BARGAINING
IN
THE
SHADOW
OF
THE
LAW:
A
TESTABLE
MODEL
OF
STRATEGIC
BEHAVIOR
ROBERT
COOTER
and
STEPHEN
MARKS
with
ROBERT
MNOOKIN*
PRETRIAL
bargaining
may
be
described
as
a
game
played
in
the
shadow
of
the
law.
There
are
two
possible
outcomes:
settlement
out
of
court
through
bargaining,
and
trial,
which
represents
a bargaining
breakdown.
The
courts
encourage
private
bargaining
but
stand
ready
to
step
from
the
shadows
and
resolve the
dispute
by
coercion
if
the
parties
cannot
agree.
Bargaining
is
successful
from
an economic
viewpoint
if an
efficient
solu-
tion
to
the
dispute
is
found
at
little
cost.
In
technical
language,
a
dispute
is
resolved
successfully
if
a
solution
is
found
on
the
contract
curve
with
little
expenditure
on
search.
The
usual
approach
to
bargaining
in
the
legal
setting
assumes
that
trial
is
caused
by
excessive
optimism
on
the
part
of
plaintiff
and
defendant.
'
If
both
parties
are
optimistic,
then
there
is
no
way
to
split
the
stakes
so
that
each
receives
as
much
as
he
or
she
expects
to
gain
from
trial.
In
these
circumstances,
trial
is
inevitable.
*
The
paper's
title
is
borrowed
from
Robert
N.
Mnookin
&
Lewis
Kornhauser,
Bargain-
ing
in
the
Shadow
of
the
Law:
The
Case
of
Divorce,
88
Yale
L.
J.
950
(1979).
We
received
helpful
criticism
of
the
paper
from
the
law
and
economics workshop
at
the
University
of
Chicago. The
ideas
in
the
paper
were
originally
presented
at
seminars
in
the economics
departments
at
California
Institute
of
Technology
and
the
University
of
California
at
San
Diego.
The
research
was
funded
by
a
grant
from the
Center
for
Law
and
Economic
Study,
Columbia
University
School
of
Law, whose
director
is
Lewis
Kadin.
I
The
theory
that
trial
in
civil
suits
occurs
because
of
optimism
is
developed
in
John
P.
Gould,
The
Economics
of
Legal
Conflicts,
2
J.
Legal
Stud.
279
(1973);
Richard
A.
Posner,
An
Economic
Approach
to Legal
Procedure
and
Judicial
Administration,
2
J.
Legal
Stud.
399
(1974),
esp.
at
419
n. 29.
This
model
of
litigation
is
expanded
in
William
M.
Landes
&
Richard
A.
Posner, Adjudication
as
a
Private
Good,
8
J.
Legal
Stud.
235
(1979).
A
clear
summary
of
the
theory
is
in
Richard
A.
Posner,
Economic
Analysis
of
Law
434-400
(2d
ed.
1977)
[Hereinafter
cited
as
Economic
Analysis].
A
similar
analysis
of
the
cause
of
trial
in
criminal
cases
is
found
in
William
M.
Landes,
An
Economic
Analysis
of
the
Courts,
14
J.
Law
&
Econ.
61 (1971).
[Journal
of
Legal
Studies,
vol.
XI
(June
1982)]
©
1982
by
The
University
of
Chicago.
All
rights
reserved.
0047-2530/82/1102-0011$01.50
HeinOnline -- 11 J. Legal Stud. 225 1982
THE
JOURNAL
OF
LEGAL
STUDIES
Our approach
is
different.
In
our
model,
excessive
optimism
is
not
the
fundamental cause
of
trials.
The
fundamental
cause
is
the
problem
of
distribution
faced
by
the
players.
The
problem
of
distribution
is
to
divide
the
stakes
in
dispute.
A
rational
bargainer
will
make
a
demand such
that
the
gain
from
settling
on
slightly
more
favorable
terms
is
offset
by
the
increased
risk
of
a
breakdown
in
negotiations.
Thus,
the
optimal bargain-
ing
strategy
of
a litigant
balances
a
larger
share
of
the
stakes
against
a
higher
probability
of
trial.
There
will
be
a
positive
risk
of
trial
when
strategies
are
optimal.
A
player's
strategy
is
optimal
when
he
maximizes
his
expected
utility.
