From f4a59983fc73f7686db93351d4f184076607e317 Mon Sep 17 00:00:00 2001
From: Tobias Eidelpes <tobias@eidelpes.info>
Date: Sat, 16 May 2020 15:26:09 +0200
Subject: [PATCH] Add proportional greedy rule

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

diff --git a/paper/termpaper.tex b/paper/termpaper.tex
index c80d96e..e2e610b 100644
--- a/paper/termpaper.tex
+++ b/paper/termpaper.tex
@@ -295,6 +295,36 @@ voters that are represented by the subset.
     voters $v_1$ and $v_5$ and project $p_4$ voters $v_2$, $v_3$ and $v_4$.
 \end{example}
 
+The third rule, which places a heavy emphasis on cost versus benefit, is similar
+to the greedy rule but instead of disregarding the satisfaction per cost that a
+project provides, it seeks to maximize the sum of satisfaction divided by cost
+for a project $p\in P$:
+\begin{equation}
+    \frac{\sum_{v\in V}sat(P_v,A\cup\{p\}) - \sum_{v\in V}sat(P_v,A)}{c(p)}
+\end{equation}
+\textcite{talmonFrameworkApprovalBasedBudgeting2019} call this type of
+aggregation rule \emph{proportional greedy rule}. Example~\ref{ex:prop greedy}
+shows how the outcome of a budgeting scenario might look like compared to using
+a simple greedy rule or a max rule. Since the proportional greedy rule is a
+variation of the simple greedy rule, it is therefore also solvable in polynomial
+time. The variation of computing the satisfaction per unit of cost does not
+change the complexity since it only adds an additional step which can be done in
+constant time.
+
+\begin{example}\label{ex:prop greedy}
+    We again have the same set of projects $P = \{ p_2,p_3,p_4,p_5,p_6 \}$, the
+    same budget limit of $B = 10$ and the five voters: $v_1 = \{ p_2,p_5,p_6
+    \}$, $v_2 = \{ p_2, p_3,p_4,p_5 \}$, $v_3 = \{ p_3,p_4,p_5 \}$, $v_4 = \{
+    p_4,p_5 \}$ and $v_5 = \{ p_6 \}$. If we combine the satisfaction function
+    $sat_\#$ from equation~\ref{eq:3} with the proportional greedy rule, we get
+    the same result as with the simple greedy rule of $\{ p_4,p_5 \}$. While the
+    simple greedy rule selects first $p_5$ and then $p_4$, the proportional
+    greedy rule first selects $p_4$ and then $p_5$. The rule
+    $\mathcal{R}_{sat_\$}^p$ yields the same result as $\mathcal{R}_{sat_\$}^g$
+    and $\mathcal{R}_{sat_\$}^m$ of $\{ p_4,p_5 \}$. $\mathcal{R}_{sat_{0/1}}^p$
+    however, gives $\{ p_2,p_3,p_4 \}$.
+\end{example}
+
 \printbibliography
 
 \end{document}