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