%Paper: ewp-le/9506003
%From: Eric Rasmusen <erasmuse@rasmusen.bus.indiana.edu>
%Date: Wed, 14 Jun 95 13:37:32 -0500

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              \titlepage
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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}
                     \vskip 15pt
                    June 8, 1995 \\
                    \bigskip                     Eric Rasmusen 

      \vskip 1in
                    {\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. 


 

                \vskip .3in
\begin{small}
          \noindent 

\hspace*{20pt}	  	  Indiana University
School of Business, Rm. 456, 

  10th Street  and Fee Lane,
  Bloomington, Indiana, 47405-1701.
  Office: (812) 855-9219.  Fax: 812-855-3354. Internet:
Erasmuse@indiana.edu.\\ 

 

 \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,'' 
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