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                     \begin{center}
             \begin{large}
    {\bf  Predictable and Unpredictable Error in  Tort Awards: The Effect of 
Plaintiff Self Selection  and Signalling}\\
             \end{large}
                     
                    June 8, 1995 \\
                    \bigskip      
               Eric Rasmusen       \\          
  
Published, {\it 
International Review of Law and Economics }(September 1995) 15: 323-345.\\


                    {\it Abstract} 
                    \end{center}
                        If a potential tort plaintiff can predict that
the court will overestimate damages he is more likely to bring suit,
  but if the court is aware of this, it  will  adjust its
awards accordingly. In general, court error 
 implies that the court should moderate extreme awards whether they
are high or low, because of regression towards the mean.
Predictable error, however, tends to push the optimal adjustment
downwards and unpredictable error pushes it upwards, because of
plaintiff selection and signalling,  respectively.  The expectation   of either 
kind  of error   leads plaintiffs to bring meritless suits. 

    
                 
\begin{small}
       

                \noindent 
\hspace*{20pt} 2000: Eric Rasmusen, 	Professor of Business Economics and 
Public Policy and Sanjay Subhedar Faculty Fellow,   Indiana University,
Kelley School of Business, BU 456,   
  1309 E 10th Street,
  Bloomington, Indiana, 47405-1701.
  Office: (812) 855-9219.   Fax: 812-855-3354. Erasmuse@indiana.edu.   
Php.indiana.edu/$\sim$erasmuse.
 
   
 \vspace{ 10pt}
 
I would like to thank A. Mitchell Polinsky and seminar participants
at George Mason Law School, the Antitrust Division of the Justice
Department, and the American Law and Economics Association 1992
Meetings for helpful comments. This work was begun  while
the author was an  Olin Faculty Fellow at Yale Law School.  

  

            \end{small}
%%-----------------------------%------------------------------------------

\newpage

 
\begin{center} {  1. INTRODUCTION} 
 \end{center}

     A fundamental asymmetry in lawsuits is that the plaintiff 
  files suit, not the defendant.  This asymmetry  is more than definitional, 
because if   no suit is filed,   losses lie where they fall,   a result that is 
satisfactory to one party but not to the other. The  selection of disagreements   
that end up in litigation is  therefore not random, because  potential 
plaintiffs are more likely to file suit when they think they will win.   The 
question to be answered in this article is whether the fact that plaintiffs 
select which  cases go to court  should make  courts 
 more generous to plaintiffs,   or less generous.

 

    The answer will  turn  on whether the plaintiff   can predict the  direction 
of the  court's
error in evaluating     evidence.  Sometimes court error is predictable; the 
plaintiff knows that he  himself is truly to blame for an accident but that a 
credible witness believes otherwise. Other times, court error is unpredictable; 
the plaintiff knows that  often the judge's attention will be wandering  at some 
point during the trial, but he does not know when.          The plaintiff's 
expected payoff from the lawsuit  are based on his knowledge of the true damage 
in this particular case  and on his estimate of the court's error in measuring 
that damage.  If the plaintiff cannot predict the court error, his filing 
decision is based only on his information about the true damages, but if he can 
predict the court error, it will be based partly on  the direction of that 
error.  How the court  adjusts its damage awards in light of    plaintiffs' 
incentives to file therefore depends on the predictability of its  measurement 
error in evaluating evidence.     

  Court error and 
asymmetric information  in lawsuits  have   been the subjects of
considerable analysis.  Cooter and Rubinfeld (1989) survey an
extensive literature on the litigation process, much of which deals
with pre-trial settlement when litigants possess
different information.\footnote{See, for example, Png [1983], Bebchuk (1984),  
Reinganum \& Wilde [1986], and Reinganum [1988].} The emphasis in this  
literature    has
been    on  the litigants'  incentives rather than  the court's,  though  any 
litigation model must include some specification of what the court does when the 
litigants fail to settle out of court. A somewhat different   literature looks 
at
the selection of cases that go to trial,   given  the  behavior of the 
court.\footnote{ The seminal paper in this
literature is Priest \& Klein (1984); a recent example is Hylton
(1993).}     Court error has also been
examined, especially in connection with the tradeoff between
punishing the innocent and not punishing the culpable.\footnote{On
court error in a variety of contexts, see  Schaefer [1978], Calfee \& Craswell 
[1984], Good \& Tullock [1984], 
 Craswell \& Calfee [1986],   Png [1986], Rubinfeld \& Sappington (1987), 
Polinsky \& Shavell [1989, 1994],  Sarath [1991], 
Kaplow (1994), and Tullock (1994).} 

  
What has   been largely ignored is the court's rational response to its
own error and its knowledge that litigants behave strategically.  An exception 
is  Daughety \& Reinganum (1995), which analyses how the court can incorporate 
its knowledge that settlement has failed to occur and    any  details of  the 
settlement
negotiations that it knows.      The present issue  is based on an even more 
basic  deduction. 
 When the court observes that the plaintiff has filed suit, it must
balance the probability that the plaintiff predicted that the court
would overestimate damages against the probability that he  actually   has
a good case.  

The model below will divide the court's
factfinding into two steps: (1) {\it measuring} the value of the
damages given the
 evidence presented for the particular case, and (2) {\it estimating}
the value of damages by incorporating not only the measured damages
but also extraneous knowledge such as typical damage levels, the 
plaintiff's incentive  to bring  suit, and the likelihood of measurement
error.  The court might measure the damage    to be \$10,000 using the evidence 
before it, but adding   its  knowledge  of  the   plaintiff's incentives to 
bring  suit  when the evidence is  favorably distorted might reduce the  best 
estimate  to \$8,000.\footnote{ Whether the court is permitted by the law
 to go beyond measuring damages to   estimate them is a
jurisprudential and  legal  question that will not be addressed
here, although      
Section 5 will briefly discuss    ``remittitur,'' a procedure   by which  the s  
court  can   threaten the plaintiff
with a new trial unless he agrees to accept a reduced award. }    
 
   The central intuition to be examined is that 
  since the plaintiff is more likely to bring a case when he knows
the court will overestimate   damages, the court should scale back
the award from what it would otherwise be.  As will become apparent,  although 
this  has
some truth to it,  other effects are also at work, and whether the intuition is 
valid will turn out to depend   (a) on  whether   the court   knows  the level 
of damages typical in the type of case at hand,  and   (b) on  the amount of 
court    error  that is predictable by the plaintiff. It will be shown that the 
court should always moderate extreme measurements of damage, and that if  court 
error is largely unpredictable,  the court should actually adjust  awards 
upwards.  

   Three effects are at work. First, regression towards the mean will  always 
justify moderating extreme awards: 
  an extreme   value of measured damage has a greater probability of being due 
to measurement error rather than high true damage.   Second,  predictability of 
the error  leads to   plaintiffs being more likely to bring suits with positive 
measurement error, and   on this account,  under   circumstances explained 
below, the court  will wish to reduce its awards.    Third,  unpredictable error 
means that sometimes  courts will observe apparently weak suits being brought, 
and the court should adjust its award upwards because the plaintiff's 
willingness to bring suit is a  credible signal that his true damages are higher 
than the court's measurement. 
 
  Section 2  of the article  lays out a  formal  model  of court error and  
derives a general proposition about adjusting extreme awards.   Sections 3  and 
4 examine situations of purely  predictable   and purely  unpredictable error.   
Section 5 illustrates these situations  with  a numerical example and relates 
the theory to the law.  
Section 6 discusses meritless suits,   and Section 7 concludes. 


%---------------------------------------------------------------
  
\bigskip
\noindent
\begin{center}
 {  2. THE MODEL } 
  \end{center}

 The decisionmakers  in the model will be  a plaintiff and a court.\footnote{For 
a discussion of the effects of  adding the possibility of settlement and 
allowing  the defendant  to be a decisionmaker in the model,       see Section 
5.}  The plaintiff is an aggrieved party who decides whether to  file suit based 
on his  cost of  litigation,  his    information about the  true damage and the 
court's measurement error, and his knowledge of   how the court  forms  its 
awards.   The word ``case'' will be used
 to refer to potential  ``lawsuits'',  because when  the plaintiff   chooses 
whether to bring  his grievance   to court or not a distinction must be made.  
The court  measures the damage with error, and uses that measurement together 
with its knowledge of how the plaintiff decides to file suit to form its award.   
The system is simultaneous, because the plaintiff's  suit-bringing strategy 
depends on the court's measurement-adjusting strategy, which in turn depends on 
the plaintiff's  suit-bringing strategy. 


Let  
  the true level of damage in a case be $d$,     where $d$ takes the
value $ \mu -1$ with probability $p$, $\mu +1$ with probability $r$, and $\mu$
with probability $q= 1-p-r$.   Let us  assume,  unless noted otherwise, that 
$p=r$,  so the damage distribution
is symmetric and $\mu$   represents the mean value of damage.
 
 The   measured value of damage   depends not only on   $d$ but on two
error terms, $\epsilon_p$ and $\epsilon_u$.  The error predictable by  
plaintiffs is $\epsilon_p$, where 
   \begin{equation} \label{e1}
  \epsilon_p = \left\{ \begin{array}{ ll}
  -1& with \; probability\; \theta\\
      0& with \; probability\; 1-2\theta\\ 
    +1& with \; probability\; \theta\\
 \end{array}
  \right.
  \end{equation}
   The error  not predictable by   plaintiffs is $\epsilon_u$, where
 \begin{equation} \label{e2}
 \epsilon_u =  \left\{
  \begin{array}{ ll}
 -1& with \; probability\; \gamma\\
      0& with \; probability\; 1-2\gamma\\ 
   +1& with \; probability\; \gamma\\
 \end{array}
  \right.
 \end{equation}
 We will assume that courts  have a positive probability of making some kind of 
error, so   at least one of the error probabilities  $\gamma$ and   $\theta$  is  
strictly positive. 
     
