1 −G)
Foster W=ye−TT
Table 4 Welfare functions.
Social welfare functions aim to define the social
welfare Wof resource allocations, where xdescribes
the input to the resource allocation process (e.g.
financial wealth), ydescribes the output of the
resource allocation process (e.g. allocated resources),
and udescribes the utility of the allocated resources
to the individuals. There are three types of welfare
functions: Leontief-Lerner, Bergson-Samuelson, and
Capability-approach welfare functions.
functions are a formal representation of individual
notions of welfare aggregated on a societal level.
Leontief-Lerner welfare functions describe welfare as
a function of available resources. Bergson-Samuelson
welfare functions describe welfare as a function of indi-
vidual utilities. Capability-approach welfare functions,
describe welfare as a function of incomes.
Table 4 shows different formalizations of the three
presented groups of welfare functions. The isoelas-
tic welfare function is a general formalization of the
Bergson-Samuelson welfare function, with the possi-
bility to weight each individuals utility with αi; for
ρ= 0 it becomes the Benthamite welfare function in
the sense of utilitarian fairness; for ρ= +∞it becomes
the Rawlsian welfare function in the sense of the Rawl-
sian fairness; for ρ→1 it becomes the Bernoulli-Nash
welfare function; for any other ρit is an intermediate
welfare function between the extremes of utilitarian
and Rawlsian fairness. Sen and Foster apply disper-
sion metrics such as the Gini-Coefficient Gand the
Theil-Index TT.
Following notes shall be mentioned in addition to
Table 2:
•The Harsanyian ideology based on the difference
principle could be reflected by the average of xias
Bergson-Samuelson Welfare function.
•The Sen and Foster Welfare function are not only
concerned with equality, but also with the average
yi. In this context, the welfare functions not only
capture relative distribution but also absolute dis-
tribution of resources. For example, this implies that
if a slightly more unequal distribution of resources
could increase the average resource received, this
would be preferred over perfect equality. This could
make sense in the context of resources, that deteri-
orate by splitting them into many small pieces.
•For the equality-of-opportunity principle could
employ dispersion metrics as Leontief-Lerner func-
tions (based on the inputs xi. Contrary to the equal-
ity principle, only the relative distribution would
play a role here, as the decision-making process is
assumed to affect the output only, but not to affect
the input.
•For the proportion principle one could use the dis-
persion of ratios as Welfare function. The proclaimer
of this principle, Aristotle, argued that any market
activity and resulting prices and outcomes are fair
per se, if these are the result of free decisions of
individuals.
•For the sufficiency principle one could use the
threshold-share of yias Welfare function.
4 Discussion
In this section we want to showcase the usage of
the proposed, quantitative fairness framework at two
exemplary case studies. The fair cake-cutting prob-
lem is used to illustrate dianemetic fairness. The daily
work of two fishermen is used to illustrate diorthotic
fairness. Afterwards, we discuss the benefits of the
proposed framework in terms of holistic, ideology-
agnostic, transitive, and quantitative discussions, and
provide some further notes on the complexities of the
equality-of-opportunities guiding principle.
