zenon/participatory-budgeting
1
0
Fork 0
Code Issues Pull Requests Projects Releases Wiki Activity

Add diff between second and final revision

Browse Source
This commit is contained in:
Tobias Eidelpes 2020-07-04 17:58:30 +02:00
parent c0a5f98586
commit da6afc8fa1
3 changed files with 853 additions and 0 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

592
final_revision/diff.tex Normal file
View File

@ -0,0 +1,592 @@
\documentclass[11pt,a4paper]{article}
%DIF LATEXDIFF DIFFERENCE FILE
%DIF DEL old.tex Sat Jul 4 17:48:08 2020
%DIF ADD ../paper/termpaper.tex Sat Jul 4 17:42:12 2020
\usepackage{termpaper}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{microtype}
\usepackage{setspace}
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage[english]{babel}
\usepackage{csquotes}
\usepackage[style=ieee,backend=biber,maxbibnames=9,mincitenames=1,maxcitenames=2]{biblatex}
\usepackage{hyperref}
\setstretch{1.07}
\addbibresource{references.bib}
% opening
\title{Participatory Budgeting: Algorithms and Complexity}
\author{
\authorname{Tobias Eidelpes} \\
\studentnumber{01527193} \\
\curriculum{033 534} \\
\email{e1527193@student.tuwien.ac.at}
}
% Numbered example environment
\newcounter{example}[section]
\newenvironment{example}[1][]{\refstepcounter{example}\par\medskip
\noindent \textbf{Example~\theexample. #1} \rmfamily}{\medskip}
%DIF PREAMBLE EXTENSION ADDED BY LATEXDIFF
%DIF UNDERLINE PREAMBLE %DIF PREAMBLE
\RequirePackage[normalem]{ulem} %DIF PREAMBLE
\RequirePackage{color}\definecolor{RED}{rgb}{1,0,0}\definecolor{BLUE}{rgb}{0,0,1} %DIF PREAMBLE
\providecommand{\DIFaddtex}[1]{{\protect\color{blue}\uwave{#1}}} %DIF PREAMBLE
\providecommand{\DIFdeltex}[1]{{\protect\color{red}\sout{#1}}} %DIF PREAMBLE
%DIF SAFE PREAMBLE %DIF PREAMBLE
\providecommand{\DIFaddbegin}{} %DIF PREAMBLE
\providecommand{\DIFaddend}{} %DIF PREAMBLE
\providecommand{\DIFdelbegin}{} %DIF PREAMBLE
\providecommand{\DIFdelend}{} %DIF PREAMBLE
\providecommand{\DIFmodbegin}{} %DIF PREAMBLE
\providecommand{\DIFmodend}{} %DIF PREAMBLE
%DIF FLOATSAFE PREAMBLE %DIF PREAMBLE
\providecommand{\DIFaddFL}[1]{\DIFadd{#1}} %DIF PREAMBLE
\providecommand{\DIFdelFL}[1]{\DIFdel{#1}} %DIF PREAMBLE
\providecommand{\DIFaddbeginFL}{} %DIF PREAMBLE
\providecommand{\DIFaddendFL}{} %DIF PREAMBLE
\providecommand{\DIFdelbeginFL}{} %DIF PREAMBLE
\providecommand{\DIFdelendFL}{} %DIF PREAMBLE
%DIF HYPERREF PREAMBLE %DIF PREAMBLE
\providecommand{\DIFadd}[1]{\texorpdfstring{\DIFaddtex{#1}}{#1}} %DIF PREAMBLE
\providecommand{\DIFdel}[1]{\texorpdfstring{\DIFdeltex{#1}}{}} %DIF PREAMBLE
%DIF LISTINGS PREAMBLE %DIF PREAMBLE
\RequirePackage{listings} %DIF PREAMBLE
\RequirePackage{color} %DIF PREAMBLE
\lstdefinelanguage{DIFcode}{ %DIF PREAMBLE
%DIF DIFCODE_UNDERLINE %DIF PREAMBLE
moredelim=[il][\color{red}\sout]{\%DIF\ <\ }, %DIF PREAMBLE
moredelim=[il][\color{blue}\uwave]{\%DIF\ >\ } %DIF PREAMBLE
} %DIF PREAMBLE
\lstdefinestyle{DIFverbatimstyle}{ %DIF PREAMBLE
language=DIFcode, %DIF PREAMBLE
basicstyle=\ttfamily, %DIF PREAMBLE
columns=fullflexible, %DIF PREAMBLE
keepspaces=true %DIF PREAMBLE
} %DIF PREAMBLE
\lstnewenvironment{DIFverbatim}{\lstset{style=DIFverbatimstyle}}{} %DIF PREAMBLE
\lstnewenvironment{DIFverbatim*}{\lstset{style=DIFverbatimstyle,showspaces=true}}{} %DIF PREAMBLE
%DIF END PREAMBLE EXTENSION ADDED BY LATEXDIFF
\begin{document}
\maketitle
\begin{abstract}
Participatory budgeting is a deliberative democratic process that allows
residents to decide how public funds should be spent. By combining a form of
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, \DIFaddbegin \DIFadd{this work presents }\DIFaddend a short
overview of useful axioms that can help select one method in practice\DIFdelbegin \DIFdel{is presented. Finally, }\DIFdelend \DIFaddbegin \DIFadd{. The
paper concludes with }\DIFaddend an outlook on future challenges surrounding
participatory budgeting\DIFdelbegin \DIFdel{is given}\DIFdelend .
