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\begin{LARGE}

\noindent
 28 August 2009    
 Eric Rasmusen, Erasmuse@indiana.edu.
http://www.rasmusen.org/.

 

\noindent
 The essential elements of a game are {\bf players}, {\bf actions},
{\bf payoffs}, and {\bf information}.

.   These are collectively
known as the {\bf rules of the game}, and the modeller's objective is to
describe a situation  in terms of the rules of a game so as to explain what
will happen in that situation.

  Trying to maximize their payoffs, the players
will devise plans known as {\bf strategies} that pick actions depending on the
information that has arrived at each moment. 

The combination of strategies
chosen by each player is known as the {\bf equilibrium}.

 Given an equilibrium,
the modeller can see what actions come out of the conjunction of all the
players'  plans, and this tells him the {\bf outcome} of the game.

 How would you describe the supply and demand of gasoline  in these terms? 

\newpage

  \noindent
 {\bf A Story to Model}
 

An  entrepreneur is  trying to decide whether to start a dry cleaning store in a   town   already served by one dry cleaner.

 We will call the two firms ``NewCleaner'' and ``OldCleaner.'' NewCleaner  is uncertain about whether the economy will be in a recession or not,  which will affect how much consumers pay for dry cleaning,   and must also worry about whether OldCleaner will respond to entry with a price war or by keeping its initial high prices.  OldCleaner is a well-established firm, and it would survive any price war, though its profits would fall. NewCleaner must itself decide whether to initiate a price war or to charge high prices, and must also decide what kind of equipment to buy, how many workers to hire, and so forth.

\newpage

\noindent {\it{\bf Players} are the individuals who make decisions. Each
player's goal is to maximize his utility by choice of actions.}
 

\bigskip

\noindent {\it An {\bf action} or {\bf move} by player $i$, denoted $a_{i},$ is
a choice he can make.}

\noindent {\it Player $i$'s {\bf action set}, $A_i = \{a_{i}\}$, is the entire
set of actions available to him.}

\noindent {\it An {\bf action combination} is a list $a =\{a_i\},\; (i=
1,\ldots,n)$ of one action for each of the $n$ players in the game.}

Newcleaner's action set:  $  \{ Enter, Stay \;
Out\}$.   Oldcleaner's action set to be  simple: it is to
choose price from   $  \{ Low,   High\}$.


\newpage

\noindent {\it {\bf Nature} is a pseudo-player who takes random actions at specified points in the game with specified probabilities.}

  In the Dry Cleaners Game, we will model  the possibility of recession as a move by Nature.  With probability 0.3, Nature decides that there will be a recession, and with probability 0.7 there will not. Even if the players always took the same actions, this random move means that the model would yield more than just one prediction. We say that there are different {\bf realizations} of a game depending on the results of random moves.

 Why is Nature not a real player? 


\newpage
 

\noindent {  By player $i$'s {\bf payoff} $\pi_i $, we mean
either:\\
   (1) The utility player $i$ receives after all players and Nature have
picked their strategies and the game has been played out; or\\ (2) His  expected
utility at the start of the game.   }



 
    \begin{center} {\bf Table 1a:  The Dry Cleaners Game: Normal Economy}

 \begin{tabular}{lllccc} &       &             &\multicolumn{3}{c}{\bf
OldCleaner}\\ &       &             &   {\it Low price}  &   & {\it High price}
\\ &   &  {\it Enter}      &     -100, -50 &    &100, 100 \\ & {\bf NewCleaner}
&& & &   \\ &  &       {\it   Stay Out}     &      0,50   &    &  0,300  \\
\multicolumn{6}{l}{ Payoffs to: (NewCleaner, OldCleaner) in thousands of dollars
} \end{tabular} \end{center}

  \begin{center} {\bf Table 1b:  The Dry Cleaners Game: Recession}

 \begin{tabular}{lllccc} &       &             &\multicolumn{3}{c}{\bf
OldCleaner}\\ &       &             &   {\it Low price}  &   & {\it High price}
\\ &   &  {\it Enter}      &     -160, -110 &    &40, 40 \\ & {\bf NewCleaner}
&& & &   \\ &  &       {\it   Stay Out}     &      0,-10   &    &  0,240  \\
\multicolumn{6}{l}{ Payoffs to: (NewCleaner, OldCleaner) in thousands of dollars
} \end{tabular} \end{center}

\newpage

    Information is modelled using the concept of the {\bf information set. }

 The elements of the information set are the different
values ofa  variable  that the player thinks are possible. If the information set has many
elements, there are many values the player cannot rule out; if it has one
element, he knows the value precisely.  

