Tobias Eidelpes
7adfcb91e1
The cost of a project is now multiplied by its degree of completion. This cost function now assumes a linear relationship between cost and degree of completion.
470 lines
27 KiB
TeX
470 lines
27 KiB
TeX
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% opening
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\title{Participatory Budgeting: Algorithms and Complexity}
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\author{
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\authorname{Tobias Eidelpes} \\
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\studentnumber{01527193} \\
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\curriculum{033 534} \\
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\email{e1527193@student.tuwien.ac.at}
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}
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\noindent \textbf{Example~\theexample. #1} \rmfamily}{\medskip}
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\begin{document}
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\maketitle
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\begin{abstract}
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Participatory budgeting is a deliberative democratic process that allows
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residents to decide how public funds should be spent. By combining a form of
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preference elicitation with an aggregation method, a set of winning projects
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is determined and funded. This paper first gives an introduction into
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participatory budgeting methods and then focuses on approval-based models to
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discuss algorithmic complexity. Furthermore, a short overview of useful
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axioms that can help select one method in practice is presented. Finally, an
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outlook on future challenges surrounding participatory budgeting is given.
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\end{abstract}
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\section{Introduction}
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\emph{Participatory Budgeting} (PB) is a process of democratic deliberation that
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allows residents of a municipality to decide how a part of the public budget is
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to be spent. It is a way to improve transparency and citizen involvement which
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are two important cornerstones of a democracy. PB was first realized in the
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1990s in Porto Alegre in Brazil by the Workers' Party to combat the growing
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divide between the rich city center and the poor living in the greater region.
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Owing to its success in the south of Brazil, PB quickly spread to North America,
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Europe, Asia and Africa.
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Although the process is heavily adapted by each municipality to suit the
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environment in which the residents live in, it generally follows the following
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stages \autocite{participatorybudgetingprojectHowPBWorks}:
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\begin{description}
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\item [Design the process] A rule book is crafted to ensure that the process
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is democratic.
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\item [Collect ideas] Residents propose and discuss ideas for projects.
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\item [Develop feasible projects] The ideas are developed into projects that
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can be undertaken by the municipality.
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\item [Voting] The projects are voted on by the residents.
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\item [Aggregating votes \& funding] The votes are combined to determine a
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set of winning projects which are then funded.
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\end{description}
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\noindent The two last stages \emph{voting} and \emph{aggregating votes} are of
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main interest for computer scientists, economists and social choice theorists
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because depending on how voters elicit their preferences (\emph{balloting} or
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\emph{input method}) and how the votes are aggregated through the use of
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algorithms, the outcome is different. To study different ways of capturing votes
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and aggregating them, the participatory process is modeled mathematically. This
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model will be called a participatory budgeting \emph{scenario}. The aim of
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studying participatory budgeting scenarios is to find ways to achieve a
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desirable outcome. A desirable outcome can be one based on fairness by making
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sure that each voter has at least one chosen project in the final set of winning
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projects for example. Other approaches are concerned with maximizing social
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welfare or discouraging \emph{gaming the voting process} (where an outcome can
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be manipulated by not voting truthfully; also called \emph{strategyproofness}).
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First, this paper will give a brief overview of common methods and show how a
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participatory budgeting scenario can be modeled mathematically. To illustrate
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these methods, one approach will be chosen and discussed in detail with respect
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to algorithmic complexity and properties. Finally, the gained insight into
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participatory budgeting algorithms will be summarized and an outlook on further
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developments will be given.
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\section{A Participatory Budgeting Framework}
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\label{sec:a participatory budgeting framework}
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\textcite{talmonFrameworkApprovalBasedBudgeting2019} define a participatory
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budgeting scenario as a tuple $E = (P,V,c,B)$, consisting of a set of projects
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$P = \{ p_1,\dots,p_m \}$ where each project $p\in P$ has an associated cost
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$c(p):P\rightarrow\mathbb{R}$, a set of voters $V = \{v_1,\dots,v_n\}$ and a
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budget limit $B$. The voters express preferences over individual projects or
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over subsets of all projects. How the preferences of voters are expressed has to
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be decided during the design phase of the process and is a choice that has to be
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made in accordance with the method that is used for aggregating the votes. After
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the voters have elicited their preferences, a set of projects $A\subseteq P$ is
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selected as \emph{winning projects} according to some rule and subject to the
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total budget limit $B$. For the case where projects are indivisible, which is
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also called discrete, the sum of the winning projects' costs is not allowed to
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exceed the limit $B$:
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\begin{equation}\label{eq:1}
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\sum_{p\in A}{c(p)\leq B}.