The court's measurement of damage is    $\hat {  d}$,     defined
by 
     \begin{equation} \label{e4}
\hat{  d} =   d + \epsilon_p + \epsilon_u . 
 \end{equation}
   The plaintiff's forecast of the measured damage   will therefore be 
 \begin{equation} \label{e4a}
 \tilde{d}  =   d + \epsilon_p.    
 \end{equation}
   True damage can be either low, medium, or
high, while measured damage can take any of the  seven values from $\mu -3$ to 
$\mu +3$. \footnote{  Some
values of $\hat{d}$ perfectly reveal $d$: if   $\hat{d}= \mu-3$, for
example, it would be clear that $d=\mu -1$, $\epsilon_p= -1$, and
$\epsilon_u= -1$.   This feature of the model is  accidental.     I have   
verified  the propositions    for other  error specifications which do not have 
the perfect-revelation property, such as, for example,  when measurement errors 
are  not cumulative and  the measured damage is constrained to lie within
$[\mu-1, \mu +1]$, the case   in the working paper version of this article, 
Rasmusen (1992b).  }
  
 The court's award will be its {\it estimated} damage, which is not
necessarily equal to the {\it measured} damage. 
    The measured damage, ${ \hat {  d} }$, is a raw measurement,
unadjusted by any considerations of equilibrium behavior or prior
knowledge of what damage is most probable. The court's award, $ a(  \hat{d} )$, 
will  equal its estimate of
the damages based on all available information, $ E(d|{ \hat{d} }, lawsuit)$.  
 The information  directly available  consists of the parameter values and the 
damage measurement, but in addition the court may be able  to deduce something 
about    the  plaintiff's private information  from his decision to file suit. 

 

 The cost of bringing suit, $c$, differs among plaintiffs and is
distributed according to a distribution $G(c)$, where $G'>0$ on the support 
$[\mu-3, \mu+3]$.  The court does not observe the  particular plaintiff's value 
of $c$, but it knows the general    distribution function, $G(c)$.   The 
plaintiff will decide whether to file suit based on his particular values of the 
litigation cost  $c$,  the measured damage forecast $\tilde{d}$,   and the   
expected award  given the plaintiff's forecast of measured damage,    $E 
(a|\tilde{d})$. 
  Let $F(\tilde{d})$ denote the
proportion of  plaintiffs whose litigation costs are low enough  that they would      
bring  suit given a forecast of measured damage of  $ \tilde{d}$.    This takes 
the value
 \begin{equation} \label{e6}
  F(\tilde{d}) = \int_0^{  E(a|{\tilde{d}})} d \cdot G(c) dc.
 \end{equation}
  Since $G$ is increasing in $c$, (\ref{e6}) implies that 
as the expected award increases, so does the fraction of
plaintiffs who bring  suit.    
 
  
 The court's objective is to      estimate the true damage $d$ as
accurately as possible, given all available information,  in deciding the award 
$a$.\footnote{Formally, let the court's payoff function be $-
[a(\hat {  d} ) -  d]^2 $, in which case it will choose an
award equal to the expected value of the damage.   Note that   I have implicitly 
assumed that  the court and the plaintiff  are uninterested in setting
precedents for future cases at the cost of   reduced payoffs in the
present case.}
This is done using
Bayes' Rule as follows:  
 \begin{equation} \label{e7}
    a( \hat{d} )   =   E(d|\hat{d}, lawsuit)   =      \sum_{i=\mu-1}^{\mu+1} 
\left( \frac{ Pr(\hat{d}, lawsuit|d=i)Pr (d=i)}{Pr(\hat{d}, lawsuit)} \right)  
i. 
  \end{equation}
 The main task is to find the component  $Pr(\hat{d}, lawsuit| d=i)$.  Since 
$\tilde{d}$ can take five possible values, from $\mu-2$ to $\mu+2$, this equals
 \begin{eqnarray} \label{e8}
 Pr(\hat{d}, lawsuit| d=i )   
 &  =& \sum_{j=\mu-2}^{\mu+2} Pr(\hat{d}, lawsuit|{\tilde{d}}=j)   Pr 
({\tilde{d}}=j| d= i) \\
  &  = &\sum_{j=\mu-2}^{\mu+2}Pr (lawsuit|{\tilde{d}}=j) 
Pr(\hat{d}|{\tilde{d}}=j)    Pr ({\tilde{d}}=j|d=i)  \nonumber   \\
 &  = &\sum_{j=\mu-2}^{\mu+2} F(\tilde{d} ) Pr(\hat{d}|{\tilde{d}}=j)    Pr 
({\tilde{d}}=j|d=i)  \nonumber 
  \end{eqnarray}
   It is then straightforward to find $Pr(\hat{d}, lawsuit)$, which equals
 \begin{equation} \label{e9}
 Pr(\hat{d}, lawsuit)   =  \sum_{i=\mu-1}^{\mu+1} Pr(\hat{d}, lawsuit|d=i)Pr 
(d=i).
 \end{equation}

 The Appendix performs the straightforward but lengthy calculations  of  
(\ref{e8}) and (\ref{e9})  necessary to find   the awards  in  equation 
(\ref{e7}) for all seven possible levels of  measured damage.       
   

{\it PROPOSITION  1: In deciding its award, the court  should increase low 
measured damages and reduce high measured damages. For any measured damage 
$\hat{ d}$, if  $\hat{ d} < \mu$  then $a(\hat{ d}) > \hat{ d}$, and if  $\hat{ 
d} > \mu$  then $a(\hat{ d})< \hat{ d}$. }

\begin{quotation}
\begin{small}
    {\it Proof:} There are seven possible values for $\hat{ d}$.  For $\hat{ d} 
=\mu -3$ and $\hat{ d} =\mu+3$, the proposition is obvious from equations 
(\ref{e9a}) and (\ref{e12a}) in the Appendix.  Inspection of equations 
(\ref{e10a}) and (\ref{e12aa}) shows that in each equation the numerator of the 
fraction is less than the denominator, proving the proposition for $\hat{ d} 
=\mu -2$ and $\hat{ d} =\mu+2$. 


 Define $z_1$ and $z_2 $ so that $a(\mu -1 ) =  \mu  - z_1/z_2.$ $z_2 = z_1 + 
2F(\mu) \gamma \theta r + F(\mu-1) (1- 2\gamma) \theta q + F(\mu) \gamma (1- 
2\theta) q $. Thus, $z_1< z_2$ and $\mu  - z_1/z_2   > \mu -1$. 

 Define $z_3$ and $z_4 $ so that  $a(\mu +1 ) =  \mu + z_3/z_4.$  Then  $z_4 = 
z_3 + 2F(\mu) \gamma \theta r + F(\mu+1) (1- 2\gamma) \theta q + F(\mu)\gamma 
(1- 2\theta) q $.   As a result,  $z_3< z_4$ and $a(\mu +1 ) = \mu  + z_3/z_4  < 
\mu +1$.
  $\Box$
 \end{small}
 \end{quotation}
 
 Proposition 1 is the effect of regression towards the mean, or, in
Bayesian terms, of combining data with the prior mean to form a
posterior mean that is between the two.\footnote{Note that Proposition 1 is true 
even if  $p \neq r$, e.g.,  even if the  true damage  is  more commonly low  
than high  and  $\mu$ is not the mean damage. For a discussion of
the characteristics of general continuous distributions that generate
regression towards the mean, see Rasmusen (1992a).} A high measured
damage $\hat{d}$ might be produced either by a high value of the true
damage $d$ or by positive values of the errors $\epsilon_u$ and $\epsilon_p$.
Placing some probability on each of these events, the court's
estimated damage is less than $\hat{d}$, although still higher than
the average true damage value, $\mu$.  The court   distrusts its own
extreme measurements and moderates them in deciding the award.

   Proposition 1   is fundamental to any analysis of court error. Even if  there 
were no plaintiff selection and all cases appeared before the court, the court 
should still moderate extreme awards. Attention in the next propositions will 
therefore be focussed on whether the court should adjust moderate awards, which 
it would not do if there were not a biased selection of cases by plaintiffs.  
Proposition 1 also establishes    a    general reason for  rational courts to  
choose an award different from the measured value of the damage.  Any 
decisionmaker cognizant of his own fallibility should adjust the award in light 
of measurement error and regression towards the mean.  Adjusting for the 
strategic behavior of the plaintiff, as will be done in the next sections,  
merely takes this  process one step further. 

 

%---------------------------------------------------------------
  


\begin{center} {  3.  AWARDS  WHEN   PLAINTIFFS   CAN  PREDICT THE COURT'S ERROR   
}
 \end{center}
  

   If plaintiffs can predict   the  court's  measurement  error, then  $\gamma 
=0$  and  measured damages range from $\mu -2$ to $\mu +2$.  The optimal award 
equations (\ref{e10a}) through (\ref{e12aa}) derived   in the Appendix become 
 
 \begin{eqnarray}  
   a(\mu -2 ) &  = &  \mu  - 1  \label{e13a}\\
   \label{e13}
 a(\mu -1 )  & = &  \mu  - \frac{   ( 1-2\theta)   p        }  {       ( 1-
2\theta)p+ \theta q   }\\
      a(\mu  )   \;\;\;\;\;\; & =&  \mu +    \frac{      \theta (r- p ) }
 {    \theta p + (1-2\theta)q + \theta r   }  \label{e14}\\
   a(\mu + 1) &= & \mu+ \frac{      (1- 2\theta) r     }           {      \theta 
q + (1- 2\theta)r   } \label{e15}\\
   a(\mu +2 ) &  = &  \mu + 1  \label{e15a} 
    \end{eqnarray} 
 

  The    function $F(\tilde{d})$ is absent from the optimal awards in equations 
(\ref{e13a}) through  (\ref{e15a}).  That is
because the court only considers the cases that appear before it, and
$\hat{d}$ summarizes all the information about those cases.  The plaintiff's   
willingness   to go to court   does not reveal 
  anything  about $d$  that  the court cannot discover by direct  evaluation of 
the evidence in forming the measurement  $\hat{d}$.    
Hence, 
  the court's award does  not  depend on plaintiff behavior.   In making his 
decision to file suit, the  plaintiff   is making use only of  the court 
measurement $\hat{d}$   and  the process by which  the court adjusts   
measurements in forming   awards,   and does not make use of his knowledge of  
how $\hat{d}$ is split   between the  true damage  $d$ and the court error 
$\epsilon_p$.    Since the plaintiff's   filing decision does not vary with his 
private information,     the court cannot deduce anything useful from  the fact 
that a suit is filed.  