            

 

       It is convenient to lay out information and actions together in an {\bf
order of play}.  Here is the  order of play we have specified for the Dry
Cleaners Game:

 \noindent
  1   Newcleaner chooses its entry decision from $  \{ Enter,  Stay \;
Out\}$.\\ 
 \noindent 2  Oldcleaner chooses its  price   from $  \{ Low,   High\}
$.  \\ \noindent 3  Nature picks demand, $D$, to be $Recession$ with probability
0.3 or $Normal$ with probability 0.7.\\

\newpage

 

\noindent {\it The {\bf outcome} of the game is a set of interesting elements
that the modeller picks from the values of actions, payoffs, and other variables
after the game is played out.}

    

  Decision theory  is like game theory  with just  one player.   
 


 \includegraphics[width=150mm]{fig01-01.jpg}



\begin{center} {\bf Figure 1:   The Dry Cleaners Game as a Decision Tree   }
\end{center}

     

 \newpage




\includegraphics[width=150mm]{fig01-02.jpg}

\begin{center} {\bf Figure  2:   The Dry Cleaners Game as a Game  Tree }
\end{center}

  
 
 
\newpage

 \noindent {\it Player $i$'s {\bf strategy} $s_i$ is a rule that tells him which
action to choose at each instant of the game, given his information set.}

 \noindent {\it Player $i$'s {\bf strategy set} or {\bf strategy space} $S_i =
\{ s_i\}$ is the set of strategies available to him. }

 \noindent {\it A {\bf strategy profile} $s=(s_1,\ldots,s_n)$ is a list
consisting of one strategy for each of the {\rm n} players in the game.}
 
 
In The  Dry Cleaners Game,  the  strategy set for NewCleaner is just {\it \{
Enter, Stay Out \} }, since NewCleaner moves first and is not reacting to any
new information.  The strategy set for  OldCleaner, though, is

 \begin{center} $$ \left\{ \begin{tabular}{l} High Price if NewCleaner Entered,
Low Price otherwise\\ Low Price if NewCleaner Entered,  High
Price otherwise\\ High Price No Matter What\\ Low Price No Matter
What\\ \end{tabular} \right\} $$ \end{center}

   
  

\newpage

\noindent {\bf Equilibrium}

\noindent  The single outcome of  {\it NewCleaner Enters} would result
from either of the following two strategy profiles:

{\it \begin{center} $$\left\{ \begin{tabular}{l} High Price if NewCleaner
Enters,  Low Price otherwise\\ Enter\\ \end{tabular} \right\} $$
\end{center} }

{\it \begin{center} $$\left\{ \begin{tabular}{l} Low Price if NewCleaner Enters,
High Price if NewCleaner otherwise\\ Enter \end{tabular} \right\} $$
\end{center} }

 Predicting what happens consists of selecting one or more strategy profiles  as
being the most rational behavior by the players acting to maximize their
payoffs.

 \noindent {\it An {\bf equilibrium} $s^* = (s_1^*,\ldots,s_n^*)$ is a strategy
profile consisting of a best strategy for each of the {\rm n} players in the
game.}

 The {\bf equilibrium strategies} are the strategies players pick in trying to
maximize their individual payoffs.
 
 Equilibrium= Equilibrium Strategy Profile = set of strategies 


 Equilibrium outcome = set of values of outcome variables 


 \newpage



 The  equilibrium concept defines ``best strategy''.

\noindent {\it An {\bf equilibrium concept} or {\bf solution concept} $F:
\{S_1,\ldots, S_n,\pi_1,\ldots,\pi_n \} \rightarrow s^*$ is a rule that defines
an equilibrium based on the possible strategy profiles and the payoff
functions.}

   A given equilibrium concept might lead to no equilibrium existing, or multiple equilibria. 
 
  \newpage

 

 \noindent {  For any vector $y= (y_1,\ldots,y_n)$, denote by $y_{-i}$ the
vector $(y_1,\ldots,y_{i-1},y_{i+1},\ldots,y_n)$, which is the portion of $y$
not associated with player $i$.}

  \noindent Using this notation, $s_{-Smith}$, for instance, is the profile of
strategies of every player except player $Smith$.  That profile is of great
interest to  Smith, because he uses it to help choose his own strategy, and the
new notation helps define his best response.