Our
assumption
that
individuals maximize
expected
utility
involves
an
innovation
in
game
theory
as
applied
to
law.
In
the
usual
analysis,
the
players
simultaneously
maximize
utility,
and
an
equilibrium
is
achieved
when
everyone
knows
exactly
what everyone
else
is
doing
(Nash
equilib-
rium).
Uncertainty
is
eliminated.
In
our
analysis,
the
players
simultane-
ously
maximize
expected
utility,
and
an
equilibrium
is
achieved
when
everyone
knows
the
distribution
of
strategies
pursued
by
others
(Bayesian
Nash
equilibrium).
2
Uncertainty
persists
about
individuals
but not
aggre-
gates.
The
persistence
of
uncertainty
allows
some
disputes
to
end
in
trial.
Notice
that,
in
equilibrium,
the
litigants
understand
the
true structure
of
the
game,
and
their
expectations
are
accurate
in
the sense
that
subjective
probabilities
correspond
to
objective
frequencies.
In
our
model, equilib-
rium
expectations
are rational,
not
biased
toward
optimism
or
pes-
simism.
3
We
wish
to
make
predictions
about
the
relation
between
observable
variables and
the
frequency
of
trials
in
equilibrium.
Bargaining
is
like
the
stock
market
in
the
sense
that
tangible
changes
have psychological
ef-
fects.
If
the
psychological
effects
overwhelm
the
tangible
changes,
then
predictions
are
difficult
to
make.
We
shall
show
that
tangible variables
dominate
psychological
effects
in
the
usual
case,
much
the
way
the
sub-
stitution
effects
dominate
the
income
effects
of
a
price
change
in
the usual
case.
Our theorems
about
the
domination
of
psychological
effects
by
tangible
variables
enable
us
to
construct
a
table
of
predictions.
We
shall
be
able
to
answer
some
perplexing
questions,
such
as:
(1)
Is
settlement
more
likely
when transaction
costs
are
low?
(2)
Is
settlement
more
likely
2
The
concept
of
a
Bayesian
Nash
equilibrium
was
introduced
by
John
Harsanyi,
Games
with
Incomplete
Information
Played by
"Bayesian
Players,"
41
J.
Manage.
Sci.
320
(1968),
esp.
pt.
2.
3
Our
model
is
consistent
with
the
optimism
model
in
predicting
that
optimism
will
lead
to
a
greater
probability
of
trial,
but
it
is
not
a
necessary
condition
for
trial
that
the
threat
points
sum
to
a
value
greater
than
the
stakes.
HeinOnline -- 11 J. Legal Stud. 226 1982
PRETRIAL
BARGAINING
when bargaining
is
risky?
(3)
Is
settlement
more
likely
when the
coopera-
tive
surplus
is
large?
Most
of
our
predictions
are
consistent
with
the
predictions
derived
from
the
optimism
model,
but
we
ground
the
predictions
in
a
more
fundamental
account of
bargaining.
A
more
fundamental
account
of
bargaining
offers
a
different
perspective
on
a
variety
of
legal
issues
as
well
as
a
few
different
predictions.
We
shall
illustrate
the
change in
perspective
by analyzing
externalities
and
legal
institutions
for
reallocating
the payoffs
from
trial.
Part
I
describes
the
framework
of
the
model.
Part
II
discusses
the
domination
of
psychological
effects
by
tangible
variables,
which
is
the
basis
for comparative
statics.
Part
III
explains
the
table
of
predictions,
and
Part
IV
applies
the
predictions
to
externalities
and rules
for
real-
locating
payoffs
from
trial.
There
is
little
mathematics
in
the
text.
The
mathematical
reader
can
refer
to
the
Appendix
for
a
brief presentation
of
our
model,
or
to
a
companion
paper
on
file
with
this
Journal
which
de-
velops
the
mathematics
at
length.
4
I.
FRAMEWORK:
OPTIMIZATION
AND
EQUILIBRIUM
In
order
to
develop
the
framework,
we
must
characterize
the
features
of
bargaining
games
which
distinguish
them
from
other
types
of
games.
A
zero-sum
game
is
a
game
in
which
total
winnings
minus
total
losses
equals
zero.
5
It
is
a
game
of
pure
redistribution
because
nothing
is
created
or
destroyed.
Poker
is
an
example.
By
contrast,
a
coordination
game
is
a
game
in
which
the
players
have
the
same
goal.
