\makeatletter
\let\ifGm@compatii\relax
\makeatother
\documentclass[12pt]{beamer} %use for presentation
% \documentclass[12pt,handout]{beamer} %use for handouts

\setbeameroption{show notes} %uncomment to see notes
\usefonttheme{serif}

\usefonttheme{structuresmallcapsserif}

\newcommand{\its}{\vskip 10pt \item}
\newcommand{\pits}{\vskip 10pt \pause \item}
\newcommand{\st}{\operatorname{st}}
\newcommand{\var}{\operatorname{var}}
\newtheorem{prop}{Proposition}
\newcommand{\R}{\mathbb{R}}
\newcommand{\E}{\mathbb{E}}
\newcommand{\AR}{\operatorname{AR}}
%\newtheorem{theorem}{Theorem}
\newcommand{\un}{\underline}
\newcommand{\ov}{\overline}
\newcommand{\gp}{\succ}
\newcommand{\lp}{\prec}
\newcommand{\gep}{\succsim}
\newcommand{\lep}{\precsim}


\newcommand{\T}{\ensuremath{\mbox{\footnotesize\textsc{occupation}}}}
\renewcommand{\S}{\ensuremath{\mbox{\footnotesize\textsc{suicide}}}}
%\newcommand{\R}{\ensuremath{\mbox{\footnotesize\textsc{religious}}}}



\usepackage{graphicx}
\usepackage{ragged2e}



% -----------------------------------------
% Center the Frame Title
% -----------------------------------------

\setbeamertemplate{frametitle} {
\begin{centering}
\vspace{0.1in} \insertframetitle
\par
\end{centering}}

% -----------------------------------------
% Number the slides
% -----------------------------------------

\setbeamertemplate{footline}[frame number]

% -----------------------------------------
% Get rid of the irritating navigation bar
% -----------------------------------------

\setbeamertemplate{navigation symbols}{}

% -----------------------------------------
% Begin Document
% -----------------------------------------

\title{Collective Goals}


\author{}
% Ethan Bueno de Mesquita\\ \small{Harris School of Public Policy Studies}\\ \small{University of Chicago}}




\date[]% (optional, should be abbreviation of conference name)

% Delete this, if you do not want the table of contents to pop up at
% the beginning of each subsection:
\AtBeginSection[]
{
 \begin{frame}<beamer>{Outline}
   \tableofcontents[currentsection,currentsubsection]
 \end{frame}
}


% Delete this, if you do not want the table of contents to pop up at
% the beginning of each subsection:
\AtBeginSubsection[]
{
 \begin{frame}<beamer>{Outline}
   \tableofcontents[currentsection,currentsubsection]
 \end{frame}
}



\begin{document}

\begin{frame}
  \titlepage
\end{frame}

\begin{frame}{The Project}
	
	Many normative frameworks are compelling
	
	\
	
	Reasonable people can disagree
	
	\
	
	Perhaps, even though we can't agree on normative principles, we can agree on procedural principles
	
	\
	
	Find an appealing \emph{aggregation rule} that takes individual preferences and returns collective preferences
\end{frame}

\begin{frame}{How to find an appealing procedure?}
	
	Identify normatively appealing conditions we want the aggregation rule to satisfy
	
	\
	
	Rule out aggregation rules that fail these conditions
	
	\
	
	Choose a procedure from the set of remaining aggregation rules
\end{frame}

\begin{frame}{Universal Domain}
	
	You don't know what issues will face your society in the future
	
	\
	
	Want the aggregation rule to perform sensibly regardless of what issue comes up
	
	\ 
	
	Aggregation rule must satisfy normative criteria for \emph{all} possible collections of alternatives and rational individual preferences
	
	\end{frame}


\section{Rational Individuals and Preference Aggregation}

\begin{frame}{Rational Preferences}
	
	Let $A$ be the set of alternatives
	
	\
	
	\textbf{Completeness:} For any $x,y \in A$ one of the following is true
	 	\begin{itemize}
			\item $ x \gp y$
			\its $y \gp x$
			\its $x \sim y$
		\end{itemize}
		
	\
	
	\textbf{Transitivity:} For any $x,y,z \in A$
		\begin{itemize}
			\item If $x \gp y$ and $y \gp z$, then $x \gp z$
			\its If $x \sim y$ and $y \sim z$, then $x \sim z$
		\end{itemize}
	\end{frame}


\begin{frame}{Majority Rule Aggregation: An Example}
	
	3 alternatives: Low, Medium, or High Taxes
	
	\
	
	3 individuals
		\[ H \gp_1 M \gp_1 L\]
		\[ M \gp_2 H \gp_2 L\]
		\[ L \gp_3 M \gp_3 H\]
		
	\
	
	Social preferences under majority rule aggregation
	
	\[ M \gp_{S,maj} H \gp_{S,maj} L\]
	