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\end{equation}
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When projects can be divisible, i.e., completed to a fractional degree, the
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authors define a function $\mu(p) : P\rightarrow [0,1]$ which maps every project
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to an interval between zero and one, representing the fractional degree to which
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this project is completed. Since the cost of each project is a function of its
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degree of completion, the goal is to select a set of projects where the cost of
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the degree of completion does not exceed the budget limit:
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\begin{equation}\label{eq:2}
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\sum_{p\in A}{\mu(p)\cdot c(p)\leq B}.
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\end{equation}
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Common ways to design the input method is to ask the voters to approve a subset
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of projects $A_v\subseteq P$ where each individual project can be either chosen
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to be in $A_v$ or not. This form is called \emph{dichotomous preferences}
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because every project is put in one of two categories: \emph{good} or
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\emph{bad}. Projects that have not been approved (are not in $A_v$) are assumed
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to be in the bad category. This type of preference elicitation is known as
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approval-based preference elicitation or balloting. It is possible to design
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variations of the described scenario by for example asking the voters to only
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specify at most $k$ projects which they want to see approved ($k$-Approval)
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\cite{goelKnapsackVotingParticipatory2019a}. These variations typically do not
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take into account the cost that is associated with each project at the voting
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stage. To alleviate this, approaches where the voters are asked to approve
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projects while factoring in the cost have been proposed. After asking the voters
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for their preferences, various aggregation methods can be used.
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Section~\ref{sec:approval-based budgeting} will go into detail about the
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complexity and axiomatic guarantees of these methods.
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One such approach, where the cost and benefit of each project is factored in, is
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described by \textcite{goelKnapsackVotingParticipatory2019a}, which they term
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\emph{knapsack voting}. It allows voters to express preferences by factoring in
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the cost as well as the benefit per unit of cost. The name stems from the
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well-known knapsack problem in which, given a set of items, their associated
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weight and value and a weight limit, a selection of items that maximize the
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value subject to the weight limit has to be chosen. In the budgeting scenario,
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the items correspond to projects, the weight limit to the budget limit and the
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value of each item to the value that a project provides to a voter. To have a
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suitable metric for the value that each voter gets from a specific project, the
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authors introduce different \emph{utility models}. These models make it possible
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to provide axiomatic guarantees such as strategyproofness or welfare
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maximization. While their model assumes fractional voting---that is each voter
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can allocate the budget in any way they see fit---utility functions are also
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used by \textcite{talmonFrameworkApprovalBasedBudgeting2019} to measure the
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total satisfaction that a winning set of projects provides under an aggregation
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rule.
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A third possibility for preference elicitation is \emph{ranked orders}. In this
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scenario, voters specify a ranking over the available choices (projects) with
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the highest ranked choice receiving the biggest amount of the budget and the
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lowest ranked one the lowest amount of the budget.
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\textcite{langPortioningUsingOrdinal2019} study a scenario in which the input
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method is ranked orders and the projects that can be chosen are divisible. The
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problem of allocating the budget to a set of winning projects under these
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circumstances is referred to as \emph{portioning}. Depending on the desired
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outcome, multiple aggregation methods can be combined with ranked orders.
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% Cite municipalities using approval-based budgeting (Paris?)
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Since approval-based methods are comparatively easy to implement and are being
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used in practice by multiple municipalities, the next section will discuss
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aggregation methods, their complexity as well as useful axioms for comparing the
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different aggregation rules.
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\section{Approval-based budgeting}
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\label{sec:approval-based budgeting}
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Although approval-based budgeting is also suitable for the case where the
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projects can be divisible, municipalities using this method generally assume
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indivisible projects. Moreover---as is the case with participatory budgeting in
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general---we not only want to select one project as a winner but multiple. This
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is called a multi-winner election and is in contrast to single-winner elections.
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Once the votes have been cast by the voters, again assuming dichotomous
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preferences, a simple aggregation rule is greedy selection. In this case the
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goal is to iteratively select one project $p\in P$ that gives the maximum
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satisfaction for all voters. Satisfaction can be viewed as a form of social
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welfare where it is not only desirable to stay below the budget limit $B$ but
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also to achieve a high score at some metric that quantifies the value that each
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voter gets from the result. \textcite{talmonFrameworkApprovalBasedBudgeting2019}
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propose three satisfaction functions which provide this metric. Formally, they
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define a satisfaction function as a function $sat : 2^P\times 2^P\rightarrow
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\mathbb{R}$, where $P$ is a set of projects. A voter $v$ selects projects to be
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in her approval set $P_v$ and a bundle $A\subseteq P$ contains the projects that
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have been selected as winners. The satisfaction that voter $v$ gets from a
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selected bundle $A$ is denoted as $sat(P_v,A)$. The set $A_v = P_v\cap A$
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denotes the set of approved items by $v$ that end up in the winning bundle $A$.
