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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}}}}



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\title{Game Theory I}

\author{}
% \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}{A Strategic Situation \\(due to Ben Polak)}
	\begin{center}
		{\setlength{\extrarowheight}{10pt}%
\begin{tabular}{*4c} 
	& &\multicolumn{2}{c}{Player 2}\\
	& & $\alpha$ &$\beta$ \\ \cline{3-4}
	\multirow{2}{*}{Player 1}& $\alpha$ & \multicolumn{1}{|c|}{B-, B-} & \multicolumn{1}{c|}{A, C} \\[10pt] \cline{3-4} 
	& $\beta$ & \multicolumn{1}{|c|}{C, A} &\multicolumn{1}{c|}{A-, A-}\\[10pt] \cline{3-4}
	\end{tabular}}
		\end{center}
	\end{frame}


	
\begin{frame}{Selfish Students}
	
	
		\begin{center}
			{\setlength{\extrarowheight}{10pt}%
	\begin{tabular}{*4c} 
		& &\multicolumn{2}{c}{Selfish 2}\\
		&  												&$\alpha$ 								&$\beta$ \\ \cline{3-4}
		\multirow{2}{*}{Selfish 1}	
									& $\alpha$ 			& \multicolumn{1}{|c|}{$1,1$} 	& \multicolumn{1}{c|}{$3,0$} \\[10pt] \cline{3-4} 
									& $\beta$ 			& \multicolumn{1}{|c|}{$0,3$}		&\multicolumn{1}{c|}{$2,2$}\\[10pt] \cline{3-4}
		\end{tabular}}
			\end{center}
	
		\pause
		\begin{itemize}
			\item No matter what Selfish 2 does, Selfish 1 wants to choose $\alpha$ (and vice versa)
			\pits $(\alpha,\alpha)$ is a sensible prediction for what will happen 
		\end{itemize}
	\end{frame}
	
	\begin{frame}{Nice Students}
	
					\begin{center}
					{\setlength{\extrarowheight}{10pt}%
			\begin{tabular}{*4c} 
				& &\multicolumn{2}{c}{Nice 2}\\
				&  												&$\alpha$ 								&$\beta$ \\ \cline{3-4}
				\multirow{2}{*}{Nice 1}	
											& $\alpha$ 			& \multicolumn{1}{|c|}{$2,2$} 	& \multicolumn{1}{c|}{$1,0$} \\[10pt] \cline{3-4} 
											& $\beta$ 			& \multicolumn{1}{|c|}{$0,1$}		&\multicolumn{1}{c|}{$3,3$}\\[10pt] \cline{3-4}
				\end{tabular}}
					\end{center}
					
				\pause
				\begin{itemize}
					\item Each nice student wants to match the behavior of the other nice student
					\pits $(\alpha,\alpha)$ or $(\beta,\beta)$ seem sensible. 
					\pits We need to know what people think about each other's behavior to have a prediction
				\end{itemize}
				
			\end{frame}


	\begin{frame}{Selfish vs.\ Nice}
	
					\begin{center}
					{\setlength{\extrarowheight}{10pt}%
			\begin{tabular}{*4c} 
				& &\multicolumn{2}{c}{Nice}\\
				&  													&$\alpha$ 								&$\beta$ \\ \cline{3-4}
				\multirow{2}{*}{Selfish}	
												& $\alpha$ 			& \multicolumn{1}{|c|}{$1,2$} 		& \multicolumn{1}{c|}{$3,0$} \\[10pt] \cline{3-4} 
												& $\beta$ 			& \multicolumn{1}{|c|}{$0,1$}		&\multicolumn{1}{c|}{$2,3$}\\[10pt] \cline{3-4}
				\end{tabular}}
					\end{center}
			
			\pause
				\begin{itemize}
					\item Nice wants to match what Selfish does
					\pits No matter what Nice does, Selfish wants to player $\alpha$ 
					\pits If Nice can think one step about Selfish, she should realize she should play $\alpha$ 
					\pits $(\alpha,\alpha)$ seems the sensible prediction
				\end{itemize}
				
	\end{frame}
			

			
\section{Strategic Form Games}

\begin{frame}{Components of a Game}
	
	 Players: Who is involved?
			
		\
		
		Strategies: What can they do?
			
