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



\usepackage{multirow}
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\usepackage{xcolor, colortbl}
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\title{Game Theory 2:\\
Extensive-Form Games and Subgame Perfection}


\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}{Dynamics in Games}
How should we think of strategic interactions that occur in sequence? 

\

 Who moves when? 

\

 And what can they do at different points in time?

\

 How do people react to different histories? 
\end{frame}

\begin{frame}{Modeling Games with Dynamics}
Players

\

 Player function
 	 \begin{itemize}
		 \item Who moves when  
		 \end{itemize}

\

 Terminal histories
 	 \begin{itemize}
		 \item Possible paths through the game
		 \end{itemize}

\

 Preferences over terminal histories
\end{frame}

\begin{frame}{Strategies}

 A strategy is a complete contingent plan

\

Player $i$'s strategy specifies her action choice at each point at which she could be called on to make a choice

\end{frame}

\begin{frame}{An Example: International Crises}

Two countries~($A$ and $B$) are competing over a piece of land that $B$ occupies

\

 Country $A$ decides whether to make a demand

\

If Country $A$ makes a demand, $B$ can either acquiesce or fight a war

\

 If $A$ does not make a demand, $B$ keeps land~(game ends)

\

 $A$'s best outcome is Demand followed by Acquiesce, worst outcome is Demand and War
 
 \
 
 $B$'s best outcome is No Demand and worst outcome is Demand and War
\end{frame}

\begin{frame}{An Example: International Crises}

$A$ can choose: Demand~($D$) or No Demand~($ND$)

\

 $B$ can choose: Fight a war~($W$) or Acquiesce~($A$)

\

 Preferences
\[ u_A(D,A) = 3 >   u_A(ND,A) = u_A(ND,W) = 2 > u_A(D,W) = 1\]


\[ u_{B}(ND, A) = u_{B}(ND, W) = 3 > u_{B}(D, A) = 2 >  u_{B}(D, W) = 1\]

\

 How can we represent this scenario as a game~(in strategic form)? 
\end{frame}


\begin{frame}{International Crisis Game: NE}
	

			\begin{center}
			{\setlength{\extrarowheight}{10pt}%
	\begin{tabular}{*4c} 
		& &\multicolumn{2}{c}{Country $B$}\\
		&  										&\only<1,3-|handout:1>{$W$} \only<2|handout:0>{\cyan $W$} 	&\only<1-2,4-|handout:1>{$A$} \only<3|handout:0>{\cyan $A$} \\ \cline{3-4}
		\multirow{2}{*}{Country $A$}	
				\only<1|handout:0>{	& $D$ 			& \multicolumn{1}{|c|}{$1,1 \text{ }$} 				& \multicolumn{1}{c|}{$3,2$} \\[10pt] \cline{3-4} 
							& $ND$ 			& \multicolumn{1}{|c|}{$2,3$}				&\multicolumn{1}{c|}{$2,3$}\\[10pt] \cline{3-4}}
				
				\only<2|handout:0>{	& $D$ 			& \multicolumn{1}{|c|}{\cyan$1,1  \text{ }$} 		& \multicolumn{1}{c|}{$3,2$} \\[10pt] \cline{3-4} 
							& $ND$ 			& \multicolumn{1}{|c|}{\cyan $2\ch,3$}			&\multicolumn{1}{c|}{$2,3$}\\[10pt] \cline{3-4}}

				\only<3|handout:0>{	& $D$ 			& \multicolumn{1}{|c|}{$1,1  \text{ }$} 			& \multicolumn{1}{c|}{\cyan $3\ch,2$} \\[10pt] \cline{3-4} 
							& $ND$ 			& \multicolumn{1}{|c|}{$2\ch,3$}				&\multicolumn{1}{c|}{\cyan $2,3$}\\[10pt] \cline{3-4}}

				\only<4|handout:0>{	& $\cyan D$ 			& \multicolumn{1}{|c|}{\cyan $1,1  \text{ }$} 	& \multicolumn{1}{c|}{\cyan $3\ch ,2\ch$} \\[10pt] \cline{3-4} 
							& $ND$ 			& \multicolumn{1}{|c|}{$2\ch ,3$}				&\multicolumn{1}{c|}{$2,3$}\\[10pt] \cline{3-4}}

				\only<5|handout:0>{	& $D$ 			& \multicolumn{1}{|c|}{$1,1  \text{ }$} 		& \multicolumn{1}{c|}{$3\ch,2\ch$} \\[10pt] \cline{3-4} 
							& $\cyan ND$ 			& \multicolumn{1}{|c|}{\cyan $2\ch,3\ch$}			&\multicolumn{1}{c|}{\cyan $2,3\ch$}\\[10pt] \cline{3-4}}

				\only<6-|handout:1>{	& $D$ 			& \multicolumn{1}{|c|}{ $1,1  \text{ }$}	& \multicolumn{1}{c|}{\orange $3\ch,2\ch$} \\[10pt] \cline{3-4} 
							& $ND$ 			& \multicolumn{1}{|c|}{\orange$2\ch,3\ch$}				&\multicolumn{1}{c|}{$2,3\ch$}\\[10pt] \cline{3-4}}

				
		\end{tabular}}
		\end{center}
\only<7|handout:1>{
\begin{itemize}
\item Is there something funny here? 
\end{itemize}}

\only<8|handout:1>{
\begin{itemize}
\item Is there something funny here? 
\its Specifically, $(ND,W)$? 
\end{itemize}}

\only<9|handout:1>{
\begin{itemize}
\item Is there something funny here? 
\its Specifically, $(ND,W)$? 
\its The threat of war deters the demand, but would $B$ follow through?
\end{itemize}}

\end{frame}

\begin{frame}{Non-Credible Threats}

The equilibrium $(ND, W)$ depends on a ``non-credible threat'' 

\

 Once $A$ makes a demand, $B$ does not want to fight a war 

\

But to rule out such behavior, we need a stronger solution concept


\

 One that incorporates the fact that actions are taken in sequence
\end{frame}

\begin{frame}{Why Rule out Non-credible Threats}
 Equilibrium as a steady state 
 
 \
 
  War is only a best-response for $B$ because when no demand is made, $B$ is indifferent 
  
  \
  
   If $A$ accidentally made a demand, war is not a sequential best-response for $B$. $B$ should acquiesce instead
   		\begin{itemize}
			\item Read the strategy $W$ as ``if $A$ makes a demand, I will go to war''
			\end{itemize}
\end{frame}

\begin{frame}{Subgame Perfect Nash Equilibrium}
 A strategy specifies what a player will do at every decision point
\begin{itemize}
\item Complete contingent plan 
\end{itemize}

\

 Strategy in a SPNE must be a best-response at each node, given the strategies of other players 

\

 Backward Induction 
\end{frame}

\begin{frame}{But First!}

 Let's introduce a way of incorporating the timing of actions into the game explicitly 
 
 \
 
Use a game tree to represent the sequential aspect of choices


\end{frame}

\begin{frame}{An Example: International Crises}
	
	\only<1|handout:0>{
	\begin{center}
	\setlength{\unitlength}{0.7in}
	\begin{picture}(5,5)

	\thicklines

	%% choices
	\put(2,4){\vector(-1,-1){1}}
	\put(2,4){\vector(1,-1){1}}
	%\put(3,3){\vector(1,-1){1}}
	%\put(3,3){\vector(-1,-1){1}}
	%\put(4,2){\vector(1,-1){1}}
	%\put(4,2){\vector(-1,-1){1}}

	%% nodes
	\put(1.85,4.1){$\fbox{A}$}
	%\put(3,3.2){$B$}
	%\put(4,2.2){$1$}

	%% utilities
	%\put(3,.8){$(0,1)$}
	%\put(2,1.8){$(1,1)$}
	\put(1,2.8){$(2,3)$}
	%\put(4,1.8){$(3,2)$}