\end{frame}

\begin{frame}{Borda Count Aggregation: An Example}
	
	Same policies and individuals
	
	\[ \mathbf{H:} \ \ \ 2 + 1+0 = 3\]
	\[ \mathbf{M:} \ \ \ 1 + 2+1 = 4\]
	\[ \mathbf{L:} \ \ \ \ 0 + 0+2 = 2\]
	
	\
	
	Social preferences under Borda count aggregation
	\[ M \gp_{S,bor} H \gp_{S,bor} L\]
\end{frame}



\section{Normative Criteria}

\subsection{Transitivity}

	\begin{frame}{Transitivity}
		
		If $x \gp_S y$ and $y \gp_S z$, then $x \gp_S z$
		
		\
		
		This is a basic standard of coherence for social preferences
		
	\end{frame}
	
	
	\begin{frame}{Majority Rule Need Not Return Transitive Social Preferences}
		
		3 alternatives ($x,y,z$) and 3 people ($1,2,3$)
		
		\
		
		Individual preferences
		
			\[ x \gp_1 y \gp_1 z\]
			\[ y \gp_2 z \gp_2 x\]
			\[ z \gp_3 x \gp_3 y\]
			
		\
		
		Social preference
		
		\[ x \gp_{S,maj} y \ \ \ \ \ y \gp_{S,maj} z  \ \ \ \ \ z \gp_{S,maj} x\]
	\end{frame}


	\subsection{Unanimity}
	
		\begin{frame}{Unanimity}
			
			If every individual in society prefers $x$ to $y$, then the social preference should prefer $x$ to $y$
			
			\
			
			This is a basic standard of the aggregation rule actually representing the views of the individuals in society
		\end{frame}
		
		\begin{frame}{Amendment Procedure and Unanimity}
			
			5 alternatives $(x^1,\ldots, x^5)$ and 3 people ($1,2,3$)
			
			\
			
			\[ x^2 \gp_1 x^1 \gp_1 x^4 \gp_1 x^3 \gp_1 x^5\]
			\[ x^3 \gp_2 x^2 \gp_2 x^1 \gp_2 x^4 \gp_2 x^5\]
			\[ x^5 \gp_3 x^1 \gp_3 x^4 \gp_3 x^3 \gp_3 x^2\]
			
			\
			
			$x^4$ wins, but $x^1$ is unanimously preferred
		\end{frame}
		
	\subsection{Independence of Irrelevant Alternatives}
	
		\begin{frame}{Independence of Irrelevant Alternatives}
			
			Social preference with respect to $x$ and $y$ should depend only on individual preferences over $x$ and $y$
			
			\
			
			Shouldn't be able to manipulate social preference by introducing new alternatives
			
			\
			
			Also a kind of basic coherence or consistency
		\end{frame}
		
		\begin{frame}{Borda Count and IIA}
					
			\[ x \gp_{1,2} y \gp_{1,2} z\]
			\[ y \gp_3 z \gp_3 x\]
			\[ y \gp_4 x \gp_4 z\]
			
			\
			
			$y$ is Borda winner with 6 points
			
			\
			
			\pause Add $w$ while holding all other preferences fixed
			
			\[ x \gp_{1,2} w \gp_{1,2} y \gp_{1,2} z\]
			\[ y \gp_3 z \gp_3 x \gp_3 w\]
			\[ w \gp_4 y \gp_4 x \gp_4 z\]
			
		
		\
		
		$x$ is the Borda winner with 8 points
	\end{frame}
	
	\section{Arrow's Theorem}
	
	\begin{frame}{Arrow's Theorem}
		Suppose there are at least 3 alternatives and at least 2 people
		
		\
		
		Only one aggregation rule satisfies
		 	\begin{itemize}
				\item Universal Domain
				\its Transitivity
				\its Unanimity
				\its Independence of Irrelevant Alternatives
				\end{itemize}
				
				\
				
		\pause That rule is dictatorship
	\end{frame}
	
	\begin{frame}
		\begin{figure}
			\includegraphics[scale = .4]{figures/arrow-cartoon.pdf}
		\end{figure}
	\end{frame}
	
		
	\section{Any Notion of the Public Interest?}
	\begin{frame}{What to do in light of Arrow's Theorem}
		
		Abandon the idea of the public interest?
		