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A simple approach is to count the number of projects that have been approved by
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a voter and which ended up being in the winning set:
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\begin{equation}\label{eq:3}
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sat_\#(P_v,A) = |A_v|
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\end{equation}
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Combined with the greedy rule for selecting projects, projects are iteratively
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added to the winning bundle $A$ where at every iteration the project that gives
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the maximum satisfaction to all voters is selected. It is assumed that the
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voters' individual satisfaction can be added together to provide the
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satisfaction that one project gives to all the voters. This gives the rule
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$\mathcal{R}_{sat_\#}^g$ which seeks to maximize $\sum_{v\in V}sat_\#(P_v,A\cup
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\{p\})$ at every iteration.
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Another satisfaction function assumes a relationship between the cost of the
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items and a voter's satisfaction. Namely, a project that has a high cost and is
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approved by a voter $v$ and ends up in the winning bundle $A$ provides more
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satisfaction than a lower cost project. Equation~\ref{eq:4} gives a definition
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of this property.
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\begin{equation}\label{eq:4}
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sat_\$(P_v,A) = \sum_{p\in A_v} c(p) = c(A_v)
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\end{equation}
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The third satisfaction function assumes that voters are content as long as there
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is at least one of the projects they have approved selected to be in the winning
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set. Therefore, a voter achieves satisfaction 1 when at least one approved
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project ends up in the winning bundle, i.e., if $|A_v| > 0$ and 0 satisfaction
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otherwise (see equation~\ref{eq:5}).
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\begin{equation}\label{eq:5}
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sat_{0/1}(P_v,A) =
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\begin{cases}
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1 & \mathsf{if}\; |A_v|>0 \\
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0 & \mathsf{otherwise}
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\end{cases}
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\end{equation}
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The satisfaction functions from equations~\ref{eq:4} and \ref{eq:5} can also be
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combined with the greedy rule, potentially giving slightly different outcomes
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than $\mathcal{R}_{sat_\#}^g$. An example demonstrating the greedy rule is given
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in example~\ref{ex:greedy}.
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\begin{example}\label{ex:greedy}
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A set of projects $P = \{ p_2,p_3,p_4,p_5,p_6 \}$ and their associated cost
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$p_i$ where project $p_i$ costs $i$ and a budget limit $B = 10$ is given.
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Futhermore, five voters vote $v_1 = \{ p_2,p_5,p_6 \}$, $v_2 = \{ p_2,
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p_3,p_4,p_5 \}$, $v_3 = \{ p_3,p_4,p_5 \}$, $v_4 = \{ p_4,p_5 \}$ and $v_5 =
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\{ p_6 \}$. Under $\mathcal{R}_{sat_\#}^g$ the winning bundle is $\{ p_4,p_5
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\}$, $\mathcal{R}_{sat_\$}^g$ gives $\{ p_4,p_5 \}$ and
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$\mathcal{R}_{sat_{0/1}}^g$ $\{ p_2,p_3,p_5 \}$.
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\end{example}
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Computing a solution to the problem of finding a winning set of projects by
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using greedy rules can be done in polynomial time due to their iterative nature.
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The downside to using a greedy selection process is that the provided solution
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might not be optimal with respect to the satisfaction.