 
  Regression to the mean is still present even when plaintiffs can predict   
court error:   high awards are adjusted down,  and low awards
adjusted up. We can also say something about the court's adjustment
when the damage measurement is moderate. 
Regression towards the mean implies that moderate awards are adjusted
up or down depending on the proportion of cases that are
meritless, with no adjustment at all if the distribution of true damage is 
symmetric,  i.e., if  $r=p$.    Inspection of equation (\ref{e14}) yields 
Proposition 2,   since the sign of the  numerator in the last term of  that 
equation depends on the sign of $(r-p)$.  
    
{\it PROPOSITION 2:  Consider a suit in which measured damage is  moderate and 
plaintiffs can predict the  court error.  The award should equal the measured 
damage  if the damage distribution is symmetric, but   be below it if  low 
damage  is generally more common than high.
If  $p = r$, then $a(\mu )=\mu $; but if $p > r$, then $a(\mu )<\mu $. }

 
   If high and low measured damages are equally likely,   the court does not 
have to make any adjustment to a moderate measured level of damages to form its   
award,  even though it  is conscious that the  error is predictable by 
plaintiffs and that    plaintiffs  are more likely to  bring suit if they know 
the error will be  in their favor. 

 If, however, low damage is more common than high damage for the type of injury  
in the case,  then the court should not trust its measurement alone, but should 
reduce it in forming the award.      If a chemical rarely causes birth defects, 
but seems to in the particular case before the court, the court should discount 
the evidence and  make only a small award.   This, like Proposition 1, is a 
result of regression towards the mean.
  
       Note that  this form of asymmetry is  completely distinct from  biased 
measurement error. If the court knows that its measurement is too  high on 
average by amount $x$, it can easily adjust by subtracting $x$ from its initial 
measurement.    The problem in the present model is   that the court should make 
use of  information  about average levels of damage, not about average levels of 
measurement error. 

 

      The situation  with an  asymmetric distribution   of true damage has 
practical importance. The  true damage  from many activities  is usually  zero 
but might be measured to be positive a certain fraction of the time.  Even if 
the court   error is unbiased, a court which uncritically awarded measured 
damage would overcompensate plaintiffs because of  the selection bias in which 
cases are filed as lawsuits.      As an example,   corporations'   decision to 
switch materials suppliers  may almost always be to the benefit of  the 
shareholders, but  if the court admits shareholder suits in the cases where the 
stock price subsequently falls, it will  often mistakenly measure the damage to 
be positive.   Knowing that most such suits are meritless, a better policy for 
the court would be to refuse to hear such suits at all,  or to adjust the awards 
downwards in recognition of the fact that the apparent merit of most suits is 
due to court error.  

  


%---------------------------------------------------------------
\bigskip
  \noindent
 {\it  Predictable Error When the Court Does Not Know the  Prior Mean}
 
   Let us now modify the informational structure   to allow for a less well-
informed court.  Propositions 1 and 2 relied on the assumption 
  that the court knows that the type of
case brought causes damage  ranging from $\mu -1$ to $\mu+1$. If  the court 
knows that $\mu=1$,  for example, the possible damages
are  0, 1,  and  2.  Often, however, the court's prior  information  will not be 
so precise, and if the court does not know the mean value of damage it cannot 
use regression  to the mean.   If  the court  does not know whether the
possible damages are (0,1,2) or (2,3,4), it does not know whether to
regard $ \hat {  d} =2$ as   high or   low   and it cannot
make the adjustments  in Propositions 1 and 2. 

 

   What the court  can do is to  make an adjustment based on its beliefs as to 
the likelihood that the estimate 
$ \hat {  d} =2$ is a low, medium, or high value.
  Let us look at the extreme case of
``diffuse priors'':    the court has no prior information  on the value of the  
mean damage, $\mu$,  but  it does know that the three possible damage values are 
$\mu-1$, $\mu$, and $\mu+1$.  The court       
regards any value  as equally likely to
be low, medium, or high, and it  will have to make the same adjustment
for any damage measurement, since it cannot tell which  
 finer adjustment  is appropriate.\footnote{  
 Note that this situation of predictable error with diffuse priors is not the 
same as a situation with both predictable and unpredictable error.   The diffuse 
priors refer to  the beliefs of the court about the true damage,  while the 
predictability of the error refers to the beliefs of the plaintiff. It is 
assumed throughout this article that the plaintiff knows that the mean value of 
court error is zero over all cases, including those that never become lawsuits.  
} 

 Let us henceforth assume that   the  distribution of true damages is symmetric, 
departing from the generality of Proposition 2.  
    If the court  knew $\mu$,  as in Propositions 1 and 2,    its awards would 
be  derived by  adapting  equations (\ref{e13}) to (\ref{e15}) to  set  $p=r$. 
Defining $Z_1 \equiv\frac{   ( 1-2\theta)   p        }  {       ( 1-2\theta)p+    
\theta (1-2p)  }$, this yields
  \begin{eqnarray} \label{e15z}
 a(\mu-2 )&  & = \mu -1  \\
   a(\mu-1 )& =   \mu  - \frac{   ( 1-2\theta)   p        }
 {       ( 1-2\theta)p+    \theta (1-2p)  }  &  = \mu - Z_1  \nonumber \\ 
  a(\mu )\;\;\;\;  & & =  \mu \nonumber\\
    a(\mu+1)&   =\mu  +\frac{      (1- 2\theta) p
    }           {      \theta (1-2p) + (1- 2\theta)p   }  &= \mu + Z_1  
\nonumber\\
 a(\mu+2)& &  = \mu  +1.  \nonumber  
 \end{eqnarray}
 
    If all cases were equally
likely to become lawsuits, the court would, on average, set the award equal to 
the measured damage.    $F(\tilde{d})$ is increasing,  however, so the more 
positive the
expected court error, the more likely a case is to appear in court.   Since 
cases with higher expected damages  generate  more   suits, the average 
adjustment by a court that  knew $\mu$
would be downwards. 
 When the court does not know $\mu$, it still knows that  suits with
positive   error are more likely than  suits with negative
  error, and it can use this information to adjust the
award, yielding Proposition 3.   

{\it PROPOSITION 3:  If plaintiffs can predict the court's  error and  the court  
does
not know the average value of damage, it should  reduce every damage measurement 
in determining the award:  $a(\hat{ d})<\hat{ d}$.} 
 
\begin{quotation}
\begin{small}
       {\it Proof.}   If the court  knew the
value of $\mu$, it would know how to adjust the damage
measurement.   In  accordance with (\ref{e15a}), on observing
$\hat{d}=\mu-1$ or $\mu-2$ it would adjust upwards, on observing $\hat{d}=\mu+1$ 
or
$\mu+2$ it would adjust downwards, and on observing $\hat{d}=\mu$ it would set
the award equal to the measured damage. 
 
 Cases with $\hat{d}=\mu+1$ are more likely to be brought in equilibrium
than cases with $\hat{d}=\mu-1$, however, because $F(\mu+1)> F(\mu -1)$.  The
probabilities of the five different values of $\tilde{d}$ are
 \begin{equation} \label{e15aaa}
 (p \theta, p (1-2\theta) + (1-2p) \theta, (1-2p)(1-2\theta)+ p \theta + 
p\theta, p (1-2\theta) + (1-2p) \theta,    p \theta).    
 \end{equation}
       Let us  invent the notation $k_5, k_6, k_7$ and rewrite vector 
(\ref{e15aaa}) as $(p \theta, k_5 , k_6 , k_5,    p \theta)$, so that the    
probability that  a suit is brought at all is  
 \begin{equation} \label{e15aa}
 k_7=F(\mu-2) p\theta + F(\mu-1)k_5 +F(\mu)k_6  +F(\mu+1)k_5+F(\mu+2) p\theta.  
 \end{equation}
   The probability that, for instance, $\tilde{d}=\mu-2$,  conditional upon a 
suit having been brought at all is, by Bayes' Rule, 
 \begin{eqnarray} \label{e15ay}
 Prob(\tilde{d}=\mu-2|suit)  & =\frac{Prob(lawsuit|\tilde{d}=\mu-2) 
Prob(\tilde{d}=\mu-2)  }{ Prob(lawsuit)} \nonumber\\ 
   \label{e15az}
    &=  \frac{F(\mu-2) p\theta}{k_7 }.
 \end{eqnarray}
 Using probabilities in the manner of  equation (\ref{e15az}), and using the 
quantity of adjustment from equation (\ref{e15a}),  the  average adjustment the 
court  would like to make is
   \begin{eqnarray} \label{e15b}
    \frac{ F(\mu-2) p\theta  (1)}{k_7} + 
  \frac{ F(\mu-1) k_5 (1-Z_1 )}{k_7}+   
\frac{  F(\mu) k_6   (0)}{k_7} +  \\
 \frac{ F(\mu+ 1) k_1  (-(1-Z_1))}{k_7} + 
 \frac{ F(\mu +  2) p\theta   (-1)   }{k_7}, 
    \end{eqnarray}
 which equals
   \begin{equation} \label{e15c}
 \frac{  [F(\mu-2)-F(\mu+2)] p\theta    +   [F(\mu-1) -F(\mu + 1) ]k_5  (1-
Z_1)}{k_7}   
        \end{equation}
        Since $F(\mu-2)<  F(\mu+2)$ and $F(\mu-1)<  F(\mu+1)$,  the court  
wishes to adjust downwards on average.     $\Box$
 \end{small}
 \end{quotation}

 When the court knew the prior mean,   the plaintiff's filing decision did not 
provide useful information, so no adjustment was made to moderate damage 
measurements.  In Proposition 3, the filing decision does convey useful 
information: the court knows that cases with   values of $\hat{d}$ above the 
mean  are  more likely to be filed, so it can deduce something about the value 
of $\mu$ from the fact of filing.  Knowing that filed cases are more likely to 
have $\hat{d}$ above $\mu$, the court adjusts its award downward from the 
measured damage. 
 

 Proposition 3 captures the  intuition motivating this article,   that plaintiff 
selection of
cases gives courts reason to scale down their initial estimates of
damages.  Not knowing the typical value of damages, the court relies  on its 
knowledge  that plaintiffs are more likely  to bring suit  when  they can 
predict that the court measurement  will err in the positive direction. 