		\
		
		Weaken our ambitions a bit more
			\begin{itemize}
				\item Let's consider a few ways
				\end{itemize}

		\end{frame}
	\subsection{Majority Rule}
	
	\begin{frame}{The Case for Majority Rule}
		
		Although we've seen majority rule can be problematic, there are two settings in which we can make a case for majority rule as a principled aggregation procedure
		
		\
		
		Let's consider them in turn
			\begin{itemize}
				\item Only two alternatives
				\its Restricted domain
				\end{itemize}
				
	\end{frame}
	
	
	
		\begin{frame}{Only Two Alternatives}
			
			Arrow's theorem applies when there are at least three alternatives under consideration
			
			
			\
			
			\emph{Anonymity}: When the preferences of two people are switched, social preference doesn't change
			
		\
		
		 \emph{Neutrality}: When everyone's preference between two alternatives flips, rank ordering of those alternatives flips
		 
		 \
		 
		 \emph{Positively responsiveness}: Suppose society is indifferent between $x$ and $y$. All else equal, if one person who preferred $y$ to $x$ moves $x$ up in her preference order, then society strictly preferes $x$ to $y$. 
				\end{frame}
			
		
		\begin{frame}{May's Theorem}
			
			Suppose society has at least two people considering exactly two alternatives. The only aggregation procedure that satisfies universal domain, anonymity, neutrality, and positive responsiveness is \textbf{simple majority rule}---if a plurality of individuals strictly prefer one alternative to the other, then so does society and otherwise society is indifferent between the two alternative.
			\end{frame}
			
			
		\begin{frame}{Restricting the Domain}
			
			Arrow's theorem applies when all preference profiles are possible (universal domain)
			
			\
			
			Suppose we think certain profiles are unrealistic? 
			
			\
			
			Let's focus on a particular, substantively motivated, domain restriction
			\end{frame}
			
		\begin{frame}{An Ideology}
			
			For any collection of alternatives, line them up according to an \emph{ideological order}
			
			\
			
			Write $a_1 < a_2 < a_3$ for $a_1$ is to the left of $a_2$ which is to the left of $a_3$
			
			\
			
			For any set of alternatives there are many possible ideological orders
			
			\
			
			Say that $i$ is more conservative than $j$ under a given ideological order if for any $a_1 < a_2$
				\begin{itemize}
					\item If $j$ likes $a_2$ at least as much as $a_1$, then so does $i$
				\end{itemize}
			 
			 \
			 
			 Define \emph{strictly more conservative} in the natural way
			 \end{frame}
			 
		\begin{frame}{Described by an Ideology}
			
			An issue is \emph{described by an ideology} if we can find some ideological order such that individuals can be rank ordered from liberal to conservative
			
			\
			
			This isn't always true 
				\begin{itemize}
					\item For example, the Condorcet Triple is not described by an ideology
				\end{itemize}

			\
			
			Substantively, an issue is described by an ideology if we agree on a single dimension that describes the issue 
				\begin{itemize}
					\item If the richer you are, the less you like redistribution, then tax policy is described by an ideology
					\end{itemize}
			
			\end{frame}
			
			\begin{frame}{The Median Voter}
				
				For any issue  described by an ideology, majority rule will yield coherent social preferences
				
				\
				
				Line people up from liberal to conservative. There is a median person (called the \emph{median voter})
				
				\
				
				Consider $x > y$
					 \begin{itemize}
						 \item If the median voter prefers $x$ to $y$, so does everyone who is more conservative than the median (a majority)
						 \its If the median voter prefers $y$ to $x$, so does everyone who is more liberal than the median (a majority)
						 \end{itemize}
						\end{frame}
						
			\begin{frame}{The Median Voter Theorem}
				
				Suppose society has an odd number of people and is considering an issue that is described by an ideology. Then social preferences under majority rule are identical to the preferences of the median voter. That is, society prefers an alternative $x$ to an alternative $y$ if and only if the median voter prefers $x$ to $y$.
			
			\
			
			\begin{itemize}
				\item If we restrict the domain to issues described by an ideology, majority rule is a coherent aggregation rule
				\end{itemize}
			\end{frame}
			
		\subsection{More Information}
		
			\begin{frame}{Intensity of Preference}
				
				Arrow's theorem only makes use of ordinal information
					\begin{itemize}
						\item I like $x$ better than $y$ better than $z$
					\end{itemize}
				\
				
				But sometimes we might also have information about
					 \begin{itemize}
						 \item \emph{Intensity of preferences}: My preference for $x$ over $y$ is much stronger than my preference for $y$ over $z$
						 \its \emph{Interpersonal comparisons}: The choice between $x$ and $y$ matters a lot more to me than it does to you
						 \end{itemize}
						 
				\
				
				With this kind of information, we can get back to arguments for things like utilitarianism
				\end{frame}
				
		\subsection{Agreement?}
			
			\begin{frame}{Agreement}
				
				Look for situations in which it is likely that any sensible rule will yield the same social choice
				
				\
				
				Suppose we all agree on certain policies
				
				\
				
				This is only a useful thought if there are plausible situations in which agreement might arise 
				
				\
				
				Thinking about how we might construct such situations is the topic of the next lecture
				\end{frame}
				
\end{document}