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To be able to compute optimal solutions,
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\textcite{talmonFrameworkApprovalBasedBudgeting2019} suggest combining the
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satisfaction functions with a maximization rule. The maximization rule always
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selects a winning set of projects that maximizes the sum of the voters'
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satisfaction:
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\begin{equation}\label{eq:6}
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\max_{A\subseteq P}\sum_{v\in V}sat(P_v,A)
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\end{equation}
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The max rule can then be used with the three satisfaction functions in the same
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way, giving: $\mathcal{R}_{sat_\#}^m$, $\mathcal{R}_{sat_\$}^m$ and
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$\mathcal{R}_{sat_{0/1}}^m$. Example~\ref{ex:max} shows that the selection of
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winning projects is not as intuitive as when using the greedy rule. Whereas it
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was still possible to compute a solution without any tools for the greedy
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selection, the max rule requires knowing the possible sets of projects
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beforehand in order to select the bundle with the maximum satisfaction. This
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hints at the complexity of the max rule being harder to solve than the greedy
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rule. The authors confirm this by identifying $\mathcal{R}_{sat_\$}^m$ as weakly
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\textsf{NP}-hard for the problem of finding a winning set that gives at least a
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specified amount of satisfaction. The proof follows from a reduction to the
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subset sum problem which asks the question of given a set of numbers (in this
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case the cost associated with each project) and a number $B$ (the budget limit)
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does any subset of the numbers sum to exactly $B$? Because the subset sum
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problem is solvable by a dynamic programming algorithm in $O(B\cdot |P|)$ where
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$P$ is the set of projects, $\mathcal{R}_{sat_\$}^m$ is solvable in
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pseudo-polynomial time. Finding a solution using the rule
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$\mathcal{R}_{sat_\#}^m$ however, is doable in polynomial time due to the
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problem's relation to the knapsack problem. If the input (either projects or
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voters) is represented in unary, a dynamic programming algorithm is bounded by a
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polynomial in the length of the input. For $\mathcal{R}_{sat_{0/1}}^m$, finding
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a set of projects that gives at least a certain amount of satisfaction is
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\textsf{NP}-hard. Assuming that the cost of all of the projects is one unit, the
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rule is equivalent to the max cover problem because we are searching for a
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subset of all projects with the number of the projects (the total cost due to
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the projects given in unit cost) smaller or equal to the budget limit $B$ and
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want to maximize the number of voters that are represented by the subset.
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\begin{example}\label{ex:max}
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Taking the initial setup from example~\ref{ex:greedy}: $P = \{
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p_2,p_3,p_4,p_5,p_6 \}$ and their associated cost $p_i$ where project $p_i$
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costs $i$, a budget limit $B = 10$ and the five voters: $v_1 = \{
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p_2,p_5,p_6 \}$, $v_2 = \{ p_2, p_3,p_4,p_5 \}$, $v_3 = \{ p_3,p_4,p_5 \}$,
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$v_4 = \{ p_4,p_5 \}$ and $v_5 = \{ p_6 \}$. We get $\{ p_2,p_3,p_5 \}$ for
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$\mathcal{R}_{sat_\#}^m$, $\{ p_4,p_5 \}$ for $\mathcal{R}_{sat_\$}^m$ and
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$\{ p_4,p_6 \}$ for $\mathcal{R}_{sat_{0/1}}^m$. Especially the last rule is
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interesting because it provides the highest amount of satisfaction possible
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by covering each voter with at least one project. Project $p_6$ covers
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voters $v_1$ and $v_5$ and project $p_4$ voters $v_2$, $v_3$ and $v_4$.
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\end{example}
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The third rule, which places a heavy emphasis on cost versus benefit, is similar
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to the greedy rule but instead of disregarding the satisfaction per cost that a
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project provides, it seeks to maximize the sum of satisfaction divided by cost
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for a project $p\in P$:
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\begin{equation}
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\frac{\sum_{v\in V}sat(P_v,A\cup\{p\}) - \sum_{v\in V}sat(P_v,A)}{c(p)}
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\end{equation}
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\textcite{talmonFrameworkApprovalBasedBudgeting2019} call this type of
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aggregation rule \emph{proportional greedy rule}. Example~\ref{ex:prop greedy}
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shows how the outcome of a budgeting scenario might look like compared to using
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a simple greedy rule or a max rule. Since the proportional greedy rule is a
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variation of the simple greedy rule, it is therefore also solvable in polynomial
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time. The variation of computing the satisfaction per unit of cost does not
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change the complexity since it only adds an additional step which can be done in
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constant time.
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\begin{example}\label{ex:prop greedy}
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We again have the same set of projects $P = \{ p_2,p_3,p_4,p_5,p_6 \}$, the
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same budget limit of $B = 10$ and the five voters: $v_1 = \{ p_2,p_5,p_6
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\}$, $v_2 = \{ p_2, p_3,p_4,p_5 \}$, $v_3 = \{ p_3,p_4,p_5 \}$, $v_4 = \{
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p_4,p_5 \}$ and $v_5 = \{ p_6 \}$. If we combine the satisfaction function
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$sat_\#$ from equation~\ref{eq:3} with the proportional greedy rule, we get
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the same result as with the simple greedy rule of $\{ p_4,p_5 \}$. While the
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simple greedy rule selects first $p_5$ and then $p_4$, the proportional
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greedy rule first selects $p_4$ and then $p_5$. The rule
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$\mathcal{R}_{sat_\$}^p$ yields the same result as $\mathcal{R}_{sat_\$}^g$
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and $\mathcal{R}_{sat_\$}^m$ of $\{ p_4,p_5 \}$. $\mathcal{R}_{sat_{0/1}}^p$
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however, gives $\{ p_2,p_3,p_4 \}$.