		\
		
		 Payoffs: What do they want?
	\end{frame}


\begin{frame}{Chicken}
	
		\begin{center}
			{\setlength{\extrarowheight}{10pt}%
	\begin{tabular}{*4c} 
		& &\multicolumn{2}{c}{Player 2}\\
		&  												& Straight  								& Swerve \\ \cline{3-4}
		\multirow{2}{*}{Player 1}	
									& Straight 			& \multicolumn{1}{|c|}{$0,0$} 	& \multicolumn{1}{c|}{$3,1$} \\[10pt] \cline{3-4} 
									& Swerve			& \multicolumn{1}{|c|}{$1,3$}		&\multicolumn{1}{c|}{$2,2$}\\[10pt] \cline{3-4}
		\end{tabular}}
			\end{center}
		\end{frame}
		
		\begin{frame}{Choosing a Restaurant}
	
				\begin{center}
					{\setlength{\extrarowheight}{10pt}%
			\begin{tabular}{*4c} 
				& &\multicolumn{2}{c}{Rebecca}\\
				&  												& P  								& V \\ \cline{3-4}
				\multirow{2}{*}{Ethan}	
										& P					& \multicolumn{1}{|c|}{$4,3$} 	& \multicolumn{1}{c|}{$1,1$} \\[10pt] \cline{3-4} 
										& V			& \multicolumn{1}{|c|}{$0,0$}		&\multicolumn{1}{c|}{$3,4$}\\[10pt] \cline{3-4}
				\end{tabular}}
					\end{center}
				\end{frame}
				
				
\begin{frame}{Working in a Team}
	
	2 players
	
	\
	
	Player $i$ chooses effort $s_i\geq 0$
	
	\
	
	Jointly produce a product. Each enjoys an amount 
	\[ \pi(s_1, s_2) = s_1 + s_2 + \frac{s_1 \times s_2}{2}\]
	
	\
	
	Cost of effort is $s_i^2$
	
	\[ u_i(s_1, s_2) = \pi(s_1, s_2) - s_i^2\]
	\end{frame}
	
	\begin{frame}{Player 1's payoffs as a function of each player's strategy}
		\begin{figure}
			\begin{center}
				\includegraphics[width = .8\textwidth]{figures/apartment.pdf}
			\end{center}
		\end{figure}
	\end{frame}
	
	\begin{frame}{Choosing a Number}
		
		$N$ players
		
		\
		
		Each player ``bids'' a real number in $[0,10]$
		
		\
		
		If the bids sum to 10 or less, each player's payoff is her bid
		
		\
		
		Otherwise players' payoffs are 0
		\end{frame}
		
\section{Solving a Game: Nash Equilibrium}



	\begin{frame}{Nash Equilibrium}
		
		A strategy profile where no individual has a unilateral incentive to change her behavior
		
		\
		
		Before we talk about why this is our central solution concept, let's formalize it
	\end{frame}
	
	\begin{frame}{Notation}
		
		Player $i$'s strategy
			\begin{itemize}
				\item $s_i$
			\end{itemize}
		
		\
		
		Set of all possible strategies for Player $i$
			\begin{itemize}
				\item $S_i$ 
			\end{itemize}
		
		\
		
		 Strategy profile (one strategy for each player)
		 	\begin{itemize}
			   	\item $\mathbf{s} = (s_1, s_2, \ldots, s_N)$
				\end{itemize}
		
		
		\
		
		Strategy profile for all players except $i$
			\begin{itemize}
				\item $\mathbf{s_{-i}} = (s_1, s_2, \ldots, s_{i-1}, s_{i+1}, \ldots, s_N)$
			\end{itemize}
		
		\
		
		Different notation for strategy profile
			\begin{itemize}
				\item $\mathbf{s} = (\mathbf{s_{-i}}, s_i)$
			\end{itemize}
		

	\end{frame}
		
		
		
		\begin{frame}{Selfish Students}
	
	
				\begin{center}
					{\setlength{\extrarowheight}{10pt}%
			\begin{tabular}{*4c} 
				& &\multicolumn{2}{c}{Player 2}\\
				&  												&$\alpha$ 							&$\beta$ \\ \cline{3-4}
				\multirow{2}{*}{Player 1}	
											& $\alpha$ 			& \multicolumn{1}{|c|}{$1,1$} 		& \multicolumn{1}{c|}{$3,0$} \\[10pt] \cline{3-4} 
											& $\beta$ 			& \multicolumn{1}{|c|}{$0,3$}		&\multicolumn{1}{c|}{$2,2$}\\[10pt] \cline{3-4}
				\end{tabular}}
					\end{center}
		