       
   
%---------------------------------------------------------------
 
 \bigskip
\noindent
\begin{center}
 {  4.  AWARDS WHEN   PLAINTIFFS   CANNOT  PREDICT THE COURT'S ERROR  }
 \end{center}
 
The next situation to consider is   when plaintiffs cannot  predict  court 
error, but they recognize it exists.    We will start by again assuming that the 
prior mean $\mu$ is known to the court. 


 Since the error is unpredictable,  $\theta=0$  and measured damages lie in the 
interval $[\mu-2, \mu+1]$.  Setting $p=r$  and defining  
   $W_1$, $W_2$, and $W_3$  appropriately,      equations (\ref{e10a}) to  
(\ref{e12aa}) in the Appendix become  
\begin{equation}   \label{e46}
  \begin{array}{llll} 
 a(\mu-2 )& && =        \mu -1  \nonumber \\
   a(\mu-1 )&   =   &   \mu  - \frac{    F(\mu-1) (1- 2\gamma)  p   }  {     
F(\mu-1) (1- 2\gamma)   p    + F(\mu)\gamma  (1-2p)  }  & =  \mu - W_1   
\nonumber  \\
 & & &\\
    a(\mu)  & =   &    \mu +    \frac{ p[F(\mu+1)- F(\mu-1)] \gamma   }
 {F(\mu-1) \gamma   p    + F(\mu) (1-2\gamma ) (1-2p)   + F(\mu+1)\gamma  p}    
&= \mu +  W_2 
\\  
 & & &\\
     a(\mu+1)& =   &     \mu+ \frac{    F(\mu+1)(1- 2\gamma)  p     }           
{F(\mu)\gamma  (1-2p)     
 + F(\mu+1)(1- 2\gamma )p  } & = \mu + W_3 \nonumber\\
 a(\mu+2)     &  &   &  =       \mu  +1 \nonumber  
 \end{array}
 \end{equation}
 Note that   $W_3 > W_1$,  because $F(\mu+1) > F(\mu-1)$. 

  These expressions exhibit the same effect of regression towards the
mean that appeared in Propositions 1 and 2.  They also exhibit a signalling 
effect,  which tends to increase the estimate, 
whatever the damage measurement may be. It is not true that  $a(\mu+1) > \mu+1$, 
because regression towards the mean is still present, but the award is adjusted 
downwards less than it would have been  if the error were predictable.  The 
signalling effect arises because  when the error is unpredictable, the   
information  the court can  extract from the fact of the plaintiff filing   is 
that the plaintiff is more likely to have higher true damages.  Since the 
plaintiff cannot predict the error, his decision to file suit is not based on 
it. 

 The signalling effect was also present 
  in Section 2, when  the model included  both predictable and unpredictable 
error. Careful inspection of equation (\ref{e11}) in the Appendix shows that 
$a(\mu)> \mu$, the court will adjust moderate measured damages 
upwards.\footnote{  If damages are symmetric, $p=q$. Since filing is more likely 
when true damages are higher, $F(\mu-1) > F(\mu+1)$. From these two facts, it 
follows that the numerator of the fraction in equation (\ref{e11}) is positive.}
 Discussion of the effect was delayed until now  to show that it is due to the 
unpredictable portion of court error; the signalling effect did not arise in 
Section 3,  where the error was entirely predictable.  Proposition 4  formalizes   
the difference in the impact of the two kinds of error.  
  

 
{\it PROPOSITION 4:   For all but   extreme levels of measured damage,    the 
court's   award will be greater if  plaintiffs cannot   predict court error than 
if they can.   For    $k \in (0,1)$  and $\hat {  d} \in [\mu-1, \mu +1]$, 
$a(\hat {  d}; \gamma =k,  \theta =0) > a(\hat{d}; \gamma =0,  \theta =k)$. }

\begin{quotation}
\begin{small}
{\it Proof}. 
 From equation (\ref{e46}) and the fact that higher awards induce more  
litigation so $F(\mu-1) <  F(\mu)$, 
 \begin{eqnarray} \label{e47}
 a(\mu -1; \gamma =k,  \theta =0)  &=   \mu  - \frac{    F(\mu-1) (1- 2k)  p   }  
{     F(\mu-1) (1- 2k)   p    + F(\mu)k  (1-2p)  }     \\
    &\hspace*{12pt}> \mu  - \frac{    F(\mu-1) (1- 2k)  p   }  {     F(\mu-1) 
(1- 2k)   p    + F(\mu-1)k  (1-2p)  } .
    \end{eqnarray}
 From equation  (\ref{e15a}), 
   \begin{equation} \label{e48}
 a(\mu  -1 ; \gamma =0,  \theta =k)=\mu  - \frac{     (1- 2k)  p   }  {    (1- 
2k)   p    + k  (1-2p)  }. 
 \end{equation}
   But  this equals the  right-hand side of (\ref{e47}), so it must be that 
$a(\mu -1; \gamma =k,  \theta =0) > a(\mu -1; \gamma =0,  \theta =k)$.  The same 
procedure can be used   straightforwardly  to show that  the proposition is also 
true for $\hat{d} = \mu$ and  $\hat{d} =\mu + 1$.
Q.E.D. 
 \end{small}
 \end{quotation}





%---------------------------------------------------------------


\bigskip
 \noindent
 {\it  Unpredictable Error When the Court Does Not Know the  Prior Mean}

 Let   us  now  assume that the court does not know $\mu$.   The analysis is 
parallel to that  for  predictable error in Section 3, and yields Proposition 5, 
which says that all damages should be adjusted upwards.      

{\it PROPOSITION 5:  If plaintiffs cannot predict court error  and the court  
does
not know the average value of damages, it   should   set  the  award  to be   
greater than the measured damage:    $a(\hat{ d}) > \hat{ d}$.  } 

\begin{quotation}
\begin{small}
{\it Proof:}  
  When the court observes a damage measurement of $ \hat{ d}$, it
knows that the true damage is within one unit of that value, so $d$ equals 
$\hat{ d}-1$, $\hat{ d} $, or $\hat{ d}+1$.   Since $G'(c) >0$,   for any 
adjustment rule  the court uses,  higher true damage  will yield a higher 
percentage of cases litigated, so
  \begin{equation} \label{e50a}
  F( \hat{ d}-1) < F( \hat{ d} ) < F( \hat{ d}+1) .
 \end{equation}
      This means that using Bayes's Rule, the court's estimated value of $d$ 
given a suit was brought   is 
 \begin{eqnarray} \label{e50}
\left( \frac{ F( \hat{ d}-1)  } {F( \hat{ d}-1) + F( \hat{ d} ) + 
  F( \hat{ d}+1) } \right) (\hat{ d}-1) + 
\left( \frac{ F( \hat{ d} )  } {F( \hat{ d}-1) + F( \hat{ d} ) + F( \hat{ d}+1) 
} \right) (\hat{ d} )   \nonumber   \\
   +  \left( \frac{ F( \hat{ d}+1)  } {F( \hat{ d}-1) + F( \hat{ d} ) + F( \hat{ 
d}+1) } \right) (\hat{ d}+1). 
  \end{eqnarray}
 Expression (\ref{e50}) is greater than $\hat{d}$ because of the inequalities in 
(\ref{e50a}), so 
$a(\hat{ d}) > \hat{ d}$. Q.E.D. 

 \end{small}
 \end{quotation}

 Proposition 5 applies to a situation where the court cannot adjust for 
regression towards the mean, because it has no information   on whether the 
measured damage is above or below the mean. Besides the measured damage itself, 
the court's only  information is the filing decision, and this tells the court 
that that the true damage is more likely to be large without conveying any 
information on the  size of the error. Thus, the court adjusts its award upwards 
from the measured damage.    
 

In the first part of this section, it was noted that when the prior mean is 
known and error  has  both predictable and unpredictable components, the average 
award is adjusted upwards, $a(\mu)> \mu$. Proposition 5, however, applies only 
when the error is entirely unpredictable.  If the prior mean is not known to the 
court and both kinds of court error are present,  the selection effect and the 
signalling effect of Proposition 5  clash with each other, and it is not clear 
whether the court should adjust awards upwards or downwards. Thus,  the policy 
conclusion depends on  the empirical issue of which kind of error is more 
important  in the type of case before the court.  
 

 
%---------------------------------------------------------------

\bigskip
\begin{center}
  {  5. NUMERICAL EXAMPLES, SETTLEMENT, AND REMITTITUR  }
 \end{center}

 
  Two 
  forces besides regression to the mean  are at work in the propositions above:      
selection and signalling.    If plaintiffs can predict the error,  the court 
knows that they have more incentive to  bring suit if the error is positive, so  
the cases that become  suits are selected nonrandomly.   If the court reduces 
its awards in response, the selection bias remains but is muted.  If  plaintiffs 
cannot  predict the error,  on the other hand,   the court knows that  the only 
selection effect at work is that plaintiffs with  higher true damages are more 
likely to bring suit.  If the court increases its awards in response, the 
selection bias remains, because  plaintiffs with high true damages have all the 
more reason to bring suit.   

Signalling to other litigants is a well-known phenomenon    and plays a  large 
part in models of settlement, where reluctance to settle can signal a strong 
case to the other side.   Signalling to the court is less common in litigation 
models. One exception is   Rubinfeld and Sappington (1987), which looks at   
legal expenditure by the litigants  as signalling  in criminal cases, but does 
not  examine  the effect on the number of
cases brought to trial.  A second  exception is Daughety \& Reinganum (1995), in 
which a litigant signals the strength of  his  case to the court by his position 
in settlement negotiations.  The signal in the present article is   a simpler 
one:  the plaintiff's   willingness to incur the cost of bringing   suit is a 
signal to the court  of his knowledge that the true damage is high. 

    
   Table 1 uses numerical examples to illustrate the propositions.  The fraction 
of cases litigated  and the damage awards are shown for the different possible 
damage measurements by the plaintiff and the court.   In every example,  the 
distribution of true damages is symmetric with $p=q=r$,  the filing cost   
$G(c)$ is uniform on [0,2], and the mean damage is  $\mu=1 $.


