From 1f37be066807444b2c5e34ebf1abdfcc06ea22bc Mon Sep 17 00:00:00 2001 From: Tobias Eidelpes Date: Tue, 28 Apr 2020 14:17:17 +0200 Subject: [PATCH] Restructure overview and include examples --- talk/talk.tex | 247 +++++++++++++++++++++++++++++--------------------- 1 file changed, 142 insertions(+), 105 deletions(-) diff --git a/talk/talk.tex b/talk/talk.tex index 22cf550..209463e 100644 --- a/talk/talk.tex +++ b/talk/talk.tex @@ -1,10 +1,14 @@ \documentclass{beamer} +\beamertemplatenavigationsymbolsempty + \usetheme{Boadilla} \usecolortheme{dolphin} \usepackage{graphicx} \usepackage{tikz} +\usepackage{dsfont} +\usepackage{comment} \usetikzlibrary{arrows} @@ -67,134 +71,167 @@ \end{frame} \begin{frame} - \frametitle{A formal model for PB} - \begin{itemize} + \frametitle{A general framework for PB} + \begin{itemize} \setlength{\itemsep}{1\baselineskip} - \item Projects can be bounded or unbounded - \item Projects can be divisible or indivisible (discrete) - \item Each project has an associated cost - \item Voters approve a subset of all projects (\emph{input method}) - \item The total cost is limited by the available budget - \item An \emph{aggregation method} provides a list of projects to fund - \end{itemize} + \item Projects $P=\{p_1,\dots,p_m\}$ + \begin{itemize} + \setlength{\itemsep}{.7\baselineskip} + \item Each project $p\in P$ has associated cost + $c(p):P\rightarrow\mathbb{R}$ + \item Projects are either divisible or indivisible (discrete) + \end{itemize} + \item Select a set $P'\subseteq P$ as \emph{winning projects} not + exceeding total budget $B$ + \begin{itemize} + \setlength{\itemsep}{.7\baselineskip} + \item Discrete case: $\sum_{p\in P'}c(p)\leq B$ + \item Divisible case: $\mu(p): P\rightarrow [0,1]$ with + $\sum_{p\in P'}c(\mu(p))\leq B$ + \end{itemize} + \end{itemize} +\end{frame} + +\begin{frame} + \frametitle{A general framework for PB ctd.} + \begin{itemize} + \setlength{\itemsep}{1\baselineskip} + \item Voters $V=\{v_1,\dots,v_n\}$ + \begin{itemize} + \setlength{\itemsep}{.5\baselineskip} + \item Express preferences over individual projects in $P$ or + over subsets in $\mathcal{P}(P) := \{P'\,|\,P'\subseteq P\}$ + \item Preference elicitation is dependent on the input method + (approval-based, ranked orders) + \end{itemize} + \item Aggregation methods + \begin{itemize} + \item Aggregation methods combine votes to determine a set + of winning projects + \item Are usually tied to the input method + \item Rules are used to select projects w.r.t. desired + properties of the outcome (fairness, welfare) + \end{itemize} + \end{itemize} \end{frame} \begin{frame} \frametitle{Input and aggregation methods} + Example input methods: \begin{itemize} - \item Approval voting - \item Ranked voting - \item Knapsack voting + \item Approval preferences + \item Ranked orders + \item Utility-based preferences \end{itemize} + \vspace{0.2cm} + Example aggregation methods: + \begin{itemize} + \item Maximizing social welfare + \item Greedy selection + \item Fairness-based selection + \end{itemize} + \vspace{0.2cm} + Aggregation methods depend on how voters elicit their preferences. +\end{frame} +\begin{frame} + \frametitle{Approval-based budgeting methods} + \begin{itemize} + \item Voters approve a subset of projects + \item Voter preferences are assumed to be \emph{dichotomous} + \item A \emph{satisfaction function} provides a metric for voter + satisfaction + \end{itemize} \begin{block}{An approval-based budgeting scenario} - A budgeting