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\title{Participatory Budgeting: Algorithms and Complexity}
\author{
 \authorname{Tobias Eidelpes} \\
 \studentnumber{01527193} \\
 \curriculum{033 534} \\
 \email{e1527193@student.tuwien.ac.at}
}

\begin{document}

\maketitle

\begin{abstract}
\end{abstract}

\section{Introduction}

\emph{Participatory Budgeting} (PB) is a process of democratic deliberation that
allows residents of a municipality to decide how a part of the public budget is
to be spent. It is a way to improve transparency and citizen involvement which
are two important cornerstones of a democracy. PB was first realized in the
1990s in Porto Alegre in Brazil by the Workers' Party to combat the growing
divide between the rich city center and the poor living in the greater region.
Owing to its success in the south of Brazil, PB quickly spread to North America,
Europe, Asia and Africa.

Although the process is heavily adapted by each municipality to suit the
environment in which the residents live in, it generally follows the following
stages \autocite{participatorybudgetingprojectHowPBWorks}:

\begin{description}
    \item [Design the process] A rule book is crafted to ensure that the process
        is democratic.
    \item [Collect ideas] Residents propose and discuss ideas for projects.
    \item [Develop feasible projects] The ideas are developed into projects that
        can be undertaken by the municipality.
    \item [Voting] The projects are voted on by the residents.
    \item [Aggregating votes \& funding] The votes are combined to determine a
        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}) and
how the votes are aggregated through the use of algorithms, the outcome is
different. For this paper it is assumed that the first three stages have already
been completed. The rules of the process have been set, ideas have been
collected and developed into feasible projects and the budget limit is known. 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}).

First, this paper will look at how a participatory budgeting scenario can be
modeled mathematically. Then, a brief overview over common models will be given.
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.

\section{Mathematical Model}
\label{sec:mathematical model}

\textcite{talmonFrameworkApprovalBasedBudgeting2019} define a participatory
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 $B$:
\begin{equation}
    \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:
\begin{equation}
    \sum_{p\in A}{c(\mu(p))\leq B}.
\end{equation}

\textcite{azizParticipatoryBudgetingModels2020} define a taxonomy of
participatory budgeting scenarios where projects can be either divisible or
indivisible and bounded or unbounded.

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