Reproduced with the permission of the *Journal
of Legal Studies*. This is based on the version on my hard disk,
and may differ in detail from the published version.

Table of Contents

*David Friedman **

It is apparent from the effort and time required to try these 160 cases, that unless this plan or some other procedure that permits damages to be adjudicated in the aggregate is approved, these cases cannot be tried. Defendants complain about the 1% likelihood that the result would be significantly different. However, plaintiffs are facing a 100% confidence level of being denied access to the courts. The Court will leave it to the academicians and legal scholars to debate whether our notion of due process has room for balancing these competing interests.--Judge Robert Parker,

Cimino v Raymark

In *Cimino v. Raymark*,^{0}
Judge Parker of the Eastern District of Texas implemented a radical
solution to the problem of litigating mass torts. Instead of
conducting individual trials for several thousand plaintiffs, he
selected a random sample of 160 of them, tried their cases, and based
the awards given to the remaining plaintiffs on the outcome of those
trials. In defending the procedure against the charge that it
deprived the parties of due process, he argued that if he had instead
required individual trials, most of the cases would never have been
resolved.

The procedure of *Cimino* was explained and
defended in a 1992 article by Michael J. Saks & Peter David
Blanck.[1]
The purpose of this essay is not to dispute either their views or
those of Judge Parker, but rather to suggest a further step along the
same path. The procedure of aggregation and sampling implemented in
*Cimino* does a reasonably good job of estimating the total
damages that the defendants would have paid if every case had been
tried separately,[2]
and does so at a cost much lower than that of individual trials. It
does a much poorer job of allocating that total among the plaintiffs.
My proposal is intended to solve that problem.

In part I of this article I explain the procedure
used by Judge Parker in *Cimino*, the improvements suggested by
Saks and Blanck, and the limitations of the procedure, even with such
improvements. Part II describes my proposal for generating an
estimate of the relative claims of the plaintiffs and incorporating
that estimate into the procedure. Part III considers the legal status
of the modified procedure, arguing that it is in some ways more
defensible than the version implemented in *Cimino*. Part IV
discusses potential problems, both those implicit in the original
idea of aggregation and sampling and additional ones created by my
proposed modifications. Part V suggests ways in which the procedure I
suggest could be extended beyond the context of class actions. Part
VI describes the results of the application of the procedure to an
explicit formal model; the mathematics are presented in an appendix.
Part VII summarizes my conclusions.

If the Court could somehow close thirty cases a month, it would take six and one-half years to try these cases and there would be pending over 5,000 untouched cases at the present rate of filing.--Judge Parker in

Cimino

*Cimino v. Raymark* went to trial as a class
action with 2,298 plaintiffs and five defendants. The trial consisted
of three phases. In Phase I, a set of issues common to all plaintiffs
and defendants were resolved.[3]
Phase II apportioned causation among the defendants and determined
which plaintiffs had had sufficient exposure to asbestos, based on
each plaintiff's workplace and craft, for such exposure to be a
producing cause of an asbestos-related injury or
disease.[4]

The purpose of Phase III was to determine damages. Instead of trying all cases, the court divided the plaintiffs into five categories, according to which asbestos related disease each suffered from. A random sample was drawn from each category; the sample was larger for categories with more plaintiffs.[5] The sample cases were tried. Plaintiffs in the sample received the damages that they were awarded. Plaintiffs not in the sample received the average of the damages awarded to the tried cases in their category.

Judge Parker argued that the result was fair to the defendants, since the total amount awarded was an accurate estimate of the total that would have been awarded if all cases had been tried. He cited confidence levels ranging from 95% to 99%, but did not explain what those numbers meant or what assumptions were used to calculate them.

The situation is not quite so clear as Judge
Parker apparently believed. The statistical conclusions reported, if
correct, depend on assumptions about the distribution of the awards
that would be produced by jury trial. It is possible to describe
distributions consistent with the observed data for which the result
of even as large a sample as was used in *Cimino* would be a
very imprecise measure of the total damages that would be awarded.
With such a distribution, the expected result of the *Cimino*
procedure would still be correct: if the procedure were repeated a
large enough number of times, the average outcome would be very close
to the result of trying every case separately. But the probability
that the result produced by the *Cimino* procedure would be
substantially different from the result of trying all cases might be
much larger than implied by the confidence levels cited by the
judge.[6]
This suggests that it might be worth looking for a procedure superior
to random sampling.

Even if, as Judge Parker argues, the procedure is
fair to the defendants, there remains the question of whether it
properly allocates the damage payment among the individual
plaintiffs. The procedure used in *Cimino* does not do so, as
Judge Parker himself conceded.[7]
He dealt with that problem by obtaining the plaintiffs' assent in
advance. In future litigation involving such procedures, however, the
question will be important for at least five reasons.

1. Many people regard justice as part of what litigation is supposed to produce. If a procedure collects the right amount of damages but gives them to the wrong people, or to the right people but in the wrong amounts, it is not just.

2. One purpose of some of the legal rules that
determine damages, such as contributory negligence, is to affect the
incentives of potential plaintiffs. In *Cimino*, some plaintiffs
whose cases were tried received no damages, possibly because their
decision to smoke was regarded by the jury as contributory
negligence.[8]
The effect of such verdicts was to reduce the award given to all
plaintiffs in the same disease category whose cases were not tried,
smokers and non-smokers alike. So the use of the procedure undercuts
the effectiveness of such a legal rule.

3. In order for a class to be certified, the judge must find that the representative parties will fairly and adequately protect the interests of the class.[9] A procedure which predictably awards some plaintiffs more than they would get from trying their case themselves and others less may not meet the requirement.

4. Even if the class is certified, individual members are free to withdraw. A procedure that predictably awards some plaintiffs less than they would receive at trial gives such plaintiffs an incentive to withdraw from the class, which reduces the benefit both of the class action and of the procedure.[10]

5. In awarding the right amount of damages to the
wrong people, the *Cimino* procedure resembles fluid recovery.
Under fluid recovery, where it is difficult to identify the members
of the plaintiff class and determine how much of the award each is
entitled to, money awarded to the plaintiffs is instead used to
benefit a group of people similar to those who were injured. That
approach has been seriously questioned by the courts.[11]

For all of these reasons, it is desirable to
construct procedures that approximate the correct result among
plaintiffs as well as between plaintiffs and defendants. In
*Cimino*, Judge Parker attempted to do so in two ways. Phase II
of the trial was designed to eliminate from the case defendants whose
exposure to asbestos was not a producing cause of an asbestos related
injury or disease. In Phase III, defendants were grouped according to
the particular sort of injury or disease they had suffered,
presumably because individuals suffering from the same disease would
have some tendency to be owed the same damages.

