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paper/references.bib
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@ -73,6 +73,18 @@
number = {2} number = {2}
} }
@article{bramsApprovalVoting1978,
title = {Approval {{Voting}}},
author = {Brams, Steven J. and Fishburn, Peter C.},
year = {1978},
month = sep,
volume = {72},
pages = {831--847},
abstract = {Approval voting is a method of voting in which voters can vote for (``approve of'') as many candidates as they wish in an election. This article analyzes properties of this method and compares it with other single-ballot nonranked voting systems. Among the theorems proved is that approval voting is the most sincere and most strategyproof of all such voting systems; in addition, it is the only system that ensures the choice of a Condorcet majority candidate if the preferences of voters are dichotomous. Its probable empirical effects would be to (1) increase voter turnout, (2) increase the likelihood of a majority winner in plurality contests and thereby both obviate the need for runoff elections and reinforce the legitimacy of first-ballot outcomes, and (3) help centrist candidates, without at the same time denying voters the opportunity to express their support for more extremist candidates. The latter effect's institutional impact may be to weaken the two-party system yet preserve middle-of-the-road public policies of which most voters approve.},
journal = {American Political Science Review},
number = {3}
}
@article{brandlFundingPublicProjects2020, @article{brandlFundingPublicProjects2020,
title = {Funding {{Public Projects}}: {{A Case}} for the {{Nash Product Rule}}}, title = {Funding {{Public Projects}}: {{A Case}} for the {{Nash Product Rule}}},
author = {Brandl, Florian and Brandt, Felix and Peters, Dominik and Stricker, Christian and Suksompong, Warut}, author = {Brandl, Florian and Brandt, Felix and Peters, Dominik and Stricker, Christian and Suksompong, Warut},

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paper/termpaper.tex
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@ -42,9 +42,10 @@
preference elicitation with an aggregation method, a set of winning projects preference elicitation with an aggregation method, a set of winning projects
is determined and funded. This paper first gives an introduction into is determined and funded. This paper first gives an introduction into
participatory budgeting methods and then focuses on approval-based models to participatory budgeting methods and then focuses on approval-based models to
discuss algorithmic complexity. Furthermore, a short overview of useful discuss algorithmic complexity. Furthermore, this work presents a short
axioms that can help select one method in practice is presented. Finally, an overview of useful axioms that can help select one method in practice. The
outlook on future challenges surrounding participatory budgeting is given. paper concludes with an outlook on future challenges surrounding
participatory budgeting.
\end{abstract} \end{abstract}
\section{Introduction} \section{Introduction}
@ -73,26 +74,26 @@ stages \autocite{participatorybudgetingprojectHowPBWorks}:
set of winning projects which are then funded. set of winning projects which are then funded.
\end{description} \end{description}
\noindent The two last stages \emph{voting} and \emph{aggregating votes} are of \noindent The last two stages \emph{voting} and \emph{aggregating votes} are of
main interest for computer scientists, economists and social choice theorists main interest for computer scientists and economists because depending on how
because depending on how voters elicit their preferences (\emph{balloting} or voters elicit their preferences (\emph{balloting} or \emph{input method}) and
\emph{input method}) and how the votes are aggregated through the use of how the votes are aggregated through the use of algorithms, the outcome is
algorithms, the outcome is different. To study different ways of capturing votes different. To study different ways of capturing votes and aggregating them, the
and aggregating them, the participatory process is modeled mathematically. This participatory process is modeled mathematically. This model will be called a
model will be called a participatory budgeting \emph{scenario}. The aim of participatory budgeting \emph{scenario}. The aim of studying participatory
studying participatory budgeting scenarios is to find ways to achieve a budgeting scenarios is to find ways to achieve a desirable outcome. A desirable
desirable outcome. A desirable outcome can be one based on fairness by making outcome can be one based on fairness by making sure that each voter has at least
sure that each voter has at least one chosen project in the final set of winning one chosen project in the final set of winning projects for example. Other
projects for example. Other approaches are concerned with maximizing social approaches are concerned with maximizing social welfare or discouraging
welfare or discouraging \emph{gaming the voting process} (where an outcome can \emph{gaming the voting process} (where an outcome cannot be manipulated by not
be manipulated by not voting truthfully; also called \emph{strategyproofness}). voting truthfully; also called \emph{strategyproofness}).
First, this paper will give a brief overview of common methods and show how a First, this paper will give a brief overview of common methods and show how a
participatory budgeting scenario can be modeled mathematically. To illustrate participatory budgeting scenario can be modeled mathematically. To illustrate
these methods, one approach will be chosen and discussed in detail with respect these methods, one approach will be chosen and discussed in detail with respect
to algorithmic complexity and properties. Finally, the gained insight into to algorithmic complexity and properties. Finally, the conclusion will summarize
participatory budgeting algorithms will be summarized and an outlook on further the gained insight into participatory budgeting algorithms and will give an
developments will be given. outlook on further developments and research directions.
