From 31065aba3c87360fab97c4a4a47aafccdcc347e9 Mon Sep 17 00:00:00 2001
From: Tobias Eidelpes <tobias@eidelpes.info>
Date: Fri, 15 May 2020 12:53:10 +0200
Subject: [PATCH] Add example for greedy selection rules

---
 paper/termpaper.tex | 35 ++++++++++++++++++++++++++++++-----
 1 file changed, 30 insertions(+), 5 deletions(-)

diff --git a/paper/termpaper.tex b/paper/termpaper.tex
index b3616f1..ae03ec9 100644
--- a/paper/termpaper.tex
+++ b/paper/termpaper.tex
@@ -18,7 +18,7 @@
 
 \addbibresource{references.bib}
 
-%opening
+% opening
 \title{Participatory Budgeting: Algorithms and Complexity}
 \author{
  \authorname{Tobias Eidelpes} \\
@@ -27,6 +27,11 @@
  \email{e1527193@student.tuwien.ac.at}
 }
 
+% Numbered example environment
+\newcounter{example}[section]
+\newenvironment{example}[1][]{\refstepcounter{example}\par\medskip
+   \noindent \textbf{Example~\theexample. #1} \rmfamily}{\medskip}
+
 \begin{document}
 
 \maketitle
@@ -210,10 +215,10 @@ of this property.
 \end{equation}
 
 The third satisfaction function assumes that voters are content as long as there
-is at least one of the projects they have approved is selected to be in the
-winning set. Therefore, a voter achieves satisfaction 1 when at least one
-approved project ends up in the winning bundle, i.e. if $|A_v| > 0$ and 0
-satisfaction otherwise (see equation~\ref{eq:5}).
+is at least one of the projects they have approved selected to be in the winning
+set. Therefore, a voter achieves satisfaction 1 when at least one approved
+project ends up in the winning bundle, i.e. if $|A_v| > 0$ and 0 satisfaction
+otherwise (see equation~\ref{eq:5}).
 \begin{equation}\label{eq:5}
     sat_{0/1}(P_v,A) =
     \begin{cases}
@@ -221,6 +226,26 @@ satisfaction otherwise (see equation~\ref{eq:5}).
         0 & \mathsf{otherwise}
     \end{cases}
 \end{equation}
+The satisfaction functions from equations~\ref{eq:4} and \ref{eq:5} can also be
+combined with the greedy rule, potentially giving slightly different outcomes
+than $\mathcal{R}_{sat_\#}^g$.
+
+\begin{example}
+    A set of projects $P = \{ p_2,p_3,p_4,p_5,p_6 \}$ and their associated cost
+    $p_i$ where project $p_i$ costs $i$ and a budget limit $B = 10$ is given.
+    Futhermore, five voters vote $v_1 = \{ p_2,p_5,p_6 \}$, $v_2 = \{ p_2,
+    p_3,p_4,p_5 \}$, $v_3 = \{ p_3,p_4,p_5 \}$, $v_4 = \{ p_4,p_5 \}$ and $v_5 =
+    \{ p_6 \}$. Under $\mathcal{R}_{sat_\#}^g$ the winning bundle is $\{ p_4,p_5
+    \}$, $\mathcal{R}_{sat_\$}^g$ gives $\{ p_4,p_5 \}$ and
+    $\mathcal{R}_{sat_{0/1}}^g$ $\{ p_2,p_3,p_5 \}$.
+\end{example}
+
+These three satisfaction functions cannot only be combined with a greedy
+selection process. A different possibility is to always select a winning set of
+projects that maximizes the sum of the voters' satisfaction:
+\begin{equation}\label{eq:6}
+    \max_{A\subseteq P}\sum_{v\in V}sat(P_v,A)
+\end{equation}
 
 \printbibliography