413 lines
16 KiB
TeX
413 lines
16 KiB
TeX
\documentclass{beamer}
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\beamertemplatenavigationsymbolsempty
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\usetheme{Boadilla}
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\usecolortheme{dolphin}
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\usepackage{graphicx}
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\usepackage{caption}
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\usepackage{tikz}
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\usepackage{dsfont}
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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?} \pause
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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} \pause
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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 \pause
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\item Budgeting part: each project requires a fixed amount of money
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\pause
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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{A bit of background information} \pause
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\begin{itemize}
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\setlength{\itemsep}{1\baselineskip}
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\item PB originated in Porto Alegre in 1990s \pause
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\item Rapid expansion and influx of immigrants increased inequality
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\pause
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\item PB as a tool to combat inequality \pause
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\item PB spread to Europe and North America \pause
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\item Today \$300M allocated
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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?} \pause
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\tikzstyle{blue} = [rectangle,rounded
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corners=3pt,draw=blue!50,fill=blue!20]
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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,node distance=1.3cm]
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\node [blue] (design) {Design the process}; \pause
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\node [blue] (collect) [below of=design] {Collect ideas};
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\draw [arrow] (design) to (collect); \pause
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\node [blue] (develop) [below of=collect] {Develop feasible
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projects};
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\draw [arrow] (collect) to (develop); \pause
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\node [red] (vote) [below of=develop] {Vote on projects};
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\draw [arrow] (develop) to (vote); \pause
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\node [red] (aggregate) [below of=vote] {Aggregate votes \& fund
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winners};
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\draw [arrow] (vote) to (aggregate);
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\onslide<1->
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\end{tikzpicture}
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\end{center}
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\end{frame}
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\section{Modeling a PB scenario}
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\begin{frame}
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\frametitle{A general framework for PB} \pause
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\begin{itemize}
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\setlength{\itemsep}{1\baselineskip}
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\item Projects $P=\{p_1,\dots,p_m\}$ \pause
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\begin{itemize}
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\setlength{\itemsep}{.7\baselineskip}
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\item Each project $p\in P$ has associated cost
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$c(p):P\rightarrow\mathbb{R}$ \pause
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\item Projects are either divisible or indivisible (discrete)
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\pause
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\end{itemize}
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\item Select a set $A\subseteq P$ as \emph{winning projects} not
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exceeding total budget $B$ \pause
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\begin{itemize}
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\setlength{\itemsep}{.7\baselineskip}
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\item Discrete case: $\sum_{p\in A}c(p)\leq B$ \pause
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\item Divisible case: $\mu(p): P\rightarrow [0,1]$ with
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$\sum_{p\in A}c(\mu(p))\leq B$
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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{A general framework for PB ctd.}
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\begin{itemize}
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\setlength{\itemsep}{1\baselineskip}
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\item Voters $V=\{v_1,\dots,v_n\}$ \pause
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\begin{itemize}
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\setlength{\itemsep}{.4\baselineskip}
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\item Express preferences over individual projects in $P$ or
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over subsets in $\mathcal{P}(P) := \{P_v\,|\,P_v\subseteq
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P\}$ \pause
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\item Preference elicitation is dependent on the input method
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(approval-based, ranked orders) \pause
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\end{itemize}
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\item Aggregation methods \pause
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\begin{itemize}
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\setlength{\itemsep}{.4\baselineskip}
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\item Aggregation methods combine votes to determine a set
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of winning projects \pause
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\item Are usually tied to the input method \pause
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\item Rules are used to select projects w.r.t. desired
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properties of the outcome (fairness, welfare)
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\end{itemize}
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\end{itemize}
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\end{frame}
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\section{Algorithms for PB}
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\begin{frame}
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\frametitle{Input and aggregation methods}
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Example input methods: \pause
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\begin{itemize}
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\item Approval preferences \pause
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\item Ranked orders \pause
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\item Utility-based preferences \pause
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\end{itemize}
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\vspace{0.2cm}
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Example aggregation methods: \pause
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\begin{itemize}
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\item Maximizing social welfare \pause
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\item Greedy selection \pause
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\item Fairness-based selection \pause
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\end{itemize}
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\vspace{0.2cm}
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Aggregation methods depend on how voters elicit their preferences.