Both of these are very imprecise ways of allocating damage payments to individual plaintiffs. Saks and Blanck offer two additional possibilities. One is to use statistical analysis to define groups with common characteristics. The other is to construct a linear model relating damages to characteristics and then use trial results to estimate the parameters of the model.[12]

While these procedures can improve on the simple
approach of giving every plaintiff the same amount or the slightly
more complicated approach implemented in *Cimino*, they suffer
from a common problem. It is neither obvious in advance nor
uncontroversial what characteristics are relevant to the damage award
or how they are related. Even if we knew the characteristics, there
is no reason to assume the relation is linear.[13]
As statisticians are aware, the same data can be fit with a multitude
of different specifications. If, after trying a few thousand, the
court finds one that happens to fit the tried cases fairly well, that
should not give us much confidence that it will also fit the untried
cases.

What we need is not a procedure for dividing the damage award among the plaintiffs-the best way of doing that will almost certainly vary from case to case. What we need is a procedure that makes it in the interest of someone to figure out, for any particular case, what the correct division among the plaintiffs in that case is. Part II describes one such procedure.

I define the strength of a plaintiff's case as the
average of what would be awarded if it were tried many times by many
separate juries; I call this average verdict (for plaintiff *i*)
<*D _{i}*>. The objective of the procedure is to
produce a damage award of about <

By examining the facts relevant to an individual
plaintiff *i*, an investigator can estimate the value of
<*D _{i}*>. The more resources are spent on the
investigation, the more accurate the estimate will be. This is true
both for an individual investigator and for a judge or jury
calculating an award in the course of a trial. I assume the cost to a
competent individual investigator of estimating
<

We start with a group of *N* plaintiffs
represented by an attorney. The procedure is as follows:

Step 1: The plaintiffs' attorney produces, for
each plaintiff *i*, a claim *C _{i}*.

Step 2: The plaintiffs' attorney gives his list of
claims *C _{i}* to the defendant's attorney.

Step 3: The defendant's attorney selects from the list a small number of cases to be tried. For simplicity in exposition, assume that ten cases are to be selected and that the cases selected turn out to be those of plaintiffs 1-10.

These cases are tried. The court awards damages
*D _{i}*

Step 4: The court calculates
*R*=(*D*_{1}/*C*_{1}+*D*_{2}/*C*_{2}+...*
D*_{10}/*C*_{10})/10, and awards damages of
*R*x*C _{i}* to each of the

Under this procedure, it should be possible to
resolve the *N* cases much more cheaply than with *N*
separate trials. Only ten cases actually have to be litigated. All
plaintiffs have their damages estimated, but the estimate is made for
everyone else by the plaintiffs' attorney.

Why does the procedure generate actual damages for
plaintiff *i* close to <*D _{i}*>? Consider
the situation first from the standpoint of the defense attorney at
Step 3. He wants to select plaintiffs whose claims

Next consider the situation from the standpoint of
the plaintiffs' attorney at step 1. Because he knows that the defense
attorney will try to select for trial plaintiffs with a high ratio of
*C _{i}* to <

The award received by a particular plaintiff may
deviate from what he ought to receive for two reasons: The court may
give the wrong verdicts for the cases tried, or the plaintiffs'
attorney may claim the wrong amount for a particular plaintiff. Since
ten cases are tried separately and their results averaged, the first
source of error should be much smaller than if each plaintiff's case
had been tried by itself.[18]
Since the attorney can estimate <*D _{i}*> much
less expensively than a court, the second source of error can be made
smaller than it would be with an actual trial, while still keeping
litigation expenses (including the expense of making such estimates)
well below those of individual trials. So it should be possible to
produce a more accurate verdict at lower cost under this procedure
than with individual trials. The cost is higher than with the

The procedure as I have described it makes sense
for a hundred plaintiffs, for a thousand, perhaps for more. In
*Cimino*, the information actually collected included medical
evaluations for about 1,400 of the 2,298 plaintiffs who eventually
went to trial, so much of the research required by my suggested
procedure had actually been done. But it makes less sense for the
sort of class action that involves a very large number of plaintiffs,
most with very small claims. In such a case, evaluating each
plaintiff's case in order to decide how much to claim for him might
cost more than the total damages awarded.

One approach to such a situation would be to allow
the plaintiffs' attorney to state *C _{i}* for classes of
plaintiffs rather than for individual plaintiffs. Thus he might claim
that each heavy smoker born before 1960 was entitled to $10, each
light smoker born between 1960 and 1970 to $2, .... The defendant's
attorney would select classes for trial; individual cases would be
selected from those classes at random. Such a variant on the
procedure might be appropriate in situations where individual claims
are low and separate estimates for each case thus unreasonably
expensive.

A more sophisticated approach would combine the procedure described here with an idea suggested by Saks and Blanck.[19] Instead of producing a claim for each plaintiff, the plaintiffs' attorney produces a statistical model showing how he believes that the amount each plaintiff is entitled to should depend on the characteristics of the plaintiff. The defendant's attorney specifies a sampling protocol, describing how plaintiffs are to be selected for trial based on their characteristics. The court then selects plaintiffs for trial at random, subject to the constraints of the sampling protocol. The verdicts for those plaintiffs are used to estimate the parameters of the model, and awards for all plaintiffs are calculated accordingly.

In the simplest version of this, the plaintiffs' attorney would specify the entire model save for one multiplicative parameter. If, for example, he believed that the amount awarded ought to depend linearly on the age of the plaintiff and the number of years he had worked at a site using asbestos, he might offer the model:

Damages* = A*[$100,000 - $1,000 x Age in
Years+ $10,000 x Years worked on site.]

The defense would then specify the range of ages
and work histories which were to be sampled, and the court would
choose plaintiffs within that range at random. Their cases would be
tried, and the results used to calculate *A*.