6
For
example,
if
a
phone
conversation
is
cut off,
then the
callers
face
a coordination
problem.
The
connection
cannot
be
restored
unless
someone
dials,
but
the
call
will
not
go
through
if
both
dial
at
once.
The
players
win
or
lose
as
a
team,
and
winning
is
productive,
so
coordination
games
are games
of
pure
produc-
tion.
A
bargaining
game
involves
distribution
and
production.
7
Typically,
there
is
something
to
be
divided
called
the
stakes.
For
example,
one
person
may
have
a
car
to
sell,
and
another
may
have
money
to
spend.
The
stakes
are
the
money
and
the
car.
If
the players
can
agree
on
how
to
'
Robert
Cooter
&
Stephen
Marks,
Bargaining
with
Rational
Expectations
Applied
to
Law:
A
Testable
Model
of
Strategic Behavior
(1981)
(unpublished
manuscript
on
file
with
J.
Legal
Stud.).
5
Zero-sum
games
were
first
explored
in
the pioneering
book
by
John
von
Neumann
&
Oskar
Morgenstern,
Theory
of
Games
and
Economic
Behavior
(1944).
6
Coordination
games
are
described
in
Thomas
C.
Schelling,
Strategy
of
Conflict
(1960).
A
definition
of
bargaining
games
and
a
discussion
of
solution
concepts
is
in
R.
Duncan
Luce
&
Howard
Raiffa,
Games
and Decisions
ch.
6
(1957).
HeinOnline -- 11 J. Legal Stud. 227 1982
THE
JOURNAL
OF
LEGAL
STUDIES
divide
the
stakes,
for
instance,
if
they
can
agree
on
a price
for
the
car,
then
both
of
them
will
benefit.
The
surplus
is
the
joint
benefits
from
cooperation,
such
as
the
gains
from
trading
the
car
for
money.
If
the
players
cannot
agree
on
how
to
divide
the
stakes,
then
the
surplus
will
be
lost.
In
brief,
bargaining
games
are
games
in
which
production
is
contin-
gent
on
agreement
about
distribution.
The
fundamental
obstacle
to
cooperation
is
the
absence
of
an au-
thoritative
rule
for
dividing
the
stakes.
It
is
up
to
the
players
to
find
a
division
of
the
stakes
acceptable
to
both
of
them.
Strategy
consists
in
trying
to
maneuver
an
opponent
into
accepting
an
unfavorable
distribu-
tion.
The
objective
of
a
skillful
bargainer
is
to
convince
others
that
he
intends
to
act
in
such
a
way
that
it
is
in
their
best
interest
to
do
what
is
in
his
best
interest.
Such
a
bargainer
transmits
information
and
makes
side
payments
in
order
to
maneuver
his
opponents
into
a
position
where
their
perceived
self-interest
compels
them
to
do
what
is
to
his
advantage.
Inefficient
outcomes
occur
when
the
players
miscalculate,
for
instance,
I
think
that
your
sincere
threat
is
a
bluff.
Bargaining
in
the
shadow
of
the
law
achieves
a
close
fit
to
our
abstract
characterization
of
bargaining
games.
The
players
are
usually
well
defined,
consisting
of
a
plaintiff
and
defendant
in
many
cases.
The
stakes
are also
well
defined,
such
as
the cost
of
an
accident,
the
damage
from
nonperformance
on
a
contract,
the
property
accumulated
in
a
marriage,
the
estate
of
the
deceased,
the
assets
of
a
bankrupt
company.
In
pretrial
negotiations,
everyone
has
an
interest
in
avoiding
a
trial. The
surplus
from
cooperation
is
usually
obvious,
for
example,
legal
fees,
cost
of
delaying
resolution
of
the
dispute,
waste
from
a
judicial
outcome
off
the
contract
curve.
The
plaintiff
and
defendant
have
an
incentive
to
avoid
trial,
but
they
have
a
disagreement
over
how
to
divide
the
stakes.
There
is
a
prob-
lem
of
efficiency
and
also
one
of
distribution.
A
legal
dispute
enters
the
public
record
when
a
complaint
is
filed.
A
trial
date
is
often
set
after
the complaint
is
filed
which
puts
a
time
limit
on
bargaining.
The
simplest
characterization
of
the
bargaining
process
is
a
sequence
of
offers
and
counteroffers
for
dividing
the
stakes.
