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\usecolortheme{dolphin} \usecolortheme{dolphin}
\usepackage{graphicx} \usepackage{graphicx}
\usepackage{tikz}
\usetikzlibrary{arrows}
\begin{document} \begin{document}
@ -28,194 +31,70 @@
Participatory Budgeting (PB) is a democratic process in which Participatory Budgeting (PB) is a democratic process in which
community members decide how to spend part of a public budget. community members decide how to spend part of a public budget.
\end{quote} \end{quote}
\vspace{1cm}
\begin{itemize}
\setlength{\itemsep}{1.1\baselineskip}
\item Participatory part: community members propose projects
\item Budgeting part: each project requires a fixed amount of money
\item Goal: Fund the \emph{best} projects without exceeding the budget
\end{itemize}
\end{frame} \end{frame}
\begin{frame} \begin{frame}
\frametitle{How does it work?} \frametitle{How does it work?}
\tikzstyle{blue} = [rectangle,rounded
corners=3pt,draw=blue!50,fill=blue!20,node distance=1cm]
\tikzstyle{red} = [rectangle,rounded corners=3pt,draw=red!50,fill=red!20]
\tikzstyle{arrow} = [line width=1pt,->,>=triangle 60,draw=black!70]
\begin{center}
\begin{tikzpicture}[shorten >= 2pt,level distance=1.3cm]
\node[blue](design){Design the process}
child { node [blue](collect){Collect ideas}
child { node [blue](develop){Develop feasible projects}
child { node [red](vote){Vote on projects}
child { node [red](aggregate){Aggregate votes \& fund winners}
}
}
}
}
;
\draw [arrow] (design) to (collect);
\draw [arrow] (collect) to (develop);
\draw [arrow] (develop) to (vote);
\draw [arrow] (vote) to (aggregate);
\end{tikzpicture}
\end{center}
\end{frame}
\begin{frame}
\frametitle{A formal model for PB}
\begin{itemize} \begin{itemize}
\item Designing the Process \setlength{\itemsep}{1\baselineskip}
\item Collecting Ideas \item Projects can be bounded or unbounded
\item Developing Proposals \item Projects can be divisible or indivisible (discrete)
\item Voting \item Each project has an associated cost
\item Funding Winners \item Voters approve a subset of all projects (\emph{input method})
\item The total cost is limited by the available budget
\item An \emph{aggregation method} provides a list of projects to fund
\end{itemize} \end{itemize}
\end{frame} \end{frame}
\begin{frame} \begin{frame}
\frametitle{Benefits of Participatory Budgeting} \frametitle{Input and aggregation methods}
\begin{itemize} \begin{itemize}
\item More efficient spending \item Approval voting
\item Diverse participants \item Ranked voting
\item Higher voter satisfaction \item Knapsack voting
\item Democratic and citizenship learning
\item Institutional and political change
\end{itemize} \end{itemize}
\end{frame}
\section{Computational Aspects} \begin{block}{An approval-based budgeting scenario}
A budgeting scenario is a tuple $E = (A,V,c,l)$ where $A =
\begin{frame} \{a_1,\dots,a_m\}$ is a set of items (projects), $V$ is a set of voters,
\frametitle{Computational Aspects of PB} $c : A\rightarrow\mathbb{N}$ is a cost function associating each project
\begin{itemize} $a\in A$ with its cost $c(a)$ and $l\in\mathbb{N}$ is a budget limit. A
\item Discrete or continuous projects? voter $v\in V$ specifies $A_v\subseteq A$, containing all approved
\item How do we model preferences mathematically? items.
\item How do we adequately capture voter's preferences?
\item How do we aggregate votes?
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Decision Space}
\begin{figure}
\centering
\includegraphics[width=\textwidth]{taxonomy.png}
\end{figure}
\end{frame}
\begin{frame}
\frametitle{Bounded Divisible PB}
\begin{itemize}
\item Projects are divisible
\item A cap for each project is defined
\item Fractional funding
\end{itemize}
\begin{exampleblock}{Example}
A project that seeks to donate a bounded amount of money to a
charity.
\end{exampleblock}
\end{frame}
\begin{frame}
\frametitle{Unbounded Divisible PB}
\begin{itemize}
\item Projects are divisible
\item No caps for projects
\item Generalizable to \emph{Portioning}
\end{itemize}
\begin{block}{Unbounded Divisible PB} $x_p = [0,1]$ denotes the
fraction of project $p\in P$ that is completed and $c_p(x_p) =
x_p$ is the cost function of project $p$. The set of feasible
budget allocations under a budget $B = 1$ is therefore defined
as \[ \{ \vec{x} : \sum_{p\in P}{x_p}\leq 1 \}. \]
\end{block} \end{block}
\begin{exampleblock}{Example}
A project that seeks to donate an unbounded amount of money to a
charity. Every additional amount can be used effectively.
