   \documentclass[12pt]{article}

%  pdflatex     auction.tex


\usepackage{graphicx}

\usepackage{amsmath}

 \usepackage{amssymb}


\reversemarginpar
  \topmargin  -1in 
\oddsidemargin .25in
 \textheight  9.4in
 \textwidth 6.4in

 


\begin{document}

\begin{small}

%\baselineskip 12pt

\parindent 24pt

 \parskip 10pt

\titlepage 

 
 

\noindent
 11 October 2005.  Eric Rasmusen, Erasmuse@indiana.edu. Http://www.rasmusen.org.   Comments    welcomed.

  \parindent 24pt \parskip 10pt

\begin{LARGE}
 \begin{center} 
 {\bf  The Repeated Prisoner's Dilemma Under Incomplete Information: A Classroom Game for  Chapter 6} 
 \end{center}
 \end{LARGE}
 

  
 Consider the  Prisoner's Dilemma in Table 3, obtained by adding 8 to each payoff in   Chapter 1's Table 2, and identical to Chapter 5's Table 10: 
 
\begin{center} 
 {\bf Table  3:  The Prisoner's Dilemma}

 \begin{tabular}{lllccc}
   &       &             &\multicolumn{3}{c}{\bf Column}\\ 
&       &             &   {\it Silence}  &   & {\it Blame}     \\
 &   &  {\it Silence}      &     7,7 & $\rightarrow$  & -2, 8 \\ 
& {\bf Row} &&$\downarrow$& & $\downarrow$ \\
 &  &       {\it   Blame}     &      8,-2  & $\rightarrow$  & {\bf 0,0} \\ 
\multicolumn{6}{l}{\it Payoffs to: (Row,Column) } 
\end{tabular}
 \end{center}

This game will be repeated five times, and your
  objective is to get as high a summed, undiscounted, payoff as possible ({\it  not } just to get a higher summed payoff than anybody else).  Remember, too, that  there are lots of pairing of Row and Column in the class, so to just beat your immediate opponent would not even be the right tournament strategy. 


  The instructor will form groups of three students each to represent $Row$, and groups of one student each to represent $Column$. Each $Row$ group will play against multiple $Columns$.

 The five-repetition games  will be  different in how $Column$ behaves. 

\noindent
Game (i) Complete Information:  Column will   seek to maximize his payoff according to Table 3. 

\noindent
Game (ii) 80\% Tit-for-Tat: With 20\% probability, Column will  seek to maximize his payoff according to Table 3.  With 80\% probability, Column is a ``Tit-for-Tat Player'' and must use the strategy of ``Tit-for-Tat,'' starting with $Silence$ in Round 1 and after that imitating what Row did in the previous round. 

\noindent
Game (iii) 10\% Tit-for-Tat: With 90\% probability, Column will   seek to maximize his payoff according to Table 3.  With 10\% probability, Column  is a ``Tit-for-Tat Player'' and must use the strategy of ``Tit-for-Tat,'' starting with $Silence$ in Round 1 and after that imitating what Row did in the previous round.  The identities of the Game (ii). 

 The probabilities are independent, so although in Game(ii)   the most likely outcome is that 8 of 10  Column players use tit-for-tat, it is possible that 7 or 9 do, or even (improbably)   0 or 10. 


 
\newpage

\noindent
{\bf Instructor's Notes   }

   This is very like the classroom game for Chapter 5, except that now (a) there are only 5 rounds, and (b) in the second and third plays, there is some probability that $Column$ must use the tit-for-tat strategy.

  


Here are some assignments for player types. N represents a normal player and T represents tit-for-tat. 


\noindent
Game (ii):  (T T T N T) (T N T N T) (8 of 10 exactly)
 


\noindent
Game (iii): (N N N  N N) (N N N N N) (0 of 10--that can happen with a 10\% probability--in fact it has probability xxx of happening)


\newpage

\noindent
{\bf  The Equilibrium   }

    Let the probability that Column is a Tit-for-Tat player be $\theta$.  In the fifth round, Row will certainly choose Blame, and so will Column if Column is a Normal player.  How much cooperation can we get up until the fifth round? There are two kinds of equilibrium outcomes, depending on $\theta$.  

