Add text for axioms of PB methods

paper/references.bib

@@ -1,4 +1,19 @@
 
+@inproceedings{azizFairMixingCase2019,
+  title = {Fair {{Mixing}}: The {{Case}} of {{Dichotomous Preferences}}},
+  shorttitle = {Fair {{Mixing}}},
+  booktitle = {Proceedings of the 2019 {{ACM Conference}} on {{Economics}} and {{Computation}}},
+  author = {Aziz, Haris and Bogomolnaia, Anna and Moulin, Herv{\'e}},
+  year = {2019},
+  month = jun,
+  pages = {753--781},
+  publisher = {{Association for Computing Machinery}},
+  address = {{Phoenix, AZ, USA}},
+  doi = {10.1145/3328526.3329552},
+  abstract = {We consider a setting in which agents vote to choose a fair mixture of public outcomes. The agents have dichotomous preferences: each outcome is liked or disliked by an agent. We discuss three outstanding voting rules. The Conditional Utilitarian rule, a variant of the random dictator, is strategyproof and guarantees to any group of like-minded agents an influence proportional to its size. It is easier to compute and more efficient than the familiar Random Priority rule. Its worst case (resp. average) inefficiency is provably (resp. in numerical experiments) low if the number of agents is low. The efficient Egalitarian rule protects individual agents but not coalitions. It is excludable strategyproof: I do not want to lie if I cannot consume outcomes I claim to dislike. The efficient Nash Max Product rule offers the strongest welfare guarantees to coalitions, who can force any outcome with a probability proportional to their size. But it even fails the excludable form of strategyproofness.},
+  series = {{{EC}} '19}
+}
+
 @article{azizParticipatoryBudgetingModels2020,
   title = {Participatory {{Budgeting}}: {{Models}} and {{Approaches}}},
   shorttitle = {Participatory {{Budgeting}}},
@@ -118,6 +133,20 @@
   series = {Lecture {{Notes}} in {{Computer Science}}}
 }
 
+@inproceedings{fainFairAllocationIndivisible2018,
+  title = {Fair {{Allocation}} of {{Indivisible Public Goods}}},
+  booktitle = {Proceedings of the 2018 {{ACM Conference}} on {{Economics}} and {{Computation}}},
+  author = {Fain, Brandon and Munagala, Kamesh and Shah, Nisarg},
+  year = {2018},
+  month = jun,
+  pages = {575--592},
+  publisher = {{Association for Computing Machinery}},
+  address = {{Ithaca, NY, USA}},
+  doi = {10.1145/3219166.3219174},
+  abstract = {We consider the problem of fairly allocating indivisible public goods. We model the public goods as elements with feasibility constraints on what subsets of elements can be chosen, and assume that agents have additive utilities across elements. Our model generalizes existing frameworks such as fair public decision making and participatory budgeting. We study a groupwise fairness notion called the core, which generalizes well-studied notions of proportionality and Pareto efficiency, and requires that each subset of agents must receive an outcome that is fair relative to its size. In contrast to the case of divisible public goods (where fractional allocations are permitted), the core is not guaranteed to exist when allocating indivisible public goods. Our primary contributions are the notion of an additive approximation to the core (with a tiny multiplicative loss), and polynomial time algorithms that achieve a small additive approximation, where the additive factor is relative to the largest utility of an agent for an element. If the feasibility constraints define a matroid, we show an additive approximation of 2. A similar approach yields a constant additive bound when the feasibility constraints define a matching. For feasibility constraints defining an arbitrary packing polytope with mild restrictions, we show an additive guarantee that is logarithmic in the width of the polytope. Our algorithms are based on the convex program for maximizing the Nash social welfare, but differ significantly from previous work in how it is used. As far as we are aware, our work is the first to approximate the core in indivisible settings.},
+  series = {{{EC}} '18}
+}
+
 @inproceedings{freemanTruthfulAggregationBudget2019,
   title = {Truthful {{Aggregation}} of {{Budget Proposals}}},
   booktitle = {Proceedings of the 2019 {{ACM Conference}} on {{Economics}} and {{Computation}}},

paper/termpaper.tex

@@ -358,6 +358,84 @@ which has the lowest cost and satisfies exactly the estimated amount of voters.
 \section{Normative Axioms}
 \label{sec:normative axioms}
 
+Axioms in the context of participatory budgeting define some kind of property of
+a budgeting method that might be desirable to have. Generally it is beneficial
+if a certain method satisfies as many axioms as possible as this gives the
+method a strong theoretical backbone. One set of axioms, discussed by
+\textcite{talmonFrameworkApprovalBasedBudgeting2019}, relates to the cost of
+projects. Another possibility is to look at the \emph{fairness} associated with
+a particular set of winning projects. Fairness captures the notion of for
+example protecting minorities and their preferences.
+\textcite{azizProportionallyRepresentativeParticipatory2018} propose axioms that
+are representative of the broad spectrum of choices which voters can make. Other
+fairness-based approaches are proposed by
+\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}.
+
+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 costs exactly the amount to which the budget was
+increased, 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.
+
 \printbibliography
 
 \end{document}