diff --git a/talk/talk.tex b/talk/talk.tex
index 135211a..22cf550 100644
--- a/talk/talk.tex
+++ b/talk/talk.tex
@@ -4,6 +4,9 @@
 \usecolortheme{dolphin}
 
 \usepackage{graphicx}
+\usepackage{tikz}
+
+\usetikzlibrary{arrows}
 
 \begin{document}
 
@@ -28,194 +31,70 @@
 		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?}
+    \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}
-		\item Designing the Process
-		\item Collecting Ideas
-		\item Developing Proposals
-		\item Voting
-		\item Funding Winners
+        \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{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}
+    \frametitle{Input and aggregation methods}
+    \begin{itemize}
+        \item Approval voting
+        \item Ranked voting
+        \item Knapsack voting
+    \end{itemize}
 
-\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}
+    \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}