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LEARNING IB EQUILIBRIUM MODELS OF ARBITRATION
by
Robert Gibbons
[
So. 485
March 1988
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
14 1988
)
%
LEARNING IB EQUILIBRIUM MODELS OF ARBITRATION
by
Robert Gibbons
No. 485
March 1988
LEARNING
IN EQUILIBRIUM
MODELS OF ARBITRATION
by
Robert Gibbons*
MIT and NBER
First version:
September 1987
This revision: March 1988
am happy
to acknowledge several forms of assistance and supoort: I have benefitted
from extensive discussions about arbitration with David Bloom, Christopher
Cavanagh, Kerry Farber. and Stanley Jacks. Many others have made helpful comments
on this paper: David Card, Vincent Crawford, Paul Klemperer, Gregory Leonard, Meg
Meyer. Kevin M. Murphy, Joel Sobel. and seminar audiences at Chicago, Harvard. MIT,
and Princeton. Finally, the NSF provided financial support through srant 1ST 86-09691.
*I
greatly
M.IXUBRARIES
JUL
1
5 1988
ABSTRACT
This paper analyzes strategic communication
and final-offer
arbitrator
known
interest arbitration.
from the
parties' offers
equilibrium models of conventional
in
Both models emphasize the role of learning by the
about the state of the employment relationship, which
is
to the parties but not to the arbitrator. In both models, the arbitrator's equilibrium
behavior
is
literature.
identical to the reduced-form decision rule typically
The paper thereby provides
assumed
in the empirical
a structural interpretation for the existing empirical
work.
The paper
it
also represents progress towards a complete theory of arbitration because
satisfies three conditions that will
be required of any such theory. First, the models'
predictions match the existing empirical evidence. Second, the models describe equilibrium
behavior.
And
third, the
models are
built
on
preferences, information, and commitment.
a
common
Tne paper
set
of assumptions about
therefore not only provides an
equilibrium foundation for the intuition that the arbitrator might learn from the parties'
offers, but also uses the idea of learning to
major forms of interest
arbitration.
develop a unified analytical treatment of the rwo
1.
Introduction
This paper analyzes strategic communication in equilibrium models of conventional
and
final-offer interest arbitration.
arbitrator
known
from the
parties' offers
Both models emphasize the role of learning by the
about the state of the employment relationship, which
is
to the parties but not to the arbitrator. In both models, the arbitrator's equilibrium
behavior
is
identical to the reduced-form decision rule typically
literature (namely, a
weighted average of the
the publicly observable facts of the case).
parties' offers
assumed
in the empirical
and a settlement motivated by
The paper thereby provides a
structural
interpretation for the existing empirical work.
The paper
it
also represents progress towards a complete theory of arbitration because
satisfies three conditions that will
be required of any such theory.
First, the
models'
predictions match the existing empirical evidence. Second, the models describe equilibrium
behavior.
And
third, the
models
are built
on
a
common
set
of assumptions about
preferences, information, and commitment. (This last condition
is
crucial for accurate
welfare comparisons of conventional and final-offer arbitration.) The paper therefore not
only provides an equilibrium foundation for the intuition that the arbitrator might learn from
the parties' offers, but also uses the idea of learning to develop a unified analytical
treatment of the two major forms of interest arbitration.
Understanding interest arbitration
is
important for several reasons.
considerable resources are allocated through interest arbitration of
the public sector (especially
where
wage
First,
disputes, both in
strikes are prohibited) as well as in private-sector
industries such as professional baseball. Second, outside the context of
wage
disputes, the
clogged judicial system has created a growing interest in alternative dispute-resolution
procedures, especially pre-trial arbitration. Third, interest arbitration offers a rare natural
experiment involving strategic behavior and significant stakes: the rules of arbitration
constitute a well defined
A fourth reason
to write an
game, and the players' choices often are observed without
to understand arbitration is that
it is
error.
an institution that allows parties
incomplete contract ex-ante while limiting opportunistic behavior ex-post.