 \noindent 
 {\bf Cooperation:  High Probability $\theta$ of a TFT Column}\\
Row: SSSSB\\
TFT Column SSSSS\\
Normal Column: SSSBB 

 \noindent 
 {\bf Defection:   Low Probability $\theta$ of a TFT Column}\\
Row: BBBBB\\
TFT Column SBBBB\\
Normal Column: BBBBB\\ 

 Let us see for what values of $\theta$ the first kind of equilibrium exists. 
  Row's expected  payoffs from various strategies would be: 
\begin{equation}\label{e1}
\pi (SSSSB) =  \theta (7+7+7+7+8) + (1-\theta) (7+7+ 7 -2 + 0) = \theta (36) + (1-\theta) (19)
\end{equation}
\begin{equation}\label{e2}
\pi (SSSBB) =  \theta (7+7+7+8+0) + (1-\theta) (7+7+ 7+   0 + 0) = \theta (29) + (1-\theta) (21)
\end{equation}
\begin{equation}\label{e3}
\pi (SSBBB) =  \theta (7+7+8+0+0) + (1-\theta) (7+7+ 8 +0 + 0) = \theta (22) + (1-\theta) (22)
\end{equation}
  Thus, if $\theta$ is high, SSSSB is Row's best response, but if 
$\theta$ is low,  it is not. $\pi (SSSSB)  > \pi (SSSBB)$ if 
\begin{equation}\label{e4}
  \theta (36) + (1-\theta) (19)>
     \theta (29) + (1-\theta) (21),
\end{equation}
 that is, if $36\theta +19- 19 \theta> 29 \theta + 21- 21\theta$ or $9\theta > 2$ or $\theta > 2/9$. 

For even lower values of $\theta$, though, 
  Row thinks cooperation is  likely enough to break down that he wants to pre-emptively break it, so at least he can get one period of the high (Blame, Silence) payoff, even though if Column is TFT he has lost one extra period of cooperation.   $\pi (SSSBB)  < \pi (SSBBB)$ if 
\begin{equation}\label{e5}
\pi (SSSBB) =    \theta (29) + (1-\theta) (21)< \pi (SSBBB) =    \theta (22) + (1-\theta) (22)
 \end{equation}
 so $ 29 \theta  +21 + -21\theta < \pi (SSBBB) =  22$ or $\theta < 1/8$.  
 
To summarize:  the Cooperation equilibrium above will be an equilibrium only if $\theta  \geq 2/9$, and otherwise Row would deviate to one or another strategy of early Blame. 

What happens if  $\theta  < 2/9$?  The strategy profile (SSSBB, SSSBB) is not an equilibrium either. There cannot be an equilibrium in which the Row  and Normal Column players first chooses Blame in the same period  (except for the first period).  Either one would then deviate to choosing Blame one period earlier. But there cannot be an equilibrium in which the Normal Column player chooses Blame later than Row. And we have just ruled out, by making $\theta$ small, situations in which Row does not choose Blame till after the Normal Column. Thus, it must be that  in equilibrium Row chooses Blame starting in the first period. 

 
 


\newpage
 \pagestyle{empty}
 \oddsidemargin -.5in
%\begin{small}


\noindent
Name: 
\begin{center}
{\bf  Scoresheet for ``The Repeated Prisoner's Dilemma Under Incomplete Information: A Classroom Game for  Chapter 6''}
\end{center} 


The instructor will assign each  pairing, Row and Column, a pairing letter (A,B, C....)

 Write the  pairing letter and name of each player or group after ``Row:'' and ``Column:''  
  Each Row group will have  one  scoresheet for each of  its members, so it will be able to keep track of its rounds against its three Column opponents. 

 For each round, write the per-round payoff, either $-2$, 0, 7, or 8. Add them up for the total. 




