343 lines
10 KiB
TeX
343 lines
10 KiB
TeX
\documentclass{beamer}
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\usetheme{Boadilla}
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\usecolortheme{dolphin}
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\usepackage{graphicx}
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\begin{document}
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\title[Participatory Budgeting]{Participatory Budgeting}
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\subtitle{Algorithms and Complexity}
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\author{Tobias Eidelpes}
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\begin{frame}
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\maketitle
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\end{frame}
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\begin{frame}
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\frametitle{Table of Contents}
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\tableofcontents
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\end{frame}
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\section{Introduction}
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\begin{frame}
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\frametitle{What is Participatory Budgeting?}
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\begin{quote}
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Participatory Budgeting (PB) is a democratic process in which
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community members decide how to spend part of a public budget.
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\end{quote}
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\end{frame}
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\begin{frame}
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\frametitle{How does it work?}
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\begin{itemize}
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\item Designing the Process
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\item Collecting Ideas
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\item Developing Proposals
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\item Voting
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\item Funding Winners
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\end{itemize}
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\end{frame}
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\begin{frame}
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\frametitle{Benefits of Participatory Budgeting}
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\begin{itemize}
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\item More efficient spending
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\item Diverse participants
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\item Higher voter satisfaction
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\item Democratic and citizenship learning
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\item Institutional and political change
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\end{itemize}
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\end{frame}
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\section{Computational Aspects}
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\begin{frame}
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\frametitle{Computational Aspects of PB}
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\begin{itemize}
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\item Discrete or continuous projects?
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\item How do we model preferences mathematically?
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\item How do we adequately capture voter's preferences?
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\item How do we aggregate votes?
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\end{itemize}
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\end{frame}
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\begin{frame}
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\frametitle{Decision Space}
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\begin{figure}
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\centering
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\includegraphics[width=\textwidth]{taxonomy.png}
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\end{figure}
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\end{frame}
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\begin{frame}
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\frametitle{Bounded Divisible PB}
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\begin{itemize}
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\item Projects are divisible
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\item A cap for each project is defined
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\item Fractional funding
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\end{itemize}
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\begin{exampleblock}{Example}
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A project that seeks to donate a bounded amount of money to a
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charity.
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\end{exampleblock}
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\end{frame}
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\begin{frame}
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\frametitle{Unbounded Divisible PB}
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\begin{itemize}
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\item Projects are divisible
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\item No caps for projects
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\item Generalizable to \emph{Portioning}
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\end{itemize}
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\begin{block}{Unbounded Divisible PB} $x_p = [0,1]$ denotes the
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fraction of project $p\in P$ that is completed and $c_p(x_p) =
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x_p$ is the cost function of project $p$. The set of feasible
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budget allocations under a budget $B = 1$ is therefore defined
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as \[ \{ \vec{x} : \sum_{p\in P}{x_p}\leq 1 \}. \]
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\end{block}
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\begin{exampleblock}{Example}
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A project that seeks to donate an unbounded amount of money to a
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charity. Every additional amount can be used effectively.
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\end{exampleblock}
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\end{frame}
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\begin{frame}
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\frametitle{Bounded Discrete PB}
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\begin{itemize}
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\item Projects are either fully implemented or not at all
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\item Degree of completion has a cap
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\item Budget is defined as subset of projects which can be
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implemented subject to budget constraints
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\end{itemize}
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\begin{exampleblock}{Example}
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A project for building a new school.
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\end{exampleblock}
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\end{frame}
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\begin{frame}
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\frametitle{Unbounded Discrete PB}
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\begin{itemize}
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\item Multiple degrees of completion
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\item Substages of projects (milestones) can be defined
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\item Still bounded by total available budget
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\end{itemize}
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\begin{exampleblock}{Example}
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A project for building public toilets. The degree of completion
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is the number of toilets that have already been built.
