participatory-budgeting/paper/termpaper.tex

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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}
}

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\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 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}
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}.

\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
    $\mathcal{R}_{sat_{0/1}}^g$ $\{ p_2,p_3,p_5 \}$.
\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.

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'
satisfaction:
\begin{equation}\label{eq:6}
    \max_{A\subseteq P}\sum_{v\in V}sat(P_v,A)
\end{equation}
The max rule can then be used with the three satisfaction functions in the same
way, giving: $\mathcal{R}_{sat_\#}^m$, $\mathcal{R}_{sat_\$}^m$ and
$\mathcal{R}_{sat_{0/1}}^m$. Example~\ref{ex:max} shows that the selection of
winning projects is not as intuitive as when using the greedy rule. Whereas it
was still possible to compute a solution without any tools for the greedy
selection, the max rule requires knowing the possible sets of projects
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
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 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 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.

\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 $v_4$.
\end{example}

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
for a project $p\in P$:
\begin{equation}
    \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.

\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
    same budget limit of $B = 10$ 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 $v_5 = \{ p_6 \}$. If we combine the satisfaction function
    $sat_\#$ from equation~\ref{eq:3} with the proportional greedy rule, we get
    the same result as with the simple greedy rule of $\{ p_4,p_5 \}$. While the
    simple greedy rule selects first $p_5$ and then $p_4$, the proportional
    greedy rule first selects $p_4$ and then $p_5$. The rule
    $\mathcal{R}_{sat_\$}^p$ yields the same result as $\mathcal{R}_{sat_\$}^g$
    and $\mathcal{R}_{sat_\$}^m$ of $\{ p_4,p_5 \}$. $\mathcal{R}_{sat_{0/1}}^p$
    however, gives $\{ p_2,p_3,p_4 \}$.
\end{example}

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