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\begin{document}

\title[Participatory Budgeting]{Participatory Budgeting}
\subtitle{Algorithms and Complexity}
\author{Tobias Eidelpes}

\begin{frame}
	\maketitle
\end{frame}

\begin{frame}
	\frametitle{Table of Contents}
	\tableofcontents
\end{frame}

\section{Introduction}

\begin{frame}
	\frametitle{What is Participatory Budgeting?}
	\begin{quote}
		Participatory Budgeting (PB) is a democratic process in which
		community members decide how to spend part of a public budget.
	\end{quote}
\end{frame}

\begin{frame}
	\frametitle{How does it work?}
	\begin{itemize}
		\item Designing the Process
		\item Collecting Ideas
		\item Developing Proposals
		\item Voting
		\item Funding Winners
	\end{itemize}
\end{frame}

\begin{frame}
	\frametitle{Benefits of Participatory Budgeting}
	\begin{itemize}
		\item More efficient spending
		\item Diverse participants
		\item Higher voter satisfaction
		\item Democratic and citizenship learning
		\item Institutional and political change
	\end{itemize}
\end{frame}

\section{Computational Aspects}

\begin{frame}
	\frametitle{Computational Aspects of PB}
	\begin{itemize}
		\item Discrete or continuous projects?
		\item How do we model preferences mathematically?
		\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}
	\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}

\begin{frame}
	\frametitle{Preference elicitation}
	\begin{block}{Range voting}
		Voters rate projects based on their utility for each project.
	\end{block}
	\begin{block}{$k$-Approval}
		Voters approve the $k$ projects they like the most.
	\end{block}
	\begin{block}{Approval voting}
		Voters approve all projects that they like.
	\end{block}
	\begin{block}{Threshold approval voting}
		Voters approve projects where their utility is above a specified
		threshold.
	\end{block}
	\begin{block}{Knapsack voting}
		Voters provide ideal allocation based on their preferences.
	\end{block}
\end{frame}

\section{Vote Aggregation}

\begin{frame}
	\frametitle{Vote Aggregation}
	\begin{itemize}
		\item Voters' preferences are aggregated to determine which
			projects to fund
		\item Main interest for research
		\item Three different approaches:
			\begin{itemize}
				\item Welfare Maximization
				\item Use of Axioms
				\item Notions of Fairness
			\end{itemize}
	\end{itemize}
\end{frame}

\begin{frame}
	\frametitle{Welfare Maximization}
	\begin{block}{Utilitarian Welfare}
		The utilitarian welfare of an allocation is the sum of utilities it gives to
		residents:
		\[ UW(\vec{x}) =
		\sum_{i\in N}{u_i(\vec{x})}\text{ for }\vec{x}\in A \]
	\end{block}
	\begin{block}{Egalitarian Welfare}
		The egalitarian welfare of an allocation is the minimum utility
		it gives to any resident:
		\[ EW(\vec{x}) =
		\mathsf{min}_{i\in N}{u_i(\vec{x})}\text{ for
	}\vec{x}\in A \]
	\end{block}
	\begin{block}{Nash Welfare}
		The Nash welfare of an allocation is the product of utilities it gives to
		residents:
		\[ NW(\vec{x}) =
		\prod_{i\in N}{u_i(\vec{x})}\text{ for }\vec{x}\in A \]
	\end{block}
\end{frame}

\begin{frame}
	\frametitle{Use of Axioms}
	\begin{block}{Exhaustiveness}
		A feasible allocation $\vec{x}$ is called exhaustive if an
		outcome $\vec{y}$ is not feasible whenever $y_p\geq x_p$ for all
		projects $p$ and a strict inequality holds for at least one
		project.
	\end{block}
	\begin{block}{Discount Monotonicity}
		Project $p$'s cost function $c_p$ is revised to $c_p'(x_p)\leq
		c_p(x_p)$ after a vote aggregation rule outputs allocation
		$\vec{x}$. For the resulting allocation $\vec{y}$, $y_p\geq
		x_p$ holds.
	\end{block}
	\begin{block}{Pareto Optimality}
		An allocation $\vec{x}\in A$ Pareto dominates another allocation
		$\vec{y}\in A$ if $u_i(\vec{x})\geq u_i(\vec{y})$ for all $i\in
		N$ and $u_i(\vec{x}) > u_i(\vec{y})$ for some $i\in N$. An
		allocation $\vec{z}\in A$ is optimal if no allocation dominates
		it.
	\end{block}
\end{frame}

\begin{frame}
	\frametitle{Notion of Fairness}
	\begin{block}{The Core of PB}
		An allocation $\vec{x} \in A$ is a core solution if there is no
		subset $S$ of voters who, given a budget of $(|S|/n)B$, could
		compute an allocation $\vec{y}\in A$ such that every voter in
		$S$ receives strictly more utility in $\vec{y}$ than in
		$\vec{x}$.
	\end{block}
	\begin{block}{Proportionality}
		An allocation $\vec{x}$ should be proportionally reflected by
		the division of voters. A majority of voters should have a
		majority of the budget under their control but a minority should
		have a minority of the budget under their control.
	\end{block}
\end{frame}

\section{Future Directions}

\begin{frame}
	\frametitle{Future Areas of Interest}
	\begin{itemize}
		\item Multi-dimensional constraints
		\item Hybrid models
		\item Complex resident preferences
		\item Market-based approaches
		\item The role of information
		\item Research spanning the entire PB process
	\end{itemize}
\end{frame}

\begin{frame}
	\centering
	\Large
	Thank you for your attention! \\
	Questions \& Answers
\end{frame}

\end{document}