Columns (1) and (2)    both  illustrate Proposition 1. In  each 
of them,  $a(0) >0$ and $a(2)<2$.   In Column (1),  the error is predictable  
and the court knows the mean true damage,  as in   Proposition 2.   Since the 
distribution of true damages is symmetric, $a(1) = 1.00$. In  Column (2),   the   
error  is unpredictable, and, as Proposition 4  predicts, the awards are greater 
for all but the extreme   levels of measured 
damage.\footnote{$F(-1)$ and $F(3) $ are left blank in Column (2) because  
$\tilde{d}$ cannot equal  $\mu -2$ or $\mu + 2$ when  no predictable error is 
added   to the true damage of $\mu-1, \mu,$ or $\mu +1$.  }   


In Column (3),  the  court does not know the mean   true damage.   It  must make 
the same adjustment whatever the level of measured damage, and it chooses to 
reduce the measured damage by 0.30,  as Proposition 3 says it should.  When the 
measured damage is 3, this    results in  a court  award  of 2.70,  exceeding 
the   largest possible true damage (which is 2). This is rational   because  a 
court which does not know the mean true damage also does not know it greatest 
possible value. 




\begin{small}
\begin{center}
\begin{tabular}{ll|c|c|c   }
         \multicolumn{5}{c}{   TABLE 1:     NUMERICAL EXAMPLES  }\\
 \multicolumn{5}{l}{ }\\
\hline
\hline
 & &    (1) &  (2)& (3)  \\
 & &   Predictable& Unpredictable   &   Predictable\\
  & &  Error Only &  Error Only & Error Only   \\
  & &    (Mean Known)&(Mean Known)  &    (Mean Unknown) \\
   & & ($\tilde{d} =  \hat{d}$)  & ($\tilde{d}=d$)  & ($\tilde{d} = \hat{d}$)  
\\
   & & $\gamma =0, \theta=.2$ & $\gamma =.2, \theta=0$ & $\gamma =0, \theta=.2$  
\\
   \hline
   &   &  &            &       \\
       & $F( -1)$ &   .000 &    ---  &           .00 \\
     Proportion & $F(0)$ &   .125&  {\bf  .16} &  .00 \\
    of  Cases & $F(1)$ &   .500 &  .61&   .35 \\
Litigated & $F(2)$ &   .875  & .87 &       .88 \\
  $\;\;F(\tilde{d})$  & $F( 3)$ &  1.000 &  ---  &    1.00 \\
  &   &  &            &       \\
  \hline
  &   &  &            &       \\
 &               $a(-1)$    &  0.00  &  0.00  &  -1.30  \\
          Adjusted & $a(0)$ &    0.25&  0.54  &    -0.30\\
           Court & $a(1)$ &   {\bf 1.00} &   1.24 &    0.70 \\
           Award & $a(2)$ &   1.75 &   1.81&  1.70  \\
 $\;\;a(\hat {  d})$  & $a(3)$ & 2.00  &  2.00  & {\bf 2.70 }  \\
   &   &  &            &       \\
  \hline
   \hline
    \multicolumn{5}{l}{Assumed: $p =q=r=1/3$. $G(c)$ is uniform on [0,2]. $\mu=1 
$. Calculations are rounded. }\\
           \end{tabular}
  \end{center}
 \end{small}

   
  
\noindent
{\it Settlement Before Trial}

       The model used in this article has assumed that    the  court makes its 
decision  without reference to the  possibility that the litigants have tried to 
settle the case and failed to reach agreement.  In reality, however, the 
majority of suits are settled before trial,   and suits  that reach trial  are 
special in some way.   The court might be able to deduce something about the 
strength of the case from the fact that it was not settled out of court. 

To see this, consider  a  settlement model in the style of   Reinganum  \& Wilde 
(1986).\footnote{In  Reinganum  \& Wilde (1986), the court does not act 
strategically, but    Daughety \& Reinganum (1995) have  extended the model to  
strategic courts.  The other major type of settlement model, deriving from 
Bebchuk (1984), assumes that the damage amount is known  but the litigants 
differ in their opinions of who will win at trial. For details, see the  survey 
by Cooter \& Rubinfeld (1989).   }  The plaintiff knows the true damage, and the 
defendant does not. The plaintiff suggests one take-it-or-leave-it settlement  
amount to the defendant, and if the defendant rejects the suggestion, the case 
goes to trial, at some cost, and the court makes an award. If the equilibrium in 
this model is separating, plaintiffs with higher true damage  request greater 
settlement amounts, but the defendant rejects them  with greater probability. 
The plaintiff's willingness to risk going to court signals high damages to the 
defendant, and also to the court, which  therefore  adjust its awards upwards. 

  Signalling  by filing suit     is distinct from   signalling by settlement 
offer.
  In the present model, the signal is the plaintiff's willingness to incur the 
cost of a  suit, while in the settlement model it is his willingness to make a  
high settlement demand and risk going to trial.  An important difference is that 
it is only the plaintiff who can signal by bringing suit, whereas it seems 
equally likely that either the plaintiff or the defendant could signal by 
holding out for a favorable settlement. In   settlement signalling, if  it is   
the defendant  who knows the true damage and makes the offer,  the  model's 
conclusions are reversed: lower settlement offers signal lower damage, they are 
rejected more often, more weak cases reach trial, and the court will adjust its 
award downwards, not upwards. The conclusions of the court error model  cannot 
be reversed so neatly. If  the defendant, not the plaintiff,  controlled the 
decision on whether to  file the  suit,   it would never be filed,    because in 
the absence of a suit he pays no damages. 
 
      In a combined model   of  pre-trial settlement  and court error, both 
kinds of signalling would be present.   They would reinforce or contradict each 
other depending on which litigant had private information and whether the court 
error was predictable or unpredictable.   Since court error would affect the 
signalling game only by changing the threat point of the expected  trial 
outcome,  I conjecture that the interaction between the two kinds of signalling 
would be relatively straightforward, if complex to model.

 

  %---------------------------------------------------------------
\bigskip
\noindent
{\it Judge, Jury, and Remittitur}      
    
       The court error model suggests that a rational court would use more than   
the evidence to decide the award.  Do courts actually  incorporate prior 
information and   recognize the implications of plaintiff selection bias?  
Viscusi (1991, p. 52) presents evidence that courts   
undercompensate large loss claims and overcompensate small loss
claims, as Proposition 1 would suggest.\footnote{ The measure of loss is purely 
monetary, so it may be that
relatively larger nonmonetary losses are associated with small loss
claims, but it might also be that courts regress damages towards the
mean. }     The extent to which courts can legally make adjustments to the 
measured damages is limited, however, by   rules of evidence and procedure and 
by the prior information available to the courts. 
 
       
     The accepted division
of labor in a jury trial is not between the use of evidence and the
use of prior information, as in the court error model, but between questions of
fact, decided by the jury, and questions of law, decided by the
judge.  The judge  in a jury trial has no part in either measuring or estimating 
damages, apart from
instructing the jury as to what evidence may be considered.
  Jurors are permitted and intended to use the priors of a
typical citizen, but jurors with special  expertise are
screened out, and the jurors are unlikely to know much about the strategic 
incentives of players in the legal system.  Even if they did, making use of that 
knowledge would be to go beyond the instructions from the judge.    

  The  problem  of court error arises even in bench trials, where the judge is 
the trier of fact,  but the 
institution of the jury  is   an obvious source both of measurement error and 
lack of  sophistication about the incentives of plaintiffs to bring suit.  Use 
of a jury  is commonly thought to  help plaintiffs with weak cases, and evidence 
supports this.    Clermont \& Eisenberg (1992) find in federal civil trials that 
plaintiffs
win a greater percentage of bench trials, in which the right to a jury is 
waived---  ratios in the winning percentages of 1.15 for motor vehicles,
1.71 for product liability, and 1.72 for medical malpractice. On its face, this 
would seem to  give plaintiffs an advantage when there is no jury, since they 
win more often, but that conclusion ignores the selection problem. If plaintiffs 
win less often in jury trials, yet they refuse to waive their right to a jury,  
the implication is that those plaintiffs think they would lose in a bench trial, 
given the weakness of their case,  but are willing to gamble on a jury. Thus, 
the fact that plaintiffs lose more often  in jury trials may   indicate that 
even very weak cases are worth  bringing before a jury.    
   
  James Blumstein,    Randall   Bovbjerg, and 
                              Frank Sloan have made two 
suggestions  which would reduce the influence of both predictable and 
unpredictable court error by increasing the amount of prior information: injury 
award schedules,  and award databases.  Courts could be provided with a  
schedule
relating the victim's age and severity of injury to the suggested
award, much like the schedule the U.S. Sentencing Commission has
provided for courts to use  in criminal sentences (Bovbjerg,
Sloan and Blumstein (1989), U.S. Sentencing Commission (1990)). The effect of 
this would be    to provide
prior information to the court with a suggestion, or perhaps an instruction, 
that
it be used in determining the award. A second suggestion is to create
a large database of awards in different types of cases, so that the
court  would  have a better idea of typical
damages as estimated by previous courts  (Blumstein,  Bovbjerg, and 
        Sloan [1991]).   

Judges  do have tools at their disposal with which they can exclude suits 
brought in the hope of jury error.  They can grant summary judgement to the 
defendant, on the grounds that there is no genuine issue of fact, and    they 
can exclude certain kinds of evidence  which might  bias the jury.  Even after 
the jury has heard the evidence, the judge can order a directed verdict, if he 
decides that the plaintiff has not presented a prima facie case.  These tools 
are extreme, and directed verdicts and summary judgement are  unsuitable for 
cases where the true damage is positive, if less than what the jury would award.  
A less blunt instrument is the use of the     procedures of ``remittitur'' 
(which reduces damages) or 
``additur'' (which increases them).\footnote{For general discussions,
see Speiser, Krause, and Gans (1985) pp. 773-797, 960-977, and Sann
(1976).}    Under {\it remittitur}, the  judge  presents the plaintiff with two 
alternatives: a reduced
award suggested by the judge, or a new trial on the grounds that no reasonable 
jury could have  found such high   damage.   Such adjustments are surprisingly 
common. 
Shanley (1991) finds that 20 percent of cases end up with a result
that differs from the jury decision, reducing the average payment to
71 percent of the jury award, and  that larger awards are 
reduced more than smaller awards.\footnote{  {\it Additur}, which offers the 
same choice on the grounds that the measured damages are unreasonably low, is 
much less common than   {\it remittitur}, 
and  is not
available in federal courts  because it is held to violate the U.S.
Constitution's Seventh Amendment's guarantee of a jury trial ({\it
Dimick v.  Scheidt}, 293 U.S. 474 (1935)).  {\it
Remittitur} is symmetric to {\it additur}, of course, but it is
allowed to stand because it was well established as part of the
common law in 1791.  State courts vary on whether they allow  {\it additur},  
as it seems to be accepted that states are not bound by the
Seventh Amendment ({\it Olesen v. Trust Co. of Chicago}, 245 F2d 522,
(7th Cir.)). Even {\it remittur} has, since 1905, been unavailable in
England (Sann, 1976, p. 301).  Oddly enough, {\it additur} is never available 
for
punitive damages, because those are entirely at the discretion of the
jury, since they need have no relation to measured damage (Speiser, Krause, and 
Gans, 1985, p. 976). }    These various rules seem for the most part to help 
defendants rather than plaintiffs, which suggests that empirically the selection 
effect of predictable error is more important than the signalling effect of 
unpredictable error. 
  