A
settlement
is
reached
if
the
plaintiff's
demand
in
some
round
of
negotiations
does
not
exceed
the
defendant's
offer.
Bargaining
terminates
and
a
trial
begins
if
the
trial
date
is
reached
before
a
settlement
occurs.
In
summary,
we
characterize
bargaining
in
the
shadow
of
the
law
as
a
game
with
the
following
characteristics:
(1)
There
is
a
dispute
between
two
players,
the
defendant
and
plaintiff,
over
how to
divide
the
stakes.
(2)
Bargaining
consists
in
an
exchange
of
demands
and
offers
for
dividing
the
stakes.
(3)
Settlement
occurs
if
the
plaintiff's
demand
does not
exceed
the
defendant's
offer.
(4)
Trial
occurs
if
a
settlement
is
not
reached
before
the
HeinOnline -- 11 J. Legal Stud. 228 1982
PRETRIAL
BARGAINING
trial
date.
(5)
The
outcome
of
a
trial
is
the
destruction
of
part
of
the
stakes
(the
surplus)
and
distribution
of
the
remainder.
Optimization
We
have
described
bargaining
in
the
shadow
of
the
law
as
a game.
Our
next
task
is
to
characterize
the
behavior
of
the
players.
Each
player
must
choose
a
bargaining
strategy.
We
can
reduce
the
choice
of
strategy
to
its
simplest
elements
by
imagining
that
the
plaintiff
writes
down
her
final
demand,
seals
it
in
an
envelope,
and
mails
it
to
the defendant.
At
the
same
time,
the
defendant
writes
down
his
final
offer,
seals
it
in
an
envelope,
and
mails
it
to
the
plaintiff.
The envelopes
are
delivered
just
before
the
trial
is
scheduled
to
begin.
If
the
offer
is
at
least
as
great
as
the
demand,
a
settlement
occurs.
If
the
demand
is
greater
than
the
offer,
then
a
trial
occurs.
The
mailing
of
the envelopes
is
a
device
for
portraying
the
uncer-
tainty
of
each
player
concerning
his
opponent's
strategy.
This
example
can
be
clarified
by
using
some
notation
to
describe
how
the
plaintiff
would
compute
her
optimal
final
demand.
Let
the
stakes
be
$100,000
and
let
x
be
the
plaintiff's
share
of
the
stakes,
where
x
is
denomi-
nated
in
hundreds
of
thousands
of
dollars.
Thus
a
settlement
will
occur
if
the
plaintiff
demands
x
for
herself
and
the defendant
demands
no
more
than
1 -
x
for
himself.
Let
P (1 -
x)
be
the
probability
that
the
defendant
demands
no
more
than
1
-
x.
Thus
the
probability
of
settlement
is
P
and
the
probability
of
trial
is
1
- P.
Let
the
plaintiff's
payoff
from
trial
be
T.
The
problem
faced
by
the
plaintiff
can
be
written"
max
P(1
-
x)x
+
{[1
-
P(1
-
x)]}T.
This
mathematical
problem
can
be
solved
intuitively.
Suppose
that
the
plaintiff
is
contemplating
whether
x
is
her
optimal
demand.
If
she
in-
creases
her
demand
by
$1.00,
then
she stands
to
gain
$1.00
with
probabil-
ity
P(1
-
x).
However,
increasing
her
demand
by
$1.00
will
increase
the
probability
of
a trial
by
the
marginal
value
of
P (1 -
x),
denoted
p.
If
a
trial
occurs,
then
she
will
lose
the difference
between
her
payoff
from
settle-
ment
x
and
her
payoff
from
trial
T.
Thus
the
risk
of
loss from
demanding
more
is
(x
-
T)p.
At
the
optimum,
the
probable
gain
is
exactly
offset
by
the
risk
of
loss:
0
=
-(x
-
T)p
+
$1P.
The
plaintiff's
optimal
demand
is
the
value
of
x
which
satisfies
this
equality.
The
same
kind
of
computation
is
made
to
find
the
defendant's
optimal
demand.
'
For
simplicity,
we
have
assumed
that
the
plaintiff
is
an
expected
income
maximizer
in
this
example.
This
is
equivalent
to
assuming
risk
neutrality.
In
general,
all
our
results
hold
for
both
risk-neutral
and
risk-averse
players.
HeinOnline -- 11 J. Legal Stud. 229 1982