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\end{exampleblock}
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\end{frame}
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\section{Preference Modeling}
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\begin{frame}
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\frametitle{Preference Modeling}
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Model preferences as a cardinal utility function or an ordinal
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preference relation:
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\begin{block}{Cardinal utility function}
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Each resident $i$ has a cardinal utility function $u_i :
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A\rightarrow \mathbb{R}$, where $A$ is the set of feasible
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allocations.
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\end{block}
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\begin{block}{Ordinal preference relation}
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$\succ_i$ over $A$
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\end{block}
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\begin{alertblock}{Problem}
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This does not adequately reflect any structural properties of
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residents' preferences.
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\end{alertblock}
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\end{frame}
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\begin{frame}
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\frametitle{Preference Modeling}
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\begin{itemize}
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\item Impose a structural assumption on the utility function:
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\[ u_i : 2^P\rightarrow\mathbb{R} \]
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and $u_i$ satisfies subadditivity or superadditivity.
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\item Use spatial models where preferences are situated in a
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metric space and the distance between them models a
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resident's utility for another allocation.
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\item Take preferences over projects and use a rule to extend
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them to allocations.
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\end{itemize}
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\end{frame}
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\begin{frame}
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\frametitle{Cardinal extensions}
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\begin{block}{Scalar separable utility function}
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A resident $i$ derives utility $u_{i,p}\cdot f_p(x_p)$ from each
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project. A resident's utility for an allocation $\vec{x}$ is
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additive across projects:
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\[ u_i(\vec{x}) = \sum_{p\in P}{u_{i,p}\cdot f_p(x_p)} \]
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\end{block}
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\begin{block}{Dichotomous preferences}
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Assuming $x_p\in\{0,1\}$ and $u_{i,p}\in\{0,1\}$, residents
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either approve or disapprove a project and care only about the
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number of projects implemented.
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\end{block}
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\begin{block}{Max set extension}
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Utility of an allocation is defined as the utility for a
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resident's most favorite project: $u_i(S) = \mathsf{max}_{p\in
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S}u_{i,p}$ for each $S\subseteq P$.
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\end{block}
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\end{frame}
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\begin{frame}
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\frametitle{Ordinal extensions}
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\begin{block}{Stochastic dominance extension}
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For two allocations $\vec{x},\vec{y}\in A$ and
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$E_i^1,\dots,E_i^{k_i}$ being the equivalence classes of the
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relation $\succ_i$ in decreasing order of preferences: \[
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\vec{x}\succ_{i}^{SD}\vec{y} \text{ iff }
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\sum_{j=1}^{l}{\sum_{p\in E_i^j}{x_p}}\geq\sum_{j=1}^{l}{\sum_{p\in
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E_i^j}{y_p}}\quad\text{for all } l\in\{1,\dots,k_i\}\]
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\end{block}
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\begin{block}{Lexicographic extension $\succ_i^{lex}$}
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A resident $i$ cares significantly more about project $p$ than
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about $p'$ whenever $p\succ_i p'$.
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\end{block}
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\begin{block}{Scoring rules}
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Convert ordinal to cardinal preferences by taking a ranking
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$\succ_i$ over projects and determining the utility as $u_{i,p}
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= s_k$ where $k$ is the rank in a scoring vector $\vec{s} =
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(s_1,\dots,s_m)$ and $s_1\geq\dots\geq s_m\geq 0$.
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\end{block}
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\end{frame}
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\section{Preference Elicitation}
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\begin{frame}
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\frametitle{Preference elicitation}
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\begin{itemize}
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\item Also known as \emph{Ballot Design}
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\item Communicating full preferences over sometimes
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exponentially many allocations is difficult
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\item Cognitive burden can lead to lower turnout rates
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\end{itemize}
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\end{frame}
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\begin{frame}
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\frametitle{Preference elicitation}
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\begin{block}{Range voting}
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Voters rate projects based on their utility for each project.
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\end{block}
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\begin{block}{$k$-Approval}
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Voters approve the $k$ projects they like the most.
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\end{block}
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\begin{block}{Approval voting}
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Voters approve all projects that they like.
