Restructure overview and include examples
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talk/talk.tex
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talk/talk.tex
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\documentclass{beamer}
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\beamertemplatenavigationsymbolsempty
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\usetheme{Boadilla}
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\usecolortheme{dolphin}
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\usepackage{graphicx}
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\usepackage{tikz}
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\usepackage{dsfont}
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\usepackage{comment}
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\usetikzlibrary{arrows}
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@ -67,134 +71,167 @@
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\end{frame}
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\begin{frame}
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\frametitle{A formal model for PB}
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\frametitle{A general framework for PB}
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\begin{itemize}
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\setlength{\itemsep}{1\baselineskip}
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\item Projects can be bounded or unbounded
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\item Projects can be divisible or indivisible (discrete)
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\item Each project has an associated cost
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\item Voters approve a subset of all projects (\emph{input method})
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\item The total cost is limited by the available budget
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\item An \emph{aggregation method} provides a list of projects to fund
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\item Projects $P=\{p_1,\dots,p_m\}$
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\begin{itemize}
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\setlength{\itemsep}{.7\baselineskip}
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\item Each project $p\in P$ has associated cost
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$c(p):P\rightarrow\mathbb{R}$
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\item Projects are either divisible or indivisible (discrete)
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\end{itemize}
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\item Select a set $P'\subseteq P$ as \emph{winning projects} not
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exceeding total budget $B$
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\begin{itemize}
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\setlength{\itemsep}{.7\baselineskip}
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\item Discrete case: $\sum_{p\in P'}c(p)\leq B$
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\item Divisible case: $\mu(p): P\rightarrow [0,1]$ with
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$\sum_{p\in P'}c(\mu(p))\leq B$
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\end{itemize}
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\end{itemize}
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\end{frame}
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\begin{frame}
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\frametitle{A general framework for PB ctd.}
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\begin{itemize}
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\setlength{\itemsep}{1\baselineskip}
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\item Voters $V=\{v_1,\dots,v_n\}$
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\begin{itemize}
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\setlength{\itemsep}{.5\baselineskip}
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\item Express preferences over individual projects in $P$ or
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over subsets in $\mathcal{P}(P) := \{P'\,|\,P'\subseteq P\}$
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\item Preference elicitation is dependent on the input method
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(approval-based, ranked orders)
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\end{itemize}
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\item Aggregation methods
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\begin{itemize}
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\item Aggregation methods combine votes to determine a set
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of winning projects
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\item Are usually tied to the input method
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\item Rules are used to select projects w.r.t. desired
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properties of the outcome (fairness, welfare)
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\end{itemize}
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\end{itemize}
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\end{frame}
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\begin{frame}
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\frametitle{Input and aggregation methods}
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Example input methods:
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\begin{itemize}
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\item Approval voting
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\item Ranked voting
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\item Knapsack voting
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\item Approval preferences
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\item Ranked orders
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\item Utility-based preferences
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\end{itemize}
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\vspace{0.2cm}
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Example aggregation methods:
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\begin{itemize}
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\item Maximizing social welfare
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\item Greedy selection
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\item Fairness-based selection
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\end{itemize}
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\vspace{0.2cm}
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Aggregation methods depend on how voters elicit their preferences.
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\end{frame}
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\begin{frame}
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\frametitle{Approval-based budgeting methods}
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\begin{itemize}
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\item Voters approve a subset of projects
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\item Voter preferences are assumed to be \emph{dichotomous}
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\item A \emph{satisfaction function} provides a metric for voter
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satisfaction
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\end{itemize}
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\begin{block}{An approval-based budgeting scenario}
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A budgeting scenario is a tuple $E = (A,V,c,l)$ where $A =
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\{a_1,\dots,a_m\}$ is a set of items (projects), $V$ is a set of voters,
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$c : A\rightarrow\mathbb{N}$ is a cost function associating each project
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$a\in A$ with its cost $c(a)$ and $l\in\mathbb{N}$ is a budget limit. A
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voter $v\in V$ specifies $A_v\subseteq A$, containing all approved
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A budgeting scenario is a tuple $E = (P,V,c,B)$ where $P =
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\{p_1,\dots,p_m\}$ is a set of projects, $V$ is a set of voters, $c :
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P\rightarrow\mathbb{N}$ is a cost function associating each project
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$p\in P$ with its cost $c(p)$ and $B\in\mathbb{N}$ is a budget limit. A
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voter $v\in V$ specifies $P_v\subseteq P$, containing all approved
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items.