\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 \DIFdelbegin \DIFdel{two last }\DIFdelend \DIFaddbegin \DIFadd{last two }\DIFaddend stages \emph{voting} and \emph{aggregating votes} are of
main interest for computer scientists \DIFdelbegin \DIFdel{, economists and social choice theorists
}\DIFdelend \DIFaddbegin \DIFadd{and economists }\DIFaddend 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 \DIFdelbegin \DIFdel{can
}\DIFdelend \DIFaddbegin \DIFadd{cannot }\DIFaddend 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 \DIFaddbegin \DIFadd{conclusion will summarize
the }\DIFaddend gained insight into participatory budgeting algorithms \DIFdelbegin \DIFdel{will be summarized and }\DIFdelend \DIFaddbegin \DIFadd{and will give }\DIFaddend an
outlook on further developments \DIFdelbegin \DIFdel{will be given}\DIFdelend \DIFaddbegin \DIFadd{and research directions}\DIFaddend .
\section{A Participatory Budgeting Framework}
\label{sec:a participatory budgeting framework}
\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 \DIFaddbegin \DIFadd{authors assume a model where the }\DIFaddend voters express
preferences over individual projects\DIFdelbegin \DIFdel{or
}\DIFdelend \DIFaddbegin \DIFadd{, although models where voters express
preferences }\DIFaddend over subsets of all projects \DIFaddbegin \DIFadd{exist}\DIFaddend . 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, \DIFdelbegin \DIFdel{the
authors define }\DIFdelend a
function $\mu(p) : P\rightarrow [0,1]$ \DIFdelbegin \DIFdel{which }\DIFdelend 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}
\DIFdelbegin \DIFdel{Common ways }\DIFdelend \DIFaddbegin \DIFadd{One way }\DIFaddend to design the input method is to ask the voters to approve a subset of
projects \DIFdelbegin \DIFdel{$A_v\subseteq P$ }\DIFdelend \DIFaddbegin \DIFadd{$P_v\subseteq P$ }\DIFaddend where each individual project can be either chosen to
be in \DIFdelbegin \DIFdel{$A_v$ }\DIFdelend \DIFaddbegin \DIFadd{$P_v$ }\DIFaddend 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 \DIFdelbegin \DIFdel{$A_v$}\DIFdelend \DIFaddbegin \DIFadd{$P_v$}\DIFaddend ) are assumed to be in the
bad category. This type of preference elicitation is known as approval-based
preference elicitation \DIFdelbegin \DIFdel{or balloting}\DIFdelend \DIFaddbegin \DIFadd{with dichotomous
preferences~\mbox{%DIFAUXCMD
\cite{bramsApprovalVoting1978}}\hspace{0pt}%DIFAUXCMD
}\DIFaddend . 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\DIFdelbegin \DIFdel{can be used. }\DIFdelend \DIFaddbegin \DIFadd{, 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{\DIFadd{additive utilities}}\DIFadd{.
}\DIFaddend Section~\ref{sec:approval-based budgeting} will go into detail about the
complexity and axiomatic guarantees of \DIFdelbegin \DIFdel{these methods }\DIFdelend \DIFaddbegin \DIFadd{a subset of aggregation methods called
}\emph{\DIFadd{approval-based aggregation methods}}\DIFaddend .
One such approach \DIFaddbegin \DIFadd{and a second way for preference elicitation}\DIFaddend , 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.
\DIFaddbegin \DIFadd{\mbox{%DIFAUXCMD
\textcite[p.~3]{goelKnapsackVotingParticipatory2019a} }\hspace{0pt}%DIFAUXCMD
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. }\DIFaddend The name stems from
the well-known knapsack problem in which, given a set of items, their associated
\DIFdelbegin \DIFdel{weight and value }\DIFdelend \DIFaddbegin \DIFadd{weights and values }\DIFaddend 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\DIFaddbegin \DIFadd{, the
weight of each item to the cost of each project }\DIFaddend 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} \DIFaddbegin \DIFadd{for the case where projects
are indivisible }\DIFaddend 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{airiauPortioningUsingOrdinal2019} 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 \DIFdelbegin \DIFdel{methods are comparatively easy to implement and are being
}\DIFdelend \DIFaddbegin \DIFadd{budgeting is }\DIFaddend 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}
\DIFaddbegin \subsection{\DIFadd{Greedy rules}}
\label{subsec:greedy rules}
\DIFaddend 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 \DIFdelbegin \DIFdel{achieve a high score at some metric that quantifies the value that each
voter gets from the result}\DIFdelend \DIFaddbegin \DIFadd{select a set of winning projects maximizing the value for the voters}\DIFaddend .
