\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}} \newcommand{\BR}{\operatorname{BR}} %\newtheorem{theorem}{Theorem} \newcommand{\un}{\underline} \newcommand{\ov}{\overline} \newcommand{\gp}{\succ} \newcommand{\lp}{\prec} \newcommand{\gep}{\succsim} \newcommand{\lep}{\precsim} \newcommand{\ch}{{^\checkmark}} \newcommand{\cyan}{\cellcolor{cyan}} \newcommand{\orange}{\cellcolor{orange}} \newcommand{\T}{\ensuremath{\mbox{\footnotesize\textsc{occupation}}}} \renewcommand{\S}{\ensuremath{\mbox{\footnotesize\textsc{suicide}}}} %\newcommand{\R}{\ensuremath{\mbox{\footnotesize\textsc{religious}}}} \usepackage{multirow} \usepackage{graphicx} \usepackage{color} \usepackage{ragged2e} \usepackage{xcolor, colortbl} % % \usepackage{array} % \renewcommand{\arraystretch}{2}% % ----------------------------------------- % 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{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}{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}{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{\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{\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{ \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{ \pause \[ s_1^* = \sqrt{s_2^*} - s_2^* \ \ \ \ \ \ \ s_2^* = \sqrt{s_1^*} - s_1^*\] } \end{frame} \begin{frame}{Solving$^3$} \only{\[ s_1^* = \sqrt{s_2^*} - s_2^* \ \ \ \ \ \ \ s_2^* = \sqrt{s_1^*} - s_1^*\] \ } Use substitution to find Player 1's equilibrium action \only{ \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{ \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}