  \noindent
Game (i): Complete Information
\begin{center}
\begin{tabular}{|l|| l|l|l|l|l||   l|  }
\hline
   & & & & &   &   \\
Round:    &     1$\;\;\;\;\;$ &2 $\;\;\;\;\;$&3 $\;\;\;\;\;$  &4$\;\;\;\;\;$ &5 $\;\;\;\;\;$   & Total \\
  & & & & &   &   \\
\hline
   & & & & &   &   \\
Row: \hspace{40pt}  & & & & &   &   \\
  & & & & &   &   \\
\hline
  & & & & &   &   \\
Column:   \hspace{40pt} & & & & &   &  \\
   & & & & &   &   \\
\hline
\end{tabular}
\end{center}


  \noindent
Game (ii): 80\% Chance of Tit-for-Tat
\begin{center}
\begin{tabular}{|l|| l|l|l|l|l||   l|  }
\hline
    & & & & &   &   \\
Round:    &     1$\;\;\;\;\;$ &2 $\;\;\;\;\;$&3 $\;\;\;\;\;$  &4$\;\;\;\;\;$ &5 $\;\;\;\;\;$   & Total \\
  & & & & &   &   \\
\hline
   & & & & &   &   \\
Row: \hspace{40pt}  & & & & &   &   \\
  & & & & &   &   \\
\hline
  & & & & &   &   \\
Column:   \hspace{40pt} & & & & &   &  \\
   & & & & &   &   \\
\hline
\end{tabular}
\end{center}


  \noindent
Game (iii):  10\% Chance of Tit-for-Tat
\begin{center}
\begin{tabular}{|l|| l|l|l|l|l||   l|  }
\hline
    & & & & &   &   \\
Round:    &     1$\;\;\;\;\;$ &2 $\;\;\;\;\;$&3 $\;\;\;\;\;$  &4$\;\;\;\;\;$ &5 $\;\;\;\;\;$   & Total \\
  & & & & &   &   \\
\hline
   & & & & &   &   \\
Row: \hspace{40pt}  & & & & &   &   \\
  & & & & &   &   \\
\hline
  & & & & &   &   \\
Column:   \hspace{40pt} & & & & &   &  \\
   & & & & &   &   \\
\hline
\end{tabular}
\end{center}


\newpage
 \pagestyle{empty}
 \oddsidemargin -.95in

 \begin{center}
{\bf GAME (i): COMPLETE INFORMATION}
\end{center}

 \begin{center}
\begin{tabular}{|l  || l | l|l|l|l| l|l|l|l|l | l| }
\hline
  & & & & &   & & & &  \\
Pairing:    &   A \hspace*{30pt} &   B \hspace*{30pt}&  C \hspace*{30pt}&    D\hspace*{30pt}&   E \hspace*{30pt}  &  F \hspace*{30pt} &   G  \hspace*{30pt}&    H \hspace*{30pt} &    I \hspace*{30pt}   \\
   & & & & &   & & & &   \\
\hline
\hline
   & & & & &   & & &  &  \\
Row's Name:    & & & & &   & & & &   \\
   & & & & &   & & &  &  \\
\hline
   & & & & &   & & &  &  \\
    & & & & &   & & &  &  \\
Row's Payoff  & & & & &   & & & &   \\
   & & & & &   & & &  &  \\
\hline
\end{tabular}
\end{center}



 \begin{center}
\begin{tabular}{|l  || l | l|l|l|l| l|l|l|l|l | l| }
\hline
  & & & & &   & & & &  \\
Pairing:     &   A \hspace*{30pt} &   B \hspace*{30pt}&  C \hspace*{30pt}&    D\hspace*{30pt}&   E \hspace*{30pt}  &  F \hspace*{30pt} &   G \hspace*{30pt}&    H \hspace*{30pt} &    I \hspace*{30pt}   \\
   & & & & &   & & & &   \\
\hline
\hline
   & & & & &   & & &  &  \\
Column's Name:     & & & & &   & & & &   \\
   & & & & &   & & &  &  \\
\hline
   & & & & &   & & &  &  \\
Column's Payoff:  & & & & &   & & & &   \\
   & & & & &   & & &  &  \\
\hline
\end{tabular}
\end{center}

\vspace*{36pt}
 
 \begin{center}
\begin{tabular}{|l  || l | l|l|l|l| l|l|l|l|l | l| }
\hline
  & & & & &   & & & &  \\
Pairing:    &  J  \hspace*{30pt} &   K \hspace*{30pt}&  L \hspace*{30pt}&   M\hspace*{30pt}&   N \hspace*{30pt}  &  O \hspace*{30pt} &   P \hspace*{30pt}&   Q \hspace*{30pt} &    R\hspace*{30pt}   \\
   & & & & &   & & & &   \\
\hline
\hline
   & & & & &   & & &  &  \\
Row's  Name:     & & & & &   & & & &   \\
   & & & & &   & & &  &  \\
\hline
   & & & & &   & & &  &  \\
Row's  Payoff:  & & & & &   & & & &   \\
   & & & & &   & & &  &  \\
\hline
\end{tabular}
\end{center}