\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 \DIFaddbegin \DIFadd{(additive utilities)}\DIFaddend . 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 \DIFdelbegin \DIFdel{slightly }\DIFdelend different outcomes than
$\mathcal{R}_{sat_\#}^g$. An example demonstrating the greedy rule is given in
example~\ref{ex:greedy} \DIFaddbegin \DIFadd{taken from
\mbox{%DIFAUXCMD
\textcite[p.~2182]{talmonFrameworkApprovalBasedBudgeting2019}}\hspace{0pt}%DIFAUXCMD
}\DIFaddend .
\begin{example}\label{ex:greedy}
A set of projects $P = \{ p_2,p_3,p_4,p_5,p_6 \}$ and their associated cost
\DIFdelbegin \DIFdel{$p_i$ where project $p_i$ costs }\DIFdelend $i$ \DIFaddbegin \DIFadd{given as subscripts (project $p_2$ costs $2$) }\DIFaddend and a budget limit $B =
10$ is given. Futhermore, five voters \DIFdelbegin \DIFdel{vote }\DIFdelend $v_1 = \{ p_2,p_5,p_6 \}$, $v_2 = \{
p_2, p_3,p_4,p_5 \}$, $v_3 = \{ p_3,p_4,p_5 \}$, $v_4 = \{ p_4,p_5 \}$ and
$v_5 = \{ p_6 \}$ \DIFaddbegin \DIFadd{vote on the five projects}\DIFaddend . 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
\DIFdelbegin \DIFdel{.
The downside to using a greedy selection process is that the provided solution
might not be optimal with respect to the satisfaction.
}\DIFdelend \DIFaddbegin \DIFadd{where each iteration takes polynomial time.
}\DIFaddend
\DIFdelbegin \DIFdel{To be able to compute optimal solutions,
}\DIFdelend \DIFaddbegin \subsection{\DIFadd{Max rules}}
\label{subsec:max rules}
\DIFaddend \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 \DIFdelbegin \DIFdel{a reduction to }\DIFdelend \DIFaddbegin \DIFadd{reducing }\DIFaddend the subset sum
problem \DIFdelbegin \DIFdel{which asks the }\DIFdelend \DIFaddbegin \DIFadd{to the problem of asking the }\DIFaddend 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. \DIFdelbegin \DIFdel{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. }\DIFdelend 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. \DIFaddbegin \DIFadd{The
bigger the resulting set of projects, the more voters are satisfied.
}\DIFaddend
\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 \DIFdelbegin \DIFdel{$p_i$ where project $p_i$
costs }\DIFdelend $i$ \DIFaddbegin \DIFadd{given as subscripts
(project $p_2$ has a cost of $2$)}\DIFaddend , 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 \DIFdelbegin \DIFdel{the highest }\DIFdelend \DIFaddbegin \DIFadd{a high }\DIFaddend amount of satisfaction \DIFdelbegin \DIFdel{possible
}\DIFdelend 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}
\DIFaddbegin \subsection{\DIFadd{Proportional greedy rules}}
\label{subsec:proportional greedy rules}
\DIFaddend 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}. \DIFdelbegin \DIFdel{Example}\DIFdelend \DIFaddbegin \DIFadd{Their example}\DIFaddend ~\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}
A benefit of the three discussed satisfaction functions is that they can be
\DIFdelbegin \DIFdel{viewed as constraint satisfaction problems (CSPs) and can thus be }\DIFdelend formulated using integer linear programming (ILP). Although integer programming
is \textsf{NP}-complete, efficient solvers are readily available for these types
of problems\DIFdelbegin \DIFdel{. }\DIFdelend \DIFaddbegin \DIFadd{, 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, }\DIFaddend \textcite{talmonFrameworkApprovalBasedBudgeting2019} show
that the rule $\mathcal{R}_{sat_{0/1}}^m$ is similar to the max cover problem
which can be approximated with \DIFdelbegin \DIFdel{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, \mbox{%DIFAUXCMD
\textcite{khullerBudgetedMaximumCoverage1999} }\hspace{0pt}%DIFAUXCMD
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}\DIFdelend \DIFaddbegin \DIFadd{an approximation ratio of $(1-\frac{1}{e})$,
giving a reasonably good solution while taking much less time to compute}\DIFaddend .
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
\DIFdelbegin \DIFdel{$p(n)$ is }\DIFdelend \DIFaddbegin \DIFadd{$n$ is the input size, $k$ some parameter (in this case the cost of each
project), $p(n)$ }\DIFaddend 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 $n^k$.