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\end{frame}
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\begin{frame}
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\frametitle{Approval-based budgeting methods} \pause
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\begin{itemize}
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\item Suitable for discrete PB \pause
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\item Voters approve a subset of projects \pause
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\item Voter preferences are assumed to be \emph{dichotomous} \pause
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\item A \emph{satisfaction function} provides a metric for voter
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satisfaction \pause
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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 $E = (P,V,c,B)$ where $P =
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\{p_1,\dots,p_m\}$ is a set of projects, $V$ is a set of voters, $c :
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P\rightarrow\mathbb{N}$ is a cost function associating each project
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$p\in P$ with its cost $c(p)$ and $B\in\mathbb{N}$ is a budget limit. A
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voter $v\in V$ specifies $P_v\subseteq P$, containing all approved
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items.
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\end{block} \pause
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\begin{block}{Budgeting method $\mathcal{R}$}
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A budgeting method $\mathcal{R}$ takes a budgeting scenario $E$ and
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returns a bundle $A\subseteq P$ where the total cost of the items in
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$A$ does not exceed the budget limit $B$.
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\end{block}
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\end{frame}
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\begin{frame}
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\frametitle{Satisfaction functions} \pause
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\begin{block}{Satisfaction function}
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A satisfaction function $sat : 2^P\times 2^P\rightarrow\mathbb{R}$ with
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a set $P$ of items, a voter $v$ and her approval set $P_v$ and a bundle
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$A\subseteq P$ provides the satisfaction $sat(P_v,A)$ of $v$ from the
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bundle $A$. The set of approved items by $v$ that end up in the winning
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bundle is denoted by $A_v = P_v\cap A$.
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\end{block} \pause
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\begin{exampleblock}{$sat_\#(P_v,A)$}
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$sat_\#(P_v,A) = |A_v|$: The satisfaction of voter $v$ is the number of
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funded items that are approved.
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\end{exampleblock} \pause
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\end{frame}
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\begin{frame}
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\frametitle{Satisfaction functions ctd.}
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\begin{exampleblock}{$sat_\$(P_v,A)$}
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$sat_{\$}(P_v,A) = \sum_{p\in A_v}{c(p) = c(A_v)}$: The satisfaction of voter
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$v$ is the total cost of her approved and funded items.
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\end{exampleblock} \pause
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\begin{exampleblock}{$sat_{0/1}(P_v,A)$}
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\[ sat_{0/1}(P_v,A) =
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\begin{cases}
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1 & \text{if } |A_v|>0 \\
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0 & \text{otherwise}
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\end{cases}
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\]
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A voter $v$ has satisfaction 1 if at least one of her approved items is
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funded and 0 otherwise.
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\end{exampleblock}
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\end{frame}
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\begin{frame}
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\frametitle{Rules for selecting a winning bundle} \pause
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{\Large Let $sat$ be a satisfaction function:} \pause
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\begin{block}{Max rules}
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The rule $\mathcal{R}_{sat}^m$ selects a bundle which maximizes the sum
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of voters' satisfaction: $\mathsf{max}_{A\subseteq P}\sum_{v\in
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V}{sat(P_v,A)}$
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\end{block} \pause
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\begin{block}{Greedy rules}
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The rule $\mathcal{R}_{sat}^g$ iteratively adds an item $p\in P$ to $A$,
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seeking to maximize $\sum_{v\in V}{sat(P_v,A\cup\{p\})}$.
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\end{block} \pause
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\begin{block}{Proportional greedy rules}
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The rule $\mathcal{R}_{sat}^p$ iteratively adds an item $p\in P$ to $A$
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seeking to maximize the sum of satisfaction per unit of cost.