In a more elaborate version, the plaintiffs' attorney would specify only the form of the model. An example might be:

Damages = *A* - *B *x Age in Years +
*C *x Years worked on site+ *D *x (Years worked on
site)^{2}

The defense would again specify the characteristics of plaintiffs to be selected for trial, and the court would choose at random plaintiffs with those characteristics.

While both variants of this approach may sound complicated, especially to non-statisticians, their logic is the same as that of the simpler version described earlier. The difference is that the plaintiffs' attorney is providing a description of how damages relate to characteristics, rather than a claim for each plaintiff. The same logic as before makes it in his interest to get the description right. If, for example, he erroneously claims that the amount plaintiffs are entitled to does not depend on their age, when a jury would actually award more to younger plaintiffs, the defense can specify a sample heavily weighted towards older plaintiffs-and the result will be to push down the total amount awarded.

The same logic applies to more subtle errors. Suppose the plaintiffs' attorney specifies a linear relation of the form:

Damages = *A *+* BL*

where *L* is (say) length of exposure to
asbestos. Further suppose that the real relation, the one that
correctly predicts jury verdicts, is quadratic of the
form:

Damages = *A *+
*BL*^{2}

The defense, if it recognizes the error, can
specify a sample containing only small values of *L*. Again the
result will be to push down the total verdict.

In each of these situations, just as with the simpler version of my proposed procedure discussed earlier, an inaccurate specification by the plaintiffs' attorney of the relative claims of different plaintiffs gives the defense an opportunity to reduce the total amount awarded, which in turn gives the plaintiffs' attorney an incentive to do an accurate job of specifying the relative claims.

So far I have assumed that the cases we are considering are ones where the plaintiffs seek money damages. The procedure can be generalized to any case with a quantitative award-one describable by some cardinal measure. An example would be a suit where the plaintiffs were employees claiming seniority.

Another assumption I have been making is that tort litigation under my procedure is always resolved by trial. What is the effect on the analysis if we include the possibility of settlement?

Even if there is some possibility of settlement, the plaintiffs' attorney still has an incentive to estimate the relative claims of the plaintiffs accurately. If the case goes to trial, inaccurate estimates will result in lower total damages, since the defense will select the overclaimed cases for trial. If the case settles, it will settle on less generous terms if the defense believes that the estimates are inaccurate, and the plaintiffs thus likely to do badly at trial.

So even when litigation leads to settlement, the procedure still provides a mechanism for allocating damage payments to plaintiffs that reflects the relative strength of their cases. By doing so, it should reduce the conflict among plaintiffs over settlement terms and so make settlement easier.

Suppose a judge wished to implement the procedure described in Part II above. What legal problems would he face?

To begin with, he would face the same problem
faced in *Cimino*: the argument that due process required that
each plaintiff have an opportunity to make his case in court, and
that the defendant should have the opportunity to rebut each
plaintiff's case. If, as in *Cimino*, the plaintiffs assented in
advance to the procedure,[20]
that argument should be no stronger here than there. If anything, the
defendant's grounds for objection are even weaker under the procedure
I have proposed. Insofar as the defense believes that some plaintiffs
have weak cases-weaker cases, relative to other plaintiffs, than
their claims indicate-the defense is free to select those cases for
trial.[21]

If the plaintiffs, instead of or in addition to
the defendant, object, the situation is somewhat more difficult.
While the procedure saves the plaintiffs the cost of litigating every
case separately, it also, for reasons I will discuss in Part IV, has
some built-in bias against the plaintiffs. The plaintiffs might
reasonably demand either that the procedure be modified to eliminate
that bias (a possibility discussed below) or that they be compensated
for accepting a biased procedure. Supposing that such objections were
met, the plaintiffs under my procedure seem to be in the same
situation as the defendants in *Cimino*; although their cases
are not all being tried, they are being given an opportunity to get
approximately the same awards they would get if they were tried, and
at a much lower cost in litigation.

There is one respect in which the procedure is
more defensible than that employed in* Cimino*-or, arguably,
than the ordinary procedure for a class action. Rule 23(a)(4) of the
Federal Rules of Civil Procedure requires that the representative
parties in a class action will fairly and adequately protect the
interests of the class. Under the procedure I have proposed, the
representative parties have a clear interest in doing so. If they
attempt to benefit themselves at the expense of other members of the
class by arranging for their attorney to overclaim on their behalf,
the defense will select their cases for trial.[22]
The representative parties will gain nothing, and their attorney will
have a lower total award out of which to compensate
himself.

There are two fundamental problems with the procedure I have described. The first is that while it could produce a more accurate result at a much lower cost than would individual trials, it is not entirely clear that it will; it might instead produce a much more accurate result at a higher cost. The second is that the procedure, as so far described, has a built-in bias in favor of the defense.

* *

The method incorporated into Phase III produces a level of economy in terms of both judicial resources and transaction cost that needs no elaboration.--Judge Parker in

Cimino.

At first glance it seems obvious that trying 160
cases costs a great deal less than trying 2,298 cases, but this is
not quite so clear as it seems. Under the procedure employed in
*Cimino*, the verdicts in the tried cases determined the outcome
for all of the other cases. The result is that the amount at stake in
each tried case was about fourteen times as much as it would have
been if each case had only determined the outcome for that plaintiff.
With more at stake, we would expect both parties to spend more on
trying to win.

Whether this eliminates the cost savings of fewer
trials depends on how litigation expenditure varies with the amount
at stake.[23]
If the increase is proportional, the total cost of trials under
either *Cimino* or the procedure I have suggested will be the
same as if every case were tried separately; the only advantage of
the procedure would then be the increased accuracy, due both to
trying cases much more carefully and to using the average of the
tried cases, rather than the result of one case, in calculating the
amount to be awarded to each plaintiff.[24]

Suppose, however, that expenditure rises less than
proportionally with the amount at stake, everything else held
constant.[25]
Under that assumption, expenditure on the tried cases becomes less
and less important as the number of plaintiffs increases, since the
larger the number of plaintiffs the smaller the fraction necessary to
provide an adequate sample. In the limit of a very large number of
plaintiffs, expenditure on trying the sample of cases is negligible
compared to the cost of trying the cases individually. That is
consistent with what actually happened in *Cimino*.

