From 3610b8e5e87bf3f24f972c9ae569aa6e867e205c Mon Sep 17 00:00:00 2001
From: Tobias Eidelpes <tobias@eidelpes.info>
Date: Fri, 15 May 2020 11:28:56 +0200
Subject: [PATCH] Add satisfaction functions

---
 paper/termpaper.tex | 66 +++++++++++++++++++++++++++++++++++++++++++--
 1 file changed, 64 insertions(+), 2 deletions(-)

diff --git a/paper/termpaper.tex b/paper/termpaper.tex
index e456e54..6c83183 100644
--- a/paper/termpaper.tex
+++ b/paper/termpaper.tex
@@ -6,6 +6,7 @@
 \usepackage{setspace}
 
 \usepackage{amssymb}
+\usepackage{amsmath}
 
 \usepackage[english]{babel}
 \usepackage{csquotes}
@@ -99,7 +100,7 @@ selected as \emph{winning projects} according to some rule and subject to the
 total budget limit $B$. For the case where projects are indivisible, which is
 also called discrete, the sum of the winning projects' costs is not allowed to
 exceed the limit $B$:
-\begin{equation}
+\begin{equation}\label{eq:1}
     \sum_{p\in A}{c(p)\leq B}.
 \end{equation}
 When projects can be divisible, i.e. completed to a fractional degree, the
@@ -108,7 +109,7 @@ to an interval between zero and one, representing the fractional degree to which
 this project is completed. Since the cost of each project is a function of its
 degree of completion, the goal is to select a set of projects where the cost of
 the degree of completion does not exceed the budget limit:
-\begin{equation}
+\begin{equation}\label{eq:2}
     \sum_{p\in A}{c(\mu(p))\leq B}.
 \end{equation}
 
@@ -157,9 +158,70 @@ problem of allocating the budget to a set of winning projects under these
 circumstances is referred to as \emph{portioning}. Depending on the desired
 outcome, multiple aggregation methods can be combined with ranked orders.
 
+% Cite municipalities using approval-based budgeting (Paris?)
+
+Since approval-based methods are comparatively easy to implement and are being
+used in practice by multiple municipalities, the next section will discuss
+complexity for these methods as well as useful axioms for comparing the
+different aggregation rules.
+
 \section{Approval-based budgeting}
 \label{sec:approval-based budgeting}
 
+Although approval-based budgeting is also suitable for the case where the
+projects can be divisible, municipalities using this method generally assume
+indivisible projects. Moreover---as is the case with participatory budgeting in
+general---we not only want to select one project as a winner but multiple. This
+is called a multi-winner election and is in contrast to single-winner elections.
+Once the votes have been cast by the voters, again assuming dichotomous
+preferences, a simple aggregation rule is greedy selection. In this case the
+goal is to iteratively select one project $p\in P$ that gives the maximum
+satisfaction for all voters. Satisfaction can be viewed as a form of social
+welfare where it is not only desirable to stay below the budget limit $B$ but
+also to achieve a high score at some metric that quantifies the value that each
+voter gets from the result. \textcite{talmonFrameworkApprovalBasedBudgeting2019}
+propose three satisfaction functions which provide this metric. Formally, they
+define a satisfaction function as a function $sat : 2^P\times 2^P\rightarrow
+\mathbb{R}$, where $P$ is a set of projects. A voter $v$ selects projects to be
+in her approval set $P_v$ and a bundle $A\subseteq P$ contains the projects that
+have been selected as winners. The satisfaction that voter $v$ gets from a
+selected bundle $A$ is denoted as $sat(P_v,A)$. The set $A_v = P_v\cap A$
+denotes the set of approved items by $v$ that end up in the winning bundle $A$.
+A simple approach is to count the number of projects that have been approved by
+a voter and which ended up being in the winning set:
+\begin{equation}\label{eq:3}
+    sat_\#(P_v,A) = |A_v|
+\end{equation}
+Combined with the greedy rule for selecting projects, projects are iteratively
+added to the winning bundle $A$ where at every iteration the project that gives
+the maximum satisfaction to all voters is selected. It is assumed that the
+voters' individual satisfaction can be added together to provide the
+satisfaction that one project gives to all the voters. This gives the rule
+$\mathcal{R}_{sat_\#}^g$ which seeks to maximize $\sum_{v\in V}sat_\#(P_v,A\cup
+\{p\})$ at every iteration.
+
+Another satisfaction function assumes a relationship between the cost of the
+items and a voter's satisfaction. Namely, a project that has a high cost and is
+approved by a voter $v$ and ends up in the winning bundle $A$ provides more
+satisfaction than a lower cost project. Equation~\ref{eq:4} gives a definition
+of this property.
+\begin{equation}\label{eq:4}
+    sat_\$(P_v,A) = \sum_{p\in A_v} c(p) = c(A_v)
+\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}).
+\begin{equation}\label{eq:5}
+    sat_{0/1}(P_v,A) =
+    \begin{cases}
+        1 & \mathsf{if}\; |A_v|>0 \\
+        0 & \mathsf{otherwise}
+    \end{cases}
+\end{equation}
+
 \printbibliography
 
 \end{document}