\section{A Participatory Budgeting Framework} \section{A Participatory Budgeting Framework}
\label{sec:a participatory budgeting framework} \label{sec:a participatory budgeting framework}
@ -101,62 +102,75 @@ developments will be given.
budgeting scenario as a tuple $E = (P,V,c,B)$, consisting of a set of projects budgeting scenario as a tuple $E = (P,V,c,B)$, consisting of a set of projects
$P = \{ p_1,\dots,p_m \}$ where each project $p\in P$ has an associated cost $P = \{ p_1,\dots,p_m \}$ where each project $p\in P$ has an associated cost
$c(p):P\rightarrow\mathbb{R}$, a set of voters $V = \{v_1,\dots,v_n\}$ and a $c(p):P\rightarrow\mathbb{R}$, a set of voters $V = \{v_1,\dots,v_n\}$ and a
budget limit $B$. The voters express preferences over individual projects or budget limit $B$. The authors assume a model where the voters express
over subsets of all projects. How the preferences of voters are expressed has to preferences over individual projects, although models where voters express
be decided during the design phase of the process and is a choice that has to be preferences over subsets of all projects exist. How the preferences of voters
made in accordance with the method that is used for aggregating the votes. After are expressed has to be decided during the design phase of the process and is a
the voters have elicited their preferences, a set of projects $A\subseteq P$ is choice that has to be made in accordance with the method that is used for
selected as \emph{winning projects} according to some rule and subject to the aggregating the votes. After the voters have elicited their preferences, a set
total budget limit $B$. For the case where projects are indivisible, which is of projects $A\subseteq P$ is selected as \emph{winning projects} according to
also called discrete, the sum of the winning projects' costs is not allowed to some rule and subject to the total budget limit $B$. For the case where projects
exceed the limit $B$: are indivisible, which is also called discrete, the sum of the winning projects'
costs is not allowed to exceed the limit $B$:
\begin{equation}\label{eq:1} \begin{equation}\label{eq:1}
\sum_{p\in A}{c(p)\leq B}. \sum_{p\in A}{c(p)\leq B}.
\end{equation} \end{equation}
When projects can be divisible, i.e., completed to a fractional degree, the When projects can be divisible, i.e., completed to a fractional degree, a
authors define a function $\mu(p) : P\rightarrow [0,1]$ which maps every project function $\mu(p) : P\rightarrow [0,1]$ maps every project to an interval between
to an interval between zero and one, representing the fractional degree to which zero and one, representing the fractional degree to which this project is
this project is completed. Since the cost of each project is a function of its completed. Since the cost of each project is a function of its degree of
degree of completion, the goal is to select a set of projects where the cost of completion, the goal is to select a set of projects where the cost of the degree
the degree of completion does not exceed the budget limit: of completion does not exceed the budget limit:
\begin{equation}\label{eq:2} \begin{equation}\label{eq:2}
\sum_{p\in A}{\mu(p)\cdot c(p)\leq B}. \sum_{p\in A}{\mu(p)\cdot c(p)\leq B}.
\end{equation} \end{equation}
Common ways to design the input method is to ask the voters to approve a subset One way to design the input method is to ask the voters to approve a subset of
of projects $A_v\subseteq P$ where each individual project can be either chosen projects $P_v\subseteq P$ where each individual project can be either chosen to
to be in $A_v$ or not. This form is called \emph{dichotomous preferences} be in $P_v$ or not. This form is called \emph{dichotomous preferences} because
because every project is put in one of two categories: \emph{good} or every project is put in one of two categories: \emph{good} or \emph{bad}.
\emph{bad}. Projects that have not been approved (are not in $A_v$) are assumed Projects that have not been approved (are not in $P_v$) are assumed to be in the
to be in the bad category. This type of preference elicitation is known as bad category. This type of preference elicitation is known as approval-based
approval-based preference elicitation or balloting. It is possible to design preference elicitation with dichotomous
variations of the described scenario by for example asking the voters to only preferences~\cite{bramsApprovalVoting1978}. It is possible to design variations
specify at most $k$ projects which they want to see approved ($k$-Approval) of the described scenario by for example asking the voters to only specify at
most $k$ projects which they want to see approved ($k$-Approval)
\cite{goelKnapsackVotingParticipatory2019a}. These variations typically do not \cite{goelKnapsackVotingParticipatory2019a}. These variations typically do not
take into account the cost that is associated with each project at the voting take into account the cost that is associated with each project at the voting
stage. To alleviate this, approaches where the voters are asked to approve stage. To alleviate this, approaches where the voters are asked to approve
projects while factoring in the cost have been proposed. After asking the voters projects while factoring in the cost have been proposed. After asking the voters
for their preferences, various aggregation methods can be used. for their preferences, various aggregation methods, which take the votes
elicited by the voters as input, aggregate them to provide a set of winning
projects. Each voter's total utility is added to the total sum of utility that a
set of winning project provides for all voters. This type of measuring total
utility is referred to as \emph{additive utilities}.
Section~\ref{sec:approval-based budgeting} will go into detail about the Section~\ref{sec:approval-based budgeting} will go into detail about the
complexity and axiomatic guarantees of these methods. complexity and axiomatic guarantees of a subset of aggregation methods called
\emph{approval-based aggregation methods}.