So far I have been considering a problem raised by
both the *Cimino* procedure and the procedure I have proposed.
There is an additional cost problem that applies only to the latter.
Under that procedure, the plaintiffs' attorney spends resources
estimating the relative claims of each plaintiff[26]
and the defendant's attorney then spends resources examining
plaintiffs in order to decide which cases to select for
trial.

The plaintiffs' attorney can, if he wishes, produce his estimates of claims more accurately and less expensively than verdicts would be produced by individual trials. The same should be true for the defense attorney. In addition, if the number of cases is large, the defense need only examine a random sample of cases in order to do a reasonably good job of locating overclaimed cases to select for trial. It follows that the attorneys can act in a way which, under the proposed procedure, produces more justice at a considerably lower cost than would individual trials.

It is not, however, clear that it is in their interest to do so. Each attorney's objective, at least in part, is to benefit his clients at the expense of the other party. By making a more accurate set of estimates, the plaintiffs' attorney not only produces a more just distribution among his clients, he also makes it harder for the defense to locate overclaimed plaintiffs for trial. The more he spends on improving the accuracy of his claims, the larger the amount his side will receive. He must balance that benefit against the associated cost. The defense attorney faces a similar situation.

Here, as elsewhere in the economics of litigation, there is no reason to assume that the level of expenditure which is privately optimal for one party to a legal dispute is also socially optimal. The amount spent on estimating claims and detecting overclaimed plaintiffs will depend on detailed assumptions about information costs and distributions of claims, as we will see in the formal analysis presented in Appendix II and discussed in Part VI.

It follows from these arguments that we cannot be sure the procedure as described will cost less than ordinary trial without aggregation and sampling. This suggests two further queries. The first is whether we can say anything interesting about the relation between the costs of alternative approaches and the number of plaintiffs. The second is whether, if experience suggests that expenditures associated with the procedure are undesirably large, there may be ways of modifying it to reduce such expenditures.

An increase in the number of plaintiffs reduces
the percentage of cases that must be tried. If expenditure per case
increases less than proportionately with the amount at stake, the
result is that trial costs for my suggested procedure (or the
*Cimino* procedure) decrease, relative to the cost of trying all
cases, as the number of plaintiffs increases.

The same is probably true for the cost to the defense of selecting cases for trial. The larger the number of plaintiffs, the smaller the fraction that must be sampled in order to find ten cases from (say) the most overclaimed five percent. We would expect defense expenditures to increase less than proportionally with the number of plaintiffs, and so become smaller and smaller, relative to the total amount at stake, as the number of plaintiffs increases. This result is demonstrated in Appendix II for the particular distributions assumed there.

The opposite result can be expected for the cost to the plaintiffs' attorney of calculating claims. The more plaintiffs there are, the easier it is for the defense to locate those who have overclaimed. The more accurately the defense can locate overclaimed plaintiffs, the greater the incentive for the plaintiffs' attorney to make accurate claims. So an increase in the number of plaintiffs will tend to increase the amount spent per plaintiff by the plaintiffs' attorney. That is one reason why it might be desirable to shift from individual claims to statistical models when the number of plaintiffs becomes sufficiently large. The per plaintiff cost of estimating the parameters of a model to a given accuracy will fall as the number of plaintiffs increases.

The size of the expenditures by the attorneys will
depend on details of the distribution of claims and on the functions
relating expenditure on investigating a claim to information
produced. We cannot predict *a priori* how large it will be, any
more than we can predict *a priori*, in the case of ordinary
litigation, how much of the damages awarded will be eaten up in
litigation costs. But if experience indicates that the attorneys are
spending more than the improved accuracy their expenditure generates
is worth, we can lower the amount they spend by a minor change in the
procedure.

The incentive for the expenditures we (hypothetically) wish to reduce comes from their influence on the damages that will be awarded.[27] A court that wishes to reduce those expenditures can do so by selecting some cases for trial in the fashion I have described and some at random. The smaller the proportion of cases selected for trial by the defense, the lower the incentive that both attorneys have to spend more money estimating claims more accurately. Thus courts have a mechanism by which they can adjust the procedure to move its outcome closer to an optimal level of cost and accuracy. An alternative approach would be to try to impose limits on the amount each party was permitted to spend on evaluating claims.[28]

In the procedure as I have described it, the plaintiffs' attorney calculates claims and the defendant's attorney selects which will be tried: the former cuts and the latter chooses, to take the obvious analogy from the incentive-compatible procedure for dividing a piece of cake. Is there any good reason to do it this way, instead of requiring the defendant's attorney to list the amount he believes each plaintiff should receive and letting the plaintiffs' attorney choose which cases will be tried?

One reason is that the attorney who is calculating claims will need information from the plaintiffs which they might be reluctant to provide to the defense attorney, for fear that it would be used against them in trial. The procedure I have described does not eliminate this problem--the defense attorney still needs enough information to decide which cases to select for trial. But, if the group is large, he can do an adequate job by examining only a small subset of the plaintiffs, and can thus afford to spend much more per case examined than the plaintiffs' attorney. That should make it possible for him to produce a reasonably accurate estimate even with less cooperation from the individual plaintiff.[29]

The defense attorney is not the only one who must
worry about being misled by individual plaintiffs. Plaintiff *i*
gains by increases in *C _{i}* above
<

One consequence of having the plaintiffs' attorney cut and the defendant's attorney choose is to give the latter a cost advantage, at least in situations where the number of plaintiffs is large. As discussed earlier, the party who chooses can use random sampling to identify overclaimed cases at a relatively low total cost. This advantage may or may not outweigh the advantage that the plaintiffs' attorney has, due to the fact that the plaintiffs, who possess private information relevant to the strength of their cases, are his clients and have agreed to make such information available to him.

A second consequence is to give the defense an
advantage in the final verdict. As I show in Appendix I, the defense
can produce an expected total damage payment equal to the expected
payment under a system of individual trials by simply selecting cases
for trial at random, with probabilities proportional to
*C _{i}*. By examining cases and selecting those that
appear to be overclaimed, the defense should be able to improve on
that result.

How significant these advantages are will depend
on the details of the underlying factfinding technology-how
accurately and at what cost each attorney can estimate
<*D _{i}*>. If the net advantage to the defense
turns out to be large,[30]
and if we wish neither to change the tort system in a way which
advantages defendants in mass torts nor to give plaintiffs an
incentive to avoid the procedure in favor of individual litigation,
we could compensate by altering other legal rules applicable to the
procedure in ways that advantage plaintiffs.

