diff --git a/paper/termpaper.tex b/paper/termpaper.tex index 91ef1ba..3a45f43 100644 --- a/paper/termpaper.tex +++ b/paper/termpaper.tex @@ -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.