One such approach, where the cost and benefit of each project is factored in, is One such approach and a second way for preference elicitation, where the cost
described by \textcite{goelKnapsackVotingParticipatory2019a}, which they term and benefit of each project is factored in, is described by
\emph{knapsack voting}. It allows voters to express preferences by factoring in \textcite{goelKnapsackVotingParticipatory2019a}, which they term \emph{knapsack
the cost as well as the benefit per unit of cost. The name stems from the voting}. It allows voters to express preferences by factoring in the cost as
well-known knapsack problem in which, given a set of items, their associated well as the benefit per unit of cost.
weight and value and a weight limit, a selection of items that maximize the \textcite[p.~3]{goelKnapsackVotingParticipatory2019a} describe a scenario
(example 1.2) where $1$-Approval voting falls short of selecting two more
valuable projects in favor of a single project even though the budget limit
would allow for the two more valuable projects to be funded. The name stems from
the well-known knapsack problem in which, given a set of items, their associated
weights and values and a weight limit, a selection of items that maximize the
value subject to the weight limit has to be chosen. In the budgeting scenario, value subject to the weight limit has to be chosen. In the budgeting scenario,
the items correspond to projects, the weight limit to the budget limit and the the items correspond to projects, the weight limit to the budget limit, the
value of each item to the value that a project provides to a voter. To have a weight of each item to the cost of each project and the value of each item to
suitable metric for the value that each voter gets from a specific project, the the value that a project provides to a voter. To have a suitable metric for the
authors introduce different \emph{utility models}. These models make it possible value that each voter gets from a specific project, the authors introduce
to provide axiomatic guarantees such as strategyproofness or welfare different \emph{utility models}. These models make it possible to provide
maximization. While their model assumes fractional voting---that is each voter axiomatic guarantees such as strategyproofness or welfare maximization. While
can allocate the budget in any way they see fit---utility functions are also their model assumes fractional voting---that is each voter can allocate the
used by \textcite{talmonFrameworkApprovalBasedBudgeting2019} to measure the budget in any way they see fit---utility functions are also used by
total satisfaction that a winning set of projects provides under an aggregation \textcite{talmonFrameworkApprovalBasedBudgeting2019} for the case where projects
rule. are indivisible to measure the total satisfaction that a winning set of projects
provides under an aggregation rule.
A third possibility for preference elicitation is \emph{ranked orders}. In this A third possibility for preference elicitation is \emph{ranked orders}. In this
scenario, voters specify a ranking over the available choices (projects) with scenario, voters specify a ranking over the available choices (projects) with
@ -170,14 +184,16 @@ outcome, multiple aggregation methods can be combined with ranked orders.
% Cite municipalities using approval-based budgeting (Paris?) % Cite municipalities using approval-based budgeting (Paris?)
Since approval-based methods are comparatively easy to implement and are being Since approval-based budgeting is used in practice by multiple municipalities,
used in practice by multiple municipalities, the next section will discuss the next section will discuss aggregation methods, their complexity as well as
aggregation methods, their complexity as well as useful axioms for comparing the useful axioms for comparing the different aggregation rules.
different aggregation rules.
\section{Approval-based budgeting} \section{Approval-based budgeting}
\label{sec:approval-based budgeting} \label{sec:approval-based budgeting}
\subsection{Greedy rules}
\label{subsec:greedy rules}
Although approval-based budgeting is also suitable for the case where the Although approval-based budgeting is also suitable for the case where the
projects can be divisible, municipalities using this method generally assume projects can be divisible, municipalities using this method generally assume
indivisible projects. Moreover---as is the case with participatory budgeting in indivisible projects. Moreover---as is the case with participatory budgeting in
@ -188,17 +204,17 @@ preferences, a simple aggregation rule is greedy selection. In this case the
goal is to iteratively select one project $p\in P$ that gives the maximum goal is to iteratively select one project $p\in P$ that gives the maximum
satisfaction for all voters. Satisfaction can be viewed as a form of social satisfaction for all voters. Satisfaction can be viewed as a form of social
welfare where it is not only desirable to stay below the budget limit $B$ but welfare where it is not only desirable to stay below the budget limit $B$ but
also to achieve a high score at some metric that quantifies the value that each also to select a set of winning projects maximizing the value for the voters.
