From 579fb445d3db106aeb360b0353789e8f293060c4 Mon Sep 17 00:00:00 2001 From: Tobias Eidelpes Date: Wed, 29 Apr 2020 16:08:03 +0200 Subject: [PATCH] Finish presentation Add `\pause` for individual bullet points, add conclusion slide and finish axioms. --- talk/talk.tex | 263 +++++++++++++++++++++++++++++++++----------------- 1 file changed, 172 insertions(+), 91 deletions(-) diff --git a/talk/talk.tex b/talk/talk.tex index 172c16d..63e6a2f 100644 --- a/talk/talk.tex +++ b/talk/talk.tex @@ -6,6 +6,7 @@ \usecolortheme{dolphin} \usepackage{graphicx} +\usepackage{caption} \usepackage{tikz} \usepackage{dsfont} @@ -29,62 +30,77 @@ \section{Introduction} \begin{frame} - \frametitle{What is Participatory Budgeting?} + \frametitle{What is Participatory Budgeting?} \pause \begin{quote} Participatory Budgeting (PB) is a democratic process in which community members decide how to spend part of a public budget. - \end{quote} + \end{quote} \pause \vspace{1cm} \begin{itemize} \setlength{\itemsep}{1.1\baselineskip} - \item Participatory part: community members propose projects + \item Participatory part: community members propose projects \pause \item Budgeting part: each project requires a fixed amount of money + \pause \item Goal: Fund the \emph{best} projects without exceeding the budget \end{itemize} \end{frame} \begin{frame} - \frametitle{How does it work?} + \frametitle{A bit of background information} \pause + \begin{itemize} + \setlength{\itemsep}{1\baselineskip} + \item PB originated in Porto Alegre in 1990s \pause + \item Rapid expansion and influx of immigrants increased inequality + \pause + \item PB as a tool to combat inequality \pause + \item PB spread to Europe and North America \pause + \item Today \$300M allocated + \end{itemize} +\end{frame} + +\begin{frame} + \frametitle{How does it work?} \pause \tikzstyle{blue} = [rectangle,rounded - corners=3pt,draw=blue!50,fill=blue!20,node distance=1cm] + corners=3pt,draw=blue!50,fill=blue!20] \tikzstyle{red} = [rectangle,rounded corners=3pt,draw=red!50,fill=red!20] \tikzstyle{arrow} = [line width=1pt,->,>=triangle 60,draw=black!70] \begin{center} - \begin{tikzpicture}[shorten >= 2pt,level distance=1.3cm] - \node[blue](design){Design the process} - child { node [blue](collect){Collect ideas} - child { node [blue](develop){Develop feasible projects} - child { node [red](vote){Vote on projects} - child { node [red](aggregate){Aggregate votes \& fund winners} - } - } - } - } - ; - \draw [arrow] (design) to (collect); - \draw [arrow] (collect) to (develop); - \draw [arrow] (develop) to (vote); + \begin{tikzpicture}[shorten >= 2pt,node distance=1.3cm] + \node [blue] (design) {Design the process}; \pause + \node [blue] (collect) [below of=design] {Collect ideas}; + \draw [arrow] (design) to (collect); \pause + \node [blue] (develop) [below of=collect] {Develop feasible + projects}; + \draw [arrow] (collect) to (develop); \pause + \node [red] (vote) [below of=develop] {Vote on projects}; + \draw [arrow] (develop) to (vote); \pause + \node [red] (aggregate) [below of=vote] {Aggregate votes \& fund + winners}; \draw [arrow] (vote) to (aggregate); + \onslide<1-> \end{tikzpicture} \end{center} \end{frame} +\section{Modeling a PB scenario} + \begin{frame} - \frametitle{A general framework for PB} + \frametitle{A general framework for PB} \pause \begin{itemize} \setlength{\itemsep}{1\baselineskip} - \item Projects $P=\{p_1,\dots,p_m\}$ + \item Projects $P=\{p_1,\dots,p_m\}$ \pause \begin{itemize} \setlength{\itemsep}{.7\baselineskip} - \item Each project $p\in P$ has associated cost + \item Each project $p\in P$ has associated cost \pause $c(p):P\rightarrow\mathbb{R}$ \item Projects are either divisible or indivisible (discrete) + \pause \end{itemize} \item Select a set $A\subseteq P$ as \emph{winning projects} not - exceeding total budget $B$ + exceeding total budget $B$ \pause \begin{itemize} \setlength{\itemsep}{.7\baselineskip} - \item Discrete case: $\sum_{p\in A}c(p)\leq B$ + \item Discrete case: $\sum_{p\in A}c(p)\leq B$ \pause \item Divisible case: $\mu(p): P\rightarrow [0,1]$ with $\sum_{p\in A}c(\mu(p))\leq B$ \end{itemize} @@ -95,51 +111,56 @@ \frametitle{A general framework for PB ctd.