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12
paper/references.bib
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@ -73,6 +73,18 @@
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,
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},

380
paper/termpaper.tex
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@ -42,9 +42,10 @@
preference elicitation with an aggregation method, a set of winning projects
is determined and funded. This paper first gives an introduction into
participatory budgeting methods and then focuses on approval-based models to
discuss algorithmic complexity. Furthermore, a short overview of useful
axioms that can help select one method in practice is presented. Finally, an
outlook on future challenges surrounding participatory budgeting is given.
discuss algorithmic complexity. Furthermore, this work presents a short
overview of useful axioms that can help select one method in practice. The
paper concludes with an outlook on future challenges surrounding
participatory budgeting.
\end{abstract}
\section{Introduction}
@ -73,26 +74,26 @@ stages \autocite{participatorybudgetingprojectHowPBWorks}:
set of winning projects which are then funded.
\end{description}
\noindent The two last stages \emph{voting} and \emph{aggregating votes} are of
main interest for computer scientists, economists and social choice theorists
because depending on how voters elicit their preferences (\emph{balloting} or
\emph{input method}) and how the votes are aggregated through the use of
algorithms, the outcome is different. To study different ways of capturing votes
and aggregating them, the participatory process is modeled mathematically. This
model will be called a participatory budgeting \emph{scenario}. The aim of
studying participatory budgeting scenarios is to find ways to achieve a
desirable outcome. A desirable outcome can be one based on fairness by making
sure that each voter has at least one chosen project in the final set of winning
projects for example. Other approaches are concerned with maximizing social
welfare or discouraging \emph{gaming the voting process} (where an outcome can
be manipulated by not voting truthfully; also called \emph{strategyproofness}).
\noindent The last two stages \emph{voting} and \emph{aggregating votes} are of
main interest for computer scientists and economists because depending on how
voters elicit their preferences (\emph{balloting} or \emph{input method}) and
how the votes are aggregated through the use of algorithms, the outcome is
different. To study different ways of capturing votes and aggregating them, the
participatory process is modeled mathematically. This model will be called a
participatory budgeting \emph{scenario}. The aim of studying participatory
budgeting scenarios is to find ways to achieve a desirable outcome. A desirable
outcome can be one based on fairness by making sure that each voter has at least
one chosen project in the final set of winning projects for example. Other
approaches are concerned with maximizing social welfare or discouraging
\emph{gaming the voting process} (where an outcome cannot be manipulated by not
voting truthfully; also called \emph{strategyproofness}).
First, this paper will give a brief overview of common methods and show how a
participatory budgeting scenario can be modeled mathematically. To illustrate
these methods, one approach will be chosen and discussed in detail with respect
to algorithmic complexity and properties. Finally, the gained insight into
participatory budgeting algorithms will be summarized and an outlook on further
developments will be given.
to algorithmic complexity and properties. Finally, the conclusion will summarize
the gained insight into participatory budgeting algorithms and will give an
outlook on further developments and research directions.
\section{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
$P = \{ p_1,\dots,p_m \}$ 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
budget limit $B$. The voters express preferences over individual projects or
over subsets of all projects. How the preferences of voters are expressed has to
be decided during the design phase of the process and is a choice that has to be
made in accordance with the method that is used for aggregating the votes. After
the voters have elicited their preferences, a set of projects $A\subseteq P$ is
selected as \emph{winning projects} according to some rule and subject to the
total budget limit $B$. For the case where projects are indivisible, which is
also called discrete, the sum of the winning projects' costs is not allowed to
exceed the limit $B$:
budget limit $B$. The authors assume a model where the voters express
preferences over individual projects, although models where voters express
preferences over subsets of all projects exist. How the preferences of voters
are expressed has to be decided during the design phase of the process and is a
choice that has to be made in accordance with the method that is used for
aggregating the votes. After the voters have elicited their preferences, a set
of projects $A\subseteq P$ is selected as \emph{winning projects} according to
some rule and subject to the total budget limit $B$. For the case where projects
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}
\sum_{p\in A}{c(p)\leq B}.
\end{equation}
When projects can be divisible, i.e., completed to a fractional degree, the
authors define a function $\mu(p) : P\rightarrow [0,1]$ which maps every project
to an interval between zero and one, representing the fractional degree to which
this project is completed. Since the cost of each project is a function of its
degree of completion, the goal is to select a set of projects where the cost of
the degree of completion does not exceed the budget limit:
When projects can be divisible, i.e., completed to a fractional degree, a
function $\mu(p) : P\rightarrow [0,1]$ maps every project to an interval between
zero and one, representing the fractional degree to which this project is
completed. Since the cost of each project is a function of its degree of
completion, the goal is to select a set of projects where the cost of the degree
of completion does not exceed the budget limit:
\begin{equation}\label{eq:2}
\sum_{p\in A}{\mu(p)\cdot c(p)\leq B}.