 \begin{center}
\begin{tabular}{|l  || l | l|l|l|l| l|l|l|l|l | l| }
\hline
  & & & & &   & & & &  \\
Pairing:    &   J \hspace*{30pt} &   K \hspace*{30pt}&  L \hspace*{30pt}&    M\hspace*{30pt}&   N \hspace*{30pt}  &  O \hspace*{30pt} &   P \hspace*{30pt}&    Q \hspace*{30pt} &    R \hspace*{30pt}   \\
   & & & & &   & & & &   \\
\hline
\hline
   & & & & &   & & & &   \\
Col's Name:   & & & & &   & & & &   \\
   & & & & &   & & &  &  \\
\hline
   & & & & &   & & &  &  \\
 Col's Payoff:  & & & & &   & & & &   \\
   & & & & &   & & &  &  \\
\hline
\end{tabular}
\end{center}


 \newpage
 \pagestyle{empty}
 \oddsidemargin -.95in

 \begin{center}
{\bf GAME (ii):   80\% TIT-FOR-TAT}
\end{center}

 \begin{center}
\begin{tabular}{|l  || l | l|l|l|l| l|l|l|l|l | l| }
\hline
  & & & & &   & & & &  \\
Pairing:    &   A \hspace*{30pt} &   B \hspace*{30pt}&  C \hspace*{30pt}&    D\hspace*{30pt}&   E \hspace*{30pt}  &  F \hspace*{30pt} &   G  \hspace*{30pt}&    H \hspace*{30pt} &    I \hspace*{30pt}   \\
   & & & & &   & & & &   \\
\hline
\hline
   & & & & &   & & &  &  \\
Row's Name:    & & & & &   & & & &   \\
   & & & & &   & & &  &  \\
\hline
   & & & & &   & & &  &  \\
    & & & & &   & & &  &  \\
Row's Payoff  & & & & &   & & & &   \\
   & & & & &   & & &  &  \\
\hline
\end{tabular}
\end{center}



 \begin{center}
\begin{tabular}{|l  || l | l|l|l|l| l|l|l|l|l | l| }
\hline
  & & & & &   & & & &  \\
Pairing:     &   A \hspace*{30pt} &   B \hspace*{30pt}&  C \hspace*{30pt}&    D\hspace*{30pt}&   E \hspace*{30pt}  &  F \hspace*{30pt} &   G \hspace*{30pt}&    H \hspace*{30pt} &    I \hspace*{30pt}   \\
   & & & & &   & & & &   \\
\hline
\hline
   & & & & &   & & &  &  \\
Column's Name:     & & & & &   & & & &   \\
   & & & & &   & & &  &  \\
\hline
   & & & & &   & & &  &  \\
Column's Payoff:  & & & & &   & & & &   \\
   & & & & &   & & &  &  \\
\hline
\end{tabular}
\end{center}

\vspace*{36pt}
 
 \begin{center}
\begin{tabular}{|l  || l | l|l|l|l| l|l|l|l|l | l| }
\hline
  & & & & &   & & & &  \\
Pairing:    &  J  \hspace*{30pt} &   K \hspace*{30pt}&  L \hspace*{30pt}&   M\hspace*{30pt}&   N \hspace*{30pt}  &  O \hspace*{30pt} &   P \hspace*{30pt}&   Q \hspace*{30pt} &    R\hspace*{30pt}   \\
   & & & & &   & & & &   \\
\hline
\hline
   & & & & &   & & &  &  \\
Row's  Name:     & & & & &   & & & &   \\
   & & & & &   & & &  &  \\
\hline
   & & & & &   & & &  &  \\
Row's  Payoff:  & & & & &   & & & &   \\
   & & & & &   & & &  &  \\
\hline
\end{tabular}
\end{center}


 \begin{center}
\begin{tabular}{|l  || l | l|l|l|l| l|l|l|l|l | l| }
\hline
  & & & & &   & & & &  \\
Pairing:    &   J \hspace*{30pt} &   K \hspace*{30pt}&  L \hspace*{30pt}&    M\hspace*{30pt}&   N \hspace*{30pt}  &  O \hspace*{30pt} &   P \hspace*{30pt}&    Q \hspace*{30pt} &    R \hspace*{30pt}   \\
   & & & & &   & & & &   \\
\hline
\hline
   & & & & &   & & & &   \\
Col's Name:   & & & & &   & & & &   \\
   & & & & &   & & &  &  \\
\hline
   & & & & &   & & &  &  \\
 Col's Payoff:  & & & & &   & & & &   \\
   & & & & &   & & &  &  \\
\hline
\end{tabular}
\end{center}