voter gets from the result. \textcite{talmonFrameworkApprovalBasedBudgeting2019} \textcite{talmonFrameworkApprovalBasedBudgeting2019} propose three satisfaction
propose three satisfaction functions which provide this metric. Formally, they functions which provide this metric. Formally, they define a satisfaction
define a satisfaction function as a function $sat : 2^P\times 2^P\rightarrow function as a function $sat : 2^P\times 2^P\rightarrow \mathbb{R}$, where $P$ is
\mathbb{R}$, where $P$ is a set of projects. A voter $v$ selects projects to be a set of projects. A voter $v$ selects projects to be in her approval set $P_v$
in her approval set $P_v$ and a bundle $A\subseteq P$ contains the projects that and a bundle $A\subseteq P$ contains the projects that have been selected as
have been selected as winners. The satisfaction that voter $v$ gets from a winners. The satisfaction that voter $v$ gets from a selected bundle $A$ is
selected bundle $A$ is denoted as $sat(P_v,A)$. The set $A_v = P_v\cap A$ denoted as $sat(P_v,A)$. The set $A_v = P_v\cap A$ denotes the set of approved
denotes the set of approved items by $v$ that end up in the winning bundle $A$. items by $v$ that end up in the winning bundle $A$. A simple approach is to
A simple approach is to count the number of projects that have been approved by count the number of projects that have been approved by a voter and which ended
a voter and which ended up being in the winning set: up being in the winning set:
\begin{equation}\label{eq:3} \begin{equation}\label{eq:3}
sat_\#(P_v,A) = |A_v| sat_\#(P_v,A) = |A_v|
\end{equation} \end{equation}
@ -206,9 +222,9 @@ Combined with the greedy rule for selecting projects, projects are iteratively
added to the winning bundle $A$ where at every iteration the project that gives added to the winning bundle $A$ where at every iteration the project that gives
the maximum satisfaction to all voters is selected. It is assumed that the the maximum satisfaction to all voters is selected. It is assumed that the
voters' individual satisfaction can be added together to provide the voters' individual satisfaction can be added together to provide the
satisfaction that one project gives to all the voters. This gives the rule satisfaction that one project gives to all the voters (additive utilities). This
$\mathcal{R}_{sat_\#}^g$ which seeks to maximize $\sum_{v\in V}sat_\#(P_v,A\cup gives the rule $\mathcal{R}_{sat_\#}^g$ which seeks to maximize $\sum_{v\in
\{p\})$ at every iteration. V}sat_\#(P_v,A\cup \{p\})$ at every iteration.
Another satisfaction function assumes a relationship between the cost of the Another satisfaction function assumes a relationship between the cost of the
items and a voter's satisfaction. Namely, a project that has a high cost and is items and a voter's satisfaction. Namely, a project that has a high cost and is
@ -232,26 +248,28 @@ otherwise (see equation~\ref{eq:5}).
\end{cases} \end{cases}
\end{equation} \end{equation}
The satisfaction functions from equations~\ref{eq:4} and \ref{eq:5} can also be The satisfaction functions from equations~\ref{eq:4} and \ref{eq:5} can also be
combined with the greedy rule, potentially giving slightly different outcomes combined with the greedy rule, potentially giving different outcomes than
than $\mathcal{R}_{sat_\#}^g$. An example demonstrating the greedy rule is given $\mathcal{R}_{sat_\#}^g$. An example demonstrating the greedy rule is given in
in example~\ref{ex:greedy}. example~\ref{ex:greedy} taken from
\textcite[p.~2182]{talmonFrameworkApprovalBasedBudgeting2019}.
\begin{example}\label{ex:greedy} \begin{example}\label{ex:greedy}
A set of projects $P = \{ p_2,p_3,p_4,p_5,p_6 \}$ and their associated cost A set of projects $P = \{ p_2,p_3,p_4,p_5,p_6 \}$ and their associated cost
$p_i$ where project $p_i$ costs $i$ and a budget limit $B = 10$ is given. $i$ given as subscripts (project $p_2$ costs $2$) and a budget limit $B =
Futhermore, five voters vote $v_1 = \{ p_2,p_5,p_6 \}$, $v_2 = \{ p_2, 10$ is given. Futhermore, five voters $v_1 = \{ p_2,p_5,p_6 \}$, $v_2 = \{
p_3,p_4,p_5 \}$, $v_3 = \{ p_3,p_4,p_5 \}$, $v_4 = \{ p_4,p_5 \}$ and $v_5 = p_2, p_3,p_4,p_5 \}$, $v_3 = \{ p_3,p_4,p_5 \}$, $v_4 = \{ p_4,p_5 \}$ and
\{ p_6 \}$. Under $\mathcal{R}_{sat_\#}^g$ the winning bundle is $\{ p_4,p_5 $v_5 = \{ p_6 \}$ vote on the five projects. Under $\mathcal{R}_{sat_\#}^g$
\}$, $\mathcal{R}_{sat_\$}^g$ gives $\{ p_4,p_5 \}$ and the winning bundle is $\{ p_4,p_5 \}$, $\mathcal{R}_{sat_\$}^g$ gives $\{
$\mathcal{R}_{sat_{0/1}}^g$ $\{ p_2,p_3,p_5 \}$. p_4,p_5 \}$ and $\mathcal{R}_{sat_{0/1}}^g$ $\{ p_2,p_3,p_5 \}$.
\end{example} \end{example}
Computing a solution to the problem of finding a winning set of projects by Computing a solution to the problem of finding a winning set of projects by
using greedy rules can be done in polynomial time due to their iterative nature. using greedy rules can be done in polynomial time due to their iterative nature
The downside to using a greedy selection process is that the provided solution where each iteration takes polynomial time.
might not be optimal with respect to the satisfaction.