\end{equation}
Common ways to design the input method is to ask the voters to approve a subset
of projects $A_v\subseteq P$ where each individual project can be either chosen
to be in $A_v$ or not. This form is called \emph{dichotomous preferences}
because every project is put in one of two categories: \emph{good} or
\emph{bad}. Projects that have not been approved (are not in $A_v$) are assumed
to be in the bad category. This type of preference elicitation is known as
approval-based preference elicitation or balloting. It is possible to design
variations 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)
One way to design the input method is to ask the voters to approve a subset of
projects $P_v\subseteq P$ where each individual project can be either chosen to
be in $P_v$ or not. This form is called \emph{dichotomous preferences} because
every project is put in one of two categories: \emph{good} or \emph{bad}.
Projects that have not been approved (are not in $P_v$) are assumed to be in the
bad category. This type of preference elicitation is known as approval-based
preference elicitation with dichotomous
preferences~\cite{bramsApprovalVoting1978}. It is possible to design variations
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
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
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
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
described by \textcite{goelKnapsackVotingParticipatory2019a}, which they term
\emph{knapsack voting}. It allows voters to express preferences by factoring in
the cost as well as the benefit per unit of cost. The name stems from the
well-known knapsack problem in which, given a set of items, their associated
weight and value and a weight limit, a selection of items that maximize the
One such approach and a second way for preference elicitation, where the cost
and benefit of each project is factored in, is described by
\textcite{goelKnapsackVotingParticipatory2019a}, which they term \emph{knapsack
voting}. It allows voters to express preferences by factoring in the cost as
well as the benefit per unit of cost.
\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,
the items correspond to projects, the weight limit to the budget limit and the
value of each item to the value that a project provides to a voter. To have a
suitable metric for the value that each voter gets from a specific project, the
authors introduce different \emph{utility models}. These models make it possible
to provide axiomatic guarantees such as strategyproofness or welfare
maximization. While their model assumes fractional voting---that is each voter
can allocate the budget in any way they see fit---utility functions are also
used by \textcite{talmonFrameworkApprovalBasedBudgeting2019} to measure the
total satisfaction that a winning set of projects provides under an aggregation
rule.
the items correspond to projects, the weight limit to the budget limit, the
weight of each item to the cost of each project and the value of each item to
the value that a project provides to a voter. To have a suitable metric for the
value that each voter gets from a specific project, the authors introduce
different \emph{utility models}. These models make it possible to provide
axiomatic guarantees such as strategyproofness or welfare maximization. While
their model assumes fractional voting---that is each voter can allocate the
budget in any way they see fit---utility functions are also used by
\textcite{talmonFrameworkApprovalBasedBudgeting2019} for the case where projects
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
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?)
Since approval-based methods are comparatively easy to implement and are being
used in practice by multiple municipalities, the next section will discuss
aggregation methods, their complexity as well as useful axioms for comparing the
different aggregation rules.
Since approval-based budgeting is used in practice by multiple municipalities,
the next section will discuss aggregation methods, their complexity as well as
useful axioms for comparing the different aggregation rules.
\section{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
projects can be divisible, municipalities using this method generally assume
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
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
also to achieve a high score at some metric that quantifies the value that each
voter gets from the result. \textcite{talmonFrameworkApprovalBasedBudgeting2019}
propose three satisfaction functions which provide this metric. Formally, they
define a satisfaction function as a function $sat : 2^P\times 2^P\rightarrow
\mathbb{R}$, where $P$ is a set of projects. A voter $v$ selects projects to be
in her approval set $P_v$ and a bundle $A\subseteq P$ contains the projects that
have been selected as winners. The satisfaction that voter $v$ gets from a
selected bundle $A$ is denoted as $sat(P_v,A)$. The set $A_v = P_v\cap A$
denotes the set of approved items by $v$ that end up in the winning bundle $A$.
A simple approach is to count the number of projects that have been approved by
a voter and which ended up being in the winning set:
also to select a set of winning projects maximizing the value for the voters.