\newpage
 \pagestyle{empty}
   \oddsidemargin -.95in

 \begin{center}
{\bf GAME (iii):   10\% TIT-FOR-TAT}
\end{center}

  \begin{center}
\begin{tabular}{|l  || l | l|l|l|l| l|l|l|l|l | l| }
\hline
  & & & & &   & & & &  \\
Pairing:    &   A \hspace*{30pt} &   B \hspace*{30pt}&  C \hspace*{30pt}&    D\hspace*{30pt}&   E \hspace*{30pt}  &  F \hspace*{30pt} &   G  \hspace*{30pt}&    H \hspace*{30pt} &    I \hspace*{30pt}   \\
   & & & & &   & & & &   \\
\hline
\hline
   & & & & &   & & &  &  \\
Row's Name:    & & & & &   & & & &   \\
   & & & & &   & & &  &  \\
\hline
   & & & & &   & & &  &  \\
    & & & & &   & & &  &  \\
Row's Payoff  & & & & &   & & & &   \\
   & & & & &   & & &  &  \\
\hline
\end{tabular}
\end{center}



 \begin{center}
\begin{tabular}{|l  || l | l|l|l|l| l|l|l|l|l | l| }
\hline
  & & & & &   & & & &  \\
Pairing:     &   A \hspace*{30pt} &   B \hspace*{30pt}&  C \hspace*{30pt}&    D\hspace*{30pt}&   E \hspace*{30pt}  &  F \hspace*{30pt} &   G \hspace*{30pt}&    H \hspace*{30pt} &    I \hspace*{30pt}   \\
   & & & & &   & & & &   \\
\hline
\hline
   & & & & &   & & &  &  \\
Column's Name:     & & & & &   & & & &   \\
   & & & & &   & & &  &  \\
\hline
   & & & & &   & & &  &  \\
Column's Payoff:  & & & & &   & & & &   \\
   & & & & &   & & &  &  \\
\hline
\end{tabular}
\end{center}

\vspace*{36pt}
 
 \begin{center}
\begin{tabular}{|l  || l | l|l|l|l| l|l|l|l|l | l| }
\hline
  & & & & &   & & & &  \\
Pairing:    &  J  \hspace*{30pt} &   K \hspace*{30pt}&  L \hspace*{30pt}&   M\hspace*{30pt}&   N \hspace*{30pt}  &  O \hspace*{30pt} &   P \hspace*{30pt}&   Q \hspace*{30pt} &    R\hspace*{30pt}   \\
   & & & & &   & & & &   \\
\hline
\hline
   & & & & &   & & &  &  \\
Row's  Name:     & & & & &   & & & &   \\
   & & & & &   & & &  &  \\
\hline
   & & & & &   & & &  &  \\
Row's  Payoff:  & & & & &   & & & &   \\
   & & & & &   & & &  &  \\
\hline
\end{tabular}
\end{center}


 \begin{center}
\begin{tabular}{|l  || l | l|l|l|l| l|l|l|l|l | l| }
\hline
  & & & & &   & & & &  \\
Pairing:    &   J \hspace*{30pt} &   K \hspace*{30pt}&  L \hspace*{30pt}&    M\hspace*{30pt}&   N \hspace*{30pt}  &  O \hspace*{30pt} &   P \hspace*{30pt}&    Q \hspace*{30pt} &    R \hspace*{30pt}   \\
   & & & & &   & & & &   \\
\hline
\hline
   & & & & &   & & & &   \\
Col's Name:   & & & & &   & & & &   \\
   & & & & &   & & &  &  \\
\hline
   & & & & &   & & &  &  \\
 Col's Payoff:  & & & & &   & & & &   \\
   & & & & &   & & &  &  \\
\hline
\end{tabular}
\end{center}

\end{small}

\end{document}
