122 lines
5.3 KiB
TeX
122 lines
5.3 KiB
TeX
\documentclass[11pt,a4paper]{article}
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\usepackage{termpaper}
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\usepackage[T1]{fontenc}
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\usepackage[utf8]{inputenc}
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\usepackage{microtype}
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\usepackage{setspace}
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\usepackage{amssymb}
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\usepackage[english]{babel}
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\usepackage{csquotes}
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\usepackage[style=ieee,backend=biber]{biblatex}
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\usepackage{hyperref}
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\setstretch{1.05}
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\addbibresource{references.bib}
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%opening
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\title{Participatory Budgeting: Algorithms and Complexity}
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\author{
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\authorname{Tobias Eidelpes} \\
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\studentnumber{01527193} \\
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\curriculum{033 534} \\
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\email{e1527193@student.tuwien.ac.at}
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}
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\begin{document}
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\maketitle
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\begin{abstract}
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\end{abstract}
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\section{Introduction}
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\emph{Participatory Budgeting} (PB) is a process of democratic deliberation that
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allows residents of a municipality to decide how a part of the public budget is
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to be spent. It is a way to improve transparency and citizen involvement which
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are two important cornerstones of a democracy. PB was first realized in the
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1990s in Porto Alegre in Brazil by the Workers' Party to combat the growing
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divide between the rich city center and the poor living in the greater region.
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Owing to its success in the south of Brazil, PB quickly spread to North America,
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Europe, Asia and Africa.
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Although the process is heavily adapted by each municipality to suit the
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environment in which the residents live in, it generally follows the following
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stages \autocite{participatorybudgetingprojectHowPBWorks}:
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\begin{description}
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\item [Design the process] A rule book is crafted to ensure that the process
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is democratic.
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\item [Collect ideas] Residents propose and discuss ideas for projects.
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\item [Develop feasible projects] The ideas are developed into projects that
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can be undertaken by the municipality.
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\item [Voting] The projects are voted on by the residents.
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\item [Aggregating votes \& funding] The votes are combined to determine a
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set of winning projects which are then funded.
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\end{description}
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\noindent The two last stages \emph{voting} and \emph{aggregating votes} are of
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main interest for computer scientists, economists and social choice theorists
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because depending on how voters elicit their preferences (\emph{balloting}) and
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how the votes are aggregated through the use of algorithms, the outcome is
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different. For this paper it is assumed that the first three stages have already
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been completed. The rules of the process have been set, ideas have been
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collected and developed into feasible projects and the budget limit is known. To
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study different ways of capturing votes and aggregating them, the participatory
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process is modeled mathematically. This model will be called a participatory
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budgeting \emph{scenario}. The aim of studying participatory budgeting scenarios
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is to find ways to achieve a desirable outcome. A desirable outcome can be one
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based on fairness by making sure that each voter has at least one chosen project
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in the final set of winning projects for example. Other approaches are concerned
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with maximizing social welfare or discouraging \emph{gaming the voting process}
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(where an outcome can be manipulated by not voting truthfully; also called
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\emph{strategyproofness}).
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First, this paper will look at how a participatory budgeting scenario can be
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modeled mathematically. Then, a brief overview over common models will be given.
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To illustrate these methods, one approach will be chosen and discussed in detail
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with respect to algorithmic complexity and properties. Finally, the gained
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insight into participatory budgeting algorithms will be summarized and an
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outlook on further developments will be given.
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\section{Mathematical Model}
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\label{sec:mathematical model}
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\textcite{talmonFrameworkApprovalBasedBudgeting2019} define a participatory
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budgeting scenario as a tuple $E = (P,V,c,B)$, consisting of a set of projects
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$P = \{ p_1,\dots,p_m \}$ where each project $p\in P$ has an associated cost
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$c(p):P\rightarrow\mathbb{R}$, a set of voters $V = \{v_1,\dots,v_n\}$ and a
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budget limit $B$. The voters express preferences over individual projects or
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over subsets of all projects. How the preferences of voters are expressed has to
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be decided during the design phase of the process and is a choice that has to be
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made in accordance with the method that is used for aggregating the votes. After
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the voters have elicited their preferences, a set of projects $A\subseteq P$ is
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selected as \emph{winning projects} according to some rule and subject to the
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total budget limit $B$. For the case where projects are indivisible, which is
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also called discrete, the sum of the winning projects' costs is not allowed to
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exceed the limit $B$:
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\begin{equation}
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\sum_{p\in A}{c(p)\leq B}.
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\end{equation}
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When projects can be divisible, i.e. completed to a fractional degree, the
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authors define a function $\mu(p) : P\rightarrow [0,1]$ which maps every project
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to an interval between zero and one, representing the fractional degree to which
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this project is completed. Since the cost of each project is a function of its
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degree of completion, the goal is to select a set of projects where the cost of
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the degree of completion does not exceed the budget limit:
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\begin{equation}
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\sum_{p\in A}{c(\mu(p))\leq B}.
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\end{equation}
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\textcite{azizParticipatoryBudgetingModels2020} define a taxonomy of
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participatory budgeting scenarios where projects can be either divisible or
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indivisible and bounded or unbounded.
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\printbibliography
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\end{document}
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