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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} or
\emph{input method}) 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 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.

\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}\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:
\begin{equation}\label{eq:2}
    \sum_{p\in A}{c(\mu(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)
\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.
Section~\ref{sec:approval-based budgeting} will go into detail about the
complexity and axiomatic guarantees of these 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
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.

A third possibility for preference elicitation is \emph{ranked orders}. In this
scenario, voters specify a ranking over the available choices (projects) with
the highest ranked choice receiving the biggest amount of the budget and the
lowest ranked one the lowest amount of the budget.
\textcite{langPortioningUsingOrdinal2019} study a scenario in which the input
method is ranked orders and the projects that can be chosen are divisible. The
problem of allocating the budget to a set of winning projects under these
circumstances is referred to as \emph{portioning}. Depending on the desired
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.

\section{Approval-based budgeting}
\label{sec:approval-based budgeting}

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
general---we not only want to select one project as a winner but multiple. This
is called a multi-winner election and is in contrast to single-winner elections.
Once the votes have been cast by the voters, again assuming dichotomous
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:
\begin{equation}\label{eq:3}
    sat_\#(P_v,A) = |A_v|
\end{equation}
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.

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
approved by a voter $v$ and ends up in the winning bundle $A$ provides more
satisfaction than a lower cost project. Equation~\ref{eq:4} gives a definition
of this property.
\begin{equation}\label{eq:4}
    sat_\$(P_v,A) = \sum_{p\in A_v} c(p) = c(A_v)
\end{equation}

The third satisfaction function assumes that voters are content as long as there
is at least one of the projects they have approved is selected to be in the
winning set. Therefore, a voter achieves satisfaction 1 when at least one
approved project ends up in the winning bundle, i.e. if $|A_v| > 0$ and 0
satisfaction otherwise (see equation~\ref{eq:5}).
\begin{equation}\label{eq:5}
    sat_{0/1}(P_v,A) =
    \begin{cases}
        1 & \mathsf{if}\; |A_v|>0 \\
        0 & \mathsf{otherwise}
    \end{cases}
\end{equation}

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