Indeed, Williamson (1975) argues that the possibility of arbitration by informed insiders
one of the central virtues of internal over market organization. Even
if
some
is
state-
contingent actions cannot be specified in a contract ex-ante, the contract can include the
provision that ex -post disputes will be resolved through arbitration.
Grossman and Hart
(1986) analyze two other responses to incomplete contracts: integration
residual rights of control) and ex-post bargaining. Arbitration
Formally,
paper considers interest
this
arbitration, but the thrust of the
is
during the
assigning
a third option.
opposed
to
grievance
paper seems applicable to grievance arbitration as well. In
wage
interest arbitration the arbitrator settles
contract
arbitration, as
is
(i.e.,
(or other) disputes that arise
when
a
new
negotiated; in grievance arbitration, the arbitrator settles disputes that arise
life
of an existing contract The two major forms of interest arbitration are
conventional arbitration and final-offer arbitration. In final-offer arbitration, the parties
simultaneously submit
wage
offers,
and then the arbitrator imposes one of the two offers as
the settlement in conventional arbitration, the arbitrator's choice of a settlement
Although
on
this
paper contains no data,
interest arbitration.
paper
is
identical to the
literature. In
As noted
it is
is
closely related to the empirical literature
briefly above, the arbitrator's equilibrium behavior in this
reduced-form decision rule typically assumed in the empirical
both the conventional and the final-offer arbitration models developed below,
the arbitrator's preferred settlement, y 2
,
is
a weighted average of the average of the parties'
offers, y
=
(y e
+
and a settlement motivated by
yu)/2,
the publicly observable facts of the
case, y*(f):
ya =
ay
+
(l-cc)y*(f).
A decision rule of this form appears in empirical studies of conventional
equation (12) in Ashenfelter and
(1986), and equation (lb) in
Bloom
Bloom
arbitration as
(1984), equation (3) in Farber and
(1986);
it
also appears in empirical studies of final-
offer arbitration as equation (7) in Ashenfelter and
Bazerman. One of the purposes of this paper
Bazerman
is
Bloom and equation
(6) in
Farber and
to provide a structural interpretation for the
estimated parameter a.
Tne paper
first
also
is
an attempt to
requirement for such a theory
strongest piece of evidence
making
(i.e.,
a is positive;
is that
see
move
is that it
A
towards a complete theory of arbitration.
match the empirical evidence. Perhaps the
the parries' offers matter in the arbitrator's decision-
Bloom and FarbeT-Bazerman).
Indeed, Stevens' (1966)
proposal of final-offer arbitration was motivated by the observation that the offers seem to
matter too
much
the difference
case. This
in conventional arbitration: the arbitrator
between the
paper uses the
parries' offers, ignoring the publicly
arbitrator's
observed importance of the offers
behavior that emerges also
importance o: tne oners
sometimes seems simply
is
need
to learn
in the arbitrator's
from the
to split
observable facts of the
parries' offers to explain the
decision-making.
The equilibrium
consistent with the Farber-Bazerman finding thai the
m tne aroitrator
s
cecision-maiang increases
wnen me oners
ar;
closer together, and with the Ashenfelter-Bioom and Farber-Bazerman finding that the
arbitrator's notion
of an ideal settlement
is identical in final-offer
and conventional
arbitration.
A second requirement for a complete theory of arbitration is that
equilibrium behavior.
arbitration is a crucial
The following
complement
discussion explains
to empirical
work on
why
it
describe
an equilibrium analysis of
the arbitrator's decision rule.
t
Consider Stevens' observation
seems simply
that in conventional arbitration the arbitrator
to split the difference
between the
splitting the difference, then the parties
Suppose instead
that the arbitrator
is
offers. If the arbitrator
would submit
committed
the
sometimes
were committed
to
most extreme offers possible.
to splitting the difference
between
reasonable offers but to ignoring (or even punishing) unreasonable ones. Such a decision
rule could cause the parties always to submit reasonable offers,
appearance of mechanical compromising by the
be only half the story: the data would be
behaved had the
silent
and so lead
arbitrator, but the
about
how
to the
observed behavior would
the arbitrator
would have
parties not submitted reasonable offers. Out-of-equilibrium responses
by
the arbitrator are never observed, and so can never be estimated, but play a crucial role in
determining what
is
observed. Only equilibrium models can elucidate the out-of-
equilibrium behavior that empirical analyses are forced to ignore.
