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\title{Coordination Problems}


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




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

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\begin{document}

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

\section{Some Simple Coordination Games}

\begin{frame}{A Simple Coordination Game: What Side of the Street?}
	
			\begin{center}
			{\setlength{\extrarowheight}{10pt}%
	\begin{tabular}{*4c} 
		& &\multicolumn{2}{c}{Driver 2}\\
		&  										&\only<1,3-|handout:1>{$L$} \only<2|handout:0>{\cyan $L$} 	&\only<1-2,4-|handout:1>{$R$} \only<3|handout:0>{\cyan $R$} \\ \cline{3-4}
		\multirow{2}{*}{Driver 1}	
				\only<1|handout:0>{	& $\ell$ 			& \multicolumn{1}{|c|}{$5,5$} 				& \multicolumn{1}{c|}{$0,0$} \\[10pt] \cline{3-4} 
							& $r$ 			& \multicolumn{1}{|c|}{$0,0$}				&\multicolumn{1}{c|}{$5,5$}\\[10pt] \cline{3-4}}
				
				\only<2|handout:0>{	& $\ell$ 			& \multicolumn{1}{|c|}{\cyan$5\ch,5$} 		& \multicolumn{1}{c|}{$0,0$} \\[10pt] \cline{3-4} 
							& $r$ 			& \multicolumn{1}{|c|}{\cyan $0,0$}			&\multicolumn{1}{c|}{$5,5$}\\[10pt] \cline{3-4}}

				\only<3|handout:0>{	& $\ell$ 			& \multicolumn{1}{|c|}{$5\ch,5$} 			& \multicolumn{1}{c|}{\cyan $0,0$} \\[10pt] \cline{3-4} 
							& $r$ 			& \multicolumn{1}{|c|}{$0,0$}				&\multicolumn{1}{c|}{\cyan $5\ch,5$}\\[10pt] \cline{3-4}}

				\only<4|handout:0>{	& $\cyan \ell$ 			& \multicolumn{1}{|c|}{\cyan $5\ch,5\ch$} 	& \multicolumn{1}{c|}{\cyan $0,0$} \\[10pt] \cline{3-4} 
							& $r$ 			& \multicolumn{1}{|c|}{$0,0$}				&\multicolumn{1}{c|}{$5\ch,5$}\\[10pt] \cline{3-4}}

				\only<5|handout:0>{	& $\ell$ 			& \multicolumn{1}{|c|}{$5\ch,5\ch$} 		& \multicolumn{1}{c|}{$0,0$} \\[10pt] \cline{3-4} 
							& $\cyan r$ 			& \multicolumn{1}{|c|}{\cyan $0,0$}			&\multicolumn{1}{c|}{\cyan $5\ch,5\ch$}\\[10pt] \cline{3-4}}

				\only<6-|handout:1>{	& $\ell$ 			& \multicolumn{1}{|c|}{\orange $5\ch,5\ch$}	& \multicolumn{1}{c|}{$0,0$} \\[10pt] \cline{3-4} 
							& $r$ 			& \multicolumn{1}{|c|}{$0,0$}				&\multicolumn{1}{c|}{$\orange 5\ch,5\ch$}\\[10pt] \cline{3-4}}
				
		\end{tabular}}
		\end{center}
\only<7-|handout:1>{
\begin{itemize}
\item Two equilibria: $(\ell,L)$ and $(r,R)$ 
\its Pure coordination game---drivers care only about choosing the same side
\end{itemize}}

	\end{frame}
	
\begin{frame}{Coordination with Distributional Concerns: Accounting Standards}
	
	
				\begin{center}
				{\setlength{\extrarowheight}{10pt}%
		\begin{tabular}{*4c} 
			& &\multicolumn{2}{c}{Britain}\\
			&  										&\only<1,3-|handout:1>{US} \only<2|handout:0>{\cyan US} 	&\only<1-2,4-|handout:1>{UK} \only<3|handout:0>{\cyan UK} \\ \cline{3-4}
			\multirow{2}{*}{America}	
					\only<1|handout:0>{	& us 			& \multicolumn{1}{|c|}{$5,1$} 				& \multicolumn{1}{c|}{$0,0$} \\[10pt] \cline{3-4} 
								& uk 			& \multicolumn{1}{|c|}{$0,0$}				&\multicolumn{1}{c|}{$1,5$}\\[10pt] \cline{3-4}}
				