\end{exampleblock}
\end{frame}
\begin{frame}
\frametitle{Bounded Discrete PB}
\begin{itemize}
\item Projects are either fully implemented or not at all
\item Degree of completion has a cap
\item Budget is defined as subset of projects which can be
implemented subject to budget constraints
\end{itemize}
\begin{exampleblock}{Example}
A project for building a new school.
\end{exampleblock}
\end{frame}
\begin{frame}
\frametitle{Unbounded Discrete PB}
\begin{itemize}
\item Multiple degrees of completion
\item Substages of projects (milestones) can be defined
\item Still bounded by total available budget
\end{itemize}
\begin{exampleblock}{Example}
A project for building public toilets. The degree of completion
is the number of toilets that have already been built.
\end{exampleblock}
\end{frame}
\section{Preference Modeling}
\begin{frame}
\frametitle{Preference Modeling}
Model preferences as a cardinal utility function or an ordinal
preference relation:
\begin{block}{Cardinal utility function}
Each resident $i$ has a cardinal utility function $u_i :
A\rightarrow \mathbb{R}$, where $A$ is the set of feasible
allocations.
\end{block}
\begin{block}{Ordinal preference relation}
$\succ_i$ over $A$
\end{block}
\begin{alertblock}{Problem}
This does not adequately reflect any structural properties of
residents' preferences.
\end{alertblock}
\end{frame}
\begin{frame}
\frametitle{Preference Modeling}
\begin{itemize}
\item Impose a structural assumption on the utility function:
\[ u_i : 2^P\rightarrow\mathbb{R} \]
and $u_i$ satisfies subadditivity or superadditivity.
\item Use spatial models where preferences are situated in a
metric space and the distance between them models a
resident's utility for another allocation.
\item Take preferences over projects and use a rule to extend
them to allocations.
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Cardinal extensions}
\begin{block}{Scalar separable utility function}
A resident $i$ derives utility $u_{i,p}\cdot f_p(x_p)$ from each
project. A resident's utility for an allocation $\vec{x}$ is
additive across projects:
\[ u_i(\vec{x}) = \sum_{p\in P}{u_{i,p}\cdot f_p(x_p)} \]
\end{block}
\begin{block}{Dichotomous preferences}
Assuming $x_p\in\{0,1\}$ and $u_{i,p}\in\{0,1\}$, residents
either approve or disapprove a project and care only about the
number of projects implemented.
\end{block}
\begin{block}{Max set extension}
Utility of an allocation is defined as the utility for a
resident's most favorite project: $u_i(S) = \mathsf{max}_{p\in
S}u_{i,p}$ for each $S\subseteq P$.
\end{block}
\end{frame}
\begin{frame}
\frametitle{Ordinal extensions}
\begin{block}{Stochastic dominance extension}
For two allocations $\vec{x},\vec{y}\in A$ and
$E_i^1,\dots,E_i^{k_i}$ being the equivalence classes of the
relation $\succ_i$ in decreasing order of preferences: \[
\vec{x}\succ_{i}^{SD}\vec{y} \text{ iff }
\sum_{j=1}^{l}{\sum_{p\in E_i^j}{x_p}}\geq\sum_{j=1}^{l}{\sum_{p\in
E_i^j}{y_p}}\quad\text{for all } l\in\{1,\dots,k_i\}\]
\end{block}
\begin{block}{Lexicographic extension $\succ_i^{lex}$}
A resident $i$ cares significantly more about project $p$ than
about $p'$ whenever $p\succ_i p'$.
\end{block}
\begin{block}{Scoring rules}
Convert ordinal to cardinal preferences by taking a ranking
$\succ_i$ over projects and determining the utility as $u_{i,p}
= s_k$ where $k$ is the rank in a scoring vector $\vec{s} =
(s_1,\dots,s_m)$ and $s_1\geq\dots\geq s_m\geq 0$.
\end{block}
\end{frame}
\section{Preference Elicitation}
\begin{frame}
\frametitle{Preference elicitation}
\begin{itemize}
\item Also known as \emph{Ballot Design}
\item Communicating full preferences over sometimes
exponentially many allocations is difficult
\item Cognitive burden can lead to lower turnout rates
\end{itemize}
\end{frame} \end{frame}
\begin{frame} \begin{frame}