	%% strategies
	%\put(3.2,1.5){$L'$}
	%\put(4.6,1.5){$R'$}
	%\put(2,2.5){$War$}
	%\put(3.6,2.5){$Acq$}
	\put(1,3.5){$ND$}
	\put(2.6,3.5){$D$}
	\end{picture}
	\end{center}}

		\only<2|handout:0>{
		\begin{center}
		\setlength{\unitlength}{0.7in}
		\begin{picture}(5,5)

		\thicklines

		%% choices
		\put(2,4){\vector(-1,-1){1}}
		\put(2,4){\vector(1,-1){1}}
		\put(3,3){\vector(1,-1){1}}
		\put(3,3){\vector(-1,-1){1}}
		%\put(4,2){\vector(1,-1){1}}
		%\put(4,2){\vector(-1,-1){1}}

		%% nodes
		\put(1.85,4.1){$\fbox{A}$}
		\put(3,3.1){$\fbox{B}$}
		%\put(4,2.2){$1$}

		%% utilities
		%\put(3,.8){$(0,1)$}
		%\put(2,1.8){$(1,1)$}
		\put(1,2.8){$(2,3)$}
		%\put(4,1.8){$(3,2)$}

		%% strategies
		%\put(3.2,1.5){$L'$}
		%\put(4.6,1.5){$R'$}
		\put(2,2.5){$War$}
		\put(3.6,2.5){$Acq$}
		\put(1,3.5){$ND$}
		\put(2.6,3.5){$D$}
		\end{picture}
		\end{center}}

			\only<3|handout:0>{
			 \begin{center}
			\setlength{\unitlength}{0.7in}
			\begin{picture}(5,5)

			\thicklines

			%% choices
			\put(2,4){\vector(-1,-1){1}}
			\put(2,4){\vector(1,-1){1}}
			\put(3,3){\vector(1,-1){1}}
			\put(3,3){\vector(-1,-1){1}}
			%\put(4,2){\vector(1,-1){1}}
			%\put(4,2){\vector(-1,-1){1}}

			%% nodes
			\put(1.85,4.1){$\fbox{A}$}
			\put(3,3.1){$\fbox{B}$}
			%\put(4,2.2){$1$}

			%% utilities
			%\put(3,.8){$(0,1)$}
			\put(2,1.8){$(1,1)$}
			\put(1,2.8){$(2,3)$}
			\put(4,1.8){$(3,2)$}

			%% strategies
			%\put(3.2,1.5){$L'$}
			%\put(4.6,1.5){$R'$}
			\put(2,2.5){$War$}
			\put(3.6,2.5){$Acq$}
			\put(1,3.5){$ND$}
			\put(2.6,3.5){$D$}
			\end{picture}
			\end{center}}


				\only<4|handout:0>{
				\begin{center}
				\setlength{\unitlength}{0.7in}
				\begin{picture}(5,5)

				\thicklines

				%% choices
				\put(2,4){\vector(-1,-1){1}}
				\put(2,4){\vector(1,-1){1}}
				\G
				\put(3,3){\vector(1,-1){1}}
				\B
				\put(3,3){\vector(-1,-1){1}}
				%\put(4,2){\vector(1,-1){1}}
				%\put(4,2){\vector(-1,-1){1}}

				%% nodes
				\put(1.85,4.1){$\fbox{A}$}
				\put(3,3.1){$\fbox{B}$}
				%\put(4,2.2){$1$}

				%% utilities
				%\put(3,.8){$(0,1)$}
				\put(2,1.8){$(1,1)$}
				\put(1,2.8){$(2,3)$}
				\put(4,1.8){$(3,2)$}

				%% strategies
				%\put(3.2,1.5){$L'$}
				%\put(4.6,1.5){$R'$}
				\put(2,2.5){$War$}
				\G
				\put(3.6,2.5){$Acq$}
				\B
				\put(1,3.5){$ND$}
				\put(2.6,3.5){$D$}
				\end{picture}
				\end{center}}

					\only<5|handout:0>{
					\begin{center}
					\setlength{\unitlength}{0.7in}
					\begin{picture}(5,5)

					\thicklines

					%% choices
					\put(2,4){\vector(-1,-1){1}}
					\put(2,4){\vector(1,-1){1}}
					%\put(3,3){\vector(1,-1){1}}
					%\put(3,3){\vector(-1,-1){1}}
					%\put(4,2){\vector(1,-1){1}}
					%\put(4,2){\vector(-1,-1){1}}

					%% nodes
					\put(1.85,4.1){$\fbox{A}$}

					%\put(3,3.2){$B$}
					%\put(4,2.2){$1$}

					%% utilities
					%\put(3,.8){$(0,1)$}
					%\put(2,1.8){$(1,1)$}
					\put(1,2.8){$(2,3)$}
					\orange
					\put(2.8,2.8){$(3,2)$}
					\cyan

					%% strategies
					%\put(3.2,1.5){$L'$}
					%\put(4.6,1.5){$R'$}
					%\put(2,2.5){$War$}
					%\put(3.6,2.5){$Acq$}
					\put(1,3.5){$ND$}
					\put(2.6,3.5){$D$}
					\end{picture}
					\end{center}}
					
					
						\only<6|handout:0>{
						\begin{center}
						\setlength{\unitlength}{0.7in}
						\begin{picture}(5,5)

						\thicklines

						%% choices
						\put(2,4){\vector(-1,-1){1}}
						\G
						\put(2,4){\vector(1,-1){1}}
						\B
						%\put(3,3){\vector(1,-1){1}}
						%\put(3,3){\vector(-1,-1){1}}
						%\put(4,2){\vector(1,-1){1}}
						%\put(4,2){\vector(-1,-1){1}}

						%% nodes
						\put(1.85,4.1){$\fbox{A}$}
						%\put(3,3.2){$B$}
						%\put(4,2.2){$1$}

						%% utilities
						%\put(3,.8){$(0,1)$}
						%\put(2,1.8){$(1,1)$}
						\put(1,2.8){$(2,3)$}
						\G
						\put(2.8,2.8){$(3,2)$}
						\B

						%% strategies
						%\put(3.2,1.5){$L'$}
						%\put(4.6,1.5){$R'$}
						%\put(2,2.5){$War$}
						%\put(3.6,2.5){$Acq$}
						\put(1,3.5){$ND$}
						\put(2.6,3.5){$D$}
						\end{picture}
						\end{center}}
						

							\only<7|handout:1>{
							\begin{center}
							\setlength{\unitlength}{0.7in}
							\begin{picture}(5,5)

							\thicklines

							%% choices
							\put(2,4){\vector(-1,-1){1}}
							\G
							\put(2,4){\vector(1,-1){1}}
							\put(3,3){\vector(1,-1){1}}
							\B
							\put(3,3){\vector(-1,-1){1}}

							%\put(4,2){\vector(1,-1){1}}
							%\put(4,2){\vector(-1,-1){1}}

							%% nodes
							\put(1.85,4.1){$\fbox{A}$}
							\put(3,3.1){$\fbox{B}$}
							%\put(4,2.2){$1$}

							%% utilities
							%\put(3,.8){$(0,1)$}
							\put(2,1.8){$(1,1)$}
							\put(1,2.8){$(2,3)$}
							\put(4,1.8){$(3,2)$}


							%% strategies
							%\put(3.2,1.5){$L'$}
							%\put(4.6,1.5){$R'$}
							\put(2,2.5){$War$}
							\G
							\put(3.6,2.5){$Acq$}
							\B
							\put(1,3.5){$ND$}
							\G
							\put(2.6,3.5){$D$}
							\B
							\end{picture}
							\end{center}}