scenario is a tuple $E = (A,V,c,l)$ where $A = - \{a_1,\dots,a_m\}$ is a set of items (projects), $V$ is a set of voters, - $c : A\rightarrow\mathbb{N}$ is a cost function associating each project - $a\in A$ with its cost $c(a)$ and $l\in\mathbb{N}$ is a budget limit. A - voter $v\in V$ specifies $A_v\subseteq A$, containing all approved + A budgeting scenario is a tuple $E = (P,V,c,B)$ where $P = + \{p_1,\dots,p_m\}$ is a set of projects, $V$ is a set of voters, $c : + P\rightarrow\mathbb{N}$ is a cost function associating each project + $p\in P$ with its cost $c(p)$ and $B\in\mathbb{N}$ is a budget limit. A + voter $v\in V$ specifies $P_v\subseteq P$, containing all approved items. \end{block} + \begin{block}{Budgeting method $\mathcal{R}$} + A budgeting method $\mathcal{R}$ takes a budgeting scenario $E$ and + returns a bundle $A\subseteq P$ where the total cost of the items in + $A$ does not exceed the budget limit $B$. + \end{block} \end{frame} \begin{frame} - \frametitle{Preference elicitation} - \begin{block}{Range voting} - Voters rate projects based on their utility for each project. - \end{block} - \begin{block}{$k$-Approval} - Voters approve the $k$ projects they like the most. - \end{block} - \begin{block}{Approval voting} - Voters approve all projects that they like. - \end{block} - \begin{block}{Threshold approval voting} - Voters approve projects where their utility is above a specified - threshold. - \end{block} - \begin{block}{Knapsack voting} - Voters provide ideal allocation based on their preferences. - \end{block} -\end{frame} - -\section{Vote Aggregation} - -\begin{frame} - \frametitle{Vote Aggregation} - \begin{itemize} - \item Voters' preferences are aggregated to determine which - projects to fund - \item Main interest for research - \item Three different approaches: - \begin{itemize} - \item Welfare Maximization - \item Use of Axioms - \item Notions of Fairness - \end{itemize} - \end{itemize} + \frametitle{Satisfaction functions} + \begin{block}{Satisfaction function} + A satisfaction function $sat : 2^P\times 2^P\rightarrow\mathbb{R}$ with + a set $P$ of items, a voter $v$ and her approval set $P_v$ and a bundle + $A\subseteq P$ provides the satisfaction $sat(P_v,A)$ of $v$ from the + bundle $A$. The set of approved items by $v$ that end up in the winning + bundle is denoted by $A_v = P_v\cap A$. + \end{block} + \begin{exampleblock}{$sat_\#(P_v,A)$} + $sat_\#(P_v,A) = |A_v|$: The satisfaction of voter $v$ is the number of + funded items that are approved. + \end{exampleblock} \end{frame} \begin{frame} - \frametitle{Welfare Maximization} - \begin{block}{Utilitarian Welfare} - The utilitarian welfare of an allocation is the sum of utilities it gives to - residents: - \[ UW(\vec{x}) = - \sum_{i\in N}{u_i(\vec{x})}\text{ for }\vec{x}\in A \] - \end{block} - \begin{block}{Egalitarian Welfare} - The egalitarian welfare of an allocation is the minimum utility - it gives to any resident: - \[ EW(\vec{x}) = - \mathsf{min}_{i\in N}{u_i(\vec{x})}\text{ for - }\vec{x}\in A \] - \end{block} - \begin{block}{Nash Welfare} - The Nash welfare of an allocation is the product of utilities it gives to - residents: - \[ NW(\vec{x}) = - \prod_{i\in N}{u_i(\vec{x})}\text{ for }\vec{x}\in A \] - \end{block} + \frametitle{Satisfaction functions ctd.