A third requirement" for a complete theory of arbitration is that models of different
forms of arbitration be
buii:
on a
common
information, and commitment. Tnis
is
set of
assumptions about preferences,
a prerequisite for accurate comparisons of
conventional and final-offer arbitration for two reasons, the
theoretical. First, if
each form
is
first
empirical and the second
adopted in environments in which
it
performs well (and
the data cannot reveal the differences in these environments), then comparisons of
performance data from
real arbitrations are likelv to suffer
from selection
bias.
And
if
second, because any form of arbitration could
environment,
a successful
model of
a given
in principle
form of
assumptions that permit other forms of arbitration
Consider Farber's (1980) model of final-offer
be adopted in any given
arbitration
to
must be
built
on
be analyzed within the same model.
arbitration, for
example. Farber's model
the leading contribution to the existing theoretical literature on that
form of
arbitration,
is
and
i
is
the basis for
that the
all
of the empirical work on the subject
Its
important
analogous model of conventional arbitration implies that the
irrelevant in the arbitrator's decision-making, an implication that
is
liability,
however,
is
parties' offers are
strongly contradicted by
the empirical evidence.
This paper
is
able to
arbitrator's attempt to learn
meet these
from the
such learning by assuming that the
by the publicly observable
relationship,
distinction
which
is
facts of the case
by the
and the
state
rate,
recent
wage
indication of the firm's financial health, and so on.
is
determined not only
state of the
employment
Tne most important
latter.
is
Examples of publicly
settlements in related industries, a coarse
Tne
state
of the employment
a collection of privately observable details of the production
environment and the labor market
that the
employer and union
inhabit.
long and close relationship, the employer and the union understand the
arbitrator, in contrast,
the possibility of
of the employment relationship
simply that the arbitrator observes the former but not the
relationship, in contrast,
emphasizes the
arbitrator's preferred settlement is
to the parties but not to the arbitrator.
observable facts are the inflation
it
The paper introduces
parties' offers.
facts of the case but also
known
between the
three requirements because
Because of
state well.
their
The
has less experience with these details but conducts an independent
invest: cation to gather information.
Two important assumptions
First, the arbitrator is
assumed
to
maintained throughout the paper deserve brief mention.
be unable to commit to a decision rule before the parties
submit their offers. This assumption seems
the
complementary case also deserves
common perception
is, it
also
attention.
observed practice accurately, but
Second, the parties are assumed to share a
of the state of the employment relationship. While
that the parties should both
arbitrator
to represent
seems
it
seems
realistic
be considerably better informed about the state than the
likely that the parties themselves will
have different beliefs about
the state. Detailed discussions of both of these assumptions are provided in Section 7.
The remainder of the paper
is
organized as follows. Section 2 describes Farber's
equilibrium model of final-offer arbitration.
As
will
become
clear, this
important role in the development of the learning models of both
model plays an
final -offer
and
conventional arbitration analyzed here. Section 3 specifies the economic environment
makes assumptions about
(i.e.,
the players' preferences and information structures) that
underlies this paper's models of both final-offer and conventional arbitration. Sections 4
and 5 analyze separating equilibria
in the final-offer
and conventional arbitration games,
respectively. In both cases, the parties' offers perfectly reveal their private information,
and the arbitrator uses
this
information to determine an ideal settlement. In final-offer
arbitration the arbitrator chooses the offer
±2:
is
closer to this ideal as the settlement: in
conventional arbitration the arbitrator simply imposes the ideal
conventional arbitration in Section 5
is
itself.
The model
of
of independent interest as a contribution to the
theory of cheap-talk communication games. Section 6 summarizes the relationship between
these learning
models and the empirical
literature.
of extensions of the models developed here.
Finally, Section 7 discusses a
program
2.