		\
		
		$S_i = \{\alpha,\beta\}$
		
		\
		
		4 strategy profiles: $(\alpha,\alpha), (\alpha,\beta), (\beta,\alpha), (\beta,\beta)$
		\end{frame}
		
		\begin{frame}{Chicken}
	
				\begin{center}
					{\setlength{\extrarowheight}{10pt}%
			\begin{tabular}{*4c} 
				& &\multicolumn{2}{c}{Player 2}\\
				&  												& Straight  					& Swerve \\ \cline{3-4}
				\multirow{2}{*}{Player 1}	
											& Straight 			& \multicolumn{1}{|c|}{$0,0$} 	& \multicolumn{1}{c|}{$3,1$} \\[10pt] \cline{3-4} 
											& Swerve			& \multicolumn{1}{|c|}{$1,3$}	&\multicolumn{1}{c|}{$2,2$}\\[10pt] \cline{3-4}
				\end{tabular}}
					\end{center}
					
					\
					
					$S_i = \{\mbox{Straight, Swerve}\}$
					
					\
					
					4 strategy profiles: (Straight, Straight), (Straight, Swerve), (Swerve, Straight), (Swerve, Swerve)
				\end{frame}
		
				\begin{frame}{Choosing a Restaurant}
	
						\begin{center}
							{\setlength{\extrarowheight}{10pt}%
					\begin{tabular}{*4c} 
						& &\multicolumn{2}{c}{Rebecca}\\
						&  									& P  								& V \\ \cline{3-4}
						\multirow{2}{*}{Ethan}	
												& P			& \multicolumn{1}{|c|}{$4,3$} 		& \multicolumn{1}{c|}{$1,1$} \\[10pt] \cline{3-4} 
												& V			& \multicolumn{1}{|c|}{$0,0$}		&\multicolumn{1}{c|}{$3,4$}\\[10pt] \cline{3-4}
						\end{tabular}}
							\end{center}
							
							$S_E = ?$  \ \ \ \ \ \ $S_R = ?$
							
							\
							
							Strategy profiles: ?
						\end{frame}
		
		
				\begin{frame}{Choosing a number with 3 players}
					
					$S_i = [0,10]$
						\begin{itemize}
							\item Player $i$ can choose any real number between 0 and 10
						\end{itemize}
					
					\
					
					$\mathbf{s} = (s_1 = 1, s_2= 4, s_3  = 7) = (1,4,7)$
						\begin{itemize}
							\item An example of a strategy profile
						\end{itemize}

					\
					
					$\mathbf{s_{-2}} = (1,7)$
						\begin{itemize}
							\item Same strategy profile, with player 2's strategy omitted
						\end{itemize}
					
					\
					
					$\mathbf{s} = (\mathbf{s_{-2}}, s_2) = ((1,7),4)$
						\begin{itemize}
							\item Reconstructing the strategy profile
						\end{itemize}
				\end{frame}
				
				
		\begin{frame}{Notating Payoffs}
			
			Players' payoffs are defined over strategy profiles
				\begin{itemize}
					\item A strategy profile implies an outcome of the game
				\end{itemize}
				
				\
				
			Player $i$'s payoff from the strategy profile $\mathbf{s}$ is 
			\[ u_i(\mathbf{s})\]
			
			\
			
			Player $i$'s payoff if she chooses $s_i$ and others play as in $\mathbf{s_{-i}}$
			\[ u_i((s_i, \mathbf{s_{-i}})) \]
			
		\end{frame}
		
	\begin{frame}{Nash Equilibrium}
			
			Consider a game with $N$ players. A strategy
			profile $\mathbf{s}^* =( s_1^*, s_2^*, \ldots, s_N^*)$ is a \textbf{Nash equilibrium} of 
			the game if, for every player $i$
				\[		u_i(s_i^*,\mathbf{s_{-i}}^*) \geq u_i(s_i', \mathbf{s_{-i}}^*)\]
			for all $s_i' \in S_i$
	\end{frame}
	
	
	\begin{frame}{Best Responses}
		A strategy, $s_i$, is a \textbf{best response} by Player $i$ to a
		profile of strategies for all other players, $\mathbf{s_{-i}}$, if 
		\[ u_i(s_i,\mathbf{s_{-i}}) \geq u_i(s_i',\mathbf{s_{-i}})\]
		for all $s_i' \in S_i$
	\end{frame}
	
