222 lines
6.8 KiB
TeX
222 lines
6.8 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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\usepackage{tikz}
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\usetikzlibrary{arrows}
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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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\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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\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{Input and aggregation methods}
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\begin{itemize}
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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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\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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\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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