From dd557b7f88c3c7f0f3deb9bf18f04e913ae1a90f Mon Sep 17 00:00:00 2001
From: Tobias Eidelpes <tobias@eidelpes.info>
Date: Wed, 13 May 2020 17:43:03 +0200
Subject: [PATCH] Finish introduction and begin with mathematical model

---
 paper/termpaper.tex | 60 ++++++++++++++++++++++++++++++++++++++-------
 1 file changed, 51 insertions(+), 9 deletions(-)

diff --git a/paper/termpaper.tex b/paper/termpaper.tex
index 2ebbf4d..21b4078 100644
--- a/paper/termpaper.tex
+++ b/paper/termpaper.tex
@@ -3,6 +3,7 @@
 \usepackage[T1]{fontenc}
 \usepackage[utf8]{inputenc}
 \usepackage{microtype}
+\usepackage{setspace}
 
 \usepackage{amssymb}
 
@@ -12,6 +13,8 @@
 
 \usepackage{hyperref}
 
+\setstretch{1.05}
+
 \addbibresource{references.bib}
 
 %opening
@@ -60,20 +63,59 @@ stages \autocite{participatorybudgetingprojectHowPBWorks}:
 main interest for computer scientists, economists and social choice theorists
 because depending on how voters elicit their preferences (\emph{balloting}) and
 how the votes are aggregated through the use of algorithms, the outcome is
-different.
+different. For this paper it is assumed that the first three stages have already
+been completed. The rules of the process have been set, ideas have been
+collected and developed into feasible projects and the budget limit is known. To
+study different ways of capturing votes and aggregating them, the participatory
+process is modeled mathematically. This model will be called a participatory
+budgeting \emph{scenario}. The aim of studying participatory budgeting scenarios
+is to find ways to achieve a desirable outcome. A desirable outcome can be one
+based on fairness by making sure that each voter has at least one chosen project
+in the final set of winning projects for example. Other approaches are concerned
+with maximizing social welfare or discouraging \emph{gaming the voting process}
+(where an outcome can be manipulated by not voting truthfully; also called
+\emph{strategyproofness}).
+
+First, this paper will look at how a participatory budgeting scenario can be
+modeled mathematically. Then, a brief overview over common models will be given.
+To illustrate these methods, one approach will be chosen and discussed in detail
+with respect to algorithmic complexity and properties. Finally, the gained
+insight into participatory budgeting algorithms will be summarized and an
+outlook on further developments will be given.
+
+\section{Mathematical Model}
+\label{sec:mathematical model}
+
+\textcite{talmonFrameworkApprovalBasedBudgeting2019} define a participatory
+budgeting scenario as a tuple $E = (P,V,c,B)$, consisting of a set of projects
+$P = \{ p_1,\dots,p_m \}$ where each project $p\in P$ has an associated cost
+$c(p):P\rightarrow\mathbb{R}$, a set of voters $V = \{v_1,\dots,v_n\}$ and a
+budget limit $B$. The voters express preferences over individual projects or
+over subsets of all projects. How the preferences of voters are expressed has to
+be decided during the design phase of the process and is a choice that has to be
+made in accordance with the method that is used for aggregating the votes. After
+the voters have elicited their preferences, a set of projects $A\subseteq P$ is
+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}
+    \sum_{p\in A}{c(p)\leq B}.
+\end{equation}
+When projects can be divisible, i.e. completed to a fractional degree, the
+authors define a function $\mu(p) : P\rightarrow [0,1]$ which maps every project
+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}
+    \sum_{p\in A}{c(\mu(p))\leq B}.
+\end{equation}
 
 \textcite{azizParticipatoryBudgetingModels2020} define a taxonomy of
 participatory budgeting scenarios where projects can be either divisible or
 indivisible and bounded or unbounded.
 
-\subsection{Participatory budgeting scenario}
-\label{subsec:Participatory budgeting scenario}
-
-Formally, a PB scenario consists of a set of projects $P = \{ p_1,\dots,p_m \}$
-where each project $p\in P$ has an associated cost $c(p):P\rightarrow\mathbb{R}$.
-
-\section{Section 2}
-
 \printbibliography
 
 \end{document}