September 9, 2001

ORIGINAL 3018 characters

 



 Another difficulty is multiple equilibria. The
dominant strategy equilibrium of any game is unique  if
it exists. Each player has at most  one strategy  whose
payoff in any strategy profile is strictly higher than
the payoff from any other strategy, so only one strategy
profile can be formed out of dominant strategies.      A   
strong  iterated dominance equilibrium is unique  if it
exists. A   weak  iterated dominance  equilibrium may not
be,   because the order in which strategies are deleted can
matter to the final solution. If  all the weakly dominated
strategies are eliminated simultaneously at each round
of elimination, the  resulting equilibrium is unique,  if
it exists, but  possibly no strategy profile will remain.   
   
 The easiest example is table 1.4's  Swiss Cheese Game.  
Every strategy is weakly dominated for every player.
Thus, one iterated dominance equilibrium is {\it (Up,
Left)}, found by first eliminating Smith's {\it Down}
and then Jones's {\it Right}, but {\it (Down, Right)} is
also an iterated dominance equilibrium. And, in fact,
every strategy combination is also a weak dominant
strategy equilibrium as well as an iterated dominance
equilibrium.  

 
 
\begin{center}
{\bf Table 1.4     The  Swiss Cheese Game   } 

  
 \begin{tabular}{lllccc}
  &       &             &\multicolumn{3}{c}{\bf Jones}\\
  &       &             &  {\it  Left}    &   &  $ Right$       \\
  &   &  $ Up$      &    {\bf  0,0} & $\leftrightarrow$  & {\bf 0,0} \\
 & {\bf Smith:} &&$\updownarrow$& & $\updownarrow$ \\
&  &    {\it Down }      &      {\bf  0,0}  & $\leftrightarrow$  & {\bf 0,0} \\  
\multicolumn{6}{l}{\it Payoffs to: (Smith, Jones) }
\end{tabular}                
\end{center}

  The Swiss Cheese Game is pathological,  but it is not hard to come up with less
obvious examples, such as the    Iteration Path Game    of  table  1.5. 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    $(r_2, c_2, c_1, r_3)$.


  
 \begin{center}
{\bf Table 1.5   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}

 
 

Despite these problems, deletion of weakly dominated strategies is a useful tool,
and it is part of
many more complicated equilibrium concepts, such as the subgame perfectness concept
of section
4.1.