\end{frame}

\begin{frame}{Another Example}

			\begin{center}
			{\setlength{\extrarowheight}{10pt}%
	\begin{tabular}{*4c} 
		& &\multicolumn{2}{c}{Player 2}\\
		&  										&$C$  	&$D$ \\ \cline{3-4}
		\multirow{2}{*}{Player 1}	
				\only<1|handout:0>{	& $A$ 			& \multicolumn{1}{|c|}{$5,3$} 				& \multicolumn{1}{c|}{$0,0$} \\[10pt] \cline{3-4} 
							& $B$ 			& \multicolumn{1}{|c|}{$0,0$}				&\multicolumn{1}{c|}{$3,5$}\\[10pt] \cline{3-4}}
				
			\only<2-|handout:1>{	& $A$ 			& \multicolumn{1}{|c|}{\orange $5\ch,3\ch$} 				& \multicolumn{1}{c|}{$0,0$} \\[10pt] \cline{3-4} 
						& $B$ 			& \multicolumn{1}{|c|}{$0,0$}				&\multicolumn{1}{c|}{\orange$3\ch,5\ch$}\\[10pt] \cline{3-4}}
				
		\end{tabular}}
		
		\end{center}

\only<3|handout:1>{Represent this as a game tree under the assumption that player 1 moves first}
\end{frame}

\begin{frame}{Another Example}

	\only<1|handout:0>{
	\begin{center}
	\setlength{\unitlength}{0.7in}
	\begin{picture}(6,2.7)

	\thicklines

	%%1's choices
	\put(3,2.7){\vector(1,-1){1}}
	\put(3,2.7){\vector(-1,-1){1}}
	\put(2.25,2.2){$A$}
	\put(3.6,2.2){$B$}

	%%2's info set and choices
	%\multiput(2.02,1.7)(0.1,0){20}{\_}
	\put(4,1.7){\vector(1,-2){0.5}}
	\put(4,1.7){\vector(-1,-2){0.5}}
	\put(4.3,1.2){$D$}
	\put(3.5,1.2){$C$}

	%%payoffs after L,l  L,r  R,l  R,r
	\put(1.4,0.3){\shortstack{ 5 \\ 3}}
	\put(2.5,0.3){\shortstack{ 0 \\ 0}}
	\put(3.4,0.3){\shortstack{ 0 \\ 0}}
	\put(4.5,0.3){\shortstack{ 3 \\ 5}}


	\put(2,1.7){\vector(-1,-2){0.5}}
	\put(2,1.7){\vector(1,-2){0.5}}
	\put(1.5,1.2){$C$}
	\put(2.3,1.2){$D$}
                         
	\put(4,1.8){\fbox{2}}
	\put(1.75,1.8){\fbox{2}}
	\put(2.9,2.8){\fbox{1}}

	\end{picture}
	\end{center}}
  
  
		\only<2|handout:0>{ 
		\begin{center}
		\setlength{\unitlength}{0.7in}
		\begin{picture}(6,2.7)

		\thicklines

		%%1's choices
		\put(3,2.7){\vector(1,-1){1}}
		\put(3,2.7){\vector(-1,-1){1}}
		\put(2.25,2.2){$A$}
		\put(3.6,2.2){$B$}

		%%2's info set and choices
		%\multiput(2.02,1.7)(0.1,0){20}{\_}
		\put(4,1.7){\vector(1,-2){0.5}}
		\put(4,1.7){\vector(-1,-2){0.5}}
		\put(4.3,1.2){$D$}
		\put(3.5,1.2){$C$}

		%%payoffs after L,l  L,r  R,l  R,r
		\put(1.4,0.3){\shortstack{ 5 \\ 3}}
		\put(2.5,0.3){\shortstack{ 0 \\ 0}}
		\put(3.4,0.3){\shortstack{ 0 \\ 0}}
		\put(4.5,0.3){\shortstack{ 3 \\ 5}}

		\G
		\put(2,1.7){\vector(-1,-2){0.5}}
		\B
		\put(2,1.7){\vector(1,-2){0.5}}
		\G
		\put(1.5,1.2){$C$}
		\B
		\put(2.3,1.2){$D$}
                         
		\put(4,1.8){\fbox{2}}
		\put(1.75,1.8){\fbox{2}}
		\put(2.9,2.8){\fbox{1}}
		\end{picture}
		\end{center}}
 
			\only<3|handout:0>{ 
			\begin{center}
			\setlength{\unitlength}{0.7in}
			\begin{picture}(6,2.7)

			\thicklines

			%%1's choices
			\put(3,2.7){\vector(1,-1){1}}
			\put(3,2.7){\vector(-1,-1){1}}
			\put(2.25,2.2){$A$}
			\put(3.6,2.2){$B$}

			%%2's info set and choices
			%\multiput(2.02,1.7)(0.1,0){20}{\_}
			\G
			\put(4,1.7){\vector(1,-2){0.5}}
			\B
			\put(4,1.7){\vector(-1,-2){0.5}}
			\G
			\put(4.3,1.2){$D$}
			\B
			\put(3.5,1.2){$C$}

			%%payoffs after L,l  L,r  R,l  R,r
			\put(1.4,0.3){\shortstack{ 5 \\ 3}}
			\put(2.5,0.3){\shortstack{ 0 \\ 0}}
			\put(3.4,0.3){\shortstack{ 0 \\ 0}}
			\put(4.5,0.3){\shortstack{ 3 \\ 5}}

			\G
			\put(2,1.7){\vector(-1,-2){0.5}}
			\B
			\put(2,1.7){\vector(1,-2){0.5}}
			\G
			\put(1.5,1.2){$C$}
			\B
			\put(2.3,1.2){$D$}

                         
			\put(4,1.8){\fbox{2}}
			\put(1.75,1.8){\fbox{2}}
			\put(2.9,2.8){\fbox{1}}
			\end{picture}
			\end{center}}
 
				\only<4-|handout:1>{
				\begin{center}
				\setlength{\unitlength}{0.7in}
				\begin{picture}(6,2.7)

				\thicklines

				%%1's choices
				\put(3,2.7){\vector(1,-1){1}}
				\G
				\put(3,2.7){\vector(-1,-1){1}}
				\put(2.25,2.2){$A$}
				\B
				\put(3.6,2.2){$B$}

				%%2's info set and choices
				%\multiput(2.02,1.7)(0.1,0){20}{\_}
				\color{red}
				\put(4,1.7){\vector(1,-2){0.5}}
				\color{black}
				\put(4,1.7){\vector(-1,-2){0.5}}
				\G
				\put(4.3,1.2){$D$}
				\B
				\put(3.5,1.2){$C$}


				%%payoffs after L,l  L,r  R,l  R,r
				\put(1.4,0.3){\shortstack{ 5 \\ 3}}
				\put(2.5,0.3){\shortstack{ 0 \\ 0}}
				\put(3.4,0.3){\shortstack{ 0 \\ 0}}
				\put(4.5,0.3){\shortstack{ 3 \\ 5}}

				\color{red}
				\put(2,1.7){\vector(-1,-2){0.5}}
				\color{black}
				\put(2,1.7){\vector(1,-2){0.5}}
				\G
				\put(1.5,1.2){$C$}
				\B
				\put(2.3,1.2){$D$}
                         
				\put(4,1.8){\fbox{2}}
				\put(1.75,1.8){\fbox{2}}
				\put(2.9,2.8){\fbox{1}}

				\end{picture}
				\end{center}}

					\only<5|handout:1>{						
						SPNE: $(A, (C, D))$
                     } 
\end{frame}

\begin{frame}{What if Player 2 Moves First?}

	\only<1|handout:0>
	{\begin{center}
	\setlength{\unitlength}{0.7in}
	\begin{picture}(6,2.7)

	\thicklines

	%%1's choices
	\put(3,2.7){\vector(1,-1){1}}
	\put(3,2.7){\vector(-1,-1){1}}
	\put(2.25,2.2){$C$}
	\put(3.6,2.2){$D$}

