Finish preliminary presentation

.gitignore

@@ -7,6 +7,8 @@
 *.pdf
 *.synctex.gz
 
+*.out
+*.toc
 
 *.nav
 *.snm

talk/talk.tex

@@ -58,8 +58,8 @@
 	\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 model these preferences?
 		\item How do we aggregate votes?
 	\end{itemize}
 \end{frame}
@@ -79,12 +79,6 @@
 		\item A cap for each project is defined
 		\item Fractional funding
 	\end{itemize}
-	\begin{block}{Bounded Divisible PB}
-		Each project has a cap $q_p = 1$ and $x_p = [0,1]$ denotes the
-		fraction of project $p\in P$ that is completed. 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 a bounded amount of money to a
 		charity.
@@ -97,33 +91,252 @@
 		\item Projects are divisible
 		\item No caps for projects
 		\item Generalizable to \emph{Portioning}
-		\item In practice still bounded by total budget
 	\end{itemize}
-	\begin{block}{Unbounded Divisible PB}
+	\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}
 
-\section{Preference Modeling}
+\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{Algorithm Axioms}
+	\frametitle{Vote Aggregation}
 	\begin{itemize}
-		\item Pareto Optimality
-		\item Monotonicity
-		\item Truthfulness
-		\item Fairness
+		\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}
 
-\section{Algorithms}
+\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}
 
-\section{Comparison}
+\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}
 
-\section{Practicality}
+\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}