4.1 Cake-cutting problem &
dianemetic fairness
The fair cake-cutting (Brams and Taylor, 1996) is a
typical division problem discussed in economics and
dianemetic, distributive justice. There are different
types of problems, including whether only the size
of the cake piece matters (homogeneous goods) or
13

The Cake The Individuals
Individual
Preferences
Shape Flavour
☆ ♡ ○ ●
Contribution
A 0.9
0.1
0.1 0.0 0.1
B 0.1
-0.1
0.1 0.1 0.2
60%
20%
20%
3
12
The Quantitative Analysis
1 2 3
1 2
2 3
1 3
1
2
3
1
2
2 3
1
1 2
1 2 3
3
3
1
2
3
4
5
6
7
8
A gets B gets
Resource Allocation Situation for A
XYU
0.9
0.9
0.9
0.9
0.9
0.9
0.9
0.9
1.0
0.8
0.4
0.8
0.6
0.2
0.2
0.0
1.4
1.1
0.7
1.0
0.7
0.4
0.3
0.0
Situation for B
XYU
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.0
0.2
0.6
0.2
0.4
0.8
0.8
1.0
0.0
0.4
0.6
0.5
0.9
1.0
1.1
1.5
Quantitative Fairness Assessment
Difference
Equality
Equality-Of-
Opportunity
Greater-Good
Proportion
Sufficiency
0.00
0.40
0.60
0.50
0.70
0.40
0.30
0.00
0.99
0.49
0.07
0.35
0.14
0.42
0.57
1.07
0.56
0.56
0.56
0.56
0.56
0.56
0.56
0.56
1.40
1.50
1.30
1.50
1.60
1.40
1.40
1.50
1.09
1.96
3.69
2.75
5.81
6.76
7.54
10.61
0.50
0.50
1.00
1.00
1.00
0.50
0.50
0.50
Fig. 4 Case Study: Fair cake-cutting problem & dianemetic fairness.
A given cake that can be cut into three heterogeneous pieces only shall be distributed across two individuals, that differ in their
contributions to paying or making the cake, and their preferences on the toppings. Eight allocations of the three pieces to the
two agents are possible. Different guiding principles on fairness result in different, fairness-optimal recommendations for how to
distribute the cakes.
also other features such as the toppings (heteroge-
neous goods), whether the cake can be cut every-
where (fully-divisible) or only to specific discrete pieces
(partially-divisible), and whether the times of cutting
creates waste of no use for anyone (non-lossy vs. lossy
division).
The fair cake-cutting problem is a metaphor for
various dianemetic, distributive fairness contexts:
•Imagine a manager that must distribute projects
to staff; staff members can usually be staffed fully
on one project only, and personal preferences and
interests will affect how much staff members can
learn from or are willing to invest into a project.
Usually, it is the manager’s decision, and therefore
a dianametic context. This could be an example
for a heterogeneous, partially-divisible cake-cutting
problem.
•Imagine a government that tries to stimulate its
stagnating economy during a recession with subsi-
dies, tax-reductions, or cash subsidies amongst its
population. How should different parts of the popu-
lation (or companies) benefit from this support pro-
gramme, e.g. it could make sense to support family
households more than single households. This could
be an example for a homogeneous, fully-divisible
cake-cutting problem.
•Imagine a traffic signal controller at an intersection.
It distributes delays to movement phases (respec-
tively green time for passage). Every time there
is a transition from one green movement phase to
another, there is a short amount of time where both
phases are red for security. Transitioning is impor-
tant so that queues and waiting times do not get too
long, but too frequent transitions will cause wasted
time, where no vehicle can pass the intersection.
How should the green time be divided to different
movement phases? This could be an example for a
lossy, fully-divisible cake-cutting problem.
For this case study, let us consider a rectangular
cake with chocolate toppings as displayed in Fig. 4.
The toppings have two features: shape (stars and
hearts), and flavour (white and dark chocolate). The
14

Two Fishermen & The Pond
Fisher (Catching rate)
Capabilities
(Work Hours)
Contribution
A 8
Equal chance 0
–6 fishes
B12
Chance of 100% to catch 5
The Quantitative Analysis
Fig. 5 Case Study: Fishermen & diorthotic fairness.
Two fishermen go fishing at a pond every day, and at this specific day they fish seven fishes together. They differ in the working
hours and catching rates due to different fishing techniques. Their outcome is stochastic. How should they distribute the total fish
catch per day? Assuming fish is divisible, there is a continuous allocation space of the fish ranging from zero to seven fishes for each
fisher. The Pareto-efficient frontier (blue line) displays this solution space. Different guiding principles on fairness result in different
heatmaps and social welfare functions (contour plots of the heatmaps), with different recommendations on how to distribute the
daily fish catch.
cake can only be cut into three pieces. This cake must
be distributed to two individuals A and B, that differ
in their contributions to making or paying the cake,
and preferences on topping features. They have in
common, that they value the amount of cake similarly.