} \begin{itemize} \setlength{\itemsep}{1\baselineskip} - \item Voters $V=\{v_1,\dots,v_n\}$ + \item Voters $V=\{v_1,\dots,v_n\}$ \pause \begin{itemize} \setlength{\itemsep}{.4\baselineskip} \item Express preferences over individual projects in $P$ or - over subsets in $\mathcal{P}(P) := \{P_v\,|\,P_v\subseteq P\}$ + over subsets in $\mathcal{P}(P) := \{P_v\,|\,P_v\subseteq + P\}$ \pause \item Preference elicitation is dependent on the input method - (approval-based, ranked orders) + (approval-based, ranked orders) \pause \end{itemize} - \item Aggregation methods + \item Aggregation methods \pause \begin{itemize} + \setlength{\itemsep}{.4\baselineskip} \item Aggregation methods combine votes to determine a set - of winning projects - \item Are usually tied to the input method + of winning projects \pause + \item Are usually tied to the input method \pause \item Rules are used to select projects w.r.t. desired properties of the outcome (fairness, welfare) \end{itemize} \end{itemize} \end{frame} +\section{Algorithms for PB} + \begin{frame} \frametitle{Input and aggregation methods} - Example input methods: + Example input methods: \pause \begin{itemize} - \item Approval preferences - \item Ranked orders - \item Utility-based preferences + \item Approval preferences \pause + \item Ranked orders \pause + \item Utility-based preferences \pause \end{itemize} \vspace{0.2cm} - Example aggregation methods: + Example aggregation methods: \pause \begin{itemize} - \item Maximizing social welfare - \item Greedy selection - \item Fairness-based selection + \item Maximizing social welfare \pause + \item Greedy selection \pause + \item Fairness-based selection \pause \end{itemize} \vspace{0.2cm} Aggregation methods depend on how voters elicit their preferences. \end{frame} \begin{frame} - \frametitle{Approval-based budgeting methods} + \frametitle{Approval-based budgeting methods} \pause \begin{itemize} - \item Voters approve a subset of projects - \item Voter preferences are assumed to be \emph{dichotomous} + \item Suitable for discrete PB \pause + \item Voters approve a subset of projects \pause + \item Voter preferences are assumed to be \emph{dichotomous} \pause \item A \emph{satisfaction function} provides a metric for voter - satisfaction + satisfaction \pause \end{itemize} \begin{block}{An approval-based budgeting scenario} A budgeting scenario is a tuple $E = (P,V,c,B)$ where $P = @@ -148,7 +169,7 @@ $p\in P$ 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} + \end{block} \pause \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 @@ -157,18 +178,18 @@ \end{frame} \begin{frame} - \frametitle{Satisfaction functions} + \frametitle{Satisfaction functions} \pause \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} + \end{block} \pause \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{exampleblock} \pause \end{frame} \begin{frame} @@ -176,7 +197,7 @@ \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} + \end{exampleblock} \pause \begin{exampleblock}{$sat_{0/1}(P_v,A)$} \[ sat_{0/1}(P_v,A) = \begin{cases} @@ -190,17 +211,17 @@ \end{frame} \begin{frame} - \frametitle{Rules for selecting a winning bundle} - {\Large Let $sat$ be a satisfaction function:} + \frametitle{Rules for selecting a winning bundle} \pause + {\Large Let $sat$ be a satisfaction function:} \pause \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} + \end{block} \pause \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} + \end{block} \pause \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. @@ -208,49 +229,51 @@ \end{frame} \begin{frame} - \frametitle{Example budgeting scenarios} + \frametitle{Example budgeting scenarios} \pause \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} + \end{block} \pause \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} + \end{exampleblock} \pause \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} + \end{exampleblock} \pause \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} +\section{Complexity of PB algorithms} + \begin{frame} - \frametitle{Complexity of budgeting algorithms} + \frametitle{Complexity of algorithms} \pause \begin{itemize} \setlength{\itemsep}{1\baselineskip} \item Computing winners using greedy rules ($\mathcal{R}^g_{sat}$) can - be done in polynomial time: + be done in polynomial time: \pause \begin{itemize} \setlength{\itemsep}{.4\baselineskip} \item these rules are defined through efficient iterative - processes + processes \pause \item however: making a series of locally optimal choices does - not always lead to a globally optimal choice + not always lead to a globally optimal choice \pause \item $\mathcal{R}^g_{|A_v|}$ is similar to $k$-Approval and - knapsack voting + knapsack voting \pause \end{itemize} - \item Max rules ($\mathcal{R}^m_{sat}$) are generally NP-hard + \item Max rules ($\mathcal{R}^m_{sat}$) are generally NP-hard \pause \begin{itemize} \setlength{\itemsep}{.4\baselineskip} \item $\mathcal{R}^m_{|A_v|}$ can be solved in polynomial time - because one dimension is fixed + because one dimension is fixed \pause \item $\mathcal{R}^m_{sat_{0/1}}$: finding a bundle with at least a - given total satisfaction is NP-hard + given total satisfaction is NP-hard \pause \item satisfaction functions can be modeled as integer linear programs \end{itemize} @@ -258,66 +281,118 @@ \end{frame} \begin{frame} - \frametitle{Complexity of budgeting algorithms ctd.} - {\large Dealing with \emph{intractability}:} + \frametitle{Complexity of algorithms ctd.} + {\large Dealing with \emph{intractability}:} \pause \vspace{.3cm} \begin{itemize} \setlength{\itemsep}{1\baselineskip} - \item Provide an approximation algorithm, sacrificing exactness + \item Provide an approximation algorithm, sacrificing exactness \pause \begin{itemize} \setlength{\itemsep}{0.4\baselineskip} \item No algorithm with approx. ratio better than $1-1/\epsilon$ - exists for $\mathcal{R}^m_{sat_{0/1}}$ + exists for $\mathcal{R}^m_{sat_{0/1}}$ \pause \end{itemize} \item Fixed-parameter tractability: fix one parameter to solve problem - in reasonable amount of time + in reasonable amount of time \pause \begin{itemize} \setlength{\itemsep}{0.4\baselineskip} - \item Fix parameter $m$ (the number of items) + \item Fix parameter $m$ (the number of items) \pause \item Fix parameter $n$ (the number of voters) \end{itemize} \end{itemize} \end{frame} +\section{Axioms for PB algorithms} + \begin{frame} - \frametitle{Comparing budgeting algorithms} - By defining desirable axioms, different budgeting algorithms can - be compared: - \begin{block}{Discount Monotonicity} - Suppose a budgeting algorithm $\mathcal{R}$ outputs a winning set of - projects $A$. The cost of project $p\in A$ is lowered (discounted) - compared to the previous cost. $\mathcal{R}$ should output another - winning set $A'$ where project $p$ is not implemented to a lesser - degree. - \end{block} + \frametitle{Comparing algorithms} \pause + \begin{itemize} + \setlength{\itemsep}{.9\baselineskip} + \item Compare algorithms by using axioms \pause + \item Axioms provide \emph{guidelines} for choosing an algorithm \pause + \item Some might be desirable, others