					\only<2|handout:0>{	& us 			& \multicolumn{1}{|c|}{\cyan$5\ch,1$} 		& \multicolumn{1}{c|}{$0,0$} \\[10pt] \cline{3-4} 
								& uk 			& \multicolumn{1}{|c|}{\cyan $0,0$}			&\multicolumn{1}{c|}{$1,5$}\\[10pt] \cline{3-4}}

					\only<3|handout:0>{	& us 			& \multicolumn{1}{|c|}{$5\ch,1$} 			& \multicolumn{1}{c|}{\cyan $0,0$} \\[10pt] \cline{3-4} 
								& uk 			& \multicolumn{1}{|c|}{$0,0$}				&\multicolumn{1}{c|}{\cyan $1\ch,5$}\\[10pt] \cline{3-4}}

					\only<4|handout:0>{	& \cyan us 			& \multicolumn{1}{|c|}{\cyan $5\ch,1\ch$} 	& \multicolumn{1}{c|}{\cyan $0,0$} \\[10pt] \cline{3-4} 
								& uk 			& \multicolumn{1}{|c|}{$0,0$}				&\multicolumn{1}{c|}{$1\ch,5$}\\[10pt] \cline{3-4}}

					\only<5|handout:0>{	& us 			& \multicolumn{1}{|c|}{$5\ch,1\ch$} 		& \multicolumn{1}{c|}{$0,0$} \\[10pt] \cline{3-4} 
								& \cyan uk 			& \multicolumn{1}{|c|}{\cyan $0,0$}			&\multicolumn{1}{c|}{\cyan $1\ch,5\ch$}\\[10pt] \cline{3-4}}

					\only<6-|handout:1>{	& us 			& \multicolumn{1}{|c|}{\orange $5\ch,1\ch$}	& \multicolumn{1}{c|}{$0,0$} \\[10pt] \cline{3-4} 
								& uk 			& \multicolumn{1}{|c|}{$0,0$}				&\multicolumn{1}{c|}{$\orange 1\ch,5\ch$}\\[10pt] \cline{3-4}}
				
			\end{tabular}}
			\end{center}
			
	\only<7-|handout:1>{
	\begin{itemize}
	\item Two equilibria: $(\text{us}, \text{US})$ and $(\text{uk}, \text{UK})$ 
	\its Coordination with distributional consequences---players want to coordinate, but disagree on preferred outcome
	\end{itemize}}
	
	\end{frame}
	
	
\begin{frame}{Coordination with Efficiency Concerns: Investing in a Developing Country}
	
	
	
	
				\begin{center}
				{\setlength{\extrarowheight}{10pt}%
		\begin{tabular}{*4c} 
			& &\multicolumn{2}{c}{Firm 2}\\
			&  										&\only<1,3-|handout:1>{Invest} \only<2|handout:0>{\cyan Invest} 	&\only<1-2,4-|handout:1>{Don't} \only<3|handout:0>{\cyan Don't} \\ \cline{3-4}
			\multirow{2}{*}{Firm 1}	
					\only<1|handout:0>{	& invest 			& \multicolumn{1}{|c|}{$20,20$} 				& \multicolumn{1}{c|}{$0,5$} \\[10pt] \cline{3-4} 
								& don't  			& \multicolumn{1}{|c|}{$5,0$}				&\multicolumn{1}{c|}{$5,5$}\\[10pt] \cline{3-4}}
				
					\only<2|handout:0>{	& invest 			& \multicolumn{1}{|c|}{\cyan$20\ch,20$} 		& \multicolumn{1}{c|}{$0,5$} \\[10pt] \cline{3-4} 
								& don't  			& \multicolumn{1}{|c|}{\cyan $5,0$}			&\multicolumn{1}{c|}{$5,5$}\\[10pt] \cline{3-4}}