\DIFdelbegin \DIFdel{The
algorithm }\DIFdelend \DIFaddbegin \DIFadd{\mbox{%DIFAUXCMD
\textcite{talmonFrameworkApprovalBasedBudgeting2019} }\hspace{0pt}%DIFAUXCMD
provide a proof }\DIFaddend for the
maximum rule \DIFdelbegin \DIFdel{tries }\DIFdelend \DIFaddbegin \DIFadd{by trying }\DIFaddend 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}
Axioms in the context of participatory budgeting define some kind of property of
a budgeting method that might be desirable to have. Generally it is beneficial
if a certain method satisfies as many axioms as possible as this gives the
method a strong theoretical backbone. One set of axioms, discussed by
\textcite{talmonFrameworkApprovalBasedBudgeting2019}, relates to the cost of
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 \DIFdelbegin \DIFdel{choices }\DIFdelend \DIFaddbegin \DIFadd{votes }\DIFaddend which voters can \DIFdelbegin \DIFdel{make}\DIFdelend \DIFaddbegin \DIFadd{cast}\DIFaddend . Other
fairness-based approaches are proposed by
\textcite{fainCoreParticipatoryBudgeting2016}, \DIFdelbegin \DIFdel{using }\DIFdelend \DIFaddbegin \DIFadd{by calculating }\DIFaddend 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\DIFdelbegin \DIFdel{, which is similar to the rules
discussed above}\DIFdelend . 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}.
\DIFaddbegin \subsection{\DIFadd{Inclusion Maximality}}
\label{subsec:inclusion Maximality}
\DIFaddend A simple axiom is termed \emph{exhaustiveness} by
\textcite{azizParticipatoryBudgetingModels2020} and \emph{inclusion maximality}
by \textcite{talmonFrameworkApprovalBasedBudgeting2019}. Inclusion maximality
encodes the requirement that if it is possible to fund more projects because the
budget is not yet exhausted, then we should. Greedy and proportional greedy
rules satisfy this axiom because of their inherent iterative process that
terminates only when the budget does not allow more projects to be funded. For
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.
\DIFaddbegin \subsection{\DIFadd{Discount Monotonicity}}
\label{subsec:discount monotonicity}
\DIFaddend 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
\DIFaddbegin \DIFadd{resulting in less budget needed for that particular project}\DIFaddend , 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
it. For the rules given in section~\ref{sec:approval-based budgeting}, the
satisfaction functions $sat_\#$ (see equation~\ref{eq:3}) and $sat_{0/1}$
(equation~\ref{eq:5}) and their combination with the three aggregation methods
(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 \DIFdelbegin \DIFdel{.
}\DIFdelend \DIFaddbegin \DIFadd{because with $sat_\$ $ a projects value is its
cost. Discounting a project under $sat_\$ $ therefore lessens its value.
}\DIFaddend
\DIFaddbegin \subsection{\DIFadd{Limit Monotonicity}}
\label{subsec:Limit monotonicity}
\DIFaddend \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
monotonicity changes the project's cost, limit monotonicity changes the total
available budget. It states that if the budget limit is increased and there
exists no project which might become affordable and give higher satisfaction
than the previous solution, then a project that was a winning project before
will still be one after the budget is increased. Not satisfying this axiom could
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 \DIFdelbegin \DIFdel{one }\DIFdelend \DIFaddbegin \DIFadd{given by \mbox{%DIFAUXCMD
\cite[p.~2185]{talmonFrameworkApprovalBasedBudgeting2019}
}\hspace{0pt}%DIFAUXCMD
}\DIFaddend 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 \DIFdelbegin \DIFdel{be bettered }\DIFdelend \DIFaddbegin \DIFadd{achieve a higher satisfaction }\DIFaddend without making someone worse off while
improving another voter\DIFaddbegin \DIFadd{'s satisfaction }\DIFaddend and that results cannot only mirror one
person's preferences, respectively.
\section{Conclusion}
\label{sec:conclusion}
We have \DIFdelbegin \DIFdel{looked at different possibilities for conducting the voting and winner
selection process }\DIFdelend \DIFaddbegin \DIFadd{ introduced different methods for preference elicitation and aggregating
a winning selection of projects }\DIFaddend for participatory budgeting. A budgeting
scenario in the mathematical sense has been described and methods for modeling
voter satisfaction are discussed. \DIFdelbegin \DIFdel{A }\DIFdelend \DIFaddbegin \DIFadd{Afterwards, a }\DIFaddend deeper view on approval-based
budgeting models has been given where the voters are assumed to have dichotomous
preferences. \DIFdelbegin \DIFdel{The
complexity }\DIFdelend \DIFaddbegin \DIFadd{In section~\ref{sec:approval-based budgeting} we summarize
complexity results }\DIFaddend of the different rules\DIFdelbegin \DIFdel{has been evaluated and contrasted }\DIFdelend \DIFaddbegin \DIFadd{. Section~\ref{sec:normative axioms}
introduces three axioms by which participatory budgeting methods can be compared
}\DIFaddend to each other \DIFdelbegin \DIFdel{. We have seen that aggregation methods cannot only be compared in terms of
complexity but also by using axioms that formulate desirable outcomes}\DIFdelend \DIFaddbegin \DIFadd{and which allow for these methods to be tested in scenarios such
as when a project gets a discount}\DIFaddend .
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.
\DIFaddbegin
\DIFaddend 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.
\DIFaddbegin
\DIFaddend 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.