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\end{block}
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\end{frame}
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\begin{frame}
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\frametitle{Preference elicitation}
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\begin{block}{Range voting}
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Voters rate projects based on their utility for each project.
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\end{block}
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\begin{block}{$k$-Approval}
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Voters approve the $k$ projects they like the most.
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\end{block}
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\begin{block}{Approval voting}
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Voters approve all projects that they like.
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\end{block}
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\begin{block}{Threshold approval voting}
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Voters approve projects where their utility is above a specified
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threshold.
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\end{block}
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\begin{block}{Knapsack voting}
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Voters provide ideal allocation based on their preferences.
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\end{block}
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\end{frame}
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\section{Vote Aggregation}
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\begin{frame}
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\frametitle{Vote Aggregation}
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\begin{itemize}
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\item Voters' preferences are aggregated to determine which
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projects to fund
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\item Main interest for research
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\item Three different approaches:
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\begin{itemize}
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\item Welfare Maximization
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\item Use of Axioms
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\item Notions of Fairness
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\end{itemize}
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\end{itemize}
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\end{frame}
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\begin{frame}
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\frametitle{Welfare Maximization}
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\begin{block}{Utilitarian Welfare}
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The utilitarian welfare of an allocation is the sum of utilities it gives to
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residents:
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\[ UW(\vec{x}) =
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\sum_{i\in N}{u_i(\vec{x})}\text{ for }\vec{x}\in A \]
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\end{block}
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\begin{block}{Egalitarian Welfare}
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The egalitarian welfare of an allocation is the minimum utility
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it gives to any resident:
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\[ EW(\vec{x}) =
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\mathsf{min}_{i\in N}{u_i(\vec{x})}\text{ for
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}\vec{x}\in A \]
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\end{block}
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\begin{block}{Nash Welfare}
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The Nash welfare of an allocation is the product of utilities it gives to
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residents:
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\[ NW(\vec{x}) =
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\prod_{i\in N}{u_i(\vec{x})}\text{ for }\vec{x}\in A \]
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\begin{block}{Budgeting method $\mathcal{R}$}
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A budgeting method $\mathcal{R}$ takes a budgeting scenario $E$ and
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returns a bundle $A\subseteq P$ where the total cost of the items in
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$A$ does not exceed the budget limit $B$.
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\end{block}
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\end{frame}
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\begin{frame}
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\frametitle{Use of Axioms}
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\begin{block}{Exhaustiveness}
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A feasible allocation $\vec{x}$ is called exhaustive if an
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outcome $\vec{y}$ is not feasible whenever $y_p\geq x_p$ for all
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projects $p$ and a strict inequality holds for at least one
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project.
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\frametitle{Satisfaction functions}
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\begin{block}{Satisfaction function}
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A satisfaction function $sat : 2^P\times 2^P\rightarrow\mathbb{R}$ with
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a set $P$ of items, a voter $v$ and her approval set $P_v$ and a bundle
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$A\subseteq P$ provides the satisfaction $sat(P_v,A)$ of $v$ from the
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bundle $A$. The set of approved items by $v$ that end up in the winning
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bundle is denoted by $A_v = P_v\cap A$.
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\end{block}
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\begin{block}{Discount Monotonicity}
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Project $p$'s cost function $c_p$ is revised to $c_p'(x_p)\leq
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c_p(x_p)$ after a vote aggregation rule outputs allocation
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$\vec{x}$. For the resulting allocation $\vec{y}$, $y_p\geq
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x_p$ holds.
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\begin{exampleblock}{$sat_\#(P_v,A)$}
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$sat_\#(P_v,A) = |A_v|$: The satisfaction of voter $v$ is the number of
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funded items that are approved.