\subsection{Max rules}
\label{subsec:max rules}
To be able to compute optimal solutions,
\textcite{talmonFrameworkApprovalBasedBudgeting2019} suggest combining the \textcite{talmonFrameworkApprovalBasedBudgeting2019} suggest combining the
satisfaction functions with a maximization rule. The maximization rule always satisfaction functions with a maximization rule. The maximization rule always
selects a winning set of projects that maximizes the sum of the voters' selects a winning set of projects that maximizes the sum of the voters'
@ -269,37 +287,40 @@ beforehand in order to select the bundle with the maximum satisfaction. This
hints at the complexity of the max rule being harder to solve than the greedy hints at the complexity of the max rule being harder to solve than the greedy
rule. The authors confirm this by identifying $\mathcal{R}_{sat_\$}^m$ as weakly rule. The authors confirm this by identifying $\mathcal{R}_{sat_\$}^m$ as weakly
\textsf{NP}-hard for the problem of finding a winning set that gives at least a \textsf{NP}-hard for the problem of finding a winning set that gives at least a
specified amount of satisfaction. The proof follows from a reduction to the specified amount of satisfaction. The proof follows from reducing the subset sum
subset sum problem which asks the question of given a set of numbers (in this problem to the problem of asking the question of given a set of numbers (in this
case the cost associated with each project) and a number $B$ (the budget limit) case the cost associated with each project) and a number $B$ (the budget limit)
does any subset of the numbers sum to exactly $B$? Because the subset sum does any subset of the numbers sum to exactly $B$? Because the subset sum
problem is solvable by a dynamic programming algorithm in $O(B\cdot |P|)$ where problem is solvable by a dynamic programming algorithm in $O(B\cdot |P|)$ where
$P$ is the set of projects, $\mathcal{R}_{sat_\$}^m$ is solvable in $P$ is the set of projects, $\mathcal{R}_{sat_\$}^m$ is solvable in
pseudo-polynomial time. Finding a solution using the rule pseudo-polynomial time. Finding a solution using the rule
$\mathcal{R}_{sat_\#}^m$ however, is doable in polynomial time due to the $\mathcal{R}_{sat_\#}^m$ however, is doable in polynomial time due to the
problem's relation to the knapsack problem. If the input (either projects or problem's relation to the knapsack problem. For $\mathcal{R}_{sat_{0/1}}^m$,
voters) is represented in unary, a dynamic programming algorithm is bounded by a finding a set of projects that gives at least a certain amount of satisfaction
polynomial in the length of the input. For $\mathcal{R}_{sat_{0/1}}^m$, finding is \textsf{NP}-hard. Assuming that the cost of all of the projects is one unit,
a set of projects that gives at least a certain amount of satisfaction is the rule is equivalent to the max cover problem because we are searching for a
\textsf{NP}-hard. Assuming that the cost of all of the projects is one unit, the
rule is equivalent to the max cover problem because we are searching for a
subset of all projects with the number of the projects (the total cost due to subset of all projects with the number of the projects (the total cost due to
the projects given in unit cost) smaller or equal to the budget limit $B$ and the projects given in unit cost) smaller or equal to the budget limit $B$ and
want to maximize the number of voters that are represented by the subset. want to maximize the number of voters that are represented by the subset. The
bigger the resulting set of projects, the more voters are satisfied.
\begin{example}\label{ex:max} \begin{example}\label{ex:max}
Taking the initial setup from example~\ref{ex:greedy}: $P = \{ Taking the initial setup from example~\ref{ex:greedy}: $P = \{
p_2,p_3,p_4,p_5,p_6 \}$ and their associated cost $p_i$ where project $p_i$ p_2,p_3,p_4,p_5,p_6 \}$ and their associated cost $i$ given as subscripts
costs $i$, a budget limit $B = 10$ and the five voters: $v_1 = \{ (project $p_2$ has a cost of $2$), a budget limit $B = 10$ and the five
p_2,p_5,p_6 \}$, $v_2 = \{ p_2, p_3,p_4,p_5 \}$, $v_3 = \{ p_3,p_4,p_5 \}$, voters: $v_1 = \{ p_2,p_5,p_6 \}$, $v_2 = \{ p_2, p_3,p_4,p_5 \}$, $v_3 = \{
$v_4 = \{ p_4,p_5 \}$ and $v_5 = \{ p_6 \}$. We get $\{ p_2,p_3,p_5 \}$ for p_3,p_4,p_5 \}$, $v_4 = \{ p_4,p_5 \}$ and $v_5 = \{ p_6 \}$. We get $\{
$\mathcal{R}_{sat_\#}^m$, $\{ p_4,p_5 \}$ for $\mathcal{R}_{sat_\$}^m$ and p_2,p_3,p_5 \}$ for $\mathcal{R}_{sat_\#}^m$, $\{ p_4,p_5 \}$ for
$\{ p_4,p_6 \}$ for $\mathcal{R}_{sat_{0/1}}^m$. Especially the last rule is $\mathcal{R}_{sat_\$}^m$ and $\{ p_4,p_6 \}$ for
interesting because it provides the highest amount of satisfaction possible $\mathcal{R}_{sat_{0/1}}^m$. Especially the last rule is interesting
by covering each voter with at least one project. Project $p_6$ covers because it provides a high amount of satisfaction by covering each voter
voters $v_1$ and $v_5$ and project $p_4$ voters $v_2$, $v_3$ and $v_4$. with at least one project. Project $p_6$ covers voters $v_1$ and $v_5$ and
project $p_4$ voters $v_2$, $v_3$ and $v_4$.