An alternative approach would be to eliminate the
bias by allowing both parties to cut and both to choose. Under such a
system, the plaintiffs' attorney produces a set of claims and
the defense produces a set of claims .
Each attorney selects a set of cases to be tried. The court
calculates two values of *R*. *R _{p}* is calculated
using the plaintiffs claims and the verdicts of the cases selected by
the defense;

My analysis so far has assumed that the procedure
I am describing will be used, as the *Cimino* procedure was, in
a class action. It might also be applied to an ordinary joint action
with a large number of plaintiffs. The use of the procedure ought to
make such a joint action easier to organize, since it provides a
mechanism for solving the problem of allocating damages among the
joint plaintiffs. After a putative mass tort had occurred, one or
more lawyers would announce that he was forming a group of plaintiffs
to litigate under the procedure; his announcement would includes the
formula by which he would be reimbursed. Plaintiffs would be free to
join his group, to join another group, or to litigate
individually.

The procedure could also be used in situations other than class actions where a single agent already controls what are really multiple cases. One example would be disputes between insurance companies, each of which controls a large number of legal claims for accidents involving its customers. In that context, the procedure would be a way of guaranteeing to each customer that the insurance company was fairly representing his interests in the litigation.

The procedure would be inappropriate if the agent
who controlled multiple cases also fully owned them. Such an agent
would care about the total awarded to all of the cases he owns, not
the distribution among them. The *Cimino *procedure would give
the correct total at a lower cost than the procedure discussed here.
Such a situation could occur in the insurance context. It might also
arise if, as some writers have suggested, tort claims were made fully
marketable, allowing legal entrepreneurs to buy up large numbers of
related claims and litigate them *en masse*.^{33} Under
such institutions the damage award would reach the victim in the form
of the price for which he sold his claim, so the distribution among
victims would be determined by the market rather than directly by the
court.

Appendix II presents a formal model, based on an error distribution that is bounded and uniform. I demonstrate that, as the number of cases goes to infinity, the defense is able to perfectly identify overclaimed cases at a cost that is vanishingly small relative to the amount at stake. The plaintiffs maximize their net return by spending the same amount in investigating each case and claiming an amount equal to their estimate of the expected return at trial.

The result becomes more complicated if we assume that some cases are more difficult to evaluate than others. The optimal strategy is then to estimate those cases less accurately, insuring against the risk that the resulting estimate may be too high by deliberately claiming less than their estimated value.

Several further points are worth noting about this situation. The first is that cases that are difficult for the plaintiffs' attorney will also be difficult for the defense attorney, so the defense has an incentive not to examine those cases. The lower the probability that a certain sort of case will be examined, the less the risk of overclaiming for such cases, so this effect will work in the opposite direction from that demonstrated in the model.[34]

A second point arises if plaintiffs are risk averse. Cases that are difficult for the attorneys are also difficult for the court, so plaintiffs with hard cases face a bigger gamble if they go to court individually, and thus gain more by replacing that gamble with the more certain outcome generated by the procedure I have proposed. In addition, hard cases are likely to be more expensive to litigate, again making the procedure particularly attractive as a substitute for individual trial to plaintiffs with hard cases. So even if the procedure gives plaintiffs with hard cases somewhat less than their expected return at trial, that may not make them less willing to join the class than plaintiffs with easy cases.

If, despite these considerations, the incentive to underclaim hard cases turns out to be a serious problem, it can be dealt with in the same way earlier suggested for dealing with the procedure's pro-defense bias. The analysis of strategies with regard to hard cases is symmetrical; if the defense cuts and the plaintiffs' attorney chooses, the defense has an incentive to overclaim hard cases. So if both parties cut and both choose, the biases will tend to cancel.

One important limitation of the formal model of
Appendix II is that its error distributions are bounded. The result
is that, as the number of cases increases, the additional gain to the
defense of more and more accurately identifying the overclaimed cases
becomes less and less; there are no cases to be found that are
overclaimed by more than a factor of 1+* _{p}*.
If the error distribution for the plaintiffs' estimates is unbounded,
and if the defense can make the error of its estimate as small as it
likes by spending enough money examining enough cases, it is in the
interest of the defense to push

How serious a problem this is likely to be with plausible numbers of cases and error distributions is an empirical issue. If it does turn out to be a problem, it might be controlled by any of several modifications to the procedure suggested earlier.

I have proposed a procedure which has the
potential to settle mass torts at a cost much less than individually
litigating each claim. Like the *Cimino *procedure, it produces
about the same outcome for the defense as would individual trials.
Unlike the *Cimino *procedure, it provides outcomes for the
individual plaintiffs tailored to the strength of their individual
cases; indeed, it may well produce a more accurate allocation of
damage payments to plaintiffs than would individual
trials.

One can imagine applying the procedure in a
variety of different contexts. In a case such as *Cimino v
Raymark*, where there are a large number of plaintiffs each with a
substantial claim, individual attorneys might compete to form groups
to litigate under the procedure, thus avoiding some of the usual
problems with class actions. Where individual claims were smaller,
the class could be formed in the usual way; the
procedure[35]
would then provide a way of allocating damages among plaintiffs. By
reducing the risk that the plaintiffs' attorney would sacrifice the
interests of the plaintiffs in general to his own interest and that
of the representative plaintiffs, the procedure makes it more likely
that a class would, and should, be certified in such
situations.

The procedure is not perfect; it provides no guarantee of an optimal expenditure on evaluating cases in order to allocate damages. This is equally true of alternatives, including the alternative of litigating each case separately. Also, although the plaintiffs' attorney will find it in his interest to make his claims roughly proportional to the strength of the individual cases, the relation will not be exact; differences in the difficulty of evaluating cases may, as demonstrated in the formal model, make it in his interest to deliberately underclaim some cases relative to others.[36] Finally, the simpler versions of the procedure are to some degree biased in favor of the defense, since the plaintiffs cut and the defense chooses. If such problems prove serious, there are ways in which the procedure can be modified to reduce them.