\textcite{talmonFrameworkApprovalBasedBudgeting2019} propose three satisfaction
functions which provide this metric. Formally, they define a satisfaction
function as a function $sat : 2^P\times 2^P\rightarrow \mathbb{R}$, where $P$ is
a set of projects. A voter $v$ selects projects to be in her approval set $P_v$
and a bundle $A\subseteq P$ contains the projects that have been selected as
winners. The satisfaction that voter $v$ gets from a selected bundle $A$ is
denoted as $sat(P_v,A)$. The set $A_v = P_v\cap A$ denotes the set of approved
items by $v$ that end up in the winning bundle $A$. A simple approach is to
count the number of projects that have been approved by a voter and which ended
up being in the winning set:
\begin{equation}\label{eq:3}
sat_\#(P_v,A) = |A_v|
\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
the maximum satisfaction to all voters is selected. It is assumed that the
voters' individual satisfaction can be added together to provide the
satisfaction that one project gives to all the voters. This gives the rule
$\mathcal{R}_{sat_\#}^g$ which seeks to maximize $\sum_{v\in V}sat_\#(P_v,A\cup
\{p\})$ at every iteration.
satisfaction that one project gives to all the voters (additive utilities). This
gives the rule $\mathcal{R}_{sat_\#}^g$ which seeks to maximize $\sum_{v\in
V}sat_\#(P_v,A\cup \{p\})$ at every iteration.
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
@ -232,26 +248,28 @@ otherwise (see equation~\ref{eq:5}).
\end{cases}
\end{equation}
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
than $\mathcal{R}_{sat_\#}^g$. An example demonstrating the greedy rule is given
in example~\ref{ex:greedy}.
combined with the greedy rule, potentially giving different outcomes than
$\mathcal{R}_{sat_\#}^g$. An example demonstrating the greedy rule is given in
example~\ref{ex:greedy} taken from
\textcite[p.~2182]{talmonFrameworkApprovalBasedBudgeting2019}.
\begin{example}\label{ex:greedy}
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.
Futhermore, five voters vote $v_1 = \{ p_2,p_5,p_6 \}$, $v_2 = \{ p_2,
p_3,p_4,p_5 \}$, $v_3 = \{ p_3,p_4,p_5 \}$, $v_4 = \{ p_4,p_5 \}$ and $v_5 =
\{ p_6 \}$. Under $\mathcal{R}_{sat_\#}^g$ the winning bundle is $\{ p_4,p_5
\}$, $\mathcal{R}_{sat_\$}^g$ gives $\{ p_4,p_5 \}$ and
$\mathcal{R}_{sat_{0/1}}^g$ $\{ p_2,p_3,p_5 \}$.
$i$ given as subscripts (project $p_2$ costs $2$) and a budget limit $B =
10$ is given. Futhermore, five voters $v_1 = \{ p_2,p_5,p_6 \}$, $v_2 = \{
p_2, p_3,p_4,p_5 \}$, $v_3 = \{ p_3,p_4,p_5 \}$, $v_4 = \{ p_4,p_5 \}$ and
$v_5 = \{ p_6 \}$ vote on the five projects. Under $\mathcal{R}_{sat_\#}^g$
the winning bundle is $\{ p_4,p_5 \}$, $\mathcal{R}_{sat_\$}^g$ gives $\{
p_4,p_5 \}$ and $\mathcal{R}_{sat_{0/1}}^g$ $\{ p_2,p_3,p_5 \}$.
\end{example}
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.
The downside to using a greedy selection process is that the provided solution
might not be optimal with respect to the satisfaction.
using greedy rules can be done in polynomial time due to their iterative nature
where each iteration takes polynomial time.
\subsection{Max rules}
\label{subsec:max rules}
To be able to compute optimal solutions,
\textcite{talmonFrameworkApprovalBasedBudgeting2019} suggest combining the
satisfaction functions with a maximization rule. The maximization rule always
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
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
specified amount of satisfaction. The proof follows from a reduction to the
subset sum problem which asks the question of given a set of numbers (in this
specified amount of satisfaction. The proof follows from reducing the subset sum
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)
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
$P$ is the set of projects, $\mathcal{R}_{sat_\$}^m$ is solvable in
pseudo-polynomial time. Finding a solution using the rule
$\mathcal{R}_{sat_\#}^m$ however, is doable in polynomial time due to the
problem's relation to the knapsack problem. If the input (either projects or
voters) is represented in unary, a dynamic programming algorithm is bounded by a
polynomial in the length of the input. For $\mathcal{R}_{sat_{0/1}}^m$, finding
a set of projects that gives at least a certain amount of satisfaction is
\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
problem's relation to the knapsack problem. For $\mathcal{R}_{sat_{0/1}}^m$,
finding a set of projects that gives at least a certain amount of satisfaction
is \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
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}
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 $p_i$
costs $i$, a budget limit $B = 10$ and the five voters: $v_1 = \{
p_2,p_5,p_6 \}$, $v_2 = \{ p_2, p_3,p_4,p_5 \}$, $v_3 = \{ p_3,p_4,p_5 \}$,
$v_4 = \{ p_4,p_5 \}$ and $v_5 = \{ p_6 \}$. We get $\{ p_2,p_3,p_5 \}$ for
$\mathcal{R}_{sat_\#}^m$, $\{ p_4,p_5 \}$ for $\mathcal{R}_{sat_\$}^m$ and
$\{ p_4,p_6 \}$ for $\mathcal{R}_{sat_{0/1}}^m$. Especially the last rule is
interesting because it provides the highest amount of satisfaction possible
by covering each voter 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 $v_4$.