The individual’s utility function for the received
pieces of cake is the sum of three components:
•amount of cake (full cake equals one unit of utility)
•utility points according to topping shape
•utility points according to topping flavour
There are eight different ways to distribute the
three pieces amongst the two individuals (assuming
that no piece is wasted). The quantitative, dianemetic
fairness measures enables the objective analysis of
this problem, and to fairness-optimally allocate the
resource amongst the individuals. Following measures
are used for the different guiding principles: mini-
mum utility (difference principle), standard deviation
of utility (equality principle), standard deviation of
contributions (equality-of-opportunity principle), sum
of utilities (greater-good principle), standard devia-
tion of utility-contribution ratios (proportion princi-
ple), threshold share of utility, with 0.50 utility as a
sufficient minimum utility (sufficiency principle).
Let us discuss one of the eight possible allocations
to outline the calculations. In this case study, we leave
the contributions constant, meaning A contributes 0.9
X to the cake, and B contributes 0.1 X to the cake.
Assume A gets piece 1 and 3, and B gets piece 2 (allo-
cation scenario 4), then A receives 0.8 Y, B receives 0.2
Y. The utility the individuals experience based on the
15

cake they get depends on the amount Y and the top-
pings. A experiences a utility of U= 1.0=0.8+2 ∗0.1,
as A receives 0.8 amount of cake, and additionally two
toppings in heart and star form which both add 0.1
up to the utility; as A is interested in dark chocolate
only but pieces 1 and 3 are of white chocolate, no addi-
tional utility can be generated for A. B experiences a
utility of U= 0.5=0.2 + 0.2 + 0.1, as B receives 0,2
amount of cake, and additionally one topping in heart
form which adds 0.1 up to the utility, and the topping
is made of dark chocolate which adds 0.2 up to the
utility.
The quantitative fairness assessment of the differ-
ent resource allocation scenarios reveals different levels
of fairness depending on the different guiding prin-
ciples. Following the difference principle, scenario 5
achieves the highest worst case with a utility of 0.7.
Following the equality principle, scenario 3 achieves
the lowest inequality / dispersion (standard deviation)
in utilities. A discussion of the equality-of-opportunity
in this case study might not be relevant, as we did not
provide details on the reasons for contributions. Fol-
lowing the greater-good principle, scenario 5 achieves
the highest total utility. Following the proportion prin-
ciple, scenario 1 achieves the highest alignment with
contributions. Following the sufficiency principle, sce-
narios 3-5 provide a sufficient minimum experienced
utility for both individuals.
These differing ratings for fairness by the different
guiding principles highlight the conflicting differences
in understanding of what fairness is. Scenario 5 for
instance, achieves the best situation in total, and even
for the poorest, while it is twice as unequal when
compared to scenario 3. From a purely merits based
perspective, A should receive everything as in scenario
1, as A contributes almost everything for buying the
cake.
4.2 Fishermen & diorthotic fairness
Let us discuss diorthotic fairness at the example of
the daily work of two fishermen, as shown in Fig. 5.
Assume there is a village with two fishermen, that go to
the same pond for fishing every day. The fishermen dif-
fer in their capabilities due to experience and catching
methods they use, but also in their working attitude
and willingness to work. Should each fisherman keep
what they catched, or should the village agree on some
form of legislation to distribute the daily catch ”fairly”
to both fishermen?
This example is a metaphor for various diorthotic
fairness contexts:
•Imagine a nation’s labor economy. People have
different capabilities to earn an income, different
chances depending on the economic situation of
markets, exports, industries, and also different levels
of discipline and perseverance. A government could
apply an income taxation to redistribute wealth
amongst the individuals of the labor force. This
could create an overall healthier economy, as work-
ers with bad luck or health in one season can still
survive based on the solidarity support of others.
However, this could also create a demotivation of the
top performers, as higher achievements mean higher
contributions to the others, that might not work as
hard as them.