 \noindent
 {\it Player $i$'s {\bf best response} or {\bf best reply} to the strategies
$s_{-i}$ chosen by the other players is the strategy $s^*_i$ that yields him the
greatest payoff; that is,}
  \begin{equation} \label{e3}
 \pi_i(s_i^*,s_{-i}) \geq \pi_i(s_i',s_{-i})\;\;\; \forall s_i' \neq s_i^*.
\end{equation}

\newpage
 
 
\noindent
 {\it  The  strategy $s_i^d$ is  a {\bf dominated strategy} if it is strictly
inferior to some other strategy no matter what strategies the other players
choose, in the sense that whatever strategies they pick, his payoff is lower
with $s_i^d$. Mathematically, $s_i^d$ is dominated if  there exists  a single
$s_i'$  such that}
\begin{equation} \label{e4}
 \pi_i(s_i^d,s_{-i}) < \pi_i(s_i',s_{-i}) \;\; \forall s_{-i}.
 \end{equation}


 $s_i^d $ is not a dominated strategy if there is no $ s_{-i}$ to
which it is the  best response, but sometimes the better strategy is $s_i'$  and
sometimes it is $s_i''$. 

In that case, $s_i^d $ could have the redeeming feature
of being a good compromise strategy for a player who  cannot predict  what the
other players are going to do

  A dominated strategy is unambiguously inferior to
some single other strategy.

   \newpage

 \noindent
{\it The strategy $s_i^*$ is a {\bf dominant strategy} if it is a player's
strictly best response to {\rm any} strategies the other players might pick, in
the sense that whatever strategies they pick, his payoff is highest with
$s_i^*$.  Mathematically,}
\begin{equation} \label{e4a}
 \pi_i(s_i^*,s_{-i}) > \pi_i(s_i',s_{-i}) \;\; \forall s_{-i},\;\; \forall s_i'
\neq s_i^*.
\end{equation}

 \noindent {\it A {\bf dominant-strategy equilibrium} is a strategy profile
consisting of each player's dominant strategy.}

 
 \newpage

\begin{center}
{\bf Table  2:  The Prisoner's Dilemma}

 \begin{tabular}{lllccc} &       &             &\multicolumn{3}{c}{\bf Column}\\
&       &             &   {\it Silence}  &   & {\it Blame}     \\ &   &  {\it
Silence}      &     -1,-1 &    & -10, 0 \\ & {\bf Row} &&  & &  \\
  &  &       {\it   Blame}     &      0,-10  &    & {\bf - 8,-8} \\
\multicolumn{6}{l}{\it Payoffs to: (Row, Column) } \end{tabular}
\end{center}
 
 



\newpage

  
\begin{center} {\bf Table  3: ITERATED DOMINANCE---  The Battle of the Bismarck Sea   }

\begin{tabular}{lcccc} &           & \multicolumn{3}{c}{\bf Imamura}\\ &
&    North    &  &  South    \\ &     North     &   {\bf 2,-2} &    & $2,-2$
\\
{\bf   Kenney} &                 &          &      &        \\ &     South     &
$1,-1$ &    &    $3,-3$ \\
  & \multicolumn{4}{l}{\it  Payoffs to: (Kenney, Imamura) } \\
  \end{tabular} \end{center}
 

\noindent {  Strategy   $s'_i$ is  {\bf weakly dominated} if  there exists
some other strategy $s''_i$ for player $i$ which is  possibly better and never
worse,  yielding a higher  payoff  in some strategy profile and never yielding a
lower payoff. Mathematically, $s'_i$ is weakly dominated if there exists $s''_i$
such that}
\begin{equation} \label{e5}
  \begin{array}{l}
 \pi_i(s_i'',s_{-i}) \geq \pi_i(s_i',s_{-i})\;\;\; \forall s_{-i}, \; \; {\rm
and}\; \\
 \\
\pi_i(s_i'',s_{-i}) > \pi_i(s_i',s_{-i})\;\;\;  \;{\rm for \;some} \; s_{-i}.
 \end{array}
 \end{equation}
 

 \noindent {\it An {\bf iterated-dominance equilibrium} is a strategy profile
found by deleting a weakly dominated strategy from the strategy set of one of
the players, recalculating to find which remaining strategies are weakly
dominated, deleting one of them, and continuing the process until only one
strategy remains for each player.}
 