p_2,p_3,p_4,p_5,p_6 \}$ and their associated cost $i$ given as subscripts
(project $p_2$ has a cost of $2$), a budget limit $B = 10$ and the five
voters: $v_1 = \{ p_2,p_5,p_6 \}$, $v_2 = \{ p_2, p_3,p_4,p_5 \}$, $v_3 = \{
p_3,p_4,p_5 \}$, $v_4 = \{ p_4,p_5 \}$ and $v_5 = \{ p_6 \}$. We get $\{
p_2,p_3,p_5 \}$ for $\mathcal{R}_{sat_\#}^m$, $\{ p_4,p_5 \}$ for
$\mathcal{R}_{sat_\$}^m$ and $\{ p_4,p_6 \}$ for
$\mathcal{R}_{sat_{0/1}}^m$. Especially the last rule is interesting
because it provides a high amount of satisfaction by covering each voter
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}
\subsection{Proportional greedy rules}
\label{subsec:proportional greedy rules}
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
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)}
\end{equation}
\textcite{talmonFrameworkApprovalBasedBudgeting2019} call this type of
aggregation rule \emph{proportional greedy rule}. Example~\ref{ex:prop greedy}
shows how the outcome of a budgeting scenario might look like compared to using
a simple greedy rule or a max rule. Since the proportional greedy rule is a
variation of the simple greedy rule, it is therefore also solvable in polynomial
time. The variation of computing the satisfaction per unit of cost does not
change the complexity since it only adds an additional step which can be done in
constant time.
aggregation rule \emph{proportional greedy rule}. Their example~\ref{ex:prop
greedy} shows how the outcome of a budgeting scenario might look like compared
to using a simple greedy rule or a max rule. Since the proportional greedy rule
is a variation of the simple greedy rule, it is therefore also solvable in
polynomial time. The variation of computing the satisfaction per unit of cost
does not change the complexity since it only adds an additional step which can
be done in constant time.
\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
@ -331,27 +352,27 @@ constant time.
\end{example}
A benefit of the three discussed satisfaction functions is that they can be
viewed as constraint satisfaction problems (CSPs) and can thus be formulated
using integer linear programming (ILP). Although integer programming is
\textsf{NP}-complete, efficient solvers are readily available for these types of
problems. \textcite{talmonFrameworkApprovalBasedBudgeting2019} show that the
rule $\mathcal{R}_{sat_{0/1}}^m$ is similar to the max cover problem which can
be approximated with a $(1-\frac{1}{\epsilon})$-approximation algorithm, where
$\epsilon > 0$ is a fixed parameter that is chosen depending on the error of the
approximation. In fact, \textcite{khullerBudgetedMaximumCoverage1999} show that
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.
formulated using integer linear programming (ILP). Although integer programming
is \textsf{NP}-complete, efficient solvers are readily available for these types
of problems, which can be an important factor when choosing a budgeting
algorithm. For the problem of finding a set of projects that achieve at least a
given satisfaction, \textcite{talmonFrameworkApprovalBasedBudgeting2019} show
that the rule $\mathcal{R}_{sat_{0/1}}^m$ is similar to the max cover problem
which can be approximated with an approximation ratio of $(1-\frac{1}{e})$,
giving a reasonably good solution while taking much less time to compute.
Instead of sacrificing exactness to get a better running time,
\textcite{talmonFrameworkApprovalBasedBudgeting2019} show that the
$\mathcal{R}_{sat_{0/1}}^m$ rule is fixed parameter tractable for the number of
voters $|V|$. A problem is fixed parameter tractable if there exists an
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
crucial to note that $f(k)$ does not admit functions of the form $n^k$. The
algorithm for the maximum rule tries 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.
$n$ is the input size, $k$ some parameter (in this case the cost of each
project), $p(n)$ a polynomial function and $f(k)$ an arbitrary function in $k$.
It is crucial to note that $f(k)$ does not admit functions of the form $n^k$.