\end{example} \end{example}
\subsection{Proportional greedy rules}
\label{subsec:proportional greedy rules}
The third rule, which places a heavy emphasis on cost versus benefit, is similar The third rule, which places a heavy emphasis on cost versus benefit, is similar
to the greedy rule but instead of disregarding the satisfaction per cost that a to the greedy rule but instead of disregarding the satisfaction per cost that a
project provides, it seeks to maximize the sum of satisfaction divided by cost project provides, it seeks to maximize the sum of satisfaction divided by cost
@ -308,13 +329,13 @@ for a project $p\in P$:
\frac{\sum_{v\in V}sat(P_v,A\cup\{p\}) - \sum_{v\in V}sat(P_v,A)}{c(p)} \frac{\sum_{v\in V}sat(P_v,A\cup\{p\}) - \sum_{v\in V}sat(P_v,A)}{c(p)}
\end{equation} \end{equation}
\textcite{talmonFrameworkApprovalBasedBudgeting2019} call this type of \textcite{talmonFrameworkApprovalBasedBudgeting2019} call this type of
aggregation rule \emph{proportional greedy rule}. Example~\ref{ex:prop greedy} aggregation rule \emph{proportional greedy rule}. Their example~\ref{ex:prop
shows how the outcome of a budgeting scenario might look like compared to using greedy} shows how the outcome of a budgeting scenario might look like compared
a simple greedy rule or a max rule. Since the proportional greedy rule is a to using a simple greedy rule or a max rule. Since the proportional greedy rule
variation of the simple greedy rule, it is therefore also solvable in polynomial is a variation of the simple greedy rule, it is therefore also solvable in
time. The variation of computing the satisfaction per unit of cost does not polynomial time. The variation of computing the satisfaction per unit of cost
change the complexity since it only adds an additional step which can be done in does not change the complexity since it only adds an additional step which can
constant time. be done in constant time.
\begin{example}\label{ex:prop greedy} \begin{example}\label{ex:prop greedy}
We again have the same set of projects $P = \{ p_2,p_3,p_4,p_5,p_6 \}$, the We again have the same set of projects $P = \{ p_2,p_3,p_4,p_5,p_6 \}$, the
@ -331,27 +352,27 @@ constant time.
\end{example} \end{example}
A benefit of the three discussed satisfaction functions is that they can be A benefit of the three discussed satisfaction functions is that they can be
viewed as constraint satisfaction problems (CSPs) and can thus be formulated formulated using integer linear programming (ILP). Although integer programming
using integer linear programming (ILP). Although integer programming is is \textsf{NP}-complete, efficient solvers are readily available for these types
\textsf{NP}-complete, efficient solvers are readily available for these types of of problems, which can be an important factor when choosing a budgeting
problems. \textcite{talmonFrameworkApprovalBasedBudgeting2019} show that the algorithm. For the problem of finding a set of projects that achieve at least a
rule $\mathcal{R}_{sat_{0/1}}^m$ is similar to the max cover problem which can given satisfaction, \textcite{talmonFrameworkApprovalBasedBudgeting2019} show
be approximated with a $(1-\frac{1}{\epsilon})$-approximation algorithm, where that the rule $\mathcal{R}_{sat_{0/1}}^m$ is similar to the max cover problem
$\epsilon > 0$ is a fixed parameter that is chosen depending on the error of the which can be approximated with an approximation ratio of $(1-\frac{1}{e})$,
approximation. In fact, \textcite{khullerBudgetedMaximumCoverage1999} show that giving a reasonably good solution while taking much less time to compute.
an approximation algorithm with the same ratio exists not only for the case
where the projects have unit cost but also for the general cost version.
Instead of sacrificing exactness to get a better running time, Instead of sacrificing exactness to get a better running time,
\textcite{talmonFrameworkApprovalBasedBudgeting2019} show that the \textcite{talmonFrameworkApprovalBasedBudgeting2019} show that the
$\mathcal{R}_{sat_{0/1}}^m$ rule is fixed parameter tractable for the number of $\mathcal{R}_{sat_{0/1}}^m$ rule is fixed parameter tractable for the number of
voters $|V|$. A problem is fixed parameter tractable if there exists an voters $|V|$. A problem is fixed parameter tractable if there exists an
algorithm that decides each instance of the problem in $O(f(k)\cdot p(n))$ where algorithm that decides each instance of the problem in $O(f(k)\cdot p(n))$ where
$p(n)$ is a polynomial function and $f(k)$ an arbitrary function in $k$. It is $n$ is the input size, $k$ some parameter (in this case the cost of each
crucial to note that $f(k)$ does not admit functions of the form $n^k$. The project), $p(n)$ a polynomial function and $f(k)$ an arbitrary function in $k$.
algorithm for the maximum rule tries to guess the number of voters that are It is crucial to note that $f(k)$ does not admit functions of the form $n^k$.
represented by the same project. The estimation is then used to pick a project \textcite{talmonFrameworkApprovalBasedBudgeting2019} provide a proof for the
which has the lowest cost and satisfies exactly the estimated amount of voters. maximum rule by trying to guess the number of voters that are represented by the
same project. The estimation is then used to pick a project which has the lowest
cost and satisfies exactly the estimated amount of voters.
