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@ -77,8 +77,8 @@ stages \autocite{participatorybudgetingprojectHowPBWorks}:
main interest for computer scientists, economists and social choice theorists main interest for computer scientists, economists and social choice theorists
because depending on how voters elicit their preferences (\emph{balloting} or because depending on how voters elicit their preferences (\emph{balloting} or
\emph{input method}) and how the votes are aggregated through the use of \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 algorithms, the outcome is different. To study different ways of capturing votes
aggregating them, the participatory process is modeled mathematically. This and aggregating them, the participatory process is modeled mathematically. This
model will be called a participatory budgeting \emph{scenario}. The aim of model will be called a participatory budgeting \emph{scenario}. The aim of
studying participatory budgeting scenarios is to find ways to achieve a studying participatory budgeting scenarios is to find ways to achieve a
desirable outcome. A desirable outcome can be one based on fairness by making 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 approval-based preference elicitation or balloting. It is possible to design
variations of the described scenario by for example asking the voters to only 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) 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 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 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 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 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 authors introduce different \emph{utility models}. These models make it possible
to provide axiomatic guarantees such as strategyproofness or welfare 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 can allocate the budget in any way they see fit---utility functions are also
used by \textcite{talmonFrameworkApprovalBasedBudgeting2019} to measure the used by \textcite{talmonFrameworkApprovalBasedBudgeting2019} to measure the
total satisfaction that a winning set of projects provides under an aggregation 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 pseudo-polynomial time. Finding a solution using the rule
$\mathcal{R}_{sat_\#}^m$ however, is doable in polynomial time due to the $\mathcal{R}_{sat_\#}^m$ however, is doable in polynomial time due to the
problem's relation to the knapsack problem. If the input (either projects or problem's relation to the knapsack problem. If the input (either projects or
voters) is represented in voters) is represented in unary, a dynamic programming algorithm is bounded by a
unary, a dynamic programming algorithm is bounded by a polynomial in the length polynomial in the length of the input. For $\mathcal{R}_{sat_{0/1}}^m$, finding
of the input. For $\mathcal{R}_{sat_{0/1}}^m$, finding a set of projects that a set of projects that gives at least a certain amount of satisfaction is
gives at least a certain amount of satisfaction is \textsf{NP}-hard. Assuming \textsf{NP}-hard. Assuming that the cost of all of the projects is one unit, the
that the cost of all of the projects is one unit, the rule is equivalent to the rule is equivalent to the max cover problem because we are searching for a
max cover problem because we are searching for a subset of all projects with the subset of all projects with the number of the projects (the total cost due to
number of the projects (the total cost due to the projects given in unit cost) the projects given in unit cost) smaller or equal to the budget limit $B$ and
smaller or equal to the budget limit $B$ and want to maximize the number of want to maximize the number of voters that are represented by the subset.
voters that are represented by the subset.
\begin{example}\label{ex:max} \begin{example}\label{ex:max}
Taking the initial setup from example~\ref{ex:greedy}: $P = \{ 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 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 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 $\epsilon > 0$ is a fixed parameter that is chosen depending on the error of the
approximation. In fact, approximation. In fact, \textcite{khullerBudgetedMaximumCoverage1999} show that
\textcite{khullerBudgetedMaximumCoverage1999} show that an approximation an approximation algorithm with the same ratio exists not only for the case
algorithm with the same ratio exists not only for the case where the projects where the projects have unit cost but also for the general cost version.
have unit cost but also for the general cost version.
Instead of sacrificing exactness to get a better running time, Instead of sacrificing exactness to get a better running time,
\textcite{talmonFrameworkApprovalBasedBudgeting2019} show that the \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 cardinal utility function. The notion of core is also studied by
\textcite{fainFairAllocationIndivisible2018} for the case where voters have \textcite{fainFairAllocationIndivisible2018} for the case where voters have
additive utilities over the selection of projects, which is similar to the rules additive utilities over the selection of projects, which is similar to the rules
discussed above. To illustrate working with axioms, the following will discussed above. To illustrate working with axioms, the following will introduce
introduce intuitive properties which are then applied to the rules discussed in intuitive properties which are then applied to the rules discussed in
section~\ref{sec:approval-based budgeting}. section~\ref{sec:approval-based budgeting}.
A simple axiom is termed \emph{exhaustiveness} by A simple axiom is termed \emph{exhaustiveness} by
\textcite{azizParticipatoryBudgetingModels2020} and \emph{inclusion maximality} \textcite{azizParticipatoryBudgetingModels2020} and \emph{inclusion maximality}
by \textcite{talmonFrameworkApprovalBasedBudgeting2019}. Inclusion maximality by \textcite{talmonFrameworkApprovalBasedBudgeting2019}. Inclusion maximality
encodes the requirement that if it is possible to fund more projects because encodes the requirement that if it is possible to fund more projects because the
the budget is not yet exhausted, then we should. Greedy and proportional greedy budget is not yet exhausted, then we should. Greedy and proportional greedy
rules satisfy this axiom because of their inherent iterative process that 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 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 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 monotonicity changes the project's cost, limit monotonicity changes the total
available budget. It states that if the budget limit is increased and there available budget. It states that if the budget limit is increased and there
exists no project which might become affordable and give higher satisfaction 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 than the previous solution, then a project that was a winning project before
after the budget is increased. Not satisfying this axiom could provoke will still be one after the budget is increased. Not satisfying this axiom could
discontent among the voters when they realize that their approved project is not provoke discontent among the voters when they realize that their approved
funded anymore because the total budget has increased, as this is somewhat project is not funded anymore because the total budget has increased, as this is
counterintuitive. Unfortunately, none of the discussed rules satisfy limit somewhat counterintuitive. Unfortunately, none of the discussed rules satisfy
monotonicity. A counterexample for the greedy and proportional greedy rules is limit monotonicity. A counterexample for the greedy and proportional greedy
one where there are three projects $a,b,c$ and $a$ gives the biggest 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 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 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. project $c$ is too expensive even though it would provide more satisfaction.