					\only<3|handout:0>{	& invest 			& \multicolumn{1}{|c|}{$20\ch,20$} 			& \multicolumn{1}{c|}{\cyan $0,5$} \\[10pt] \cline{3-4} 
								& don't  			& \multicolumn{1}{|c|}{$5,0$}				&\multicolumn{1}{c|}{\cyan $5\ch,5$}\\[10pt] \cline{3-4}}

					\only<4|handout:0>{	& \cyan invest 			& \multicolumn{1}{|c|}{\cyan $20\ch,20\ch$} 	& \multicolumn{1}{c|}{\cyan $0,5$} \\[10pt] \cline{3-4} 
								& don't  			& \multicolumn{1}{|c|}{$5,0$}				&\multicolumn{1}{c|}{$5\ch,5$}\\[10pt] \cline{3-4}}

					\only<5|handout:0>{	& invest 			& \multicolumn{1}{|c|}{$20\ch,20\ch$} 		& \multicolumn{1}{c|}{$0,5$} \\[10pt] \cline{3-4} 
								& \cyan don't  			& \multicolumn{1}{|c|}{\cyan $5,0$}			&\multicolumn{1}{c|}{\cyan $0\ch,5\ch$}\\[10pt] \cline{3-4}}

					\only<6-|handout:1>{	& invest 			& \multicolumn{1}{|c|}{\orange $20\ch,20\ch$}	& \multicolumn{1}{c|}{$0,5$} \\[10pt] \cline{3-4} 
								& don't  			& \multicolumn{1}{|c|}{$5,0$}				&\multicolumn{1}{c|}{$\orange 5\ch,5\ch$}\\[10pt] \cline{3-4}}
				
			\end{tabular}}
			\end{center}
			
	\only<7-|handout:1>{
	\begin{itemize}
	\item Two equilibria: $(\text{invest}, \text{Invest})$ and $(\text{don't}, \text{Don't})$ 
	\its  $(\text{invest}, \text{Invest})$ Pareto dominates $(\text{don't}, \text{Don't})$ 
	\end{itemize}}
	


	\end{frame}


\section{Coordination Traps}

\begin{frame}{Coordination Trap}
 Multiple equilibria 

\

 One equilibrium is more desirable than another

\


 Players can become ``trapped'' in an undesirable equilibrium

\

 How? --- reinforcing expectations
\end{frame}

\begin{frame}{Some Examples}
 Social Conventions
\begin{itemize}
\item Foot binding
\item Honor killings
\item Private vs public schools \pause
\end{itemize}

\

 Economic

\begin{itemize}
\item Underdevelopment
\item Agglomeration economies
\item Technology adoption \pause
\end{itemize}

\

 Political 
\begin{itemize}
\item Failure of accountability
\item Revolutions/Protests
\end{itemize}
\end{frame}

\begin{frame}{Investment}

2 investors who choose an investment level: $1 \geq e_{i} \geq 0$

\

Payoff to investing is
\begin{equation*}
\pi_{i}(e_{1},e_{2}) = e_{1} \cdot e_{2} - \frac{e^{2}_{i}}{2}
\end{equation*}

\

 What is the best-response level of investment? \pause
\begin{equation*}
e_{2} = e_{1}
\end{equation*}

\

Investor 1 wants to match investor 2's investment and vice-versa
\end{frame}

\begin{frame}{Equilibrium Investment}

Any strategy profile where $e_{1} = e_{2}$ is an equilibrium! 

\

Equilibrium payoffs: Let $e = e_{1} = e_{2}$
\begin{equation*}
e^{2} - \frac{e^{2}}{2} =  \frac{e^{2}}{2}
\end{equation*}

\

Increasing in $e$

\

Greater joint investments lead to Pareto improvements \emph{and are equilibria}
\end{frame}

\begin{frame}\frametitle{The Coordination Trap}

Each player investing 0 is an equilibrium: $u_i(0,0) = 0$

\

Each player investing $\frac{1}{2}$ is an equilibrium: $u_i(\frac{1}{2},\frac{1}{2}) = \frac{1}{8}$