 

\bigskip
\begin{center}
  {  6.   MERITLESS  SUITS} 
 \end{center}

        What level of litigation is efficient, and whether the United States has 
exceeded that level or not, are questions beyond the scope of this article, 
involving as they do the issues of optimal deterrence and the size of 
transaction costs.   Where the court error model can be helpful, however, is in 
clarifying  what is meant by excessive litigation.  Much of the  present-day 
concern  about excessive litigation   seems to   arise   from  a perception that 
(a)  too many plaintiffs  bring suits that have little chance of winning large 
awards and do not deserve to win them, and (b) some of these  plaintiffs  win   
large awards  anyway.   

  One interpretation  is that these are 
  suits in which  the expected
value of the court award is less than the plaintiff's  transaction costs of
obtaining the award---  what I will call ``nuisance suits''  or ``frivolous 
suits'' .    A nuisance suit  is
brought to extract a settlement offer, or from the plaintiff's  malice towards 
the
defendant, or because  the plaintiff is mistaken about his probability of 
winning.\footnote{For models
of nuisance    suits, see Rosenberg and Shavell (1985), Bebchuk (1988),
and the general discussion in Cooter and Rubinfeld (1989).   }  

  The court error model  points out the need to be careful in defining frivolous 
suits, because  it would be misleading to  define a frivolous suit as  a suit 
that both plaintiff and defendant recognize has no merit, as is sometimes 
done.\footnote{E.g.,  the definition of nuisance suit by  Cooter and Rubinfeld 
(1989, p.
1083) .}  When   courts make mistakes,  it is not just the true merits that 
affect incentives, but  the court's view of the merits, and the litigants' views 
of the
court's view.  
 
 
 The court error model thus  suggests a second interpretation of the problem of 
excessive litigation: that the problem is     ``meritless suits,''  in which the 
true damage is     low, but the expected award is greater than the cost of 
bringing suit.      A  frivolous suit 
might  not be
meritless. The  plaintiff might be able to predict that though the
damage is large, the court error will be negative. Likewise, a  meritless
suit need not be frivolous. The plaintiff may know his case is
meritless but be confident of fooling the court.    Both kinds of  suits create 
inefficiency by
generating litigation costs and deterring potential targets from harmless 
behavior that  might generate lawsuits.
   In addition,  to the extent that courts adjust their awards as described in 
the present model, the presence of  meritless suits   reduces the number of 
meritorious suits. 
An immediate implication of Propositions 2 and  3 is that when court error is
predictable, meritless suits impose a negative externality on
plaintiffs with meritorious suits.  If  the fraction of meritless
suits is high, the court reduces even moderate awards substantially.   Depending 
on the
distribution   of litigation costs, it is even possible that a
majority of meritorious cases will not be brought.\footnote{This externality is
also noted in Bebchuk (1988).}

Both predictable and unpredictable error generate 
 meritless  but non-frivolous suits.   If   positive  error is predictable, the 
plaintiff can bring a meritless suit confident that he  will be overcompensated.   
Even sizeable litigation costs will not deter these suits, and  ``loser pays'' 
rules would only encourage them.  
  If the court  error is unpredictable,  the plaintiff  runs  a risk  in 
bringing suit,  
but if  litigation costs are small   relative to the potential award, it   is  
worthwhile even if the probability of winning is also small.    This would 
generate the pattern described above of many  seemingly frivolous suits but a  
certain number of  absurdly high awards.   Thus, either systematic  and 
predictable court bias in interpreting  certain kinds of evidence  or erratic 
and unpredictable court error can generate meritless suits.      
  
 This raises the question of whether   the degree of 
  predictability of court
error   increases or decreases  the number of meritless  suits.  It can do 
either,   as      the following two examples will show.   

\bigskip
 \noindent
 {\it Example 1: Predictability increases  the number of meritless suits.}
 Let the parameters be  those  of    Table 1.   First, suppose the  error is 
predictable, as in Column (1) of Table 1. If $d=0$,  the predicted
damage measurement  of suits that are filed is either  $\hat{d}=\tilde{d} = -1 $ 
(with probability .2),  $\hat{d}=\tilde{d} = 0$ (with
probability .6) or $\hat{d}=\tilde{d}=1$ (with probability .2). If $\tilde{d}=-
1$,
suit is brought with probability 0; if  $\tilde{d}=0$,
  with probability .125; and if $\tilde{d}=1$  with probability .5. Thus, the 
overall probability of a meritless suit    is
(.2) (0) + (.6)(.125) + .2(.5) =  .175. 
 
     
Next,  suppose the error is   unpredictable, as in Column (2) of Table 1.  If 
$d=0$,   then
$\tilde{d}=0$ also.  Suit  is brought with probability .16 when $\tilde{d}=0$. 
Thus, the probability of a meritless suit is  .16 when error is unpredictable. 
  This is less than .175, so
predictability {\it increases} the number of meritless suits.

 The  intuition  behind   Example  1 is that when the error is predictable, the 
plaintiff  feels safe in bringing meritless suits  with evidence that 
exaggerates the amount of damage, but if it  is  unpredictable, he is    more 
reluctant because  the court error may go against him rather than in his favor.   

  
\bigskip
 \noindent
 {\it Example 2: Predictability  reduces  the number of meritless suits.}
Let the parameters  be those of those of Table 2, which modifies     Table 1  by 
putting a  probability atom  of weight .5 on $c=.30$, so   half of all potential 
plaintiffs face costs of $c=.30$ from a lawsuit, and the rest are distributed 
with costs from 0 to 2.       

 If the error is predictable, the court's equilibrium
awards are  the same as in Example 1, because  $F(\tilde{d})$ plays no role in 
the determination of the final awards.    Column (2.1) of  Table 2 shows that   
a meritless suit ($d=0$)   results in $\tilde{d}=-1$ with probability .2, 
$\tilde{d}=0$ with probability .6,  and $\tilde{d}=1$ with probability .2.  The 
probability of  suit  being  brought, given a meritless case, is .2(.00)+ 
.8(.06) + .2(.75), or  .20. 

  

\begin{small}
\begin{center}
\begin{tabular}{ll|c|c  }
          \multicolumn{4}{c}{   TABLE 2: MERITLESS SUITS }\\
        \multicolumn{4}{l}{  }\\
\hline
\hline
 & &    (2.1) &  (2.2)   \\
 & &   Predictable &   Unpredictable \\
  & &  Error Only &  Error Only   \\
    & & ($\tilde{d} =  \hat{d}$) &   ($d=\tilde{d}$)   \\
   & & $\gamma =0, \theta=.2$&  $\gamma =.2, \theta=0$   \\
   \hline
   & &    &      \\
    & $F( -1)$ &  {\bf .00}&   .00     \\
     Proportion & $F(0)$ &    {\bf .06 } &  {\bf .60} \\
    of  lawsuits & $F(1)$ &   {\bf .75 } &  .77 \\
Brought & $F(2)$ &    .95&   .92  \\
  $\;\;F(\tilde{d})$ & $F(  3)$ &  1.00 &  1.00   \\
  & &    &      \\
  \hline
  & &    &      \\
 & $a(-1)$ &  0.00& {\bf 0.00}    \\
  Final & $a(0)$ &    0.25 &    {\bf 0.30}  \\
   Court & $a(1)$ &   1.00 & {\bf 1.08}\\
   Award & $a(2)$ &   1.75&   1.78 \\
 $\;\;a(\hat {  d})$ & $a(3)$ & 2.00&  2.00   \\
   & &    &      \\
  \hline
   \hline
    \multicolumn{4}{l}{Assumed: $p =q=r=1/3$. $G(c)$ is uniform on [0,2],} \\
   \multicolumn{4}{l}{ except for an atom of weight .5 on $c=.30$. $\mu=1$. }\\
 \multicolumn{4}{l}{Calculations are rounded.   }\\
   \end{tabular}
  \end{center}
 \end{small}

 
 If the error is unpredictable,  as in Column (2.2) of Table 2,  then a 
plaintiff with a meritless suit  ($d=0$) knows that the measured damage  is  
$\hat{d}=-1$ with probability .2, $\hat{d}=0$ with probability .6, and  
$\hat{d}=1$ with probability .2, which yield awards of $a(-1)=0.00$,  
$a(0)=0.30$, and  $a(1)=1.08$,   for  an expected award of 0.40.  This expected 
award exceeds 0.30,  the modal  litigation cost  of plaintiffs, so   a large 
number of plaintiffs decide to bring  suit,  and the proportion of meritless 
cases that become lawsuits is    0.60. This is higher than the proportion of 
meritless cases which become lawsuits when error is predictable (0.20), so 
predictability {\it reduces} litigation. 


When   the error is predictable in Example 2,  the plaintiff  knows from the 
start whether his  meritless suit will lead to  a high award, so often he will  
not bring  suit. If the error is unpredictable, however, and the cost of 
bringing suit is low enough, it is worth bringing suit  in the hope that the 
court will  make misjudge the evidence.  