	%%2's info set and choices
	%\multiput(2.02,1.7)(0.1,0){20}{\_}
	\put(4,1.7){\vector(1,-2){0.5}}
	\put(4,1.7){\vector(-1,-2){0.5}}
	\put(4.3,1.2){$B$}
	\put(3.5,1.2){$A$}

	%%payoffs after L,l  L,r  R,l  R,r
	\put(1.4,0.3){\shortstack{ 5 \\ 3}}
	\put(2.5,0.3){\shortstack{ 0 \\ 0}}
	\put(3.4,0.3){\shortstack{ 0 \\ 0}}
	\put(4.5,0.3){\shortstack{ 3 \\ 5}}


	\put(2,1.7){\vector(-1,-2){0.5}}
	\put(2,1.7){\vector(1,-2){0.5}}
	\put(1.5,1.2){$A$}
	\put(2.3,1.2){$B$}
                         
	\put(4,1.8){\fbox{1}}
	\put(1.75,1.8){\fbox{1}}
	\put(2.9,2.8){\fbox{2}}

	\end{picture}
	\end{center}
	               }       

		\only<2|handout:0>
		{\begin{center}
		\setlength{\unitlength}{0.7in}
		\begin{picture}(6,2.7)

		\thicklines

		%%1's choices
		\put(3,2.7){\vector(1,-1){1}}
		\put(3,2.7){\vector(-1,-1){1}}
		\put(2.25,2.2){$C$}
		\put(3.6,2.2){$D$}

		%%2's info set and choices
		%\multiput(2.02,1.7)(0.1,0){20}{\_}
		\put(4,1.7){\vector(1,-2){0.5}}
		\put(4,1.7){\vector(-1,-2){0.5}}
		\put(4.3,1.2){$B$}
		\put(3.5,1.2){$A$}

		%%payoffs after L,l  L,r  R,l  R,r
		\put(1.4,0.3){\shortstack{ 5 \\ 3}}
		\put(2.5,0.3){\shortstack{ 0 \\ 0}}
		\put(3.4,0.3){\shortstack{ 0 \\ 0}}
		\put(4.5,0.3){\shortstack{ 3 \\ 5}}

		\G
		\put(2,1.7){\vector(-1,-2){0.5}}
		\B
		\put(2,1.7){\vector(1,-2){0.5}}
		\G
		\put(1.5,1.2){$A$}
		\B
		\put(2.3,1.2){$B$}
                         
		\put(4,1.8){\fbox{1}}
		\put(1.75,1.8){\fbox{1}}
		\put(2.9,2.8){\fbox{2}}

		\end{picture}
		\end{center}}
 
\only<3|handout:0>
			{\begin{center}
			\setlength{\unitlength}{0.7in}
			\begin{picture}(6,2.7)

			\thicklines

			%%1's choices
			\put(3,2.7){\vector(1,-1){1}}
			\put(3,2.7){\vector(-1,-1){1}}
			\put(2.25,2.2){$C$}
			\put(3.6,2.2){$D$}

			%%2's info set and choices
			%\multiput(2.02,1.7)(0.1,0){20}{\_}
			\G
			\put(4,1.7){\vector(1,-2){0.5}}
			\B
			\put(4,1.7){\vector(-1,-2){0.5}}
			\G
			\put(4.3,1.2){$B$}
			\B
			\put(3.5,1.2){$A$}

			%%payoffs after L,l  L,r  R,l  R,r
			\put(1.4,0.3){\shortstack{ 5 \\ 3}}
			\put(2.5,0.3){\shortstack{ 0 \\ 0}}
			\put(3.4,0.3){\shortstack{ 0 \\ 0}}
			\put(4.5,0.3){\shortstack{ 3 \\ 5}}

			\G
			\put(2,1.7){\vector(-1,-2){0.5}}
			\B
			\put(2,1.7){\vector(1,-2){0.5}}
			\G
			\put(1.5,1.2){$A$}
			\B
			\put(2.3,1.2){$B$}
                         
			\put(4,1.8){\fbox{1}}
			\put(1.75,1.8){\fbox{1}}
			\put(2.9,2.8){\fbox{2}}

			\end{picture}
			\end{center}}
    
				\only<4-|handout:1>
				{\begin{center}
				\setlength{\unitlength}{0.7in}
				\begin{picture}(6,2.7)

				\thicklines

				%%1's choices
				\G
				\put(3,2.7){\vector(1,-1){1}}
				\B
				\put(3,2.7){\vector(-1,-1){1}}
				\put(2.25,2.2){$C$}
				\G
				\put(3.6,2.2){$D$}
				\B

				%%2's info set and choices
				%\multiput(2.02,1.7)(0.1,0){20}{\_}
				\G
				\put(4,1.7){\vector(1,-2){0.5}}
				\B
				\put(4,1.7){\vector(-1,-2){0.5}}
				\G
				\put(4.3,1.2){$B$}
				\B
				\put(3.5,1.2){$A$}

				%%payoffs after L,l  L,r  R,l  R,r
				\put(1.4,0.3){\shortstack{ 5 \\ 3}}
				\put(2.5,0.3){\shortstack{ 0 \\ 0}}
				\put(3.4,0.3){\shortstack{ 0 \\ 0}}
				\put(4.5,0.3){\shortstack{ 3 \\ 5}}

				\G
				\put(2,1.7){\vector(-1,-2){0.5}}
				\B
				\put(2,1.7){\vector(1,-2){0.5}}
				\G
				\put(1.5,1.2){$A$}
				\B
				\put(2.3,1.2){$B$}
                         
				\put(4,1.8){\fbox{1}}
				\put(1.75,1.8){\fbox{1}}
				\put(2.9,2.8){\fbox{2}}

				\end{picture}
				\end{center}}

						\only<5|handout:1>{SPNE: $((A, B), D)$}
                      
\end{frame}

\begin{frame}{The Order of Action Matters}
The outcome depends on who chooses first
\begin{itemize}
\item When 1 moves first: $(A,C)$ with payoffs $5,3$
\its When 2 moves first: $(D,B)$ with payoffs $3,5$
\end{itemize}

\

 In this game, there is an advantage to moving first
\end{frame}

\begin{frame}{The Centipede Game}

	\only<1|handout:0>{
	\begin{center}
	\setlength{\unitlength}{0.7in}
	\begin{picture}(6,2.7)

	\thicklines

	%% choices
	\put(1.2,2){\vector(0,-1){1}}
	\put(1.2,2){\vector(1,0){1}}
	\put(2.2,2){\vector(0,-1){1}}
	\put(2.2,2){\vector(1,0){1}}
	\put(3.2,2){\vector(0,-1){1}}
	\put(3.2,2){\vector(1,0){1}}
	\put(4.2,2){\vector(0,-1){1}}
	\put(4.2,2){\vector(1,0){1}}


	%Player labels
	\put(1,2.2){$\fbox{1}$}
	\put(2,2.2){$\fbox{2}$}
	\put(3,2.2){$\fbox{1}$}
	\put(4,2.2){$\fbox{2}$}


	%Strategies
	\put(1,1.5){$E$}
	\put(1.5,2.1){$C$}
	\put(2,1.5){$E$}
	\put(2.5,2.1){$C$}
	\put(3,1.5){$E$}
	\put(3.5,2.1){$C$}
	\put(4,1.5){$E$}
	\put(4.5,2.1){$C$}


	%Payoffs
	\put(5.3,1.8){\shortstack{ $6$ \\ $6$}}
	\put(1.15,.5){\shortstack{ $1$ \\ $1$}}
	\put(2.15,.5){\shortstack{ $0$ \\ $3$}}
	\put(3.15,.5){\shortstack{ $5$ \\ $2$}}
	\put(4.15,.5){\shortstack{ $4$ \\ $7$}}