\

Each player investing 1 is an equilibrium: $u_i(1,1) = \frac{1}{2}$

\

Self fulfilling expectations can create Pareto inefficient equilibrium outcomes
\begin{itemize}
\item Underdevelopment 
\item Underinvestment in education
\end{itemize}
\end{frame}

\begin{frame}{Policy Responses to Coordination Traps}

Insurance
	\begin{itemize}
		\item Suppose policy maker promises to ``top up'' other player's investment
		\item Never end up having to actually do so
	\end{itemize}


\


\phantom{\parbox{\linewidth}{%
Communication and Leadership:
\begin{itemize}
\item Create a mutual expectation that others will invest
\end{itemize}}}

\

\phantom{\parbox{\linewidth}{%
Short-run interventions
	\begin{itemize}
		\item Fundamentally different than externalities because new behavior is also an equilibrium
		\item This tells us about the scope of leadership
		\end{itemize}}}
\end{frame}

\begin{frame}{FDIC}
	
	Banking Acts of 1933 and 1935 create Federal Deposit Insurance Corporation (FDIC) in response to bank runs of the Grat Depression
	
	\
	
	Guarantees depositors won't lose money, even if bank is insolvent
	
	\
	
	All but eliminates bank runs in the United States
	
	\
	
	But also creates a moral hazard problem---banks can take bigger investment risks
		\begin{itemize}
			\item Financial crisis of 2007
			\its There are \emph{always} second best concerns
			\end{itemize}
		\end{frame}
	
	\begin{frame}{Policy Responses to Coordination Traps}

	Insurance
		\begin{itemize}
			\item Suppose policy maker promises to ``top up'' other player's investment
			\item Never end up having to actually do so
		\end{itemize}


	\

	Communication and Leadership:
	\begin{itemize}
	\item Create a mutual expectation that others will invest
	\end{itemize}

	\

	\phantom{\parbox{\linewidth}{%
	Short-run interventions
		\begin{itemize}
			\item Fundamentally different than externalities because new behavior is also an equilibrium
			\item This tells us about the scope of leadership
			\end{itemize}}}
	\end{frame}
	
	
		
		\begin{frame}{Footbinding}
			
			Footbinding appears in 11th century
			
			\
			
			
			 Spreads gradually from royalty to all of society over 300 years
			
			
			\
			
			  Becomes more extreme over time
			  
			  \
			  
			   Social Norm by Ming Dynasty (1368-1644)

			   \
  			 	
				In 1835, 50-80 percent of women (depending on locale)
		\end{frame}
		
		\begin{frame}{Ending Footbinding}
			
			Foot binding is a coordination trap
				\begin{itemize}
					\item Coordination on bad outcome
				\end{itemize}
				
			\
			
			 Late 19th century societies
			 	 \begin{itemize}
					 \item  Education
					 \item  Public relations
					 \item Advocacy for ``natural feet''
				\end{itemize}
				
			
			\
			
			Tighsien (rural area south of Beijing)
				\begin{itemize}
					\item 1889: 99\%
					\item 1899: 94\%
					\item 1919: 0\%
				\end{itemize}
		\end{frame}

	\begin{frame}{Policy Responses to Coordination Traps}

	Insurance
		\begin{itemize}
			\item Suppose policy maker promises to ``top up'' other player's investment
			\item Never end up having to actually do so
		\end{itemize}


	\


	Communication and Leadership:
	\begin{itemize}
	\item Create a mutual expectation that others will invest
	\end{itemize}

	\

	Short-run interventions
		\begin{itemize}
			\item Fundamentally different than externalities because new behavior is also an equilibrium
			\item This tells us about the scope of leadership
			\end{itemize}
	\end{frame}
	
	\begin{frame}{The Tennessee Valley Authority}
		
		Starting in 1930s, TVA is a massive federal investment to modernize the economy of one of the poorest and least developed areas of the country
			\begin{itemize}
				\item Hydroelectric dams
				\item Canals
				\item Road networks
				\item Schools
			\end{itemize}

		\

		1940s and 1950s, spent over \$14 billion (2000 dollars)
		