The key difference between the examples is in   the cost of bringing suit. In 
Example 1, plaintiffs have an even distribution of  litigation costs, so 
predictability of the error  substantially increases the number of plaintiffs 
for whom suits are profitable.  In    Example 2,  a large bloc of plaintiffs 
have low litigation costs and are willing to gamble on what an unpredictable  
court will do, but if the error is predictable many of them realize that while 
the cost of a suit is low, the benefit is even lower. 


 The  two examples prove  Proposition 6. 
 
{\it PROPOSITION 6: Increased predictability of court error can
either increase or decrease the number of meritless  suits.  }
 
     Meritless suits will be most common when numerous cases of damage occur but 
only a few are due to tortious
behavior. Product liability and employment law face this problem. 
Many people become sick or injured, and many lose existing jobs or fail to
acquire new ones, but the great majority of harm is  not caused by
torts.  Even unpredictable court error may induce lawsuits to be brought
in the hope of a lucky award,  and  predictable error has an even stronger 
effect.  It is easy, for example,  to find  misleading statistical evidence for
employment discrimination. Even if no employer is discriminatory,  half of them
will employ less than the median proportion of racial minorities, and some of   
them will appear highly discriminatory.  It is the applicants for jobs at those 
companies who will choose to file suit.\footnote{See Epstein (1992) p. 210, 
citing  
Follett
\& Welch (1983).} Knowing this, the court should discount such evidence. 
  
  If the distribution
of cases is asymmetric and the probability of meritless cases is
high, the court's optimal policy may be to  scale back damages so much
that no lawsuits of any kind, meritless or meritorious, are brought.  The 
problem is one of false positives.  When a large proportion of cases are
meritless and court error is sizeable,  then even if meritorious suits
also exist it may be efficient to block all suits.  This argument    supports 
the exclusion of   pain and suffering from damage awards, for example,  because  
measurement error is  particularly great  for that category of damage.      
    

 
%---------------------------------------------------------------
\newpage
 \bigskip
\begin{center} { 7.  CONCLUDING REMARKS}
 \end{center}
 
 
The model has shown that the effect  of court error  is  not simply to
bias damage awards upwards or downwards, because different biases go
in different directions. 
Court error has
three effects: 

{\it 1. Regression to the mean.} Both predictable and unpredictable
error introduce regression towards the mean:  extreme measured damage  is
more likely to have been produced by  court  error.  The court
should compensate   by increasing small damage awards and reducing
large ones.


{\it 2. Plaintiff selection.} Predictable error encourages the
plaintiff to file suit if the error is positive. If the court  does
not know whether to classify an award as large or small,  it should
adjust the award downwards.


{\it 3. Plaintiff signalling.} Unpredictable error introduces a
signalling effect because the willingness of a plaintiff to bring a
suit with low apparent damages shows that he thought measured damages
would be higher. The court  should   adjust the award upwards.


  Both kinds of  error
encourage the filing of meritless suits, in which the true damage 
is  zero.   Predictable positive error   creates the possibility of a suit that 
is both meritless and riskless for the plaintiff, while unpredictable error 
allows   the plaintiff to gamble that his suit, while meritless, will 
nonetheless generate a positive award. 
 If abundant opportunities are available to bring  meritless  suits,   courts 
should adjust even
moderate damage measurements downwards. 

 
   The analysis   has assumed that the court's goal is to make
the award match the true damage  in the particular case as closely as
possible. This is a much narrower issue than that of  what level of
award is optimal.  Optimality depends on the law's goal, and if the goal is to 
deter harmful behavior, trying to match
awards to damages  on a case by case basis may not be the best policy.       If 
a potential tortfeasor does not know whether his action will
cause  a minor or a major injury,   it may be optimal to
overcompensate major injuries because minor injuries do not generate
lawsuits and receive zero compensation.  If, on the other hand,  potential 
tortfeasors fear heavy legal costs of defense, they may
be overcautious and all awards should be scaled down (Polinsky and
Rubinfeld [1988]). Or, it may be that courts should reduce the amount
of litigation while keeping the damages paid out high by raising both
the standard of proof and the size of awards (Polinsky and Che
[1991]).  Moreover, court error has important implications for  the question of 
what level of penalty is optimal.    Polinsky and Shavell (1994), for example,  
note that if the court awards penalties that are inefficiently and predictably  
small, then  penalties based on the   harm to the injured are  superior  to 
penalties  based on the 
benefit to the injuror. 

  The present model ignores these considerations, and its
conclusions must be considered as one more set of effects to add to
the tangle.  It does, however, address a question that most people
think is at the heart of justice---- How  can the court most accurately 
compensate plaintiffs for the damage done by defendants?---    and concludes 
that     a court which recognizes its own fallibility should  use that knowledge 
in deciding its awards.  

%---------------------------------------------------------------
 \begin{small}
\bigskip
\noindent
\begin{center}
 {APPENDIX}
\end{center}


The appendix   uses the  model in   Section 2 to calculate the relevant
probabilities used to calculate the expected value of $d$ given $\hat{d}$. 
   For notational convenience, let 
$\hat{d}_i$ denote $\hat{d}=\mu + i$, $\tilde{d}_i$ denote $\tilde{d}=\mu +i$,  
and $d_i$ denote $d=\mu +i$.  
  From equations (\ref{e1}) to (\ref{e4}) one can find
$Pr(\hat{d}|{\tilde{d}})$ and $Pr ({\tilde{d}}|d)$ for different values of $d$,
${\tilde{d}}$, and $\hat{d}$.  

 For $\hat{d}=\mu-3$  the Bayesian updating is very
simple: $a(\mu-3) =\mu- 1$.  This is so because the only way
that $\hat{d}=\mu-3$   could arise is if $d=\mu-1$ and  both errors were 
negative. Similarly, $a(\mu+3)=  \mu + 1$.
  This leaves the intermediate values of $\hat{d}$, which do not
perfectly reveal $d$.  These can be broken down depending on which of
the five values of $\tilde{d}$ has arisen. 

  For $\hat{d}=\mu-2$, equation (\ref{e8}) becomes 
 \begin{eqnarray} \label{e19a}
    Pr(\hat{d}_{-2}, lawsuit|d_i) & = & \sum_{j=-2}^{2}   F(j) Pr(\hat{d}_{-
2}|{\tilde{d}}_j)Pr (\tilde{d}_j|d_i)  \nonumber\\
  & = &  F(\mu-2)(1-2\gamma ) Pr({\tilde{d}}_{-2}|d_i)  +F(\mu-1) \gamma 
Pr({\tilde{d}}_{-1}|d_i) + F(\mu)(0) \nonumber  \\ 
   &&  +  F(\mu+1)(0)     + F(\mu+2)(0)   . 
 \end{eqnarray}
  Applying equation (\ref{e19a})  to $i= \mu-1,\mu, \mu + 1$ , the three 
possible true values of damage,  yields
 \begin{eqnarray}  
   \label{e20a}
 Pr(\hat{d}_{-2}, lawsuit |d_{-1}) &  = &   F(\mu-2)(1-2\gamma)  \theta +  
F(\mu-1)\gamma  (1-2\theta) \nonumber   \\
   \label{e21a}
 Pr(\hat{d}_{-2}, lawsuit |d_0)   & = & F(\mu-2)(1-2\gamma) ( 0)+  F(\mu-
1)\gamma   \theta  \nonumber  \\
   \label{e22a}
 Pr(\hat{d}_{-2}, lawsuit |d_1)   &=&   F(\mu-2)(1-2\gamma)  (0 )+  F(\mu-
1)\gamma ( 0)    \nonumber
 \end{eqnarray} 
   and,  from equation (\ref{e9}),
   \begin{eqnarray}   \nonumber
  Pr (\hat{d}_{-2},  lawsuit )   = F(\mu-2)(1-2\gamma)  \theta p +  F(\mu-
1)\gamma  (1-2\theta)p +  F(\mu-1)\gamma   \theta q 
    \end{eqnarray}



 For $\hat{d}= \mu -  1 $, equation (\ref{e8}) becomes 
 \begin{eqnarray} \label{e19}
    Pr(\hat{d}_{-1}, lawsuit|d_i) & = & \sum_{j=-2}^{2}   F(j) Pr(\hat{d}_{-
1}|{\tilde{d}}_j)Pr (\tilde{d}_j|d_i) \nonumber \\
  & = &  F(\mu-2)\gamma  Pr({\tilde{d}}_{-2}|d_i)  +F(\mu-1) (1-2\gamma ) 
Pr({\tilde{d}}_{-1}|d_i) + F(\mu)\gamma Pr({\tilde{d}}_0|d_i)  \nonumber\\ 
   &&   +  F(\mu+1)(0) Pr({\tilde{d}}_1|d_i)   + F(\mu+2)(0) 
Pr({\tilde{d}}_2|d_i) . 
 \end{eqnarray}
  Applying equation (\ref{e19})  to $i= \mu-1,\mu, \mu + 1$ , the three possible 
true values of damage,  yields
  \begin{eqnarray}  
   \nonumber
 Pr(\hat{d}_{-1}, lawsuit |d_{-1}) &  = &   F(\mu-2)\gamma  \theta +  F(\mu-
1)(1-2\gamma) (1-2\theta)  + F(\mu)\gamma  \theta \;\;\;\;\\
    \nonumber
 Pr(\hat{d}_{-1}, lawsuit |d_0)   & = & F(\mu-2)\gamma  ( 0)+  F(\mu-1) (1-
2\gamma) ( \theta)  + F(\mu) \gamma  (1-2\theta)\;\;\;\; \\
   \nonumber
 Pr(\hat{d}_{-1}, lawsuit |d_1)   &=&   F(\mu-2)\gamma  (0 )+  F(\mu-1)((1-
2\gamma)) ( 0)  + F(\mu)\gamma   \theta  \;\;\;\;
 \end{eqnarray} 
   and,  from equation (\ref{e9}),
   \begin{eqnarray}   \nonumber
  Pr (\hat{d}_{-1},  lawsuit )   =F(\mu-2)\gamma  \theta p +  F(\mu-1) (1-
2\gamma)  [( 1-2\theta)p+ \theta q]  + F(\mu)\gamma    [\theta p + ( 1-
2\theta)q+ \theta r]   
   \end{eqnarray}
  