Suppose we are dividing a cake under the conventional rule of "I cut, you choose." Further suppose that we have identical tastes; each of us prefers the larger slice. It seems obvious that, if there is any inaccuracy in cutting cakes, the party who moves second has the advantage. One way of seeing this is to note that if he selects his slice at random he will, on average, get half the cake; if he has any ability at all to recognize the larger piece, he will do better than that. An analogous argument implies that, under the procedure described in this article, the defense can always do at least as well as it would with individual trials, and may be able to do better. The analysis goes as follows:

Suppose that, instead of examining cases and trying to select the ones that are overclaimed, the defense simply selects cases by a random process, with a probability

*p _{i}* = of
selecting case

Expected total damage payment = <*R*> x
Total Claims* *

*=
= =*

= Expected total damage payment with individual trials.

So a random procedure, with no examination at all, produces as good a result for the defense as individual trials. By selecting cases that are overclaimed, the defense can get a better result than that-a lower expected total damage payment-at some cost. If the cost is less than the gain, the defense does better under this procedure than with individual trials. If the cost is greater than the gain for all levels of expenditure on examining cases, the defense follows the strategy described above and does as well as it would with individual trials.

There are *N* plaintiffs. Each plaintiff
*i* has an expected result at trial *d _{i}*. This
is the same as <

Either attorney can generate an estimate of
*d _{i}* by spending an amount

*=d _{i}*(1

Here *e _{i}* is a random error,
uniformly distributed between

The more the attorney spends on investigating the case, the more accurate the estimate:

< 0. (Assumption 1)

Investigation is subject to diminishing returns-additional expenditures yield less and less reduction in error:

> 0. (Assumption 2)

There is no limit to how accurate the investigation can be, if the attorney is willing to spend enough-he could, for example, stage repeated dummy trials. So:

= 0. (Assumption 3)

Finally, I assume that the prior distribution
(*d*)
is sufficiently flat and (*E _{i}*)
sufficiently small so that the conditional distribution (

My objective is to describe a Nash equilibrium, a
pair of strategies such that each party's strategy is optimal against
the strategy of the other party. I will start by deriving the
plaintiffs' strategy in the limit of large *N*, then derive the
defense strategy in the general case, then use that argument to
derive the plaintiffs' strategy in the general case. My objective is
to provide something more than a sketch but short of a full blown
proof; I will not, for example, demonstrate that the solution I offer
is unique.

The
Plaintiffs' Strategy: The Limit of Large *N*Consider the
situation from the standpoint of the plaintiffs' attorney deciding
how much to spend examining each case. He plans to spend an amount
*E _{i}* examining each case

As *N* goes to infinity, the defense, as we
will see below, can perfectly identify the most overclaimed
cases,[38]
so the cases selected for trial will be those for which
*d _{i}* /

The plaintiffs' attorney wishes to maximize the net gain to his clients. Since the number of cases being tried is determined by the rules of the procedure, not by the attorneys, we take expenditure for trial as fixed.[39] So the attorney minimizes:

Damage payment received - expenditure on examination

= = .

Where *D _{i}* is the verdict from a
jury trial of case

Expected value of damages - expenditure on examination

= = Net Benefit to Plaintiffs = nbp.

In the large number situation, we can think of the
cases as grouped; each group contains cases with common
values[40]
of *E _{i}*,
and

( ,*E _{i}*)
is /(1+(

Imagine that there are two groups, *i* and
*j*, such that *R _{i}*>

* *

*R _{i}*=[ /

Since *E _{i}* is chosen and spent
before

set

*C _{i}*=[ /

thus making *R _{i}*=

<nbp>= . (Equation 3)

There is some value of *E _{i}* that
maximizes the expression

;

call that value *E**. We have, for
*E*=*E**:

. (Equation 4)

The plaintiffs' attorney maximizes the net
expected return for his clients by setting
*E _{i}*=

The Defense
StrategyWe retain assumptions 1-3 above. We further assume that the
plaintiffs' strategy is as described earlier; *C _{i}*=
,

Figure 1

Suppose the plaintiffs have made a claim
*C _{i}* for case

(Equation 5a)

if

and

(Equation 5b)

otherwise.

Next consider the distribution of outcomes that
the defense can expect from spending *E _{i}* examining a
case

Figure 2

Suppose the defense wishes to examine a number of
cases *N _{d}* in order to select

Suppose that, as shown,

* *

*R ^{m}* < .

We then have, with a little manipulation:

. (Equation 6)

Where the expected value, here and below, is taken
over the cases selected for trial. Combining Equation 5a with
Equation 6, we have, for a given value of
*R ^{m}*:

. (Equation 7)

We also have, with a little more manipulation:

. (Equation 8)

Substituting Equation 8 into Equation 7 gives us:

=<*R*>.

The defense chooses *N _{d}*,

Expected Damage payments plus expenditure examining cases

=<*R*> +*N _{d}E_{d}*=<

_{=} _{.}(Equation
9)

Setting the derivatives wrt *N _{d}*
and

(Equation 10a)

. (Equation 10b)

Combining the last two equations and solving for
*N _{d}* yields:

.

From which it follows that:

. (Equation 11)

The solution to equation 11 is a value
*E _{d}** independent of

Hence:

As the number of cases goes to infinity, the
defense perfectly identifies overclaimed cases at a cost that is
vanishing small compared to the amount at stake. This confirms the
verbal analysis earlier used to derive the plaintiffs' strategy in
the limit of large *N*.

The Plaintiffs' Strategy: Finite *N*We can
use our results for the defense to learn more about the plaintiffs'
strategy for finite *N*.[42]
The equivalent of Equation 8, seen from the plaintiffs' side,
is:

Net Benefit to Plaintiffs

= .

Here *E _{p}* is the expenditure by
the plaintiffs on examining each case. The Plaintiffs maximize their
net benefit by choosing

In the limit as *N *goes to infinity, this
gives us back Equation
4.Some
ComplicationsWe have assumed, so far, that all cases are identical ex
ante. Suppose we instead assume that there are two sorts of cases:
easy cases and hard cases. For easy cases, the distribution of error
is * _{e}*(

The argument implying that
*R _{i}*=

In my earlier verbal analysis, I asserted that the plaintiffs' attorney would find it in his interest to make his claims proportional to his estimate of the strength of each case. We now see that this conclusion must be qualified. If some cases are known to be more difficult than others, meaning that it is more costly to estimate the average verdict if they are tried, the plaintiffs' attorney has an incentive to hold down his costs by making a less accurate estimate for those cases, and make up for it by somewhat underclaiming them.

