\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{\G}{\color{red}} \newcommand{\B}{\color{black}} \renewcommand{\(}{\left(} \renewcommand{\)}{\right)} \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 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}{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}{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}