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\end{block}
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\begin{block}{Threshold approval voting}
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Voters approve projects where their utility is above a specified
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threshold.
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\end{block}
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\begin{block}{Knapsack voting}
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Voters provide ideal allocation based on their preferences.
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\end{block}
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\end{frame}
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\section{Vote Aggregation}
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\begin{frame}
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\frametitle{Vote Aggregation}
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\begin{itemize}
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\item Voters' preferences are aggregated to determine which
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projects to fund
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\item Main interest for research
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\item Three different approaches:
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\begin{itemize}
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\item Welfare Maximization
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\item Use of Axioms
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\item Notions of Fairness
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\end{itemize}
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\end{itemize}
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\end{frame}
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\begin{frame}
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\frametitle{Welfare Maximization}
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\begin{block}{Utilitarian Welfare}
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The utilitarian welfare of an allocation is the sum of utilities it gives to
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residents:
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\[ UW(\vec{x}) =
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\sum_{i\in N}{u_i(\vec{x})}\text{ for }\vec{x}\in A \]
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\end{block}
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\begin{block}{Egalitarian Welfare}
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The egalitarian welfare of an allocation is the minimum utility
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it gives to any resident:
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\[ EW(\vec{x}) =
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\mathsf{min}_{i\in N}{u_i(\vec{x})}\text{ for
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}\vec{x}\in A \]
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\end{block}
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\begin{block}{Nash Welfare}
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The Nash welfare of an allocation is the product of utilities it gives to
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residents:
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\[ NW(\vec{x}) =
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\prod_{i\in N}{u_i(\vec{x})}\text{ for }\vec{x}\in A \]
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\end{block}
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\end{frame}
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\begin{frame}
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\frametitle{Use of Axioms}
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\begin{block}{Exhaustiveness}
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A feasible allocation $\vec{x}$ is called exhaustive if an
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outcome $\vec{y}$ is not feasible whenever $y_p\geq x_p$ for all
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projects $p$ and a strict inequality holds for at least one
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project.
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\end{block}
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\begin{block}{Discount Monotonicity}
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Project $p$'s cost function $c_p$ is revised to $c_p'(x_p)\leq
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c_p(x_p)$ after a vote aggregation rule outputs allocation
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$\vec{x}$. For the resulting allocation $\vec{y}$, $y_p\geq
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x_p$ holds.
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\end{block}
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\begin{block}{Pareto Optimality}
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An allocation $\vec{x}\in A$ Pareto dominates another allocation
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$\vec{y}\in A$ if $u_i(\vec{x})\geq u_i(\vec{y})$ for all $i\in
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N$ and $u_i(\vec{x}) > u_i(\vec{y})$ for some $i\in N$. An
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allocation $\vec{z}\in A$ is optimal if no allocation dominates
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it.
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\end{block}
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\end{frame}
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\begin{frame}
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\frametitle{Notion of Fairness}
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\begin{block}{The Core of PB}
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An allocation $\vec{x} \in A$ is a core solution if there is no
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subset $S$ of voters who, given a budget of $(|S|/n)B$, could
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compute an allocation $\vec{y}\in A$ such that every voter in
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$S$ receives strictly more utility in $\vec{y}$ than in
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$\vec{x}$.
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\end{block}
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\begin{block}{Proportionality}
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An allocation $\vec{x}$ should be proportionally reflected by
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the division of voters. A majority of voters should have a
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majority of the budget under their control but a minority should
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have a minority of the budget under their control.
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\end{block}
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\end{frame}
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\section{Future Directions}
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\begin{frame}
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\frametitle{Future Areas of Interest}
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\begin{itemize}
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\item Multi-dimensional constraints
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\item Hybrid models
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\item Complex resident preferences
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\item Market-based approaches
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\item The role of information
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\item Research spanning the entire PB process
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\end{itemize}
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\end{frame}
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\begin{frame}
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\centering
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\Large
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Thank you for your attention! \\
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Questions \& Answers
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\end{frame}
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\end{document}
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