Add text for the preference elicitation models

paper/references.bib

@@ -40,6 +40,20 @@
   series = {{{AAAI}}'17}
 }
 
+@article{bogomolnaiaCollectiveChoiceDichotomous2005,
+  title = {Collective Choice under Dichotomous Preferences},
+  author = {Bogomolnaia, Anna and Moulin, Herv{\'e} and Stong, Richard},
+  year = {2005},
+  month = jun,
+  volume = {122},
+  pages = {165--184},
+  doi = {10.1016/j.jet.2004.05.005},
+  abstract = {Agents partition deterministic outcomes into good or bad. A mechanism selects a lottery over outcomes (time-shares). The probability of a good outcome is the canonical utility. The utilitarian mechanism averages over outcomes with largest ``approval''. It is efficient, strategyproof, anonymous and neutral. We reach an impossibility if, in addition, each agent's utility is at least 1n, where n is the number of agents; or is at least the fraction of good to feasible outcomes. We conjecture that no ex ante efficient and strategyproof mechanism guarantees a strictly positive utility to all agents, and prove a weaker statement.},
+  journal = {Journal of Economic Theory},
+  language = {en},
+  number = {2}
+}
+
 @article{brandlFundingPublicProjects2020,
   title = {Funding {{Public Projects}}: {{A Case}} for the {{Nash Product Rule}}},
   author = {Brandl, Florian and Brandt, Felix and Peters, Dominik and Stricker, Christian and Suksompong, Warut},
@@ -59,7 +73,7 @@
 }
 
 @article{cabannesParticipatoryBudgetingSignificant2004,
-  title = {Participatory Budgeting: A Significant Contribution to Participatory Democracy:},
+  title = {Participatory Budgeting: A Significant Contribution to Participatory Democracy},
   shorttitle = {Participatory Budgeting},
   author = {Cabannes, Yves},
   year = {2004},
@@ -74,6 +88,21 @@
   number = {1}
 }
 
+@article{duddyElectingRepresentativeCommittee2014,
+  title = {Electing a Representative Committee by Approval Ballot: {{An}} Impossibility Result},
+  shorttitle = {Electing a Representative Committee by Approval Ballot},
+  author = {Duddy, Conal},
+  year = {2014},
+  month = jul,
+  volume = {124},
+  pages = {14--16},
+  doi = {10.1016/j.econlet.2014.04.009},
+  abstract = {We consider methods of electing a fixed number of candidates, greater than one, by approval ballot. We define a representativeness property and a Pareto property and show that these jointly imply manipulability.},
+  journal = {Economics Letters},
+  language = {en},
+  number = {1}
+}
+
 @inproceedings{fainCoreParticipatoryBudgeting2016,
   title = {The {{Core}} of the {{Participatory Budgeting Problem}}},
   booktitle = {Web and {{Internet Economics}}},

paper/termpaper.tex

@@ -13,7 +13,7 @@
 
 \usepackage{hyperref}
 
-\setstretch{1.05}
+\setstretch{1.07}
 
 \addbibresource{references.bib}
 
@@ -61,27 +61,27 @@ stages \autocite{participatorybudgetingprojectHowPBWorks}:
 
 \noindent The two last stages \emph{voting} and \emph{aggregating votes} are of
 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. 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}).
+because depending on how voters elicit their preferences (\emph{balloting} or
+\emph{input method}) and how the votes are aggregated through the use of
+algorithms, the outcome is 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.
+First, this paper will give a brief overview of common methods and show how a
+participatory budgeting scenario can be modeled mathematically. 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}
@@ -112,9 +112,53 @@ the degree of completion does not exceed the budget limit:
     \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.
+Common ways to design the input method is to ask the voters to approve a subset
+of projects $A_v\subseteq P$ where each individual project can be either chosen
+to be in $A_v$ or not. This form is called \emph{dichotomous preferences}
+because every project is put in one of two categories: \emph{good} or
+\emph{bad}. Projects that have not been approved (are not in $A_v$) are assumed
+to be in the bad category. This type of preference elicitation is known as
+approval-based preference elicitation or balloting. It is possible to design
+variations of the described scenario by for example asking the voters to only
+specify at most $k$ projects which they want to see approved ($k$-Approval)
+\cite{goelKnapsackVotingParticipatory2019a}.  These variations typically do not
+take into account the cost that is associated with each project at the voting
+stage. To alleviate this, approaches where the voters are asked to approve
+projects while factoring in the cost have been proposed. After asking the voters
+for their preferences, various aggregation methods can be used.
+Section~\ref{sec:approval-based budgeting} will go into detail about the
+complexity and axiomatic guarantees of these methods.
+
+One such approach, where the cost and benefit of each project is factored in, is
+described by \textcite{goelKnapsackVotingParticipatory2019a}, which they term
+\emph{knapsack voting}. It allows voters to express preferences by factoring in
+the cost as well as the benefit per unit of cost. The name stems from the
+well-known knapsack problem in which, given a set of items, their associated
+weight and value and a weight limit, a selection of items that maximize the
+value subject to the weight limit has to be chosen. In the budgeting scenario,
+the items correspond to projects, the weight limit to the budget limit and the
+value of each item to the value that a project provides to a voter. To have a
+suitable metric for the value that each voter gets from a specific project, the
+authors introduce different \emph{utility models}. These models make it possible
+to provide axiomatic guarantees such as strategyproofness or welfare
+maximization.  While their model assumes fractional voting---that is each voter
+can allocate the budget in any way they see fit---utility functions are also
+used by \textcite{talmonFrameworkApprovalBasedBudgeting2019} to measure the
+total satisfaction that a winning set of projects provides under an aggregation
+rule.
+
+A third possibility for preference elicitation is \emph{ranked orders}. In this
+scenario, voters specify a ranking over the available choices (projects) with
+the highest ranked choice receiving the biggest amount of the budget and the
+lowest ranked one the lowest amount of the budget.
+\textcite{langPortioningUsingOrdinal2019} study a scenario in which the input
+method is ranked orders and the projects that can be chosen are divisible. The
+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.
+
+\section{Approval-based budgeting}
+\label{sec:approval-based budgeting}
 
 \printbibliography