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\end{block}
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\end{frame}
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\begin{frame}
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\frametitle{Example budgeting scenarios} \pause
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\begin{block}{A budgeting scenario}
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Items $P = \{p_2,p_3,p_4,p_5,p_6\}$ and their associated cost $p_i$
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where $p_i$ costs $i$. Budget limit $B=10$ and 5 voters vote
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$v_1=\{p_2,p_5,p_6\}$, $v_2=\{p_2,p_3,p_4,p_5\}$, $v_3=\{p_3,p_4,p_5\}$,
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$v_4=\{p_4,p_5\}$ and $v_5=\{p_6\}$.
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\end{block} \pause
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\begin{exampleblock}{Combining max rule with $sat_\#$}
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Under $\mathcal{R}_{|A_v|}^m$ the winning bundle is $\{p_2,p_3,p_5\}$.
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The total satisfaction is 8.
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\end{exampleblock} \pause
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\begin{exampleblock}{Combining greedy rule with $sat_\#$}
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Under $\mathcal{R}_{|A_v|}^g$ the winning bundle is $\{p_4,p_5\}$ (first
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selecting $p_5$). The total satisfaction is 7.
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\end{exampleblock} \pause
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\begin{exampleblock}{Combining max rule with $sat_{0/1}$}
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Under $\mathcal{R}^m_{sat_{0/1}}$ the winning bundle is $\{p_4,p_6\}$,
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achieving max satisfaction.
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\end{exampleblock}
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\end{frame}
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\section{Complexity of PB algorithms}
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\begin{frame}
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\frametitle{Complexity of algorithms} \pause
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\begin{itemize}
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\setlength{\itemsep}{1\baselineskip}
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\item Computing winners using greedy rules ($\mathcal{R}^g_{sat}$) can
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be done in polynomial time: \pause
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\begin{itemize}
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\setlength{\itemsep}{.4\baselineskip}
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\item these rules are defined through efficient iterative
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processes \pause
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\item however: making a series of locally optimal choices does
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not always lead to a globally optimal choice \pause
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\item $\mathcal{R}^g_{|A_v|}$ is similar to $k$-Approval and
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knapsack voting \pause
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\end{itemize}
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\item Max rules ($\mathcal{R}^m_{sat}$) are generally NP-hard \pause
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\begin{itemize}
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\setlength{\itemsep}{.4\baselineskip}
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\item $\mathcal{R}^m_{|A_v|}$ can be solved in polynomial time
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because one dimension is fixed \pause
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\item $\mathcal{R}^m_{sat_{0/1}}$: finding a bundle with at least a
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given total satisfaction is NP-hard \pause
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\item satisfaction functions can be modeled as integer linear
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programs
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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{Complexity of algorithms ctd.}
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{\large Dealing with \emph{intractability}:} \pause
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\vspace{.3cm}
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\begin{itemize}
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\setlength{\itemsep}{1\baselineskip}
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\item Provide an approximation algorithm, sacrificing exactness \pause
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\begin{itemize}
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\setlength{\itemsep}{0.4\baselineskip}
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\item No algorithm with approx. ratio better than $1-1/\epsilon$
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exists for $\mathcal{R}^m_{sat_{0/1}}$ \pause
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\end{itemize}
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\item Fixed-parameter tractability: fix one parameter to solve problem
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in reasonable amount of time \pause
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\begin{itemize}
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\setlength{\itemsep}{0.4\baselineskip}
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\item Fix parameter $m$ (the number of items) \pause
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\item Fix parameter $n$ (the number of voters)
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\end{itemize}
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\end{itemize}
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\end{frame}
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\section{Axioms for PB algorithms}
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\begin{frame}
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\frametitle{Comparing algorithms} \pause
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\begin{itemize}
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\setlength{\itemsep}{.9\baselineskip}
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\item Compare algorithms by using axioms \pause
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\item Axioms provide \emph{guidelines} for choosing an algorithm \pause
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\item Some might be desirable, others are not a drawback, if not
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satisfied \pause
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\item Desirable properties: \emph{fairness}, \emph{strategyproofness}
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\end{itemize}
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\end{frame}
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\begin{frame}
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\frametitle{Comparing algorithms—Discount Monotonicity} \pause
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\begin{block}{Discount Monotonicity}
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Suppose a budgeting algorithm $\mathcal{R}$ outputs a winning set of
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projects $A$. The cost of project $p\in A$ is lowered (discounted)
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compared to the previous cost. $\mathcal{R}$ should output another
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winning set $A'$ where project $p$ is not implemented to a lesser
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degree.