are not a drawback, if not + satisfied \pause + \item Desirable properties: \emph{fairness}, \emph{strategyproofness} + \end{itemize} \end{frame} \begin{frame} - \frametitle{Axiom Examples} + \frametitle{Comparing algorithms—Discount Monotonicity} \pause + \begin{block}{Discount Monotonicity} + Suppose a budgeting algorithm $\mathcal{R}$ outputs a winning set of + projects $A$. The cost of project $p\in A$ is lowered (discounted) + compared to the previous cost. $\mathcal{R}$ should output another + winning set $A'$ where project $p$ is not implemented to a lesser + degree. + \end{block} \pause \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} + \end{block} \pause \begin{exampleblock}{Discount Monotonicity Example} Under $\mathcal{R}^m_{|A_v|}$ the winning bundle is $\{p_2,p_3,p_5\}$. - After discounting $p_2$ to $p_1$, we still get $\{p_1,p_3,p_5\}$. The - total cost is one unit less but the total satisfaction remains the same. + After discounting $p_2$ to $p_1$, we still get $\{p_1,p_3,p_5\}$. \end{exampleblock} \end{frame} -\section{Future Directions} +\begin{frame} + \frametitle{Comparing algorithms—Limit Monotonicity} \pause + \begin{block}{Limit Monotonicity} + A budgeting method $\mathcal{R}$ satisfies Limit Monotonicity if for a + pair of budgeting scenarios $E=(P,V,c,B)$, $E'=(P,V,c,B+1)$ and with no + project that costs exactly $B+1$, for each project $p\in P$ + $p\in\mathcal{R}(E)\implies p\in\mathcal{R}(E')$ holds. + \end{block} \pause + \begin{block}{A budgeting scenario} + Items $P=\{a_1,b_1,c_1\}$ (all with unit cost), budget limit $B=1$ and 4 + voters vote $v_1=\{a_1\}$, $v_2=\{a_1,b_1\}$, $v_3=\{b_1,c_1\}$ and + $v_4=\{c_1\}$. + \end{block} \pause + \begin{exampleblock}{Limit Monotonicity Example} + Under $\mathcal{R}^m_{sat_{0/1}}$ a winning bundle might be $\{b_1\}$. + Increasing the budget limit to 2 results in $\{a_1,c_1\}$. + \end{exampleblock} +\end{frame} \begin{frame} - \frametitle{Future Areas of Interest} + \frametitle{Applying the two axioms to the example algorithms} \pause + \begin{itemize} + \setlength{\itemsep}{1\baselineskip} + \item Discount Monotonicity: \pause + \begin{itemize} + \setlength{\itemsep}{.4\baselineskip} + \item All 3 algorithms ($\mathcal{R}^m_{|A_v|}$, + $\mathcal{R}^g_{|A_v|}$, $\mathcal{R}^m_{sat_{0/1}}$) + satisfy discount monotonicity \pause + \item Decreasing the cost increases the attractiveness of an + item \pause + \item Not true for algorithms that measure satisfaction by + maximizing the cost of winning projects \pause + \end{itemize} + \item Limit Monotonicity: \pause + \begin{itemize} + \setlength{\itemsep}{.4\baselineskip} + \item No discussed algorithm satisfies limit monotonicity \pause + \item Greedy heuristics fail when a project fits into the new + budget, providing higher satisfaction \pause + \item Increasing the budget might lead to projects being dropped + \end{itemize} + \end{itemize} +\end{frame} + +\section{Conclusion} + +\begin{frame} + \frametitle{Conclusion} \pause \begin{itemize} - \item Multi-dimensional constraints - \item Hybrid models - \item Complex resident preferences - \item Market-based approaches - \item The role of information - \item Research spanning the entire PB process + \setlength{\itemsep}{1\baselineskip} + \item PB: What is it? \pause + \item Computational aspects \pause + \item Aggregation algorithms \pause + \item Complexity of algorithms \pause + \item Comparison of algorithms using axioms \end{itemize} \end{frame} @@ -326,6 +401,12 @@ \Large Thank you for your attention! \\ Questions \& Answers + \begin{figure} + \centering + \includegraphics[width=.5\textwidth]{voting_referendum.png} + \captionsetup{labelformat=empty} + \caption{\tiny [\url{https://xkcd.com/2225}]} + \end{figure} \end{frame} \end{document}