	\begin{frame}{Best Response Correspondence}
		Player $i$'s \textbf{best response correspondence}, $\BR_i$, is a mapping from strategies for all players other than $i$ into subsets of $S_i$ satisfying the following 
		condition:
		 \begin{itemize}
			 \item For each $\mathbf{s_{-i}}$, the mapping yields a set of strategies for Player $i$, $\BR_i(\mathbf{s_{-i}})$, such that $s_i$ is in $\BR_i(\mathbf{s_{-i}})$ 
			 if and only if $s_i$ is a best response to $\mathbf{s_{-i}}$
			 \end{itemize}
	\end{frame}
	
	\begin{frame}{An Equivalent Definition of NE}
		
		Consider a game with $N$ players. A strategy
		profile $\mathbf{s}^* =( s_1^*, s_2^*, \ldots, s_N^*)$ is a \textbf{Nash equilibrium} of 
		the game if $s_i^*$ is a best response to $\mathbf{s_{-i}}^*$ for each $i = 1, 2, \ldots, N$

	\end{frame}
	
	
	\begin{frame}{Selfish vs.\ Nice}
	
					\begin{center}
					{\setlength{\extrarowheight}{10pt}%
			\begin{tabular}{*4c} 
				& &\multicolumn{2}{c}{Nice}\\
				&  										&\only<1,3-|handout:1>{$\alpha$} \only<2|handout:0>{\cyan $\alpha$} 	&\only<1-2,4-|handout:1>{$\beta$} \only<3|handout:0>{\cyan $\beta$} \\ \cline{3-4}
				\multirow{2}{*}{Selfish}	
						\only<1|handout:0>{	& $\alpha$ 			& \multicolumn{1}{|c|}{$1,2$} 				& \multicolumn{1}{c|}{$3,0$} \\[10pt] \cline{3-4} 
									& $\beta$ 			& \multicolumn{1}{|c|}{$0,1$}				&\multicolumn{1}{c|}{$2,3$}\\[10pt] \cline{3-4}}
						
						\only<2|handout:0>{	& $\alpha$ 			& \multicolumn{1}{|c|}{\cyan$1\ch,2$} 		& \multicolumn{1}{c|}{$3,0$} \\[10pt] \cline{3-4} 
									& $\beta$ 			& \multicolumn{1}{|c|}{\cyan $0,1$}			&\multicolumn{1}{c|}{$2,3$}\\[10pt] \cline{3-4}}

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

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

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

						\only<6|handout:1>{	& $\alpha$ 			& \multicolumn{1}{|c|}{\orange $1\ch,2\ch$}	& \multicolumn{1}{c|}{$3\ch,0$} \\[10pt] \cline{3-4} 
									& $\beta$ 			& \multicolumn{1}{|c|}{$0,1$}				&\multicolumn{1}{c|}{$2,3\ch$}\\[10pt] \cline{3-4}}

						
				\end{tabular}}
				\end{center}
			\end{frame}
	
	
	
	
	\begin{frame}{Chicken}

		\begin{center}
			{\setlength{\extrarowheight}{10pt}%
			\begin{tabular}{*4c} 
			& &\multicolumn{2}{c}{Player 2}\\
			&  												& \only<1,4-|handout:1> {Straight} \only<2|handout:0>{\cyan Straight} 	& \only<1-2,4-|handout:1>{Swerve}\only<3|handout:0> {\cyan Swerve} \\ \cline{3-4}
			\multirow{2}{*}{Player 1}	
			
			\only<1|handout:0>{& Straight 	& \multicolumn{1}{|c|}{$0,0$} 	& \multicolumn{1}{c|}{$3,1$} \\[10pt] \cline{3-4} 
					 & Swerve		& \multicolumn{1}{|c|}{$1,3$}	&\multicolumn{1}{c|}{$2,2$}  \\[10pt] \cline{3-4}}
			
			
			\only<2|handout:0>{& Straight 	& \multicolumn{1}{|c|}{\cyan $0,0$} 	& \multicolumn{1}{c|}{$3,1$} \\[10pt] \cline{3-4} 
					 & Swerve		& \multicolumn{1}{|c|}{ \cyan $1\ch,3$}	&\multicolumn{1}{c|}{$2,2$}  \\[10pt] \cline{3-4}}
			
			\only<3|handout:0>{& Straight 	& \multicolumn{1}{|c|}{$0,0$} 	& \multicolumn{1}{c|}{\cyan $3\ch,1$} \\[10pt] \cline{3-4} 
					 & Swerve		& \multicolumn{1}{|c|}{$1\ch,3$}	&\multicolumn{1}{c|}{\cyan $2,2$}  \\[10pt] \cline{3-4}}
			
