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September 9, 2001


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 REVISED 3027 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 combination is strictly higher than
the payoff from any other strategy, so only one strategy
combination 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. Nor does it help uniqeness to   eliminate    weakly
dominated
strategies simultaneously at each round. 
 

Table 1.4's   Iteration Path Game     illustrates this. The strategy combinations
$(r_1, c_1)$ and  $(r_1, c_3)$   are both   iterated dominance equilibria  because
each   can be generated by iterated deletion. One order the  deletion can proceed
in  is  $(r_2, c_2, c_1, r_3)$, leaving $(r_1, c_3)$.  First $r_2$ is deleted,
since it is dominated by $r_1$, then  $c_2$, since it is dominated by $c_3$.  This
leaves the 2-by-2 game of table 1.5. In that game, $c_1$ is dominated by $c_3$, and
so would next be deleted. Finally, $r_3$ is deleted because it is dominated by
$r_1$, and  we are left with $(r_1, c_3)$. If, however,    deletion   proceeds in
the order $(r_3, c_3, c_2, r_2)$    it leaves $(r_1, c_1)$ as the game shrinks to
its northwest corner. And if  dominated strategies are deleted  simultaneously at
each iteration, $r_3$, $r_2$, and $c_2$ would be deleted in the first round,
leaving    $(r_1, c_1)$ and $(r_1, c_3)$  with no further iterations possible.

  
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 \begin{center}
{\bf Table 1.4 }  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}


 \begin{center}
{\bf Table 1.5}   The    Iteration Path Game After Two Iterations    

  
 \begin{tabular}{lllccc}
  &       &             &\multicolumn{3}{c}{\bf Column}\\
  &       &             &    $c_1$    &    &  $ c_3$       \\
 \multicolumn{6}{l}{ }\\
   &   &  $  r_1$      &    2,12    &     & {\bf  1,12} \\
 \multicolumn{6}{l}{ }\\
  & {\bf Row:} &   &  & &    \\
 \multicolumn{6}{l}{ }\\
 &  &     $ r_3$      &   0,12      &    &  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.    
    
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