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\end{block} \pause
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\begin{block}{A budgeting scenario}
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Items $P = \{p_2,p_3,p_4,p_5,p_6\}$ and their associated cost $p_i$
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where $p_i$ costs $i$. Budget limit $B=10$ and 5 voters vote
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$v_1=\{p_2,p_5,p_6\}$, $v_2=\{p_2,p_3,p_4,p_5\}$, $v_3=\{p_3,p_4,p_5\}$,
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$v_4=\{p_4,p_5\}$ and $v_5=\{p_6\}$.
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\end{block} \pause
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\begin{exampleblock}{Discount Monotonicity Example}
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Under $\mathcal{R}^m_{|A_v|}$ the winning bundle is $\{p_2,p_3,p_5\}$.
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After discounting $p_2$ to $p_1$, we still get $\{p_1,p_3,p_5\}$.
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\end{exampleblock}
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\end{frame}
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\begin{frame}
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\frametitle{Comparing algorithms—Limit Monotonicity} \pause
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\begin{block}{Limit Monotonicity}
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A budgeting method $\mathcal{R}$ satisfies Limit Monotonicity if for a
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pair of budgeting scenarios $E=(P,V,c,B)$, $E'=(P,V,c,B+1)$ and with no
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project that costs exactly $B+1$, for each project $p\in P$
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$p\in\mathcal{R}(E)\implies p\in\mathcal{R}(E')$ holds.
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\end{block} \pause
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\begin{block}{A budgeting scenario}
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Items $P=\{a_1,b_1,c_1\}$ (all with unit cost), budget limit $B=1$ and 4
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voters vote $v_1=\{a_1\}$, $v_2=\{a_1,b_1\}$, $v_3=\{b_1,c_1\}$ and
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$v_4=\{c_1\}$.
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\end{block} \pause
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\begin{exampleblock}{Limit Monotonicity Example}
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Under $\mathcal{R}^m_{sat_{0/1}}$ a winning bundle might be $\{b_1\}$.
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Increasing the budget limit to 2 results in $\{a_1,c_1\}$.
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\end{exampleblock}
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\end{frame}
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\begin{frame}
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\frametitle{Applying the two axioms to the example algorithms} \pause
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\begin{itemize}
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\setlength{\itemsep}{1\baselineskip}
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\item Discount Monotonicity: \pause
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\begin{itemize}
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\setlength{\itemsep}{.4\baselineskip}
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\item All 3 algorithms ($\mathcal{R}^m_{|A_v|}$,
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$\mathcal{R}^g_{|A_v|}$, $\mathcal{R}^m_{sat_{0/1}}$)
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satisfy discount monotonicity \pause
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\item Decreasing the cost increases the attractiveness of an
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item \pause
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\item Not true for algorithms that measure satisfaction by
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maximizing the cost of winning projects \pause
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\end{itemize}
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\item Limit Monotonicity: \pause
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\begin{itemize}
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\setlength{\itemsep}{.4\baselineskip}
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\item No discussed algorithm satisfies limit monotonicity \pause
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\item Greedy heuristics fail when a project fits into the new
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budget, providing higher satisfaction \pause
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\item Increasing the budget might lead to projects being dropped
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\end{itemize}
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\end{itemize}
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\end{frame}
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\section{Conclusion}
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\begin{frame}
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\frametitle{Conclusion} \pause
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\begin{itemize}
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\setlength{\itemsep}{1\baselineskip}
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\item PB: What is it? \pause
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\item Computational aspects \pause
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\item Aggregation algorithms \pause
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\item Complexity of algorithms \pause
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\item Comparison of algorithms using axioms
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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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\begin{figure}
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\centering
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\includegraphics[width=.5\textwidth]{voting_referendum.png}
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\captionsetup{labelformat=empty}
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\caption{\tiny [\url{https://xkcd.com/2225}]}
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\end{figure}
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\end{frame}
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\end{document}
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