For $\hat{d}=\mu $,  equation (\ref{e8}) becomes
  \begin{eqnarray}  \label{e24}
 Pr(\hat{d}_0, lawsuit|d_i) &  = &  \sum_{j=-2}^{2}F(j) 
Pr(\hat{d}_0|{\tilde{d}}_j)Pr ({\tilde{d}}_j|d_i) \nonumber\\
  & = &
F(\mu-2) (0)  Pr({\tilde{d}}_{-2}|d_i) + F(\mu-1) \gamma Pr({\tilde{d}}_{-
1}|d_i)+ F(\mu) (1-2\gamma )Pr({\tilde{d}}_0|d_i)\nonumber \\ 
   &&   + F(\mu+1) \gamma Pr({\tilde{d}}_{ 1}|d_i)+ F(\mu+2)  
(0)Pr({\tilde{d}}_{2}|d_i)  
  \end{eqnarray}
   Applying this to $i= \mu-1,\mu, \mu + 1$  gives
 \begin{eqnarray} \label{e25}
 Pr(\hat{d}_0, lawsuit |d_{-1}) =  F(\mu-1) \gamma (1-2\theta) + F(\mu) (1-
2\gamma )  \theta + F(\mu+1)\gamma (0)  \nonumber
\\
 \nonumber
 Pr(\hat{d}_0, lawsuit |d_0) =   F(\mu-1) \gamma \theta  + F(\mu) (1-2\gamma )  
(1-2\theta) + F(\mu+1)\gamma  (0) \nonumber
\\
 \nonumber
 Pr(\hat{d}_0, lawsuit |d_1) =   F(\mu-1) \gamma  (0)  + F(\mu) (1-2\gamma ) 
\theta + F(\mu+1)\gamma (1-2\theta)
   \end{eqnarray}
 and,  from equation (\ref{e9}),
   \begin{eqnarray} \nonumber
 Pr (\hat{d}_0,  lawsuit)   = F(\mu-1) \gamma  [(1-2\theta)p + \theta q ] + \\
   F(\mu) (1-2\gamma )[\theta p + (1-2\theta)q + \theta r]  + F(\mu+1)\gamma  
(1-2\theta)r
  \end{eqnarray}
 


 For $\hat{d}= \mu + 1$, equation (\ref{e8}) becomes
   \begin{eqnarray} \label{e29}
  Pr(\hat{d}_1, lawsuit |d_i) &  =  & \sum_{j=-2}^{2} F(j) 
Pr(\hat{d}_1|{\tilde{d}}_j)Pr ({\tilde{d}}_j|d_i)\nonumber \\
  & =&  F(\mu-2) (0 ) Pr({\tilde{d}}_{-2}|d_i)  +F(\mu-1) (0) Pr({\tilde{d}}_{-
1}|d_i)+ F(\mu)\gamma Pr({\tilde{d}}_0|d_i)\nonumber \\ 
   &&   
 + F(\mu+1)(1-2\gamma ) Pr({\tilde{d}}_1|d_i) + F(\mu+2) (\gamma) 
Pr({\tilde{d}}_{ 2}|d_i)  .  \end{eqnarray}
   Applying this to $i= \mu-1,\mu, \mu + 1$  yields
 \begin{eqnarray}\nonumber
 Pr(\hat{d}_1, lawsuit |d_{-1}) =   F(\mu)\gamma \theta   
 + F(\mu+1)(1-2\gamma ) (0) + F(\mu+2)\gamma(0 )   \\
\nonumber
 Pr(\hat{d}_1, lawsuit |d_0) =  F(\mu)\gamma  (1-2\theta)  
 + F(\mu+1)(1-2\gamma ) \theta + F(\mu+2)  \gamma(0 )  \\
\nonumber
 Pr(\hat{d}_1, lawsuit |d_1) =  F(\mu)\gamma  \theta  
 + F(\mu+1)(1-2\gamma ) (1-2\theta) + F(\mu+2)   \gamma \theta
 \end{eqnarray}
 and,  from equation (\ref{e9}),
 \begin{equation} \nonumber
 Pr (\hat{d}_1, lawsuit ) =   F(\mu)\gamma [ \theta p + (1-2\theta)q + \theta r   
]
 + F(\mu+1)(1-2\gamma ) [ \theta q + (1- 2\theta)r ]+ F(\mu+2)   \gamma \theta r
  \end{equation}
 
For $\hat{d}= \mu + 2$, equation (\ref{e8}) becomes
   \begin{eqnarray} \label{e29a}
  Pr(\hat{d}_2, lawsuit |d_i) &  =  & \sum_{j=-2}^{2} F(j) 
Pr(\hat{d}_2|{\tilde{d}}_j)Pr ({\tilde{d}}_j|d_i)  \nonumber\\
  & =&  F(\mu-2) (0 ) Pr({\tilde{d}}_{-2}|d_i)  +F(\mu-1) (0) Pr({\tilde{d}}_{-
1}|d_i)+ F(\mu) (0) Pr({\tilde{d}}_0|d_i)  \nonumber\\ 
   &&  
 + F(\mu+1) \gamma Pr({\tilde{d}}_1|d_i) + F(\mu+2) (1-2\gamma) Pr({\tilde{d}}_{ 
2}|d_i)  . 
 \end{eqnarray}
   Applying this to $i= \mu-1,\mu, \mu + 1$  yields
 \begin{eqnarray} \nonumber
 Pr(\hat{d}_2, lawsuit |d_{-1}) =   F(\mu+1) \gamma  (0) + F(\mu+2)(1-2\gamma)(0 
)   \\
\nonumber
 Pr(\hat{d}_2, lawsuit |d_0) =    F(\mu+1) \gamma  \theta + F(\mu+2)  (1-
2\gamma)(0 )  \\
\nonumber
 Pr(\hat{d}_2, lawsuit |d_1) =  F(\mu+1) \gamma  (1-2\theta) + F(\mu+2) (1-
2\gamma)\theta
 \end{eqnarray}
 and,  from equation (\ref{e9}),
 \begin{equation} \nonumber
 Pr (\hat{d}_2, lawsuit ) =     F(\mu+1) \gamma  \theta q +  F(\mu+1) \gamma  
(1-2\theta) r + F(\mu+2) (1-2\gamma)\theta r
  \end{equation}

 
Combining  the  calculations above to fill in the terms in equation (\ref{e7}) 
yields the court's adjusted awards for different
levels of  deviations from the average measured damage:
     \begin{footnotesize} \begin{equation} \label{e9a}
 a(\mu -3 ) =    \mu -1  
 \end{equation}

 \begin{equation} \label{e10a}
 a(\mu -2 ) =  \mu  -  \frac{ [F(\mu-2)(1-2\gamma)  \theta   +F(\mu-1)\gamma  
(1-2\theta)]p    }
   { [F(\mu-2)(1-2\gamma)  \theta   +  F(\mu-1)\gamma  (1-2\theta)]p +  F(\mu-
1)\gamma   \theta q }
 \end{equation}

   
 \begin{equation} \label{e10}
 a(\mu -1 ) =  \mu  - \frac{   [F(\mu-2) \gamma \theta+F(\mu-1) (1- 2\gamma) (1-
2\theta) +F(\mu)\gamma    \theta  ] p  -      F(\mu)\gamma    \theta r }
 {  F(\mu-2) \gamma \theta  p +  F(\mu-1) (1- 2\gamma)  [( 1-2\theta)p+ \theta 
q]  +
 F(\mu)\gamma    [\theta p + ( 1-2\theta)q+ \theta r] }
 \end{equation}
 
 \begin{equation} \label{e11}
    a(\mu  ) 
   =  \mu +    \frac{ -[F(\mu-1) \gamma  (1-2\theta)    + F(\mu) (1-2\gamma ) 
\theta ]p + 
   [F(\mu)(1-2\gamma)   \theta +  F(\mu+1)\gamma  (1-2\theta) ]     r }
 {F(\mu-1) \gamma  [(1-2\theta)p + \theta q ] + F(\mu) (1-2\gamma )[\theta p + 
(1-2\theta)q + \theta r]  + F(\mu+1)\gamma  (1-2\theta)r}
  \end{equation}
 
 \begin{equation} \label{e12}
 a(\mu + 1) =  \mu+ \frac{ -[F(\mu)\gamma \theta     ]p + 
  [F(\mu)\gamma  \theta + F(\mu+1)(1- 2\gamma) (1- 2\theta) + F(\mu+2)  \gamma   
\theta]r
    }
          {F(\mu)\gamma [ \theta p + (1-2\theta)q + \theta r   ]
 + F(\mu+1)(1- 2\gamma ) [ \theta q + (1- 2\theta)r ]+ F(\mu+2)   \gamma \theta 
r}
 \end{equation}

   \begin{equation} \label{e12aa}
a(\mu + 2)   =  \mu+  \frac{[F(\mu+1) \gamma  (1-2\theta)  + F(\mu+2) (1-
2\gamma)\theta]r}{ F(\mu+1) \gamma  \theta q +  [F(\mu+1) \gamma  (1-2\theta ) + 
F(\mu+2) (1-2\gamma)\theta ]r}
  \end{equation}


 \begin{equation} \label{e12a}
a(\mu + 3)   =  \mu+1  
 \end{equation}


Equations (\ref{e9a}) to  (\ref{e12a}) that are used to  prove the propositions 
in the main text. 

 \end{footnotesize}


\end{small}

%---------------------------------------------------------------

\newpage
 \noindent
 {\bf References}

BEBCHUK, LUCIEN,   ``Litigation and Settlement Under Imperfect Information,'' 
{\it RAND Journal of Economics},  15 (1984) , 404-415. 


BEBCHUK, LUCIEN,   ``Suing Solely to Extract a Settlement Offer,'' {\it Journal 
of Legal
Studies},    17  (1988), 437-450.

BLUMSTEIN, JAMES, RANDALL BOVBJERG, AND FRANK SLOAN, 
   ``Beyond Tort Reform: Developing Better Tools for Assessing Damages for 
Personal Injury,'' {\it Yale Journal on Regulation},    8  (1991),  171-212.

BOVBJERG,  RANDALL, FRANK SLOAN,  AND JAMES BLUMSTEIN,   ``Valuing
Life and Limb in Tort,''  {\it Northwestern University Law Review},    83 
(1989), 
908-976.  
                          
CALFEE, JOHN AND RICHARD CRASWELL, 
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