			
			\only<4|handout:0>{& \cyan Straight 	& \multicolumn{1}{|c|}{\cellcolor{cyan}$0,0$}	& \multicolumn{1}{c|}{\cellcolor{cyan}$3\ch,1\ch$} \\[10pt] \cline{3-4} 
					& Swerve						& \multicolumn{1}{|c|}{$1\ch,3$}					&\multicolumn{1}{c|}{$2,2$}					\\[10pt] \cline{3-4}}


			\only<5|handout:0>{& Straight 			& \multicolumn{1}{|c|}{$0,0$} 		& \multicolumn{1}{c|}{$3\ch,1\ch$} 		\\[10pt] \cline{3-4} 
					 &  \cyan Swerve		& \multicolumn{1}{|c|}{\cyan$1\ch,3\ch$}	&\multicolumn{1}{c|}{\cyan $2,2$}  	\\[10pt] \cline{3-4}}
			
			\only<6|handout:1>{& Straight 			& \multicolumn{1}{|c|}{$0,0$} 		& \multicolumn{1}{c|}{\cellcolor{orange} $3\ch,1\ch$} 		\\[10pt] \cline{3-4} 
					 &   Swerve		& \multicolumn{1}{|c|}{\cellcolor{orange} $1\ch,3\ch$}	&\multicolumn{1}{c|}{$2,2$}  	\\[10pt] \cline{3-4}}
			\end{tabular}}
		\end{center}
				
	\end{frame}
	
	
			\begin{frame}{You Solve Choosing a Restaurant}
						\begin{center}
							{\setlength{\extrarowheight}{10pt}%
					\begin{tabular}{*4c} 
						& &\multicolumn{2}{c}{Rebecca}\\
						&  									& P  								& V \\ \cline{3-4}
						\multirow{2}{*}{Ethan}	
												& P			& \multicolumn{1}{|c|}{$4,3$} 		& \multicolumn{1}{c|}{$1,1$} \\[10pt] \cline{3-4} 
												& V			& \multicolumn{1}{|c|}{$0,0$}		&\multicolumn{1}{c|}{$3,4$}\\[10pt] \cline{3-4}
						\end{tabular}}
							\end{center}
				
						
					\end{frame}
		
		\begin{frame}{Another Practice Game}
					\begin{center}
						{\setlength{\extrarowheight}{10pt}%
				\begin{tabular}{*4c} 
					& &\multicolumn{2}{c}{Player 2}\\
					&  									& L  								& R \\ \cline{3-4}
					\multirow{2}{*}{Player 1}	
											& U			& \multicolumn{1}{|c|}{$10,2$} 		& \multicolumn{1}{c|}{$3,4$} \\[10pt] \cline{3-4} 
											& D			& \multicolumn{1}{|c|}{$-1,0$}		&\multicolumn{1}{c|}{$5,7$}\\[10pt] \cline{3-4}
					\end{tabular}}
						\end{center}
			
			
					
				\end{frame}
				
		\begin{frame}{Working in a Team}
			
			\[ u_1(s_1, s_2) = \pi(s_1,s_2) - s_1^2 = s_1 + s_2 + \frac{s_1s_2}{2} - s_1^2\]
			
			\
			
			Find Player $i$'s best response by maximizing for each $s_2$
			
			\[ \frac{\partial u_1(s_1, s_2)}{\partial s_1} = 1 + \frac{s_2}{2} - 2s_1 \]
			
			\
			
			\pause First-order condition sets this equal to 0 to get $\BR_1(s_2)$
			\[ 1 + \frac{s_2}{2} - 2\BR_1(s_2)= 0\]
			
			\pause	\[\BR_1(s_2) = \frac{1}{2} + \frac{s_2}{4} \ \ \ \ \ \ \BR_2(s_1) = \frac{1}{2} + \frac{s_1}{4}\]
		\end{frame}
		
		\begin{frame}{Player 1's Best Response}
			
			\begin{figure}
				\begin{center}
					\includegraphics[width = .95\textwidth]{figures/apartment_br.pdf}
				\end{center}
			\end{figure}
		\end{frame}	
		
		\begin{frame}{Nash Equilibrium}
				\begin{figure}
					\begin{center}
						\includegraphics[width = .8\textwidth]{figures/br.pdf}
					\end{center}
				\end{figure}
		\end{frame}

		\begin{frame}{Solving for NE}
			
			Since best responses are unique, a NE is a profile, $(s_1^*,s_2^*)$ satisfying
				\[ s_1^* = \BR_1(s_2^*)= \frac{1}{2} + \frac{s_2^*}{4} \ \ \ \ \ \ \ \ s_2^* = \BR_2(s_1^*) = \frac{1}{2} + \frac{s_1^*}{4}\]
				
