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paper/termpaper.tex
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@ -77,8 +77,8 @@ stages \autocite{participatorybudgetingprojectHowPBWorks}:
main interest for computer scientists, economists and social choice theorists
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. To study different ways of capturing votes and
aggregating them, the participatory process is modeled mathematically. This
algorithms, the outcome is different. 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
@ -132,7 +132,7 @@ 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
\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
@ -152,7 +152,7 @@ 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
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
@ -278,15 +278,14 @@ $P$ is the set of projects, $\mathcal{R}_{sat_\$}^m$ is solvable in
pseudo-polynomial time. Finding a solution using the rule
$\mathcal{R}_{sat_\#}^m$ however, is doable in polynomial time due to the
problem's relation to the knapsack problem. If the input (either projects or
voters) is represented in
unary, a dynamic programming algorithm is bounded by a polynomial in the length
of the input. For $\mathcal{R}_{sat_{0/1}}^m$, finding a set of projects that
gives at least a certain amount of satisfaction is \textsf{NP}-hard. Assuming
that the cost of all of the projects is one unit, the rule is equivalent to the
max cover problem because we are searching for a subset of all projects with the
number of the projects (the total cost due to the projects given in unit cost)
smaller or equal to the budget limit $B$ and want to maximize the number of
voters that are represented by the subset.
voters) is represented in unary, a dynamic programming algorithm is bounded by a
polynomial in the length of the input. For $\mathcal{R}_{sat_{0/1}}^m$, finding
a set of projects that gives at least a certain amount of satisfaction is
\textsf{NP}-hard. Assuming that the cost of all of the projects is one unit, the
rule is equivalent to the max cover problem because we are searching for a
subset of all projects with the number of the projects (the total cost due to
the projects given in unit cost) smaller or equal to the budget limit $B$ and
want to maximize the number of voters that are represented by the subset.
\begin{example}\label{ex:max}
Taking the initial setup from example~\ref{ex:greedy}: $P = \{
@ -339,10 +338,9 @@ problems. \textcite{talmonFrameworkApprovalBasedBudgeting2019} show that the
rule $\mathcal{R}_{sat_{0/1}}^m$ is similar to the max cover problem which can
be approximated with a $(1-\frac{1}{\epsilon})$-approximation algorithm, where
$\epsilon > 0$ is a fixed parameter that is chosen depending on the error of the
approximation. In fact,
\textcite{khullerBudgetedMaximumCoverage1999} show that an approximation
algorithm with the same ratio exists not only for the case where the projects
have unit cost but also for the general cost version.
approximation. In fact, \textcite{khullerBudgetedMaximumCoverage1999} show that
an approximation algorithm with the same ratio exists not only for the case
where the projects have unit cost but also for the general cost version.
Instead of sacrificing exactness to get a better running time,
\textcite{talmonFrameworkApprovalBasedBudgeting2019} show that the
@ -374,15 +372,15 @@ 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
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}.
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
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
@ -409,13 +407,13 @@ 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
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.