\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}