•Imagine a nation’s public health system. People
have different chances to get sick based on genetic
and socio-economic dispositions that they cant
affect, but also do they make life choices that lead to
different health-affecting life-styles (e.g. smoking).
Should the government enforce a mandatory health
insurance system, so that everyone pays into the sys-
tem, even when not using it? The solidarity could
lead to less extreme poverty, as often severe diseases,
accompanied by job loss and divorce, lead to home-
lessness. However, many diseases could be prevented
by a healthier life style, and a governmentally-
enforced safety net could incentivize a free-riding
behaviour, in which individuals have even less incen-
tivizes to take care of them, as they can get cured
on the costs of others. If such a public health system
and insurance would be implemented, how should
people pay for it? Based on their income, based on
their lifestyle, or even based on their genetic and
socio-economic dispositions?
For this case study, let us consider two fishermen, A
and B. Fisherman A works 8 hours per day, and has an
equal chance to find 0 to 6 fishes per day. This means,
on average, A catches 3 fishes per working day, and
0.375 fishes per hour. Another fisherman B works 12
hours per day, and due to his experience and catching
method, catches exactly 5 fishes per day. This means,
on average B catches 5 fishes per working day, and
0.417 fishes per hour.
Let us assume one day, A catches 2 fishes, and B
catches 5 fishes, so there are 7 fishes in total to be dis-
tributed. As fish is a perishable good, it does not make
sense to not allocate all fishes, so the sum of all fishes
16

of A and B (5+2=7) forms the Pareto-efficient fron-
tier. Moreover, each fisher has the same utility for the
amount of fish received, however when walking back
home, fisher A looses around 5%, and fisher B looses
around 15%.
Fish can be considered as a divisible good, one
could assume continuous distribution based on weight.
This means there is an infinitely large solution space
for allocations along the Pareto-efficient frontier. The
quantitative, diorthotic fairness measures enable us
to objectively analyse this problem and to fairness-
optimally allocate the resources. Following measures
are used for the different guiding principles: Rawlsian
welfare function (difference principle), Foster wel-
fare function (equity principle), standard deviation of
working time (equality-of-opportunity principle), sum
of all utilities (greater-good principle), standard devi-
ation of received fish-working time ratios (proportion
principle), threshold share of received fish, with 2 units
as a sufficient minimum (sufficiency principle).
The quantitative fairness assessment of the differ-
ent resource allocation scenarios reveals different levels
of fairness depending on the different guiding principle
of fairness. Following the difference principle, it would
be most fair to equally distribute the catched fish,
meaning both fishermen get 3.5 units of fish. Following
the equality principle, it would be most fair to equally
distribute as well. A discussion of the equality-of-
opportunity in this case study might not be relevant,
as we did not provide details on the reasons for work-
ing times. It might be, that A is older and cannot
work that long any more as B does. In this case, one
would need to discuss ways to account for that. Follow-
ing the greater-good principle all fish should be given
to A, as A can bring most of fish home where it can
actually be cooked, while giving fish to B would gen-
erate slightly more waste. If the greater-good principle
would be applied to resources y rather than utilities,
then the greater-good principle would be indifferent on
any allocation along the Pareto-efficient frontier. Fol-
lowing the proportion principle, A should receive 2.8
and B should receive 4.2 units of fish, according to
their contributed working time. One could argue, that
not working time, but actual contribution matters as
Aristotle argues; in this case each fisher should keep as
much as they catched without sharing. The sufficiency
principle would be indifferent to all allocations along
the Pareto-efficient front between 2 to 5 fishes for A
resp. B.
The different social welfare functions exhibit differ-
ent shapes. Difference and equality principle share the
same optima and similar gradients. The greater-good
principle is almost parallel, and the proportion prin-
ciple is even almost orthogonal to the Pareto-efficient
front. The sufficiency principle rather distributes the
solution space into distinct areas.