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\end{example}
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A benefit of the three discussed satisfaction functions is that they can be
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viewed as constraint satisfaction problems (CSPs) and can thus be formulated
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using integer linear programming (ILP). Although integer programming is
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\textsf{NP}-complete, efficient solvers are readily available for these types of
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problems. \textcite{talmonFrameworkApprovalBasedBudgeting2019} show that the
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rule $\mathcal{R}_{sat_{0/1}}^m$ is similar to the max cover problem which can
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be approximated with a $(1-\frac{1}{\epsilon})$-approximation algorithm, where
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$\epsilon > 0$ is a fixed parameter that is chosen depending on the error of the
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approximation. In fact, \textcite{khullerBudgetedMaximumCoverage1999} show that
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an approximation algorithm with the same ratio exists not only for the case
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where the projects have unit cost but also for the general cost version.
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Instead of sacrificing exactness to get a better running time,
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\textcite{talmonFrameworkApprovalBasedBudgeting2019} show that the
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$\mathcal{R}_{sat_{0/1}}^m$ rule is fixed parameter tractable for the number of
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voters $|V|$. A problem is fixed parameter tractable if there exists an
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algorithm that decides each instance of the problem in $O(f(k)\cdot p(n))$ where
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$p(n)$ is a polynomial function and $f(k)$ an arbitrary function in $k$. It is
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crucial to note that $f(k)$ does not admit functions of the form $n^k$. The
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algorithm for the maximum rule tries to guess the number of voters that are
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represented by the same project. The estimation is then used to pick a project
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which has the lowest cost and satisfies exactly the estimated amount of voters.
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\section{Normative Axioms}
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\label{sec:normative axioms}
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Axioms in the context of participatory budgeting define some kind of property of
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a budgeting method that might be desirable to have. Generally it is beneficial
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if a certain method satisfies as many axioms as possible as this gives the
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method a strong theoretical backbone. One set of axioms, discussed by
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\textcite{talmonFrameworkApprovalBasedBudgeting2019}, relates to the cost of
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projects. Another possibility is to look at the \emph{fairness} associated with
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a particular set of winning projects. Fairness captures the notion of for
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example protecting minorities and their preferences.
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\textcite{azizProportionallyRepresentativeParticipatory2018} propose axioms that
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are representative of the broad spectrum of choices which voters can make. Other
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|
fairness-based approaches are proposed by
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|
\textcite{fainCoreParticipatoryBudgeting2016}, using the core of a solution,
|
|
although they focus on cases where voters elicit their preferences via a
|
|
cardinal utility function. The notion of core is also studied by
|
|
\textcite{fainFairAllocationIndivisible2018} for the case where voters have
|
|
additive utilities over the selection of projects, which is similar to the rules
|
|
discussed above. To illustrate working with axioms, the following will introduce
|
|
intuitive properties which are then applied to the rules discussed in
|
|
section~\ref{sec:approval-based budgeting}.
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|
|
|
A simple axiom is termed \emph{exhaustiveness} by
|
|
\textcite{azizParticipatoryBudgetingModels2020} and \emph{inclusion maximality}
|
|
by \textcite{talmonFrameworkApprovalBasedBudgeting2019}. Inclusion maximality
|
|
encodes the requirement that if it is possible to fund more projects because the
|
|
budget is not yet exhausted, then we should. Greedy and proportional greedy
|
|
rules satisfy this axiom because of their inherent iterative process that
|
|
terminates only when the budget does not allow more projects to be funded. For
|
|
the maximum rules inclusion maximality still holds because for two feasible sets
|
|
of projects where one set is a subset of the other and the smaller set is
|
|
winning then also the bigger set is winning.
|
|
|
|
An axiom which is not met by all the discussed aggregation rules is
|
|
\emph{discount monotonicity}. Discount monotonicity states that if an already
|
|
selected project which is going to be funded receives a revised cost function,
|
|
then that project should not be implemented to a lesser degree
|
|
\cite[p.~11]{azizParticipatoryBudgetingModels2020}. This is an important
|
|
property because if a rule were to fail discount monotonicity, the outcome may
|
|
be manipulated by increasing the cost of a project instead of trying to minimize
|
|
it. For the rules given in section~\ref{sec:approval-based budgeting}, the
|
|
satisfaction functions $sat_\#$ (see equation~\ref{eq:3}) and $sat_{0/1}$
|
|
(equation~\ref{eq:5}) and their combination with the three aggregation methods
|
|
(greedy, proportional greedy and maximum rule) satisfy discount monotonicity.