\newpage

 

 \begin{center} {\bf Table  4:   The    Iteration Path Game  }

 \begin{tabular}{lllccc} &       &             &\multicolumn{3}{c}{\bf Column}\\
&       &             &    $c_1$    &  $c_2$ &  $ c_3$       \\ \multicolumn{6}
{l}{ }\\ &   &  $  r_1$      & {\bf  2,12 }   &  1,10  & {\bf  1,12} \\
\multicolumn{6}{l}{ }\\ & {\bf Row} & $r_2$ &0,12 & 0,10 & 0,11  \\
\multicolumn{6}{l}{ }\\ &  &     $ r_3$      &   0,12      & 1,10  &  0,13 \\
\multicolumn{6}{l}{ }\\ \multicolumn{6}{l}{  }\\ \multicolumn{6}{l}{\it Payoffs
to: (Row, Column) }\\ \end{tabular}
  \end{center}

The strategy profiles  $(r_1, c_1)$
and $(r_1, c_3)$ are both   iterated dominance equilibria, because each of those
strategy profiles can be found by iterated deletion.

 The deletion can proceed in
the order  $(r_3, c_3, c_2, r_2)$,  or in the order    $(r_2, c_2, c_1, r_3)$.



\newpage

 \noindent
  {\bf Zero-Sum Games}

 

 \noindent
 { A {\bf zero-sum game} is a game in which the sum of the payoffs of all the
players is zero whatever strategies they choose.  }

  

\begin{center} {\bf      MATCHING PENNIES (not in chapter 1)  }

Smith wins the penny  if the two pennies match; otherwise Jones wins. 

 \begin{tabular}{lllccc} &       &             &\multicolumn{3}{c}{\bf Jones}\\
&       &             &    $Heads$  & & $Tails$  \\
 &   &  $Heads$   &     
1, -1 &    & $-1,1$ \\
  & {\bf Smith} &     &    & &   \\
  &  &       $Tails$     &
$-1, 1$ &    &  1,1  \\ 
& & & &\\
 \end{tabular} \end{center}
 \vspace{-24pt}
 {\it Payoffs  to:  (Smith, Jones).    }
   
 
 \newpage
 

   

\begin{center} {\bf Table  5: { Boxed Pigs} (NASH EQUILIBRIUM)  }

 \begin{tabular}{lllcc c} &       &       &\multicolumn{3}{c}{\bf Small Pig}\\
 &       &           & Press &  & Wait      \\
         & &Press& $ 5,1 $ & $\rightarrow$ &{\bf \fbox{4}, \fbox{4}} \\
  &{\bf Big Pig}&         &   $\downarrow$  &   & $\uparrow$    \\
 &       &   Wait  &  $\fbox{9}, -1 $ & $\rightarrow$  &  0, \fbox{0 } \\
 &       &       &\multicolumn{3}{c}{   }\\
  \end{tabular}  \end{center}
 \vspace{-24pt}
 {\it Payoffs  to: (Big Pig, Small Pig). Arrows show how a player can increase
his payoff. Best-response payoffs are boxed.  }

 

 

\noindent
  {\it The strategy profile $s^*$ is a {\bf Nash equilibrium} if no player has
incentive to deviate from his strategy given that the other players do not
deviate.  Formally, }
  \begin{equation} \label{e6}
\forall i,\;\; \pi_i(s_i^*,s_{-i}^*) \geq \pi_i(s_i',s_{-i}^*),\;\; \forall
s_i'.
  \end{equation}

 
\newpage
   

 

\begin{center} {\bf Table  6:  The  Modeller's Dilemma   }\\
 \begin{tabular}{lllccc} &       &             &\multicolumn{3}{c}{\bf Column}\\
&       &             &    $Silence$   &   &  $Blame $     \\
 &   &  $Silence$       &      \begin{picture}(10,15) \put(0,-2){\dashbox{3}(8,
12) {\bf 0}} \end{picture}, \begin{picture}(10,15) \put(0,-2){\dashbox{3}(8,12)
{\bf 0}} \end{picture} & $\leftrightarrow$  & $-10$,     \begin{picture}(10,15)
\put(0,-2){\dashbox{3}(8,12) {0}} \end{picture}  \\
 & {\bf Row} &&$\updownarrow$& & $\downarrow$ \\
 &  &         $ Blame $        &        \begin{picture}(10,15) \put(0,-2)
{\dashbox{3}(8,12) {0}} \end{picture},-10  & $\rightarrow$  & {\bf \fbox{ -8},
\fbox{-8}} \\
 \end{tabular} 
 \end{center}
\vspace{-24pt}
 {\it Payoffs  to: (Row, Column) . Arrows show how a player can increase his
payoff.    }

What are the equilibria?