\textcite{talmonFrameworkApprovalBasedBudgeting2019} provide a proof for the
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}
\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
example protecting minorities and their preferences.
\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
\textcite{fainCoreParticipatoryBudgeting2016}, using the core of a solution,
although they focus on cases where voters elicit their preferences via a
cardinal utility function. The notion of core is also studied by
\textcite{fainCoreParticipatoryBudgeting2016}, by calculating the core of a
solution, although they focus on cases where voters elicit their preferences via
a cardinal utility function. The notion of core is also studied by
\textcite{fainFairAllocationIndivisible2018} for the case where voters have
additive utilities over the selection of projects, which is similar to the rules
discussed above. To illustrate working with axioms, the following will introduce
intuitive properties which are then applied to the rules discussed in
section~\ref{sec:approval-based budgeting}.
additive utilities over the selection of projects. To illustrate working with
axioms, the following will introduce intuitive properties which are then applied
to the rules discussed in section~\ref{sec:approval-based budgeting}.
\subsection{Inclusion Maximality}
\label{subsec:inclusion Maximality}
A simple axiom is termed \emph{exhaustiveness} by
\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
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
\emph{discount monotonicity}. Discount monotonicity states that if an already
selected project which is going to be funded receives a revised cost function,
then that project should not be implemented to a lesser degree
selected project which is going to be funded receives a revised cost function
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
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
@ -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.
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 $sat_\$$
(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
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
somewhat counterintuitive. Unfortunately, none of the discussed rules satisfy
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
satisfaction. Project $a$ is therefore selected first. For the case where the
budget limit has not yet been increased, project $b$ is selected second because
project $c$ is too expensive even though it would provide more satisfaction.
When the budget limit is increased, project $c$ can now be funded instead of $b$
and will provide a higher total satisfaction. Voters which have approved project
$b$ will thus lose some of their satisfaction. This example is also applicable
to the maximum rules because the maximum satisfaction before the budget is
increased is provided by $\{ a,b \}$. Because $c$ can be funded additionally to
$a$ after increasing the budget and provides a higher total satisfaction, the
winning set is $\{ a,c \}$.
rules is given by \cite[p.~2185]{talmonFrameworkApprovalBasedBudgeting2019}
where there are three projects $a,b,c$ and $a$ gives the biggest satisfaction.
Project $a$ is therefore selected first. For the case where the budget limit has
not yet been increased, project $b$ is selected second because project $c$ is
too expensive even though it would provide more satisfaction. When the budget
limit is increased, project $c$ can now be funded instead of $b$ and will
provide a higher total satisfaction. Voters which have approved project $b$ will
thus lose some of their satisfaction. This example is also applicable to the
maximum rules because the maximum satisfaction before the budget is increased is
provided by $\{ a,b \}$. Because $c$ can be funded additionally to $a$ after
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
rules by their fulfillment of axiomatic properties. The social choice theory
often uses axioms such as \emph{strategyproofness}, \emph{pareto efficiency} and
\emph{non-dictatorship} to classify voting schemes. These properties are
concerned with making sure that each voter votes truthfully, that a solution
cannot be bettered without making someone worse off while improving another
voter and that results cannot only mirror one person's preferences,
respectively.
cannot achieve a higher satisfaction without making someone worse off while
improving another voter's satisfaction and that results cannot only mirror one
person's preferences, respectively.
\section{Conclusion}
\label{sec:conclusion}
We have looked at different possibilities for conducting the voting and winner
selection process for participatory budgeting. A budgeting scenario in the
mathematical sense has been described and methods for modeling voter
satisfaction are discussed. A deeper view on approval-based budgeting models has
been given where the voters are assumed to have dichotomous preferences. The
complexity of the different rules has been evaluated and contrasted to each
other. We have seen that aggregation methods cannot only be compared in terms of
complexity but also by using axioms that formulate desirable outcomes.
We have introduced different methods for preference elicitation and aggregating
a winning selection of projects for participatory budgeting. A budgeting
scenario in the mathematical sense has been described and methods for modeling
voter satisfaction are discussed. Afterwards, a deeper view on approval-based
budgeting models has been given where the voters are assumed to have dichotomous
preferences. In section~\ref{sec:approval-based budgeting} we summarize
complexity results of the different rules. Section~\ref{sec:normative axioms}
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
satisfaction into aggregating winning projects but also other factors such as
environmental costs, practicability of participatory budgeting methods as well
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
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
that have been theorized by researchers have not yet been implemented in
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
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.
Exploring more axioms and rule configurations is important for achieving a
complete picture of the possibilities within the field of computational social
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
works and what does not.
Another area of research could focus on allowing 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.
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.
\printbibliography