		
			\
			
			Substituting
			
				\[ s_1^* =  \frac{1}{2} + \frac{ \frac{1}{2} + \frac{s_1^*}{4}}{4}\]
				
				\
				
			\[ s_1^* = \frac{2}{3} \ \ \ \ \ \ \ \ s_2^* = \frac{2}{3}\]
		\end{frame}
		
		
		\begin{frame}{Practice Game with Continuous Choices}
			
			2 players
			
			\
			
			Each player, $i$, chooses a real number  $s_i$
			
			\
			
			There is a benefit of value 1 to be divided between the players
			
			\
			\
			At a strategy profile $(s_i, s_{-i})$, Player $i$ wins a share 
			 \[ \frac{s_i}{s_i + s_{-i}}\]
			 
	
			\
			
			The cost of $s_i$ is $s_i$
			\end{frame}
			
		\begin{frame}{Solving}
			Write down Player $1$'s payoff from $(s_1, s_2)$
			
			\only<beamer>{\pause 
			\[ u_1(s_1,s_2) = \frac{s_1}{s_1+s_2} \times 1 - s_1\]
			
									}
									
			\
			
			\pause 
			
			Calculate Player $1$'s best response correspondence
			
			
			\only<beamer>{\pause
			\[ \frac{\partial u_1(s_1,s_2)}{\partial s_1} = \frac{s_1+s_2 - s_1}{(s_1+s_2)^2}\times 1 - 1  = \frac{s_2}{(s_1+s_2)^2} - 1 \]
			
			\
			
			Set equal to zero to maximize
			\[ \frac{s_2}{(\BR_1(s_2) + s_2)^2} - 1 = 0 \Rightarrow \BR_1(s_2) = \sqrt{s_2} - s_2\]
			}
		\end{frame}

		\begin{frame}{Solving$^2$}
			Player 2 is symmetric to Player 1, so write down both players' best response correspondences
				\only<beamer>{
				\pause 
					\[ \BR_1(s_2) = \sqrt{s_2} - s_2 \ \ \ \ \ \ \ \BR_2(s_1) = \sqrt{s_1} - s_1\]
									}
					
				\
				\pause 
			
			At a NE each player is playing a best response to the other. Write down two equations that characterize equilibrium.
				\only<beamer>{
					\pause
						\[ s_1^* = \sqrt{s_2^*} - s_2^* \ \ \ \ \ \ \ s_2^* = \sqrt{s_1^*} - s_1^*\]
				}
		\end{frame}
		
		\begin{frame}{Solving$^3$}

			\only<beamer>{\[ s_1^* = \sqrt{s_2^*} - s_2^* \ \ \ \ \ \ \ s_2^* = \sqrt{s_1^*} - s_1^*\]

			\
				
				}

			Use substitution to find Player 1's equilibrium action
				\only<beamer>{
					\pause
						\[ s_1^* = \sqrt{\sqrt{s_1^*} - s_1^*} - \left(\sqrt{s_1^*} - s_1^*\right) \Rightarrow s_1^* = \frac{1}{4} \]
						}

			\

			Now substitute this in to find Player 2's equilibrium action
				\only<beamer>{
					\pause
					\[ s_2^* = \sqrt{\frac{1}{4}} - \frac{1}{4} = \frac{1}{2} - \frac{1}{4} = \frac{1}{4}\]
					}
	
		\end{frame}
		
		\begin{frame}{Why Nash Equilibrium?}
		
			No regrets
		
			\
		
			 Social learning
		
			\
		
			Self-enforcing agreements
		
			\
		
			Analyst humility
		\end{frame}
	
		

		\begin{frame}{Take Aways}
			
			A Nash Equilibrium is a strategy profile where each player is best responding to what all other players are doing
			
			\
			
			You find a NE by calculating each player's best response correspondence and seeing where they intersect
			
			\
			
			NE is our main \emph{solution concept} for strategic situations
			\end{frame}
			
		
\end{document}
