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january 29, 1996
  
\section*{  4 DYNAMIC GAMES WITH SYMMETRIC INFORMATION} 
 
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 {\bf The Open-Set Problem in ``Nuisance Suits II''} 

   The equilibrium in ``Nuisance Suits II'' is  only  a  weak Nash equilibrium.   If the parameters are such that $s =\gamma x + d= 60$, for example, why does the plaintiff risk holding out for 60  when he might be rejected and  most likely receive  0 at trial, when he could offer 59 and give the  defendant a strong incentive to accept? 

One  answer  is that no other   equilibrium exists besides   $s=60$. Offering 59 cannot be part of an equilibrium, because it is dominated by offering 59.9;   a set must be bounded {\it and closed} to guarantee that a maximum exists.  

 A second answer is that under the assumptions of rationality and Nash equilibrium the  objection's premise is false because  the plaintiff   bears no risk whatsoever in offering $s=60$. It is fundamental to Nash equilibrium that each players  believes that the others will follow equilibrium behavior. 


A third answer is that the problem is an artifact of using a model with a continuous strategy space, and it disappears if the strategy space is made discrete.  Assume that $s$ can only take values in multiples of 0.01, so it could be 59.0, 59.01, 59.02, and so forth, but not 59.001 or 59.002.    The settlement part of the game will now have two perfect  equilibria. In the strong equilibrium E1, $s=59.99$ and the defendant rejects any offer  of $s <60$. In the weak equilibrium E2, $s=60$ and the defendant rejects any offer  of $s \leq 60$. The difference is trivial, so the   discrete strategy space  has made the model more complicated without any extra insight.\footnote{A good example  of the ideas of discrete money values and sequential rationality is in Robert Louis Stevenson's story, ``The Bottle Imp''  (Stevenson (1987).  The   imp grants the wishes of the bottle's owner  but will seize his soul if he dies in possession of  it. Although the  bottle cannot be  given away,   it can be sold, but only at a price less than that for which it was purchased.} 

   
  
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