		\
		
		1930: Tennessee Valley is almost entirely agricultural
		
		\
		
		1945: Largest supplier of electricity in the country
		\end{frame}
		
	\begin{frame}{Long Run Effects}
		
		TVA subsidies decline starting in 1960s
		
		\
		
		Gains in agricultural sector disappears
			\begin{itemize}
				\item After 1960, TVA counties have a 13-16 percentage point decrease in ten-year agricultural employment growth
				\end{itemize}
			
		\
		
		Manufacturing gains persist
			\begin{itemize}
				\item After 1960, TVA counties continue to have a growth rate that is 3 percentage points higher than non-TVA counties
				\end{itemize}
				
		\
		
		
		In sectors with agglomeration economies (and, so, coordination traps), a short run intervention creates a new equilibrium that persists after the intervention
	\end{frame}
	
	
	\begin{frame}{Summing Up Coordination Traps}
		Multiple equilibria with one Pareto dominated by another
		
		\
		
		Communication and leadership
		
		\
		
		Insurance
		
		\
		
		Short-run rather than long-run policy interventions
	\end{frame}


\section{Coordination Failures}

\begin{frame}{Coordination Failure}
 Primary motive is for individuals to coordinate 

\

 Players have uncertainty about each other

\

 Uncertainty creates a friction when trying to coordinate
\end{frame}

\begin{frame}{Some Examples}

 Social Conventions
\begin{itemize}
\item Private vs public schools
\item Fashion \pause
\end{itemize}

\

 Economic
\begin{itemize}
\item Bank runs
\item Underdevelopment \pause
\end{itemize}

\

Political 
\begin{itemize}
\item Revolutions/Protests
\item Voting
\end{itemize}
\end{frame}

\begin{frame}{A Simple Model of Rebellion}
	
	2 citizens

	\
	
	Each can rebel or not; rebelling costs $c=1$
	
	\
	
	If both rebel the regime is defeated 

	\
	
	Each citizen gets 5 if regime falls and 0 otherwise
	
\end{frame}
\begin{frame}{Equilibrium}
	
			\begin{center}
			{\setlength{\extrarowheight}{10pt}%
	\begin{tabular}{*4c} 
		& &\multicolumn{2}{c}{Citizen 2}\\
		&  										&\only<1,3-|handout:1>{$R$} \only<2|handout:0>{\cyan $R$} 	&\only<1-2,4-|handout:1>{$NR$} \only<3|handout:0>{\cyan $NR$} \\ \cline{3-4}
		\multirow{2}{*}{Citizen 1}	
				\only<1|handout:0>{	& $r$ 			& \multicolumn{1}{|c|}{$4,4$} 				& \multicolumn{1}{c|}{$-1,0$} \\[10pt] \cline{3-4} 
							& $nr$ 			& \multicolumn{1}{|c|}{$0,-1$}				&\multicolumn{1}{c|}{$0,0$}\\[10pt] \cline{3-4}}
			
				\only<2|handout:0>{	& $r$ 			& \multicolumn{1}{|c|}{\cyan$4\ch,4$} 		& \multicolumn{1}{c|}{$-1,0$} \\[10pt] \cline{3-4} 
							& $nr$ 			& \multicolumn{1}{|c|}{\cyan $0,-1$}			&\multicolumn{1}{c|}{$0,0$}\\[10pt] \cline{3-4}}

				\only<3|handout:0>{	& $r$ 			& \multicolumn{1}{|c|}{$4\ch,4$} 			& \multicolumn{1}{c|}{\cyan $-1,0$} \\[10pt] \cline{3-4} 
							& $nr$ 			& \multicolumn{1}{|c|}{$0,-1$}				&\multicolumn{1}{c|}{\cyan $0\ch,0$}\\[10pt] \cline{3-4}}

				\only<4|handout:0>{	& \cyan $r$ 			& \multicolumn{1}{|c|}{\cyan $4\ch,4\ch$} 	& \multicolumn{1}{c|}{\cyan $-1,0$} \\[10pt] \cline{3-4} 
							& $nr$ 			& \multicolumn{1}{|c|}{$0,-1$}				&\multicolumn{1}{c|}{$0\ch,0$}\\[10pt] \cline{3-4}}