[*] John M. Olin Law and Economics Fellow, University of Chicago. I would like to thank my colleagues at the University of Chicago and Cornell Law Schools, especially Jonathan Macey, Geoffrey Miller, and Richard Posner, for many helpful suggestions.

[0]751 F.Supp. 649, Claude Cimino, et al. v. Raymark Industries, Inc., et al.

[1]Michael J. Saks &
Peter David Blanck, "Justice Improved: The Unrecognized Benefits of
Aggregation and Sampling in the Trial of Mass Torts," *Stan L.
Rev*. 44 (1992) 815-851.

[2]This assumes, of
course, that the cases would have been tried. As Judge Parker pointed
out in his opinion, the defendants "assert a right to individual
trials in each case and assert the right to repeatedly contest in
each case every contestable issue involving the same products, the
same warnings, and the same conduct. The strategy is a sound one; the
defendants know that if the procedure in *Cimino* is not
affirmed, these cases will never be tried."

[3] The issues were whether each asbestos containing insulation product manufactured by each defendant, settling and non-settling, was defective and unreasonably dangerous, the adequacy of warnings, the state of the art defense and the fiber type defense. The question of punitive damages in the entire case of the 2,298 class representatives was also submitted for jury determination.

[4]Phase II was resolved by stipulation by the parties.

[5]The increase in sample size was less than proportional, as one would expect if the objective was to get equally reliable results for each category. The opinion states that "When setting the sample size for each disease category, the Court sought a confidence level of 95%, in other words +- 2.00 standard deviations." The numbers (samples of 50 each for two categories with 1,050 and 972 plaintiffs) suggest that the court did not apply any very precise statistical rule.

[6]Judge Parker's statement that "Defendants complain about the 1% likelihood that the result would be significantly different" suggests that he interprets a 99% confidence level as a probability of 99% that the procedure will yield a result within some (unspecified) significant error--where "significant" means "important" not "statistically significant."

Whatever error he did use, what ought he to have used? One possibility would be to compare the procedure to the result of individual trials, taking account of the difference in litigation costs. Suppose, for example, that aggregation saves the defense a million dollars in legal expenses. One might then ask how likely it is that the award is more than a million dollars greater than what would have been awarded if all cases were tried. If the answer is .01, there is then only one chance in a hundred that the procedure has made the defendants worse off. While that approach solves the problem of picking an appropriate error, it still leaves the problem that statistics cannot generate such probabilities without making assumptions about the characteristics of the sample.

[7] "Individual members of a disease
category who will receive an award that might be different from one
they would have received had their individual case been decided by a
jury have waived any objections." Judge Parker in
*Cimino.*

[8]The opinion discusses under what circumstances smoking would constitute contributory negligence and notes that some plaintiffs received awards of zero, but does not say whether any received zero awards for that reason.

[9]Federal Rules of Civil Procedure 23a(4).

[10]Suppose the court
uses an aggregation process that awards every plaintiff the average
of what all plaintiffs in the class are entitled to. Plaintiffs who
can expect an above average return withdraw from the class. That
lowers the average that the remainder can expect to get, causing more
plaintiffs to withdraw. Under some circumstances, the entire class
may come apart in this way. This is a form of adverse selection, more
familiar in the context of insurance. See G. Akerlof, "The Market for
`Lemons'," Vol. No. 336 *QJE* 488-500 (1970).

[11]It was permitted in
*Daar v. Yellow Cab Co*., 67 Cal. 2d 695, 433 P.2d 732, 63 Cal.
Rptr. 724, rejected by the Second Circuit in *Eisen v. Carlisle and
Jacquelin*, 479 F. 2d 1005, and has not been ruled upon by the
Supreme Court.

[12]*Stan. L. Rev.*
44 (1992) 851. "Collective Justice in Tort Law," by Glen O. Robinson
& Kenneth S. Abraham, 78 Va. L. Rev. suggests and discusses
several other statistical approaches to dealing with mass torts,
using information from the outcomes of similar cases to determine, or
at least affect, awards.

[13]As Saks and Blanck
point out, average jury awards seem to increase less than linearly
with the amount of injury suffered by the plaintiff (*Stan. L.
Rev.* 44 (1992) p. 840).

[14]In explaining my proposed procedure, I assume that it is being applied to a case with many plaintiffs and one defendant; the application to the less common case of one plaintiff and many defendants should be straightforward.

[15]One reason such rules are necessary is that the decision-maker in a trial has only weak incentives to reach the correct decision, and can therefore not be trusted to do unless severely constrained. Under the proposed procedures, it is in the private interest of the decision maker (the plaintiffs' attorney) to estimate the strength of claims accurately, making such constraints less necessary.

[16] The plaintiffs
attorney can achieve the same objective by attempting to set
*C _{i}* proportional to <

[17]As we will see later, this statement is only approximately true. If some cases are harder to evaluate than others, the optimal strategy for the plaintiffs' attorney may deviate somewhat from that described here.

[18]This is one of the
central points made by Saks and Blanck in defending the *Cimino
*procedure. *Stan. L. Rev.* 44 (1992) pp. 833-836.

[19]*Stan. L. Rev.*
44 (1992) 851.

[20]Since the class was certified before the procedure was proposed, the assent was presumably by the representative plaintiffs controlling the litigation rather than by the unanimous decision of all plaintiffs. But the procedure created a conflict of interest among members of the class, which arguably called into question the ability of the representative plaintiffs to represent the interests of the remaining plaintiffs.

[21]For an extensive
discussion of legal issues associated with aggregation, see
"Collective Justice in Tort Law," by Glen O. Robinson & Kenneth
S. Abraham, 78 *Va. L. Rev.*

[22]If it is not obvious that they are overclaiming, the defense may miss some of their cases, in which case some of the overclaimed representative defendants will get more than they should. On the other hand, given that possibility, one would expect the defense to take special care in examining the claims made for the representative plaintiffs. I am assuming here that plaintiffs whose cases are actually tried get the amount awarded to them, rather than having their award calculated from their claim in the same fashion as plaintiffs whose cases are not tried. Without that assumption, representative plaintiffs gain by overclaiming even if they are sure their cases will be among those tried--although the attorney for the class of plaintiffs loses, if his recompense is an increasing function of the total amount awarded.