\section{Normative Axioms} \section{Normative Axioms}
\label{sec:normative axioms} \label{sec:normative axioms}
@ -365,16 +386,18 @@ projects. Another possibility is to look at the \emph{fairness} associated with
a particular set of winning projects. Fairness captures the notion of for a particular set of winning projects. Fairness captures the notion of for
example protecting minorities and their preferences. example protecting minorities and their preferences.
\textcite{azizProportionallyRepresentativeParticipatory2018} propose axioms that \textcite{azizProportionallyRepresentativeParticipatory2018} propose axioms that
are representative of the broad spectrum of choices which voters can make. Other are representative of the broad spectrum of votes which voters can cast. Other
fairness-based approaches are proposed by fairness-based approaches are proposed by
\textcite{fainCoreParticipatoryBudgeting2016}, using the core of a solution, \textcite{fainCoreParticipatoryBudgeting2016}, by calculating the core of a
although they focus on cases where voters elicit their preferences via a solution, although they focus on cases where voters elicit their preferences via
cardinal utility function. The notion of core is also studied by a cardinal utility function. The notion of core is also studied by
\textcite{fainFairAllocationIndivisible2018} for the case where voters have \textcite{fainFairAllocationIndivisible2018} for the case where voters have
additive utilities over the selection of projects, which is similar to the rules additive utilities over the selection of projects. To illustrate working with
discussed above. To illustrate working with axioms, the following will introduce axioms, the following will introduce intuitive properties which are then applied
intuitive properties which are then applied to the rules discussed in to the rules discussed in section~\ref{sec:approval-based budgeting}.
section~\ref{sec:approval-based budgeting}.
\subsection{Inclusion Maximality}
\label{subsec:inclusion Maximality}
A simple axiom is termed \emph{exhaustiveness} by A simple axiom is termed \emph{exhaustiveness} by
\textcite{azizParticipatoryBudgetingModels2020} and \emph{inclusion maximality} \textcite{azizParticipatoryBudgetingModels2020} and \emph{inclusion maximality}
@ -387,10 +410,14 @@ the maximum rules inclusion maximality still holds because for two feasible sets
of projects where one set is a subset of the other and the smaller set is of projects where one set is a subset of the other and the smaller set is
winning then also the bigger set is winning. winning then also the bigger set is winning.
\subsection{Discount Monotonicity}
\label{subsec:discount monotonicity}
An axiom which is not met by all the discussed aggregation rules is An axiom which is not met by all the discussed aggregation rules is
\emph{discount monotonicity}. Discount monotonicity states that if an already \emph{discount monotonicity}. Discount monotonicity states that if an already
selected project which is going to be funded receives a revised cost function, selected project which is going to be funded receives a revised cost function
then that project should not be implemented to a lesser degree resulting in less budget needed for that particular project, then that project
should not be implemented to a lesser degree
\cite[p.~11]{azizParticipatoryBudgetingModels2020}. This is an important \cite[p.~11]{azizParticipatoryBudgetingModels2020}. This is an important
property because if a rule were to fail discount monotonicity, the outcome may property because if a rule were to fail discount monotonicity, the outcome may
be manipulated by increasing the cost of a project instead of trying to minimize be manipulated by increasing the cost of a project instead of trying to minimize
@ -400,7 +427,11 @@ satisfaction functions $sat_\#$ (see equation~\ref{eq:3}) and $sat_{0/1}$
(greedy, proportional greedy and maximum rule) satisfy discount monotonicity. (greedy, proportional greedy and maximum rule) satisfy discount monotonicity.
This is the case because decreasing a project's cost makes it more attractive This is the case because decreasing a project's cost makes it more attractive
for selection, which is not the case when the satisfaction function $sat_\$$ for selection, which is not the case when the satisfaction function $sat_\$$
(equation~\ref{eq:4}) is used. (equation~\ref{eq:4}) is used because with $sat_\$$ a projects value is its
cost. Discounting a project under $sat_\$$ therefore lessens its value.
\subsection{Limit Monotonicity}
\label{subsec:Limit monotonicity}
\emph{Limit monotonicity} is similar to discount monotonicity in that the \emph{Limit monotonicity} is similar to discount monotonicity in that the
relation of a project's cost to the budget limit is modified. Whereas discount relation of a project's cost to the budget limit is modified. Whereas discount
@ -413,55 +444,60 @@ provoke discontent among the voters when they realize that their approved
project is not funded anymore because the total budget has increased, as this is project is not funded anymore because the total budget has increased, as this is
somewhat counterintuitive. Unfortunately, none of the discussed rules satisfy somewhat counterintuitive. Unfortunately, none of the discussed rules satisfy
limit monotonicity. A counterexample for the greedy and proportional greedy limit monotonicity. A counterexample for the greedy and proportional greedy
rules is one where there are three projects $a,b,c$ and $a$ gives the biggest rules is given by \cite[p.~2185]{talmonFrameworkApprovalBasedBudgeting2019}
satisfaction. Project $a$ is therefore selected first. For the case where the where there are three projects $a,b,c$ and $a$ gives the biggest satisfaction.