	\end{picture}
	\end{center}
	}

		\only<2|handout:0>{
		\begin{center}
		\setlength{\unitlength}{0.7in}
		\begin{picture}(6,2.7)

		\thicklines

		%% choices
		\put(1.2,2){\vector(0,-1){1}}
		\put(1.2,2){\vector(1,0){1}}
		\put(2.2,2){\vector(0,-1){1}}
		\put(2.2,2){\vector(1,0){1}}
		\put(3.2,2){\vector(0,-1){1}}
		\put(3.2,2){\vector(1,0){1}}
		\G
		\put(4.2,2){\vector(0,-1){1}}
		\B
		\put(4.2,2){\vector(1,0){1}}


		%Player labels
		\put(1,2.2){$\fbox{1}$}
		\put(2,2.2){$\fbox{2}$}
		\put(3,2.2){$\fbox{1}$}
		\put(4,2.2){$\fbox{2}$}


		%Strategies
		\put(1,1.5){$E$}
		\put(1.5,2.1){$C$}
		\put(2,1.5){$E$}
		\put(2.5,2.1){$C$}
		\put(3,1.5){$E$}
		\put(3.5,2.1){$C$}
		\G
		\put(4,1.5){$E$}
		\B
		\put(4.5,2.1){$C$}


		%Payoffs
		\put(5.3,1.8){\shortstack{ $6$ \\ $6$}}
		\put(1.15,.5){\shortstack{ $1$ \\ $1$}}
		\put(2.15,.5){\shortstack{ $0$ \\ $3$}}
		\put(3.15,.5){\shortstack{ $5$ \\ $2$}}
		\put(4.15,.5){\shortstack{ $4$ \\ $7$}}


		\end{picture}
		\end{center}}

			\only<3|handout:0>{
			\begin{center}
			\setlength{\unitlength}{0.7in}
			\begin{picture}(6,2.7)

			\thicklines

			%% choices
			\put(1.2,2){\vector(0,-1){1}}
			\put(1.2,2){\vector(1,0){1}}
			\put(2.2,2){\vector(0,-1){1}}
			\put(2.2,2){\vector(1,0){1}}
			\G
			\put(3.2,2){\vector(0,-1){1}}
			\B
			\put(3.2,2){\vector(1,0){1}}
			\G
			\put(4.2,2){\vector(0,-1){1}}
			\B
			\put(4.2,2){\vector(1,0){1}}


			%Player labels
			\put(1,2.2){$\fbox{1}$}
			\put(2,2.2){$\fbox{2}$}
			\put(3,2.2){$\fbox{1}$}
			\put(4,2.2){$\fbox{2}$}


			%Strategies
			\put(1,1.5){$E$}
			\put(1.5,2.1){$C$}
			\put(2,1.5){$E$}
			\put(2.5,2.1){$C$}
			\G
			\put(3,1.5){$E$}
			\B
			\put(3.5,2.1){$C$}
			\G
			\put(4,1.5){$E$}
			\B
			\put(4.5,2.1){$C$}


			%Payoffs
			\put(5.3,1.8){\shortstack{ $6$ \\ $6$}}
			\put(1.15,.5){\shortstack{ $1$ \\ $1$}}
			\put(2.15,.5){\shortstack{ $0$ \\ $3$}}
			\put(3.15,.5){\shortstack{ $5$ \\ $2$}}
			\put(4.15,.5){\shortstack{ $4$ \\ $7$}}


			\end{picture}
			\end{center}}

				\only<4|handout:0>{
				\begin{center}
				\setlength{\unitlength}{0.7in}
				\begin{picture}(6,2.7)

				\thicklines

				%% choices
				\put(1.2,2){\vector(0,-1){1}}
				\put(1.2,2){\vector(1,0){1}}
				\G
				\put(2.2,2){\vector(0,-1){1}}
				\B
				\put(2.2,2){\vector(1,0){1}}
				\G
				\put(3.2,2){\vector(0,-1){1}}
				\B
				\put(3.2,2){\vector(1,0){1}}
				\G
				\put(4.2,2){\vector(0,-1){1}}
				\B
				\put(4.2,2){\vector(1,0){1}}


				%Player labels
				\put(1,2.2){$\fbox{1}$}
				\put(2,2.2){$\fbox{2}$}
				\put(3,2.2){$\fbox{1}$}
				\put(4,2.2){$\fbox{2}$}


				%Strategies
				\put(1,1.5){$E$}
				\put(1.5,2.1){$C$}
				\G
				\put(2,1.5){$E$}
				\B
				\put(2.5,2.1){$C$}
				\G
				\put(3,1.5){$E$}
				\B
				\put(3.5,2.1){$C$}
				\G
				\put(4,1.5){$E$}
				\B
				\put(4.5,2.1){$C$}


				%Payoffs
				\put(5.3,1.8){\shortstack{ $6$ \\ $6$}}
				\put(1.15,.5){\shortstack{ $1$ \\ $1$}}
				\put(2.15,.5){\shortstack{ $0$ \\ $3$}}
				\put(3.15,.5){\shortstack{ $5$ \\ $2$}}
				\put(4.15,.5){\shortstack{ $4$ \\ $7$}}


				\end{picture}
				\end{center}
				}

					\only<5-|handout:1>{
					\begin{center}
					\setlength{\unitlength}{0.7in}
					\begin{picture}(6,2.7)

					\thicklines

					%% choices
					\G
					\put(1.2,2){\vector(0,-1){1}}
					\B
					\put(1.2,2){\vector(1,0){1}}
					\G
					\put(2.2,2){\vector(0,-1){1}}
					\B
					\put(2.2,2){\vector(1,0){1}}
					\G
					\put(3.2,2){\vector(0,-1){1}}
					\B
					\put(3.2,2){\vector(1,0){1}}
					\G
					\put(4.2,2){\vector(0,-1){1}}
					\B
					\put(4.2,2){\vector(1,0){1}}


					%Player labels
					\put(1,2.2){$\fbox{1}$}
					\put(2,2.2){$\fbox{2}$}
					\put(3,2.2){$\fbox{1}$}
					\put(4,2.2){$\fbox{2}$}


					%Strategies
					\G
					\put(1,1.5){$E$}
					\B
					\put(1.5,2.1){$C$}
					\G
					\put(2,1.5){$E$}
					\B
					\put(2.5,2.1){$C$}
					\G
					\put(3,1.5){$E$}
					\B
					\put(3.5,2.1){$C$}
					\G
					\put(4,1.5){$E$}
					\B
					\put(4.5,2.1){$C$}


					%Payoffs
					\put(5.3,1.8){\shortstack{ $6$ \\ $6$}}
					\put(1.15,.5){\shortstack{ $1$ \\ $1$}}
					\put(2.15,.5){\shortstack{ $0$ \\ $3$}}
					\put(3.15,.5){\shortstack{ $5$ \\ $2$}}
					\put(4.15,.5){\shortstack{ $4$ \\ $7$}}


					\end{picture}
					\end{center}}

						\only<6|handout:1>{
						Unique SPNE: $((E,E),(E,E))$ 
						
						\
						
						Equilibrium payoffs $(1,1)$

						\

						Pareto dominated by 3 outcomes!}
\end{frame}



\begin{frame}{Multiple Equilibria}

	\only<1|handout:1>{
	\setlength{\unitlength}{0.7in}
	\begin{picture}(6,4)

	\thicklines

	%%Origin and label that 1 moves there
	\put(1,3){\circle{0.1}}
	\put(.9,3.15){$\fbox{1}$}

	%%1's two choices
	\put(1.05,3){\vector(1,0){2}}
	\put(1,2.95){\vector(0,-1){1.0}}

	%%labels for 1's choices
	\put(2,3.15){$U$}
	\put(0.7,2.35){$D$}

	%%2's two choices and label that 2 moves there
	\put(3.05,3){\vector(3,1){1.5}}
	\put(3.05,3){\vector(0,-1){1.05}}
	\put(2.9,3.15){$\fbox{2}$}

