From da6afc8fa1676f37c3692c7bc3a83070100811d2 Mon Sep 17 00:00:00 2001 From: Tobias Eidelpes Date: Sat, 4 Jul 2020 17:58:30 +0200 Subject: [PATCH] Add diff between second and final revision --- final_revision/diff.tex | 592 ++++++++++++++++++++++++++++++++++ final_revision/references.bib | 232 +++++++++++++ final_revision/termpaper.sty | 29 ++ 3 files changed, 853 insertions(+) create mode 100644 final_revision/diff.tex create mode 100644 final_revision/references.bib create mode 100644 final_revision/termpaper.sty diff --git a/final_revision/diff.tex b/final_revision/diff.tex new file mode 100644 index 0000000..523e2ee --- /dev/null +++ b/final_revision/diff.tex @@ -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} diff --git a/final_revision/references.bib b/final_revision/references.bib new file mode 100644 index 0000000..abeacb4 --- /dev/null +++ b/final_revision/references.bib @@ -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} +} + + diff --git a/final_revision/termpaper.sty b/final_revision/termpaper.sty new file mode 100644 index 0000000..6f1b1aa --- /dev/null +++ b/final_revision/termpaper.sty @@ -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% +} \ No newline at end of file