				\only<5|handout:0>{	& $r$ 			& \multicolumn{1}{|c|}{$4\ch,4\ch$} 		& \multicolumn{1}{c|}{$-1,0$} \\[10pt] \cline{3-4} 
							& \cyan $nr$ 			& \multicolumn{1}{|c|}{\cyan $0,-1$}			&\multicolumn{1}{c|}{\cyan $0\ch,0\ch$}\\[10pt] \cline{3-4}}

				\only<6-|handout:1>{	& $r$ 			& \multicolumn{1}{|c|}{\orange $4\ch,4\ch$}	& \multicolumn{1}{c|}{$-1,0$} \\[10pt] \cline{3-4} 
							& $nr$ 			& \multicolumn{1}{|c|}{$0,-1$}				&\multicolumn{1}{c|}{$\orange 0\ch,0\ch$}\\[10pt] \cline{3-4}}
			
		\end{tabular}}
		\end{center}

\

\only<7-|handout:1>{
There's a coordination trap here as well}
\end{frame}

\begin{frame}{Introducing some uncertainty}
 Suppose citizens don't know each other that well 
 
 \
 
 Two types of citizens:
\begin{enumerate}
\item Low cost: $c_{L} =1$ with probability $p$
\item High cost: $c_{H} = 6$ with probability $1-p$
\end{enumerate}

\

 Citizens know their own cost but not the other citizens

\

 Citizen strategies depend on their cost

\end{frame}

\begin{frame}{High cost best response}
	
 For a high cost citizen rebelling is dominated
 
 \
 
 So high cost types will never rebel
 \end{frame}
 
 \begin{frame}{Low cost best response}
 
Suppose a low cost player believes that both high and low cost players will not rebel
	 \begin{itemize}
		 \item Her best response is to not rebel
		 \its There is an equilibrium where all types do not rebel
	\end{itemize}
		 
\

Suppose a low cost player believes high cost players will not rebel but low cost players will rebel?

\

Expected utility of rebelling is 
\begin{equation*}
4 p + (-1)(1-p) 
\end{equation*}

\

\pause
It is a best response to rebel if this is greater than 0
\[  p \geq \frac{1}{5}\]

\

In this event, the game has two equilibria
	\begin{itemize}
		\item No one rebels
		\its High cost types do not rebel and low cost types do rebel
	\end{itemize}
\end{frame}

\begin{frame}{Equilibria}
	
	If $p < \frac{1}{5}$ all players don't rebel
	
	\
	
	If $p  \geq \frac{1}{5}$ there are two equilibria
		\begin{itemize}
			\item All players don't rebel
			\its High cost players don't rebel, low cost players do rebel
		\end{itemize}

	\
	
	This latter equilibrium creates the possibility of coordination failure 
		\begin{itemize}
			\item One player rebels, the other player does not
		\end{itemize}
\end{frame}
\begin{frame}{Coordination Failure}
	
	How often does a coordination failure occur? 
	
	\
	
	If no one rebels, there is never coordination failure
		\begin{itemize}
			\item But with probability $p^2$ there are two low types in a coordination trap
		\end{itemize}
	
	\
	
	If high types do not rebel and low types do rebel, there is coordination failure when there is one of each type
		 \begin{itemize}
			 \item This happens with probability $2p(1-p)$
		 \end{itemize}

\end{frame}


\begin{frame}{Policy Interventions}
Same ideas as coordination traps, but perhaps a bit trickier

\

 Individuals need to know things about each other to reassure risky actions
 
 \
 
Leadership may be insufficient because mutual expectations are harder to establish
\end{frame}

\begin{frame}{Take-Aways}
Coordination Traps
\begin{itemize}
\item Reinforcing expectations can lead to bad outcomes
\its One-shot policy interventions can get things on track
\end{itemize}

\

 Coordination Failures
\begin{itemize}
\item Result from strategic uncertainty about others
\its Different types of policy interventions are needed
\end{itemize}
\end{frame}



\end{document}