	%%labels for 2's choices
	\put(2.75,2.35){$B$}
	\put(3.55,3.35){$A$}

	%%3's info set and the label for 3's info set
	%\multiput(1.03,1.95)(0.1,0){20}{\_}
	\put(.7,2.0){$\fbox{3}$}
	\put(3.1,2.0){$\fbox{3}$}

	%%3's actions after D_1 and the associated labels
	\put(1,1.95){\vector(1,-2){0.5}}
	\put(0.4,1.35){$L$}
	\put(1,1.95){\vector(-1,-2){0.5}}
	\put(1.4,1.35){$R$}

	%%3's actions after A_1 then D_2 and the associated labels
	\put(3.05,1.95){\vector(1,-2){0.5}}
	\put(2.45,1.35){$L$}
	\put(3.05,1.95){\vector(-1,-2){0.5}}
	\put(3.45,1.35){$R$}

	%%payoffs after D_1
	\put(0.4,0.35){\shortstack[r]{ 1 \\ 1 \\ 0 } }
	\put(1.4,0.35){\shortstack[r]{ 3  \\ 1 \\  5 } }

	%%payoffs after A_2 then D_2
	\put(2.45,0.35){\shortstack[r]{  2 \\ 0 \\  2 } }
	\put(3.45,0.35){\shortstack[r]{ 4 \\ 4 \\  2 } }

	%%payoffs after A_1 then A_2
	\put(4.65,3.30){\shortstack[r]{ 2 \\ 2 \\ 2 } }

	\end{picture} }


		\only<2|handout:0>{
		\setlength{\unitlength}{0.7in}
		\begin{picture}(6,4)
 
		\thicklines
 
		%%Origin and label that 1 moves there
		\put(1,3){\circle{0.1}}
		\put(.9,3.15){$\fbox{1}$}
 
		%%1's two choices
		\put(1.05,3){\vector(1,0){2}}
		\put(1,2.95){\vector(0,-1){1.0}}
 
		%%labels for 1's choices
		\put(2,3.15){$U$}
		\put(0.7,2.35){$D$}
 
		%%2's two choices and label that 2 moves there
		\put(3.05,3){\vector(3,1){1.5}}
		\put(3.05,3){\vector(0,-1){1.05}}
		\put(2.9,3.15){$\fbox{2}$}
 
		%%labels for 2's choices
		\put(2.75,2.35){$B$}
		\put(3.55,3.35){$A$}
 
		%%3's info set and the label for 3's info set
		%\multiput(1.03,1.95)(0.1,0){20}{\_}
		\put(.7,2.0){$\fbox{3}$}
		\put(3.1,2.0){$\fbox{3}$}
 
		%%3's actions after D_1 and the associated labels
		\G
		\put(1,1.95){\vector(1,-2){0.5}}
		\B
		\put(0.4,1.35){$L$}
		\put(1,1.95){\vector(-1,-2){0.5}}
		\G
		\put(1.4,1.35){$R$}
		\B
 
		%%3's actions after A_1 then D_2 and the associated labels
		\put(3.05,1.95){\vector(1,-2){0.5}}
		\put(2.45,1.35){$L$}
		\put(3.05,1.95){\vector(-1,-2){0.5}}
		\put(3.45,1.35){$R$}
 
		%%payoffs after D_1
		\put(0.4,0.35){\shortstack[r]{ 1 \\ 1 \\ 0 } }
		\put(1.4,0.35){\shortstack[r]{ 3  \\ 1 \\  5 } }
 
		%%payoffs after A_2 then D_2
		\put(2.45,0.35){\shortstack[r]{  2 \\ 0 \\  2 } }
		\put(3.45,0.35){\shortstack[r]{ 4 \\ 4 \\  2 } }
 
		%%payoffs after A_1 then A_2
		\put(4.65,3.30){\shortstack[r]{ 2 \\ 2 \\ 2 } }
 
		\end{picture} }

			\only<3|handout:0>{
			\setlength{\unitlength}{0.7in}
			\begin{picture}(6,4)
 
			\thicklines
 
			%%Origin and label that 1 moves there
			\put(1,3){\circle{0.1}}
			\put(.9,3.15){$\fbox{1}$}
 
			%%1's two choices
			\put(1.05,3){\vector(1,0){2}}
			\put(1,2.95){\vector(0,-1){1.0}}
 
			%%labels for 1's choices
			\put(2,3.15){$U$}
			\put(0.7,2.35){$D$}
 
			%%2's two choices and label that 2 moves there
			\put(3.05,3){\vector(3,1){1.5}}
			\put(3.05,3){\vector(0,-1){1.05}}
			\put(2.9,3.15){$\fbox{2}$}
 
			%%labels for 2's choices
			\put(2.75,2.35){$B$}
			\put(3.55,3.35){$A$}
 
			%%3's info set and the label for 3's info set
			%\multiput(1.03,1.95)(0.1,0){20}{\_}
			\put(.7,2.0){$\fbox{3}$}
			\put(3.1,2.0){$\fbox{3}$}
 
			%%3's actions after D_1 and the associated labels
			\G
			\put(1,1.95){\vector(1,-2){0.5}}
			\B
			\put(0.4,1.35){$L$}
			\put(1,1.95){\vector(-1,-2){0.5}}
			\G
			\put(1.4,1.35){$R$}
			\B
 
			%%3's actions after A_1 then D_2 and the associated labels
			\put(3.05,1.95){\vector(1,-2){0.5}}
			\G
			\put(2.45,1.35){$L$}
			\put(3.05,1.95){\vector(-1,-2){0.5}}
			\B
			\put(3.45,1.35){$R$}
 
			%%payoffs after D_1
			\put(0.4,0.35){\shortstack[r]{ 1 \\ 1 \\ 0 } }
			\put(1.4,0.35){\shortstack[r]{ 3  \\ 1 \\  5 } }
 
			%%payoffs after A_2 then D_2
			\put(2.45,0.35){\shortstack[r]{  2 \\ 0 \\  2 } }
			\put(3.45,0.35){\shortstack[r]{ 4 \\ 4 \\  2 } }
 
			%%payoffs after A_1 then A_2
			\put(4.65,3.30){\shortstack[r]{ 2 \\ 2 \\ 2 } }
 
			\end{picture} }


				\only<4|handout:0>{
				\setlength{\unitlength}{0.7in}
				\begin{picture}(6,4)
 
				\thicklines
 
				%%Origin and label that 1 moves there
				\put(1,3){\circle{0.1}}
				\put(.9,3.15){$\fbox{1}$}
 
				%%1's two choices
				\put(1.05,3){\vector(1,0){2}}
				\put(1,2.95){\vector(0,-1){1.0}}
 
				%%labels for 1's choices
				\put(2,3.15){$U$}
				\put(0.7,2.35){$D$}
 
				%%2's two choices and label that 2 moves there
				\G
				\put(3.05,3){\vector(3,1){1.5}}
				\B
				\put(3.05,3){\vector(0,-1){1.05}}
				\put(2.9,3.15){$\fbox{2}$}
 
				%%labels for 2's choices
				\put(2.75,2.35){$B$}
				\G
				\put(3.55,3.35){$A$}
				\B
 
				%%3's info set and the label for 3's info set
				%\multiput(1.03,1.95)(0.1,0){20}{\_}
				\put(.7,2.0){$\fbox{3}$}
				\put(3.1,2.0){$\fbox{3}$}
 
				%%3's actions after D_1 and the associated labels
				\G
				\put(1,1.95){\vector(1,-2){0.5}}
				\B
				\put(0.4,1.35){$L$}
				\put(1,1.95){\vector(-1,-2){0.5}}
				\G
				\put(1.4,1.35){$R$}
				\B
 
