Restructure talk
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talk/talk.tex
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talk/talk.tex
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\usecolortheme{dolphin}
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\usepackage{graphicx}
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\usepackage{tikz}
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\usetikzlibrary{arrows}
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\begin{document}
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@ -28,194 +31,70 @@
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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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\vspace{1cm}
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\begin{itemize}
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\setlength{\itemsep}{1.1\baselineskip}
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\item Participatory part: community members propose projects
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\item Budgeting part: each project requires a fixed amount of money
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\item Goal: Fund the \emph{best} projects without exceeding the budget
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\end{itemize}
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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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\tikzstyle{blue} = [rectangle,rounded
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corners=3pt,draw=blue!50,fill=blue!20,node distance=1cm]
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\tikzstyle{red} = [rectangle,rounded corners=3pt,draw=red!50,fill=red!20]
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\tikzstyle{arrow} = [line width=1pt,->,>=triangle 60,draw=black!70]
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\begin{center}
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\begin{tikzpicture}[shorten >= 2pt,level distance=1.3cm]
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\node[blue](design){Design the process}
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child { node [blue](collect){Collect ideas}
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child { node [blue](develop){Develop feasible projects}
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child { node [red](vote){Vote on projects}
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child { node [red](aggregate){Aggregate votes \& fund winners}
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}
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}
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}
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}
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;
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\draw [arrow] (design) to (collect);
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\draw [arrow] (collect) to (develop);
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\draw [arrow] (develop) to (vote);
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\draw [arrow] (vote) to (aggregate);
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\end{tikzpicture}
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\end{center}
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\end{frame}
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\begin{frame}
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\frametitle{A formal model for PB}
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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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\setlength{\itemsep}{1\baselineskip}
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\item Projects can be bounded or unbounded
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\item Projects can be divisible or indivisible (discrete)
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\item Each project has an associated cost
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\item Voters approve a subset of all projects (\emph{input method})
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\item The total cost is limited by the available budget
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\item An \emph{aggregation method} provides a list of projects to fund
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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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\frametitle{Input and aggregation methods}
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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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\item Approval voting
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\item Ranked voting
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\item Knapsack voting
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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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\begin{block}{An approval-based budgeting scenario}
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A budgeting scenario is a tuple $E = (A,V,c,l)$ where $A =
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\{a_1,\dots,a_m\}$ is a set of items (projects), $V$ is a set of voters,
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$c : A\rightarrow\mathbb{N}$ is a cost function associating each project
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$a\in A$ with its cost $c(a)$ and $l\in\mathbb{N}$ is a budget limit. A
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voter $v\in V$ specifies $A_v\subseteq A$, containing all approved
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items.
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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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