|
|
This is the case because decreasing a project's cost makes it more attractive
|
|
for selection, which is not the case when the satisfaction function $sat_\$$
|
|
(equation~\ref{eq:4}) is used.
|
|
|
|
\emph{Limit monotonicity} is similar to discount monotonicity in that the
|
|
relation of a project's cost to the budget limit is modified. Whereas discount
|
|
monotonicity changes the project's cost, limit monotonicity changes the total
|
|
available budget. It states that if the budget limit is increased and there
|
|
exists no project which might become affordable and give higher satisfaction
|
|
than the previous solution, then a project that was a winning project before
|
|
will still be one after the budget is increased. Not satisfying this axiom could
|
|
provoke discontent among the voters when they realize that their approved
|
|
project is not funded anymore because the total budget has increased, as this is
|
|
somewhat counterintuitive. Unfortunately, none of the discussed rules satisfy
|
|
limit monotonicity. A counterexample for the greedy and proportional greedy
|
|
rules is one where there are three projects $a,b,c$ and $a$ gives the biggest
|
|
satisfaction. Project $a$ is therefore selected first. For the case where the
|
|
budget limit has not yet been increased, project $b$ is selected second because
|
|
project $c$ is too expensive even though it would provide more satisfaction.
|
|
When the budget limit is increased, project $c$ can now be funded instead of $b$
|
|
and will provide a higher total satisfaction. Voters which have approved project
|
|
$b$ will thus lose some of their satisfaction. This example is also applicable
|
|
to the maximum rules because the maximum satisfaction before the budget is
|
|
increased is provided by $\{ a,b \}$. Because $c$ can be funded additionally to
|
|
$a$ after increasing the budget and provides a higher total satisfaction, the
|
|
winning set is $\{ a,c \}$.
|
|
|
|
These three examples provide a rudimentary introduction to comparing aggregation
|
|
rules by their fulfillment of axiomatic properties. The social choice theory
|
|
often uses axioms such as \emph{strategyproofness}, \emph{pareto efficiency} and
|
|
\emph{non-dictatorship} to classify voting schemes. These properties are
|
|
concerned with making sure that each voter votes truthfully, that a solution
|
|
cannot be bettered without making someone worse off while improving another
|
|
voter and that results cannot only mirror one person's preferences,
|
|
respectively.
|
|
|
|
\section{Conclusion}
|
|
\label{sec:conclusion}
|
|
|
|
We have looked at different possibilities for conducting the voting and winner
|
|
selection process for participatory budgeting. A budgeting scenario in the
|
|
mathematical sense has been described and methods for modeling voter
|
|
satisfaction are discussed. A deeper view on approval-based budgeting models has
|
|
been given where the voters are assumed to have dichotomous preferences. The
|
|
complexity of the different rules has been evaluated and contrasted to each
|
|
other. We have seen that aggregation methods cannot only be compared in terms of
|
|
complexity but also by using axioms that formulate desirable outcomes.
|
|
|
|
Future research might focus on not only incorporating monetary cost and
|
|
satisfaction into aggregating winning projects but also other factors such as
|
|
environmental costs, practicability of participatory budgeting methods as well
|
|
as scalability of these methods to a very high amount of projects and voters.
|
|
Interesting further questions are posed by the possibility to combine projects
|
|
that are indivisible with projects that are divisible under one aggregation
|
|
rule, leading to a host of \emph{hybrid models}. Because a lot of the methods
|
|
that have been theorized by researchers have not yet been implemented in
|
|
practice, research on feasibility could lead to a better understanding of what
|
|
works and what does not. Another area of research could focus on allowing
|
|
projects to be related to each other and reflecting those inter-relations in the
|
|
outcome while still maintaining a grip on the explosion of possible solutions.
|
|
Exploring more axioms and rule configurations is important for achieving a
|
|
complete picture of the possibilities within the field of computational social
|
|
choice. As a final point, research into user interface design during the voting
|
|
phase might uncover previously unknown impacts of ballot design on the resulting
|
|
selection of winning projects.
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|
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|
\printbibliography
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\end{document}
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