\DIFdelbegin \DIFdel{Exploring more axioms and rule configurations is important for achieving a
complete picture of the possibilities within the field of computational social
choice. }\DIFdelend \DIFaddbegin
\DIFaddend 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
\end{document}

232
final_revision/references.bib Normal file
View File

@ -0,0 +1,232 @@
@inproceedings{airiauPortioningUsingOrdinal2019,
title = {Portioning {{Using Ordinal Preferences}}: {{Fairness}} and {{Efficiency}}},
shorttitle = {Portioning {{Using Ordinal Preferences}}},
booktitle = {Proceedings of the 28th {{International Joint Conference}} on {{Artificial Intelligence}}},
author = {Airiau, St{\'e}phane and Aziz, Haris and Caragiannis, Ioannis and Kruger, Justin and Lang, J{\'e}r{\^o}me and Peters, Dominik},
year = {2019},
month = jul,
pages = {11--17},
abstract = {A public divisible resource is to be divided among projects. We study rules that decide on a distribution of the budget when voters have ordinal preference rankings over projects. Examples of such portioning problems are participatory budgeting, time shares, and parliament elections. We introduce a family of rules for portioning, inspired by positional scoring rules. Rules in this family are given by a scoring vector (such as plurality or Borda) associating a positive value with each rank in a vote, and an aggregation function such as leximin or the Nash product. Our family contains well-studied rules, but most are new. We discuss computational and normative properties of our rules. We focus on fairness, and introduce the SD-core, a group fairness notion. Our Nash rules are in the SD-core, and the leximin rules satisfy individual fairness properties. Both are Pareto-efficient.}
}
@inproceedings{azizFairMixingCase2019,
title = {Fair {{Mixing}}: The {{Case}} of {{Dichotomous Preferences}}},
shorttitle = {Fair {{Mixing}}},
booktitle = {Proceedings of the 2019 {{ACM Conference}} on {{Economics}} and {{Computation}}},
author = {Aziz, Haris and Bogomolnaia, Anna and Moulin, Herv{\'e}},
year = {2019},
month = jun,
pages = {753--781},
address = {{Phoenix, AZ, USA}},
abstract = {We consider a setting in which agents vote to choose a fair mixture of public outcomes. The agents have dichotomous preferences: each outcome is liked or disliked by an agent. We discuss three outstanding voting rules. The Conditional Utilitarian rule, a variant of the random dictator, is strategyproof and guarantees to any group of like-minded agents an influence proportional to its size. It is easier to compute and more efficient than the familiar Random Priority rule. Its worst case (resp. average) inefficiency is provably (resp. in numerical experiments) low if the number of agents is low. The efficient Egalitarian rule protects individual agents but not coalitions. It is excludable strategyproof: I do not want to lie if I cannot consume outcomes I claim to dislike. The efficient Nash Max Product rule offers the strongest welfare guarantees to coalitions, who can force any outcome with a probability proportional to their size. But it even fails the excludable form of strategyproofness.},
series = {{{EC}} '19}
}
@article{azizParticipatoryBudgetingModels2020,
title = {Participatory {{Budgeting}}: {{Models}} and {{Approaches}}},
shorttitle = {Participatory {{Budgeting}}},
author = {Aziz, Haris and Shah, Nisarg},
year = {2020},
month = mar,
url = {http://arxiv.org/abs/2003.00606},
urldate = {2020-04-22},
abstract = {Participatory budgeting is a democratic approach to deciding the funding of public projects, which has been adopted in many cities across the world. We present a survey of research on participatory budgeting emerging from the computational social choice literature, which draws ideas from computer science and microeconomic theory. We present a mathematical model for participatory budgeting, which charts existing models across different axes including whether the projects are treated as "divisible" or "indivisible" and whether there are funding limits on individual projects. We then survey various approaches and methods from the literature, giving special emphasis on issues of preference elicitation, welfare objectives, fairness axioms, and voter incentives. Finally, we discuss several directions in which research on participatory budgeting can be extended in the future.},
archivePrefix = {arXiv},
journal = {arXiv:2003.00606 [cs]},
primaryClass = {cs}
}
@inproceedings{azizProportionallyRepresentativeParticipatory2018,
title = {Proportionally {{Representative Participatory Budgeting}}: {{Axioms}} and {{Algorithms}}},
shorttitle = {Proportionally {{Representative Participatory Budgeting}}},
booktitle = {Proceedings of the 17th {{International Conference}} on {{Autonomous Agents}} and {{MultiAgent Systems}}},
author = {Aziz, Haris and Lee, Barton E. and Talmon, Nimrod},
year = {2018},
month = jul,
pages = {23--31},
abstract = {Participatory budgeting is one of the exciting developments in deliberative grassroots democracy. We concentrate on approval elections and propose proportional representation axioms in participatory budgeting, by generalizing relevant axioms for approval-based multi-winner elections. We observe a rich landscape with respect to the computational complexity of identifying proportional budgets and computing such, and present budgeting methods that satisfy these axioms by identifying budgets that are representative to the demands of vast segments of the voters.},
series = {{{AAMAS}} '18}
}
@inproceedings{benadePreferenceElicitationParticipatory2017,