				%%3's actions after A_1 then D_2 and the associated labels
				\put(3.05,1.95){\vector(1,-2){0.5}}
				\G
				\put(2.45,1.35){$L$}
				\put(3.05,1.95){\vector(-1,-2){0.5}}
				\B
				\put(3.45,1.35){$R$}

				%%payoffs after D_1
				\put(0.4,0.35){\shortstack[r]{ 1 \\ 1 \\ 0 } }
				\put(1.4,0.35){\shortstack[r]{ 3  \\ 1 \\  5 } }
 
				%%payoffs after A_2 then D_2
				\put(2.45,0.35){\shortstack[r]{  2 \\ 0 \\  2 } }
				\put(3.45,0.35){\shortstack[r]{ 4 \\ 4 \\  2 } }
 
				%%payoffs after A_1 then A_2
				\put(4.65,3.30){\shortstack[r]{ 2 \\ 2 \\ 2 } }
 
				\end{picture} 
				}

					\only<5|handout:0>{
					\setlength{\unitlength}{0.7in}
					\begin{picture}(6,4)
 
					\thicklines
 
					%%Origin and label that 1 moves there
					\put(1,3){\circle{0.1}}
					\put(.9,3.15){$\fbox{1}$}
 
					%%1's two choices
					\put(1.05,3){\vector(1,0){2}}
					\G
					\put(1,2.95){\vector(0,-1){1.0}}
					\B
 
					%%labels for 1's choices
					\put(2,3.15){$U$}
					\G
					\put(0.7,2.35){$D$}
					\B
 
					%%2's two choices and label that 2 moves there
					\G
					\put(3.05,3){\vector(3,1){1.5}}
					\B
					\put(3.05,3){\vector(0,-1){1.05}}
					\put(2.9,3.15){$\fbox{2}$}
 
					%%labels for 2's choices
					\put(2.75,2.35){$B$}
					\G
					\put(3.55,3.35){$A$}
					\B
 
					%%3's info set and the label for 3's info set
					%\multiput(1.03,1.95)(0.1,0){20}{\_}
					\put(.7,2.0){$\fbox{3}$}
					\put(3.1,2.0){$\fbox{3}$}
 
					%%3's actions after D_1 and the associated labels
					\G
					\put(1,1.95){\vector(1,-2){0.5}}
					\B
					\put(0.4,1.35){$L$}
					\put(1,1.95){\vector(-1,-2){0.5}}
					\G
					\put(1.4,1.35){$R$}
					\B
 
					%%3's actions after A_1 then D_2 and the associated labels
					\put(3.05,1.95){\vector(1,-2){0.5}}
					\G
					\put(2.45,1.35){$L$}
					\put(3.05,1.95){\vector(-1,-2){0.5}}
					\B
					\put(3.45,1.35){$R$}
 
					%%payoffs after D_1
					\put(0.4,0.35){\shortstack[r]{ 1 \\ 1 \\ 0 } }
					\put(1.4,0.35){\shortstack[r]{ 3  \\ 1 \\  5 } }
 
					%%payoffs after A_2 then D_2
					\put(2.45,0.35){\shortstack[r]{  2 \\ 0 \\  2 } }
					\put(3.45,0.35){\shortstack[r]{ 4 \\ 4 \\  2 } }
 
					%%payoffs after A_1 then A_2
					\put(4.65,3.30){\shortstack[r]{ 2 \\ 2 \\ 2 } }
 
					\end{picture} }

						\only<5|handout:0>{
						SPNE 1: (D, A, (R,L))}



						\only<6|handout:0>{
						\setlength{\unitlength}{0.7in}
						\begin{picture}(6,4)

						\thicklines

						%%Origin and label that 1 moves there
						\put(1,3){\circle{0.1}}
						\put(.9,3.15){$\fbox{1}$}

						%%1's two choices
						\put(1.05,3){\vector(1,0){2}}
						\put(1,2.95){\vector(0,-1){1.0}}

						%%labels for 1's choices
						\put(2,3.15){$U$}
						\put(0.7,2.35){$D$}

						%%2's two choices and label that 2 moves there
						\put(3.05,3){\vector(3,1){1.5}}
						\put(3.05,3){\vector(0,-1){1.05}}
						\put(2.9,3.15){$\fbox{2}$}

						%%labels for 2's choices
						\put(2.75,2.35){$B$}
						\put(3.55,3.35){$A$}

						%%3's info set and the label for 3's info set
						%\multiput(1.03,1.95)(0.1,0){20}{\_}
						\put(.7,2.0){$\fbox{3}$}
						\put(3.1,2.0){$\fbox{3}$}

						%%3's actions after D_1 and the associated labels
						\G
						\put(1,1.95){\vector(1,-2){0.5}}
						\B
						\put(0.4,1.35){$L$}
						\put(1,1.95){\vector(-1,-2){0.5}}
						\G
						\put(1.4,1.35){$R$}
						\B

						%%3's actions after A_1 then D_2 and the associated labels
						\G
						\put(3.05,1.95){\vector(1,-2){0.5}}
						\B
						\put(2.45,1.35){$L$}
						\put(3.05,1.95){\vector(-1,-2){0.5}}
						\G
						\put(3.45,1.35){$R$}
						\B

						%%payoffs after D_1
						\put(0.4,0.35){\shortstack[r]{ 1 \\ 1 \\ 0 } }
						\put(1.4,0.35){\shortstack[r]{ 3  \\ 1 \\  5 } }

						%%payoffs after A_2 then D_2
						\put(2.45,0.35){\shortstack[r]{  2 \\ 0 \\  2 } }
						\put(3.45,0.35){\shortstack[r]{ 4 \\ 4 \\  2 } }

						%%payoffs after A_1 then A_2
						\put(4.65,3.30){\shortstack[r]{ 2 \\ 2 \\ 2 } }

						\end{picture} 
						}

						\only<6|handout:0>{
						SPNE 1: (D, A, (R,L))}

							\only<7|handout:0>{
							\setlength{\unitlength}{0.7in}
							\begin{picture}(6,4)
 
							\thicklines
 
							%%Origin and label that 1 moves there
							\put(1,3){\circle{0.1}}
							\put(.9,3.15){$\fbox{1}$}
 
							%%1's two choices
							\put(1.05,3){\vector(1,0){2}}
							\put(1,2.95){\vector(0,-1){1.0}}
 
							%%labels for 1's choices
							\put(2,3.15){$U$}
							\put(0.7,2.35){$D$}
 
							%%2's two choices and label that 2 moves there
							\put(3.05,3){\vector(3,1){1.5}}
							\G
							\put(3.05,3){\vector(0,-1){1.05}}
							\B
							\put(2.9,3.15){$\fbox{2}$}
 
							%%labels for 2's choices
							\G
							\put(2.75,2.35){$B$}
							\B
							\put(3.55,3.35){$A$}
 
							%%3's info set and the label for 3's info set
							%\multiput(1.03,1.95)(0.1,0){20}{\_}
							\put(.7,2.0){$\fbox{3}$}
							\put(3.1,2.0){$\fbox{3}$}
 
							%%3's actions after D_1 and the associated labels
							\G
							\put(1,1.95){\vector(1,-2){0.5}}
							\B
							\put(0.4,1.35){$L$}
							\put(1,1.95){\vector(-1,-2){0.5}}
							\G
							\put(1.4,1.35){$R$}
							\B
 
							%%3's actions after A_1 then D_2 and the associated labels
							\G
							\put(3.05,1.95){\vector(1,-2){0.5}}
							\B
							\put(2.45,1.35){$L$}
							\put(3.05,1.95){\vector(-1,-2){0.5}}
							\G
							\put(3.45,1.35){$R$}
							\B
 
