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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}
    \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}

\begin{frame}
    \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}
        \setlength{\itemsep}{1\baselineskip}
		\item Projects can be bounded or unbounded
        \item Projects can be divisible or indivisible (discrete)
        \item Each project has an associated cost
        \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{frame}

\begin{frame}
    \frametitle{Input and aggregation methods}
    \begin{itemize}
        \item Approval voting
        \item Ranked voting
        \item Knapsack voting
    \end{itemize}

    \begin{block}{An approval-based budgeting scenario}
        A budgeting scenario is a tuple $E = (A,V,c,l)$ where $A =
        \{a_1,\dots,a_m\}$ is a set of items (projects), $V$ is a set of voters,
        $c : A\rightarrow\mathbb{N}$ is a cost function associating each project
        $a\in A$ with its cost $c(a)$ and $l\in\mathbb{N}$ is a budget limit. A
        voter $v\in V$ specifies $A_v\subseteq A$, containing all approved
        items.
    \end{block}
\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}