title = {Preference Elicitation for Participatory Budgeting},
booktitle = {Proceedings of the {{Thirty}}-{{First AAAI Conference}} on {{Artificial Intelligence}}},
author = {Benade, Gerdus and Nath, Swaprava and Procaccia, Ariel D. and Shah, Nisarg},
year = {2017},
month = feb,
pages = {376--382},
address = {{San Francisco, California, USA}},
abstract = {Participatory budgeting enables the allocation of public funds by collecting and aggregating individual preferences; it has already had a sizable real-world impact. But making the most of this new paradigm requires a rethinking of some of the basics of computational social choice, including the very way in which individuals express their preferences. We analytically compare four preference elicitation methods \textemdash{} knapsack votes, rankings by value or value for money, and threshold approval votes \textemdash{} through the lens of implicit utilitarian voting, and find that threshold approval votes are qualitatively superior. This conclusion is supported by experiments using data from real participatory budgeting elections.},
series = {{{AAAI}}'17}
}
@article{bogomolnaiaCollectiveChoiceDichotomous2005,
title = {Collective Choice under Dichotomous Preferences},
author = {Bogomolnaia, Anna and Moulin, Herv{\'e} and Stong, Richard},
year = {2005},
month = jun,
volume = {122},
pages = {165--184},
abstract = {Agents partition deterministic outcomes into good or bad. A mechanism selects a lottery over outcomes (time-shares). The probability of a good outcome is the canonical utility. The utilitarian mechanism averages over outcomes with largest ``approval''. It is efficient, strategyproof, anonymous and neutral. We reach an impossibility if, in addition, each agent's utility is at least 1n, where n is the number of agents; or is at least the fraction of good to feasible outcomes. We conjecture that no ex ante efficient and strategyproof mechanism guarantees a strictly positive utility to all agents, and prove a weaker statement.},
journal = {Journal of Economic Theory},
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},
year = {2020},
pages = {20}
}
@book{brandtHandbookComputationalSocial2016,
title = {Handbook of Computational Social Choice},
author = {Brandt, Felix and Conitzer, Vincent and Endriss, Ulle and Lang, J{\'e}r{\^o}me and Procaccia, Ariel D.},
year = {2016},
abstract = {The rapidly growing field of computational social choice, at the intersection of computer science and economics, deals with the computational aspects of collective decision making. This handbook, written by thirty-six prominent members of the computational social choice community, covers the field comprehensively. Chapters devoted to each of the field's major themes offer detailed introductions. Topics include voting theory (such as the computational complexity of winner determination and manipulation in elections), fair allocation (such as algorithms for dividing divisible and indivisible goods), coalition formation (such as matching and hedonic games), and many more. Graduate students, researchers, and professionals in computer science, economics, mathematics, political science, and philosophy will benefit from this accessible and self-contained book.},
lccn = {HB846.8 .H33 2016}
}
@article{cabannesParticipatoryBudgetingSignificant2004,
title = {Participatory Budgeting: A Significant Contribution to Participatory Democracy},
shorttitle = {Participatory Budgeting},
author = {Cabannes, Yves},
year = {2004},
month = apr,
volume = {16},
pages = {27--46},
abstract = {This paper describes participatory budgeting in Brazil and elsewhere as a significant area of innovation in democracy and local development. It draws on the exp...},
journal = {Environment and Urbanization},
number = {1}
}
@article{duddyElectingRepresentativeCommittee2014,
title = {Electing a Representative Committee by Approval Ballot: {{An}} Impossibility Result},
shorttitle = {Electing a Representative Committee by Approval Ballot},
author = {Duddy, Conal},
year = {2014},
month = jul,
volume = {124},
pages = {14--16},
abstract = {We consider methods of electing a fixed number of candidates, greater than one, by approval ballot. We define a representativeness property and a Pareto property and show that these jointly imply manipulability.},
journal = {Economics Letters},
number = {1}
}
@inproceedings{fainCoreParticipatoryBudgeting2016,
title = {The {{Core}} of the {{Participatory Budgeting Problem}}},
booktitle = {Web and {{Internet Economics}}},
author = {Fain, Brandon and Goel, Ashish and Munagala, Kamesh},
editor = {Cai, Yang and Vetta, Adrian},
year = {2016},
pages = {384--399},
address = {{Berlin, Heidelberg}},
abstract = {In participatory budgeting, communities collectively decide on the allocation of public tax dollars for local public projects. In this work, we consider the question of fairly aggregating preferences to determine an allocation of funds to projects. We argue that the classic game theoretic notion of core captures fairness in the setting. To compute the core, we first develop a novel characterization of a public goods market equilibrium called the Lindahl equilibrium. We then provide the first polynomial time algorithm for computing such an equilibrium for a broad set of utility functions. We empirically show that the core can be efficiently computed for utility functions that naturally model data from real participatory budgeting instances, and examine the relation of the core with the welfare objective. Finally, we address concerns of incentives and mechanism design by developing a randomized approximately dominant-strategy truthful mechanism building on the Exponential Mechanism from differential privacy.},