[23]I do not know of any
definitive analysis of this question. One possible approach would be
to assume Nash equilibrium. *S* is the amount at stake. The
probability that the plaintiff will win the case depends on
expenditures *L _{p}* (by the plaintiff) and

Under these assumptions, the question of how
expenditure increases with amount at stake becomes the question of
how rapidly decreases
as *L _{p}* and

,

then expenditure increases more (less) than in
proportion to the amount at stake if *b*<1*
*(*b*>1).

One objection to this approach is that Nash equilibrium is not very plausible in a game involving only two parties, and still less plausible in a situation where the two parties can and do bargain with each other.

[24]In the case of the
*Cimino* procedure, that must be balanced against the decreased
accuracy from awarding plaintiffs whose cases are not selected for
trial average verdicts even though the particular plaintiff may not
have average characteristics.

[25]The comparison is between two cases whose only difference is the amount at stake; each of my ten cases is simply one of the thousands of cases that might be tried individually. I am not assuming that the ratio of litigation cost to the amount at stake for the typical large case is smaller than for the typical small case; presumably the typical large case not only has more at stake but also a more complicated set of legal and factual issues than the typical small case.

[26]Or spends resources determining how the amount a plaintiff is entitled to is related to the plaintiffs characteristics, under the alternative version that I proposed for cases with very large numbers of plaintiffs and small average claims.

[27]It is possible that the plaintiffs' attorney may have additional incentives, due to concerns with either justice or risk among his clients. They might prefer that claims be proportional to the actual injury each client has suffered, even if claims did not affect the total amount paid out.

[28]In a class action, a judge could limit expenditure by one side simply by limiting the expenses he was willing to permit the class attorney to claim. Limiting expenditure by the other party, or by both parties if the procedure was being used outside of a class action, would be more difficult.

[29]Presumably there would be legal rules requiring some cooperation from the plaintiffs. "(M)ost courts have taken the view that reasonably necessary discovery against individual class members should be allowed as a matter of judicial discretion, but that discovery is not allowable of right as it would be against a party to a nonclass suit. (e.g. Brennan v. Midwestern United Life Ins. Col, 450 F.2d 999 (7th Cir. 1971)."Civil Procedure, 3d ed, Fleming James Jr. and Geoffrey C. Hazard, Little, Brown and Co. Boston, Toronto 1985 p. 579.

[30]This does not require
the defense attorney to be better at estimating
<*D _{i}*> than the plaintiffs' attorney, as should
be clear from the analysis above. If, for example, accurate estimates
are very expensive and the number of plaintiffs is large, the
plaintiffs' attorney will produce very inaccurate estimates and the
defendant's attorney, spending much more per case on a small fraction
of the cases, will be able to find cases that are greatly
overclaimed, thus greatly reducing the total amount paid out in
damages.

[31]That conclusion depends on assuming that both sides are equally able to generate the relevant information. If, as suggested earlier, the plaintiffs' attorney has better access to information about plaintiffs, the version of the procedure described here is biased in favor of the plaintiffs. If one knew how great the informational advantage was, one could compensate for it by using an appropriately weighted average of the awards calculated from the two different sets of claims.

[32]If we assumed that each party aimed at the same level of accuracy as under the earlier version of the procedure, expenditure would be increased but not doubled by requiring each party to both cut and choose. The information generated in cutting can also be used in choosing. The defense attorney's first step in identifying overclaimed cases will be to compare the plaintiffs' attorney's claims with his own.

[33]For discussions of the idea of marketable claims, see Marc J. Shukaitis, "A Market in Personal Injury Tort Claims," XVI J.L.Stud. pp. 329-349 (June 1987), David Friedman, "Private Creation and Enforcement of Law-A Historical Case," VIII J.L.Stud. (March 1979), pp. 399-415, David Friedman, "What is Fair Compensation for Death or Injury?" IRLE 2 (1982), pp. 81-93, Jonathan Macey and Geoffrey Miller, "The Plaintiffs' Attorneys Role in Class Action and Derivative Litigation: Economic Analysis and Recommendations for Reform.," U. Chi. L. Rev. 58 (Winter 1991) pp. 1-118.

[34] That is not true for
the formal model of Appendix II in the limit of large *N*,
because in that situation the defense is able to perfectly identify
overclaimed cases at negligible cost.

[35]Possibly the statistical version discussed above.

[36]I have presented this as a fault of the procedure, but it could be viewed as a desirable consequence. It may be desirable, on grounds of either efficiency or justice, for parties who insist on litigating difficult cases to bear part or all of the cost of doing so. Underclaiming difficult cases costs plaintiffs less than estimating the strength of their case as accurately as easy cases are estimated. Thus this feature of the procedure has consequences similar to those of the usual (American) rule that each party must bear his own litigation costs: plaintiffs with cases that are expensive to litigate take home less, net of litigation costs, than parties who have suffered similar damage but have easy cases..

[37]The reason we need to
assume a sufficiently flat prior in order for this to be true is that
the conditional probability will depend both on the distribution of
the error and on the prior distribution, in a fashion described by
Bayes' theorem. The reason we must assume that
*e*(*E _{i}*)<<1 is that otherwise a uniform
distribution of e will not give something close to a uniform
distribution of 1/(1+

[38]More precisely, the
defense can identify the most overclaimed cases that exist with
positive probability. There could be (say) a single case that was
overclaimed by more than any other but that the defense missed
because it was not in the sample examined. As *N* goes to
infinity, the probability and hence the effect on average damages
collected of any single case goes to zero. Any kind of case with
positive probability will be represented an infinite number of times
in the total (as *N* goes to infinity), and thus will be
included in the sample selected for examination by the
defense.

[39]The strategy followed
in selecting cases might have some effect on the two sides'
incentives to spend money litigating them. Since I have no theory of
litigation expenditures, and in any case expect them to become
insignificant relative to the total amount at stake when *N*
becomes sufficiently large, I ignore this possibility in the
analysis.

[40]Or, more precisely, a
narrow range of values. As *N* goes to infinity, we can make the
range as narrow as we wish.

[41]The defense is
ignoring *C _{i}* in picking cases to examine since, as
we see by Equations 6 and 7, it can get the same <

[42]This is not a full
analysis, since I have not redone, for finite *N*, the proof
that the plaintiffs spend the same amount on every case and choose
claims proportional to the estimated strength of each
case.

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