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\end{exampleblock}
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\end{frame}
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\begin{frame}
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\frametitle{Satisfaction functions ctd.}
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\begin{exampleblock}{$sat_\$(P_v,A)$}
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$sat_{\$}(P_v,A) = \sum_{p\in A_v}{c(p) = c(A_v)}$: The satisfaction of voter
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$v$ is the total cost of her approved and funded items.
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\end{exampleblock}
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\begin{exampleblock}{$sat_{0/1}(P_v,A)$}
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\[ sat_{0/1}(P_v,A) =
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\begin{cases}
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1 & \text{if } |A_v|>0 \\
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0 & \text{otherwise}
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\end{cases}
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\]
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A voter $v$ has satisfaction 1 if at least one of her approved items is
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funded and 0 otherwise.
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\end{exampleblock}
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\end{frame}
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\begin{frame}
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\frametitle{Rules for selecting a winning bundle}
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{\Large Let $sat$ be a satisfaction function:}
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\begin{block}{Max rules}
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The rule $\mathcal{R}_{sat}^m$ selects a bundle which maximizes the sum
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of voters' satisfaction: $\mathsf{max}_{A\subseteq P}\sum_{v\in
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V}{sat(P_v,A)}$
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\end{block}
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\begin{block}{Pareto Optimality}
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An allocation $\vec{x}\in A$ Pareto dominates another allocation
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$\vec{y}\in A$ if $u_i(\vec{x})\geq u_i(\vec{y})$ for all $i\in
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N$ and $u_i(\vec{x}) > u_i(\vec{y})$ for some $i\in N$. An
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allocation $\vec{z}\in A$ is optimal if no allocation dominates
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it.
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\begin{block}{Greedy rules}
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The rule $\mathcal{R}_{sat}^g$ iteratively adds an item $p\in P$ to $A$,
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seeking to maximize $\sum_{v\in V}{sat(P_v,A\cup\{p\})}$.
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\end{block}
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\begin{block}{Proportional greedy rules}
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The rule $\mathcal{R}_{sat}^p$ iteratively adds an item $p\in P$ to $A$
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seeking to maximize the sum of satisfaction per unit of cost.
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\end{block}
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\end{frame}
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\begin{frame}
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\frametitle{Notion of Fairness}
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\begin{block}{The Core of PB}
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An allocation $\vec{x} \in A$ is a core solution if there is no
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subset $S$ of voters who, given a budget of $(|S|/n)B$, could
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compute an allocation $\vec{y}\in A$ such that every voter in
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$S$ receives strictly more utility in $\vec{y}$ than in
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$\vec{x}$.
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\end{block}
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\begin{block}{Proportionality}
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An allocation $\vec{x}$ should be proportionally reflected by
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the division of voters. A majority of voters should have a
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majority of the budget under their control but a minority should
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have a minority of the budget under their control.
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\frametitle{Example budgeting scenarios}
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\begin{block}{A budgeting scenario}
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Items $P = \{p_2,p_3,p_4,p_5,p_6\}$ and their associated cost $p_i$
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where $p_i$ costs $i$. Budget limit $B=10$ and 5 voters vote
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$v_1=\{p_2,p_5,p_6\}$, $v_2=\{p_2,p_3,p_4,p_5\}$, $v_3=\{p_3,p_4,p_5\}$,
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$v_4=\{p_4,p_5\}$ and $v_5=\{p_6\}$.
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\end{block}
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\begin{exampleblock}{Combining max rule with $sat_\#$}
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Under $\mathcal{R}_{|A_v|}^m$ the winning bundle is $\{p_2,p_3,p_5\}$.
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The total satisfaction is 8.
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\end{exampleblock}
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\begin{exampleblock}{Combining greedy rule with $sat_\#$}
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Under $\mathcal{R}_{|A_v|}^g$ the winning bundle is $\{p_4,p_5\}$ (first
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selecting $p_5$). The total satisfaction is 7.
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\end{exampleblock}
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\begin{exampleblock}{Combining max rule with $sat_{0/1}$}
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Under $\mathcal{R}^m_{sat_{0/1}}$ the winning bundle is $\{p_4,p_6\}$,
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achieving max satisfaction.
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\end{exampleblock}
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\end{frame}
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\begin{frame}
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\frametitle{}
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\end{frame}
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\section{Future Directions}
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