series = {Lecture {{Notes}} in {{Computer Science}}}
}
@inproceedings{fainFairAllocationIndivisible2018,
title = {Fair {{Allocation}} of {{Indivisible Public Goods}}},
booktitle = {Proceedings of the 2018 {{ACM Conference}} on {{Economics}} and {{Computation}}},
author = {Fain, Brandon and Munagala, Kamesh and Shah, Nisarg},
year = {2018},
month = jun,
pages = {575--592},
abstract = {We consider the problem of fairly allocating indivisible public goods. We model the public goods as elements with feasibility constraints on what subsets of elements can be chosen, and assume that agents have additive utilities across elements. Our model generalizes existing frameworks such as fair public decision making and participatory budgeting. We study a groupwise fairness notion called the core, which generalizes well-studied notions of proportionality and Pareto efficiency, and requires that each subset of agents must receive an outcome that is fair relative to its size. In contrast to the case of divisible public goods (where fractional allocations are permitted), the core is not guaranteed to exist when allocating indivisible public goods. Our primary contributions are the notion of an additive approximation to the core (with a tiny multiplicative loss), and polynomial time algorithms that achieve a small additive approximation, where the additive factor is relative to the largest utility of an agent for an element. If the feasibility constraints define a matroid, we show an additive approximation of 2. A similar approach yields a constant additive bound when the feasibility constraints define a matching. For feasibility constraints defining an arbitrary packing polytope with mild restrictions, we show an additive guarantee that is logarithmic in the width of the polytope. Our algorithms are based on the convex program for maximizing the Nash social welfare, but differ significantly from previous work in how it is used. As far as we are aware, our work is the first to approximate the core in indivisible settings.},
series = {{{EC}} '18}
}
@inproceedings{freemanTruthfulAggregationBudget2019,
title = {Truthful {{Aggregation}} of {{Budget Proposals}}},
booktitle = {Proceedings of the 2019 {{ACM Conference}} on {{Economics}} and {{Computation}}},
author = {Freeman, Rupert and Pennock, David M. and Peters, Dominik and Wortman Vaughan, Jennifer},
year = {2019},
month = jun,
pages = {751--752},
address = {{Phoenix, AZ, USA}},
abstract = {We study a participatory budgeting setting in which a single divisible resource (such as money or time) must be divided among a set of projects. For example, participatory budgeting could be used to decide how to divide a city's tax surplus between its departments of health, education, infrastructure, and parks. A voter might propose a division of the tax surplus among the four departments into the fractions (30\%, 40\%, 20\%, 10\%). The city could invite each citizen to submit such a budget proposal, and they could then be aggregated by a suitable mechanism. In this paper, we seek mechanisms of this form that are resistant to manipulation by the voters. In particular, we require that no voter can, by lying, move the aggregate division toward her preference on one alternative without moving it away from her preference by at least as much on other alternatives. In other words, we seek budget aggregation mechanisms that are incentive compatible when each voter's disutility for a budget division is equal to the 1 distance between that division and the division she prefers most. Goel et al. [4] showed that choosing an aggregate budget division that maximizes the welfare of the voters-that is, a division that minimizes the total 1 distance from each voter's report-is both incentive compatible and Pareto-optimal under this voter utility model. However, this utilitarian aggregate has a tendency to overweight majority preferences, creeping back towards all-or-nothing allocations. For example, imagine that a hundred voters prefer (100\%, 0\%) while ninety-nine prefer (0\%, 100\%). The utilitarian aggregate is (100\%, 0\%) even though the mean is close to (50\%, 50\%). In many participatory budgeting scenarios, the latter solution is more in the spirit of consensus. To capture this idea of fairness, we define a notion of proportionality, requiring that when voters are single-minded (as in this example), the fraction of the budget assigned to each alternative is equal to the proportion of voters who favor that alternative. Do there exist aggregators that are both incentive compatible and proportional?},
series = {{{EC}} '19}
}
@article{goelKnapsackVotingParticipatory2019a,
title = {Knapsack {{Voting}} for {{Participatory Budgeting}}},
author = {Goel, Ashish and Krishnaswamy, Anilesh K. and Sakshuwong, Sukolsak and Aitamurto, Tanja},
year = {2019},
month = jul,
volume = {7},
abstract = {We address the question of aggregating the preferences of voters in the context of participatory budgeting. We scrutinize the voting method currently used in practice, underline its drawbacks, and introduce a novel scheme tailored to this setting, which we call ``Knapsack Voting.'' We study its strategic properties\textemdash we show that it is strategy-proof under a natural model of utility (a dis-utility given by the {$\mathscr{l}$}1 distance between the outcome and the true preference of the voter) and ``partially'' strategy-proof under general additive utilities. We extend Knapsack Voting to more general settings with revenues, deficits, or surpluses and prove a similar strategy-proofness result. To further demonstrate the applicability of our scheme, we discuss its implementation on the digital voting platform that we have deployed in partnership with the local government bodies in many cities across the nation. From voting data thus collected, we present empirical evidence that Knapsack Voting works well in practice.},