budget limit has not yet been increased, project $b$ is selected second because Project $a$ is therefore selected first. For the case where the budget limit has
project $c$ is too expensive even though it would provide more satisfaction. not yet been increased, project $b$ is selected second because project $c$ is
When the budget limit is increased, project $c$ can now be funded instead of $b$ too expensive even though it would provide more satisfaction. When the budget
and will provide a higher total satisfaction. Voters which have approved project limit is increased, project $c$ can now be funded instead of $b$ and will
$b$ will thus lose some of their satisfaction. This example is also applicable provide a higher total satisfaction. Voters which have approved project $b$ will
to the maximum rules because the maximum satisfaction before the budget is thus lose some of their satisfaction. This example is also applicable to the
increased is provided by $\{ a,b \}$. Because $c$ can be funded additionally to maximum rules because the maximum satisfaction before the budget is increased is
$a$ after increasing the budget and provides a higher total satisfaction, the provided by $\{ a,b \}$. Because $c$ can be funded additionally to $a$ after
winning set is $\{ a,c \}$. increasing the budget and provides a higher total satisfaction, the winning set
is $\{ a,c \}$.
These three examples provide a rudimentary introduction to comparing aggregation These three examples provide a rudimentary introduction to comparing aggregation
rules by their fulfillment of axiomatic properties. The social choice theory rules by their fulfillment of axiomatic properties. The social choice theory
often uses axioms such as \emph{strategyproofness}, \emph{pareto efficiency} and often uses axioms such as \emph{strategyproofness}, \emph{pareto efficiency} and
\emph{non-dictatorship} to classify voting schemes. These properties are \emph{non-dictatorship} to classify voting schemes. These properties are
concerned with making sure that each voter votes truthfully, that a solution concerned with making sure that each voter votes truthfully, that a solution
cannot be bettered without making someone worse off while improving another cannot achieve a higher satisfaction without making someone worse off while
voter and that results cannot only mirror one person's preferences, improving another voter's satisfaction and that results cannot only mirror one
respectively. person's preferences, respectively.
\section{Conclusion} \section{Conclusion}
\label{sec:conclusion} \label{sec:conclusion}
We have looked at different possibilities for conducting the voting and winner We have introduced different methods for preference elicitation and aggregating
selection process for participatory budgeting. A budgeting scenario in the a winning selection of projects for participatory budgeting. A budgeting
mathematical sense has been described and methods for modeling voter scenario in the mathematical sense has been described and methods for modeling
satisfaction are discussed. A deeper view on approval-based budgeting models has voter satisfaction are discussed. Afterwards, a deeper view on approval-based
been given where the voters are assumed to have dichotomous preferences. The budgeting models has been given where the voters are assumed to have dichotomous
complexity of the different rules has been evaluated and contrasted to each preferences. In section~\ref{sec:approval-based budgeting} we summarize
other. We have seen that aggregation methods cannot only be compared in terms of complexity results of the different rules. Section~\ref{sec:normative axioms}
complexity but also by using axioms that formulate desirable outcomes. introduces three axioms by which participatory budgeting methods can be compared
to each other and which allow for these methods to be tested in scenarios such
as when a project gets a discount.
Future research might focus on not only incorporating monetary cost and Future research might focus on not only incorporating monetary cost and
satisfaction into aggregating winning projects but also other factors such as satisfaction into aggregating winning projects but also other factors such as
environmental costs, practicability of participatory budgeting methods as well environmental costs, practicability of participatory budgeting methods as well
as scalability of these methods to a very high amount of projects and voters. as scalability of these methods to a very high amount of projects and voters.
Interesting further questions are posed by the possibility to combine projects Interesting further questions are posed by the possibility to combine projects
that are indivisible with projects that are divisible under one aggregation that are indivisible with projects that are divisible under one aggregation
rule, leading to a host of \emph{hybrid models}. Because a lot of the methods rule, leading to a host of \emph{hybrid models}. Because a lot of the methods
that have been theorized by researchers have not yet been implemented in that have been theorized by researchers have not yet been implemented in
practice, research on feasibility could lead to a better understanding of what practice, research on feasibility could lead to a better understanding of what
works and what does not. Another area of research could focus on allowing works and what does not.
projects to be related to each other and reflecting those inter-relations in the
outcome while still maintaining a grip on the explosion of possible solutions. Another area of research could focus on allowing projects to be related to each
Exploring more axioms and rule configurations is important for achieving a other and reflecting those inter-relations in the outcome while still
complete picture of the possibilities within the field of computational social maintaining a grip on the explosion of possible solutions.
choice. As a final point, research into user interface design during the voting
phase might uncover previously unknown impacts of ballot design on the resulting As a final point, research into user interface design during the voting phase
might uncover previously unknown impacts of ballot design on the resulting
selection of winning projects. selection of winning projects.
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