							%%payoffs after D_1
							\put(0.4,0.35){\shortstack[r]{ 1 \\ 1 \\ 0 } }
							\put(1.4,0.35){\shortstack[r]{ 3  \\ 1 \\  5 } }
 
							%%payoffs after A_2 then D_2
							\put(2.45,0.35){\shortstack[r]{  2 \\ 0 \\  2 } }
							\put(3.45,0.35){\shortstack[r]{ 4 \\ 4 \\  2 } }
 
							%%payoffs after A_1 then A_2
							\put(4.65,3.30){\shortstack[r]{ 2 \\ 2 \\ 2 } }
 
							\end{picture} }


								\only<8|handout:0>{
								\setlength{\unitlength}{0.7in}
								\begin{picture}(6,4)
 
								\thicklines
 
								%%Origin and label that 1 moves there
								\put(1,3){\circle{0.1}}
								\put(.9,3.15){$\fbox{1}$}
 
								%%1's two choices
								\G
								\put(1.05,3){\vector(1,0){2}}
								\B
								\put(1,2.95){\vector(0,-1){1.0}}
 
								%%labels for 1's choices
								\G
								\put(2,3.15){$U$}
								\B
								\put(0.7,2.35){$D$}
 
								%%2's two choices and label that 2 moves there
								\put(3.05,3){\vector(3,1){1.5}}
								\G
								\put(3.05,3){\vector(0,-1){1.05}}
								\B
								\put(2.9,3.15){$\fbox2$}
 
								%%labels for 2's choices
								\G
								\put(2.75,2.35){$B$}
								\B
								\put(3.55,3.35){$A$}
 
								%%3's info set and the label for 3's info set
								%\multiput(1.03,1.95)(0.1,0){20}{\_}
								\put(.7,2.0){$\fbox{3}$}
								\put(3.1,2.0){$\fbox{3}$}
 
								%%3's actions after D_1 and the associated labels
								\G
								\put(1,1.95){\vector(1,-2){0.5}}
								\B
								\put(0.4,1.35){$L$}
								\put(1,1.95){\vector(-1,-2){0.5}}
								\G
								\put(1.4,1.35){$R$}
								\B
 
								%%3's actions after A_1 then D_2 and the associated labels
								\G
								\put(3.05,1.95){\vector(1,-2){0.5}}
								\B
								\put(2.45,1.35){$L$}
								\put(3.05,1.95){\vector(-1,-2){0.5}}
								\G
								\put(3.45,1.35){$R$}
								 \B
								%%payoffs after D_1
								\put(0.4,0.35){\shortstack[r]{ 1 \\ 1 \\ 0 } }
								\put(1.4,0.35){\shortstack[r]{ 3  \\ 1 \\  5 } }
 
								%%payoffs after A_2 then D_2
								\put(2.45,0.35){\shortstack[r]{  2 \\ 0 \\  2 } }
								\put(3.45,0.35){\shortstack[r]{ 4 \\ 4 \\  2 } }
 
								%%payoffs after A_1 then A_2
								\put(4.65,3.30){\shortstack[r]{ 2 \\ 2 \\ 2 } }
 
								\end{picture} }

									\only<8|handout:1>{
									SPNE 1: (D, A, (R,L))
									
									SPNE 2: (U,B,(R,R))}

\end{frame}

\begin{frame}{A Familiar Example: Public Good in a Team}
Two players: 1 \& 2 

\

Each can choose a level to contribute to a public good: $s_{i}$

\

Payoff for individual $i$ are
\begin{equation*}
u_{i}(s_{1},s_{2})= s_{1} + s_{2} + \frac{s_{1}s_{2}}{2} - \frac{s_{i}^{2}}{2}
\end{equation*}

\end{frame}

\begin{frame}{Nash Equilibrium}
\begin{equation*}
s^{*}_{1} = 2  \ \ \ \ \   s^{*}_{2} = 2 
\end{equation*}

\

Individual player's equilibrium payoff: 
\begin{equation*}
 2 + 2 + \frac{ 2 \cdot 2}{2} - \frac{2^2}{2} = 4
\end{equation*}
\end{frame}

\begin{frame}{Consider an extensive form version}

  Player 1 must make her choice first 
 
 \
 
 Before Player 2 decides how much to put in, she observes how much Player 1 puts in 
 
 \
 
  How might this change contributions? 

 \
 
We will use backward induction
\end{frame}

\begin{frame}{Best Response for Player 2}
The payoff function for player $2$:
\begin{equation*}
u_{2}(s_{1},s_{2}) = s_{1} + s_{2} + \frac{s_{1}s_{2}}{2} - \frac{s_{2}^{2}}{2}
\end{equation*}

\

How do we determine the best response of player $2$? 
\pause
\begin{equation*}
\frac{\partial u_{2}(s_{1}, s_{2})}{\partial s_{2}} = 1+ \frac{s_{1}}{2} - s_{2} 
\end{equation*}
\pause

\

Setting equal to zero~($\tfrac{\partial u_{2}(s_{1}, s_{2})}{\partial s_{1}}=0$), Player $2$'s best-response to $s_{1}$ is
\begin{equation*}
\BR_2(s_{1}) = 1+\frac{s_{1}}{2} \equiv s_2\ch(s_1)
\end{equation*}
\end{frame}

\begin{frame}{Best Response for Player 1}
Player $1$'s best response must account for how Player $2$ will respond to whatever she chooses \pause:
\begin{equation*}
u_{1}(s_{1},s_{2}^{\checkmark}(s_{1}))
\end{equation*}
\pause
\begin{equation*}
u_{1}(s_{1},s_{2}^{\checkmark}(s_{1})) = s_{1} + s_{2}^{\checkmark}(s_{1}) + \frac{s_{1}s_{2}^{\checkmark}(s_{1})}{2} - \frac{s_{1}^{2}}{2}
\end{equation*}
\pause
\begin{equation*}
u_{1}(s_{1},s_{2}^{\checkmark}(s_{1})) = s_{1} + \(1+\frac{s_{1}}{2}\) + \frac{s_{1}}{2} \(1+\frac{s_{1}}{2}\)  - \frac{s_{1}^{2}}{2}
\end{equation*}
\pause
\begin{equation*}
u_{1}(s_{1},s_{2}^{\checkmark}(s_{1})) =1+ \frac{3}{2} s_{1} + \frac{s_{1}}{2} + \frac{s_{1}^{2}}{4} - \frac{s_{1}^{2}}{2}
\end{equation*}
\end{frame}

\begin{frame}{Best Response for Player 1}
We can write Player 1's problem as:
\begin{equation*}
u_{1}(s_{1},s_{2}^{\checkmark}(s_{1})) =1+ 2 s_{1}  - \frac{s_{1}^{2}}{4}
\end{equation*}

\

Solve for Player 1's optimal choice:
\begin{equation*}
2 - \frac{s_{1}}{2} = 0
\end{equation*}

\

\begin{equation*}
s^{*}_{1} = 4
\end{equation*}
\pause
Go back to Player 2:
\begin{equation*}
s^{*}_{2} = s_2\ch(4) = 1 + \frac{1}{2} \(4\) = 3
\end{equation*}
\end{frame}

\begin{frame}{Public Good in a Team}
So each player contributes more:
\begin{equation*}
s^{*}_{1} = 4 \ \ \ \ \ \ s^{*}_{2} = 3
\end{equation*}
\pause
and equilibrium utilities:
\begin{equation*}
u^{*}_{1} = 6 \ \ \ \ \ \ \ u^{*}_{2} = 8.5\end{equation*}

\

They each are better off, but it's better to move second
\end{frame}

\begin{frame}{Subgame Perfect Nash Equilibrium}
 Subgame Perfect Nash Equilibrium is a refinement of Nash Equilibrium 
 
 \
 
 It rules out equilibria that rely on incredible threats in a dynamic environment 

\ 

 All SPNE are identified by backward induction
\end{frame}





\end{document}