journal = {ACM Transactions on Economics and Computation},
number = {2}
}
@article{khullerBudgetedMaximumCoverage1999,
title = {The Budgeted Maximum Coverage Problem},
author = {Khuller, Samir and Moss, Anna and Naor, Joseph},
year = {1999},
month = apr,
volume = {70},
pages = {39--45},
abstract = {The budgeted maximum coverage problem is: given a collection S of sets with associated costs defined over a domain of weighted elements, and a budget L, find a subset of S{${'}\subseteqq$}S such that the total cost of sets in S{${'}$} does not exceed L, and the total weight of elements covered by S{${'}$} is maximized. This problem is NP-hard. For the special case of this problem, where each set has unit cost, a (1-1/e)-approximation is known. Yet, prior to this work, no approximation results were known for the general cost version. The contribution of this paper is a (1-1/e)-approximation algorithm for the budgeted maximum coverage problem. We also argue that this approximation factor is the best possible, unless NP{$\subseteqq$}DTIME(nO(loglogn)).},
journal = {Information Processing Letters},
number = {1}
}
@misc{participatorybudgetingprojectHowPBWorks,
title = {How {{PB Works}} \textendash{} {{Participatory Budgeting Project}}},
author = {Participatory Budgeting Project},
url = {https://www.participatorybudgeting.org/how-pb-works/},
urldate = {2020-05-12},
journal = {Participatory Budgeting Project}
}
@article{shapiroParticipatoryDemocraticBudgeting2018,
title = {A {{Participatory Democratic Budgeting Algorithm}}},
author = {Shapiro, Ehud and Talmon, Nimrod},
year = {2018},
month = jun,
url = {http://arxiv.org/abs/1709.05839},
urldate = {2020-04-03},
abstract = {The budget is the key means for effecting policy in democracies, yet its preparation is typically an excluding, opaque, and arcane process. We aim to rectify this by providing for the democratic creation of complete budgets --- for cooperatives, cities, or states. Such budgets are typically (i) prepared, discussed, and voted upon by comparing and contrasting with last-year's budget, (ii) quantitative, in that items appear in quantities with potentially varying costs, and (iii) hierarchical, reflecting the organization's structure. Our process can be used by a budget committee, the legislature or the electorate at large. We allow great flexibility in vote elicitation, from perturbing last-year's budget to a complete ranked budget proposal. We present a polynomial-time algorithm which takes such votes, last-year's budget, and a budget limit as input and produces a budget that is provably "democratically optimal" (Condorcet-consistent), in that no proposed change to it has majority support among the votes.},
archivePrefix = {arXiv},
journal = {arXiv:1709.05839 [cs]},
primaryClass = {cs}
}
@article{suksompongFairlyAllocatingContiguous2019,
title = {Fairly Allocating Contiguous Blocks of Indivisible Items},
author = {Suksompong, Warut},
year = {2019},
month = may,
volume = {260},
pages = {227--236},
abstract = {In this paper, we study the classic problem of fairly allocating indivisible items with the extra feature that the items lie on a line. Our goal is to find a fair allocation that is contiguous, meaning that the bundle of each agent forms a contiguous block on the line. While allocations satisfying the classical fairness notions of proportionality, envy-freeness, and equitability are not guaranteed to exist even without the contiguity requirement, we show the existence of contiguous allocations satisfying approximate versions of these notions that do not degrade as the number of agents or items increases. We also study the efficiency loss of contiguous allocations due to fairness constraints.},
journal = {Discrete Applied Mathematics}
}
@article{talmonFrameworkApprovalBasedBudgeting2019,
title = {A {{Framework}} for {{Approval}}-{{Based Budgeting Methods}}},
author = {Talmon, Nimrod and Faliszewski, Piotr},
year = {2019},
month = jul,
volume = {33},
pages = {2181--2188},
abstract = {We define and study a general framework for approval-based budgeting methods and compare certain methods within this framework by their axiomatic and computational properties. Furthermore, we visualize their behavior on certain Euclidean distributions and analyze them experimentally.},
copyright = {Copyright (c) 2019 Association for the Advancement of Artificial Intelligence},
journal = {Proceedings of the AAAI Conference on Artificial Intelligence},
number = {01}
}

29
final_revision/termpaper.sty Normal file
View File

@ -0,0 +1,29 @@
\NeedsTeXFormat{LaTeX2e}
\ProvidesPackage{termpaper}
\RequirePackage[top=2cm, left=2cm, right=2cm, bottom=3cm]{geometry}
\RequirePackage{needspace}
%% Title formatting
\def\authorname#1{{\large#1}}
\def\studentnumber#1{{\large Matr.: #1}}
\def\curriculum#1{{\large Curriculum: #1}}
\def\email#1{\tt{#1}}
\makeatletter
\makeatletter
\def\@maketitle{%
\newpage%
\null%
\begin{center}%
\huge{\@title}\par%
\vskip 1em%
\large{\@author}%
\vskip 1em%
\normalsize{\@date}%
\end{center}%
\par\par%
}