4.3 Holistic, ideology-agnostic fairness
discussions
The proposed quantitative framework purposely does
not advocate one over another guiding principle, but
rather enables a holistic, and integrative analysis.
Deviating from the fairness-optimal allocations from
one guiding principle by just a bit, can achieve sig-
nificant improvements form the perspective of another
guiding principle. A compromise solution might be
derived based on the preferences and weights the deci-
sion maker provides to the different guiding principles.
Another way could be: rankings of all alternatives
based on the different guiding principles could be cre-
ated, and then aggregated to a combined ranking for
a final decision making.
4.4 Transitive and quantitative, rather
than transcendental and normative
fairness discussions
Contrary to previous, rather transcendental, quali-
tative and mostly normative, philosophic works, our
framework can be used to analyse situations from a
more transitive, and quantitative perspective. Not only
can two situations be compared to decide which one
is more fair (transitive), but also can the framework
be used to assess how much more fair it is (quanti-
tative). This enables a more systematic discussion of
the deviation from strict optima, and encourages an
inclusive discussion that allows for the combination
of different goals, including different fairness and effi-
ciency definitions. Similar to quantitative definitions
of efficiency, quantitative definitions of fairness can be
used as a goal metric for the design and optimization
of algorithms.
4.5 Equality-of-opportunities guiding
principle and fair resource
allocating algorithms
The equality-of-opportunities guiding principle is
related to the chance aspect of procedural fairness
and a guiding principle for the distributive fairness. In
the case studies we excluded a discussion, as further
17

assumptions must be taken. Besides, the question that
needs to be answered for a discussion from this guid-
ing principles perspective includes how the resource
allocation can affect the opportunities and chances
individuals have.
If there is a clear relationship between the allocated
resources in a cycle, and the chances of individuals in
the next cycles, then equality-of-opportunity is clearly
relevant to discussing the allocation of resources. For
instance, one could assume that having more money at
the beginning of market opening will allow to generate
more trading profits, which will then enable even more
chances and opportunities at the beginning of the next
day. This could be considered as a feedback loop, and
therefore redistribution of shares of profits that are
due to luck rather than capabilities or efforts, might
be considered as fair, when they enable more chances
for everyone else.
If there is no clear relationship between the allo-
cated resources and the chances, a different discussion
is necessary (e.g. how much fish you get as fisherman
in the case study wont affect how much fish you can
get on the next day). One could rather focus on which
decision criteria an algorithm uses, or how it weights
different inputs to the decision making. Doing so, one
could aim for inputs that reflect more equal opportu-
nities for individuals to participate and actually affect
the outcome.
Rather than focusing on pure contribution of mak-
ing or buying the cake in the dianemetic case study
above, one could try to adjust and normalize the inputs
for capability (how much pocket money do you have
available to pay for the cake) or ability (how much
knowledge and tools do you have to make a cake).
Rather than focusing on pure contribution of fish, or
working time in the diorthotic case study above, one
could try to adjust and normalize these inputs for
capability (catching rate of fish) or ability (age, gender,
size), to better reflect the pure willingness to work.
5 Conclusion
This work set out to propose an useful, holistic, quan-
titative, transactional, distributive fairness framework,
which enables the systematic design of socially-feasible
decision-making systems in the context of equitable
cybernetic societies.
After the review of distributive and procedural fair-
ness theories from relevant literature and domains,
algorithmic fairness and the importance of trans-
parency and explainable AI were elaborated.
The proposed quantitative fairness framework
offers measures for dianemetic and diorthotic fairness
discussions based on statistic metrics, dispersion met-
rics, and social welfare functions. Two case studies
on fair cake-cutting and fishermen demonstrate the
usefulness and flexibility of the proposed framework.
Future work could focus not only on situational
quantification of fairness at a specific time, but to
include a temporal component for repeated settings.
For example, forms of aggregation over many itera-
tions of the same algorithm could be part of investiga-
tion. A useful way could be the probabilistic, stochastic
discussion of the distributive effects of algorithms.
Declaration of competing interest. None.
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