} + \begin{exampleblock}{$sat_\$(P_v,A)$} + $sat_{\$}(P_v,A) = \sum_{p\in A_v}{c(p) = c(A_v)}$: The satisfaction of voter + $v$ is the total cost of her approved and funded items. + \end{exampleblock} + \begin{exampleblock}{$sat_{0/1}(P_v,A)$} + \[ sat_{0/1}(P_v,A) = + \begin{cases} + 1 & \text{if } |A_v|>0 \\ + 0 & \text{otherwise} + \end{cases} + \] + A voter $v$ has satisfaction 1 if at least one of her approved items is + funded and 0 otherwise. + \end{exampleblock} \end{frame} \begin{frame} - \frametitle{Use of Axioms} - \begin{block}{Exhaustiveness} - A feasible allocation $\vec{x}$ is called exhaustive if an - outcome $\vec{y}$ is not feasible whenever $y_p\geq x_p$ for all - projects $p$ and a strict inequality holds for at least one - project. - \end{block} - \begin{block}{Discount Monotonicity} - Project $p$'s cost function $c_p$ is revised to $c_p'(x_p)\leq - c_p(x_p)$ after a vote aggregation rule outputs allocation - $\vec{x}$. For the resulting allocation $\vec{y}$, $y_p\geq - x_p$ holds. - \end{block} - \begin{block}{Pareto Optimality} - An allocation $\vec{x}\in A$ Pareto dominates another allocation - $\vec{y}\in A$ if $u_i(\vec{x})\geq u_i(\vec{y})$ for all $i\in - N$ and $u_i(\vec{x}) > u_i(\vec{y})$ for some $i\in N$. An - allocation $\vec{z}\in A$ is optimal if no allocation dominates - it. - \end{block} + \frametitle{Rules for selecting a winning bundle} + {\Large Let $sat$ be a satisfaction function:} + \begin{block}{Max rules} + The rule $\mathcal{R}_{sat}^m$ selects a bundle which maximizes the sum + of voters' satisfaction: $\mathsf{max}_{A\subseteq P}\sum_{v\in + V}{sat(P_v,A)}$ + \end{block} + \begin{block}{Greedy rules} + The rule $\mathcal{R}_{sat}^g$ iteratively adds an item $p\in P$ to $A$, + seeking to maximize $\sum_{v\in V}{sat(P_v,A\cup\{p\})}$. + \end{block} + \begin{block}{Proportional greedy rules} + The rule $\mathcal{R}_{sat}^p$ iteratively adds an item $p\in P$ to $A$ + seeking to maximize the sum of satisfaction per unit of cost. + \end{block} \end{frame} \begin{frame} - \frametitle{Notion of Fairness} - \begin{block}{The Core of PB} - An allocation $\vec{x} \in A$ is a core solution if there is no - subset $S$ of voters who, given a budget of $(|S|/n)B$, could - compute an allocation $\vec{y}\in A$ such that every voter in - $S$ receives strictly more utility in $\vec{y}$ than in - $\vec{x}$. - \end{block} - \begin{block}{Proportionality} - An allocation $\vec{x}$ should be proportionally reflected by - the division of voters. A majority of voters should have a - majority of the budget under their control but a minority should - have a minority of the budget under their control. - \end{block} + \frametitle{Example budgeting scenarios} + \begin{block}{A budgeting scenario} + Items $P = \{p_2,p_3,p_4,p_5,p_6\}$ and their associated cost $p_i$ + where $p_i$ costs $i$. Budget limit $B=10$ and 5 voters vote + $v_1=\{p_2,p_5,p_6\}$, $v_2=\{p_2,p_3,p_4,p_5\}$, $v_3=\{p_3,p_4,p_5\}$, + $v_4=\{p_4,p_5\}$ and $v_5=\{p_6\}$. + \end{block} + \begin{exampleblock}{Combining max rule with $sat_\#$} + Under $\mathcal{R}_{|A_v|}^m$ the winning bundle is $\{p_2,p_3,p_5\}$. + The total satisfaction is 8. + \end{exampleblock} + \begin{exampleblock}{Combining greedy rule with $sat_\#$} + Under $\mathcal{R}_{|A_v|}^g$ the winning bundle is $\{p_4,p_5\}$ (first + selecting $p_5$). The total satisfaction is 7. + \end{exampleblock} + \begin{exampleblock}{Combining max rule with $sat_{0/1}$} + Under $\mathcal{R}^m_{sat_{0/1}}$ the winning bundle is $\{p_4,p_6\}$, + achieving max satisfaction. + \end{exampleblock} +\end{frame} + +\begin{frame} + \frametitle{} \end{frame} \section{Future Directions}