What would you predict to happen? 

 
\newpage

 
\begin{center} {\bf Table  7:  Battle  of the Sexes   }\\
 \begin{tabular}{lllccc}
 &       &             &\multicolumn{3}{c}{\bf Woman}\\ &       &             &
$Prize \; Fight$  & & $Ballet$  \\ &   &  $Prize \; Fight$   &  {\bf 2,1} &
$\leftarrow$  & $0$, 0 \\ & {\bf Man} &     & $\uparrow$  & & $\downarrow$ \\
 &  &       $Ballet$     &      $0$, $0$ & $\rightarrow$  & {\bf 1,2} \\ & & &
&\\
 \end{tabular}
 \end{center}
 
 {\it Payoffs  to: (Man, Woman). Arrows show how a player can increase his
payoff.    }

 

\newpage

\begin{center} {\bf Table  8:     Ranked Coordination    }

 \begin{tabular}{lllccc}
 &       &             &\multicolumn{3}{c}{\bf Jones}\\
&       &             &    $Large$  & & $Small$  \\
 &   &  $Large$   &     {\bf 2,2} & $\leftarrow$  & $-1,-1$ \\
 & {\bf Smith} &     & $\uparrow$  & & $\downarrow$ \\
 &  &       $Small$     &      $-1,-1$ & $\rightarrow$  & {\bf 1,1} \\ & & & &\\
 \end{tabular}
 \end{center}
 \vspace{-24pt}
 {\it Payoffs  to:  (Smith, Jones). Arrows show how a player can increase his
payoff.   }

\newpage


 
\begin{center} {\bf Table  9:   Dangerous  Coordination  (ASSURANCE GAME) (STAG HUNT) }

 \begin{tabular}{lllccc} &       &             &\multicolumn{3}{c}{\bf Jones}\\
&       &             &    $Large$  & & $Small$  \\ &   &  $Large$   &     {\bf
2,2} & $\leftarrow$  & $-1000, -1$ \\
  & {\bf Smith} &     & $\uparrow$  & & $\downarrow$ \\ &  &       $Small$     &
$-1, -1$ & $\rightarrow$  & {\bf 1,1} \\ & & & &\\
 \end{tabular} \end{center}
 \vspace{-24pt}
 {\it Payoffs  to:  (Smith, Jones). Arrows show how a player can increase his
payoff.    }

(Large, Large) is Nash and Pareto superior (payoff dominant) but not ``risk dominant'': give a 50-50 chance of the other player choosing each strategy, each player would choose SMALL, as ``safer.'' 

\newpage

 
   Two firms are choosing outputs $Q_1$ and  $Q_2$  simultaneously.
The Nash equilibrium is a pair of numbers ($Q_1^*, Q_2^*$) such that neither
firm would deviate unilaterally. 


This troubles the beginner, who says to
himself,   \begin{quotation}
\noindent ``Sure, Firm 1  will pick  $Q_1^*$  if it thinks Firm 2 will pick
$Q_2^*$. But Firm 1  will realize that  if it makes  $Q_1$  bigger, then Firm 2
will react by making  $Q_2$  smaller.  So the situation is much more
complicated, and  ($Q_1^*, Q_2^*$) is not a Nash equilibrium. Or, if it is, Nash
equilibrium is a bad equilibrium concept.''
 \end{quotation} 

  
\newpage

 \noindent
   {\bf Focal Points}
 

\noindent 1  Circle one of the following numbers:  100, 14, 15, 16, 17, 18.

\noindent 2 Circle one of the following numbers 7, 100, 13, 261, 99, 666.

\noindent 3  Name Heads or Tails.

\noindent 4  Name Tails or Heads.

\noindent 5 You are to split a pie, and get nothing if your proportions add to
more than 100 percent.

\noindent 6  You are to meet somebody in New York City.  When?  Where?

 

 

    
 


 

 

            
 \end{LARGE}

\end{document}