Final-Offer Arbitration without Learning
This section presents the key points
The
model of
in Farber's
final-offer arbitration.
analysis of learning in equilibrium models of final-offer and conventional arbitration
4 and 5 builds on the work described here.
that follows in Sections
Farber characterizes an arbitrator by a number,
most preferred settlement
in the case in question.
To
with those that follow, suppose that the arbitrator's
actual settlement
is
y
is
given by the
utility
Assuming
that the employer's offer, y e is
,
chooses the employer's offer
if
and only
facilitate
if
=
function v a (y,z)
<
y,
where y =
-(y-z) 2
settlement
is
z and the
In final-offer
.
parties' offers as a settlement.
below the union's
z
comparison of this model
utility if the ideal
choose one of the
arbitration, the arbitrator is constrained to
describes the arbitrator's
z, that
offer,
(y e
y u the arbitrator
,
+ y u )/2
is
the average of
the parties' offers.
The
parties are
assumed
to
be uncertain about the value of
described by the probability distribution F(z).
strictly positive
strictly
density ffz) with connected support
opposed preferences and are
arbitrator's
the arbitration
submit their offers to the
the utility function
-ax
v„
risk-neutral: the
arbitrator.
va (y,z). Tne
game
Then
parties'
y £ F(y)-y u [l-F(y)],
is
Let
that F(z) has a
Assume
this uncertainty
to
be
continuous and
also that the parties have
employer simply seeks
expected settlement, while the union seeks
The timing of
(1)
Assume
z.
maximize
to
minimize the
it.
as follows. Firs:, the parties simultaneously
the arbitrator chooses the offer that
Nash equilibrium
and
offers therefore
maximizes
must solve
y e F(y) + y u [l-F(y)].
min
(2)
The first-order conditions
(3)
l-F(y)
(4)
F(y)
which imply
for these optimization problems are
= j(y u -y e )f(y)
and
= j(y u -y e)f(y),
that F(y)
=
1/2 and y u
-
yc =
clearly cannot be false in equilibrium.)
off: in
These
equilibrium, each party considers
the gain
from having such an
f(y)' ]
.
(Note
that the
assumption
that y u
> ye
first-order conditions express a simple trade-
making
a
more aggressive
offer but
must balance
offer accepted against the reduced probability that
it
will
be
accepted.
Much
precision
of what follows assumes that z
H (i.e., variance
(5)
yu
=M + /^H
(6)
ye
=M--\/^
-
N
is
Normally distributed with mean
M and
1/H). In this case, (3) and (4) imply that the equilibrium offers
and
(Checking for the global optimality of these and subsequent offers
relegated to
Appendix
1.)
Note
is
tedious and so
that the equilibrium offers are centered
about the
the parties' belief about the arbitrator's ideal settlement, and that the distance
is
mean of
between the
equilibrium offers decreases as this belief becomes more precise.
3.
An Economic Environment
This section describes the preferences and information structure that underlie both
the final-offer and the conventional arbitration
novel feature introduced here
(or
simply the
state)
is
models analyzed
an unknown called the
and denoted by
s.
The
state of the
state
in Sections
4 and
5.
The
of the employment relationship
employment
relationship represents
the details of the production environment and the labor market that the
employer and the
union inhabit Because of their long and close relationship, the employer and the union
understand the state well. The arbitrator, in contrast, has less experience with these details
but conducts an independent investigation to gather information.
The
knew
central
theme of the models analyzed below
the state precisely then the ideal settlement
is
would be
as follows. If the arbitrator
clear.
Because the
arbitrator is
comparatively poorly informed, however, any information about the stare that can be
extracted from the parties' offers
may
be useful. Such inferences must of course account
for the natural incentive of each rjarrv to mislead the arbitrator.
Assume
relationship: z
=
that the arbitrator's ideal settlement
z(s).
depends on the
state
of the employment
This dependence could arise because the arbitrator wishes to be
fair,
10
or to be rehired. For simplicity,
Morgenstern)
utility
Unlike the
let
z(s)=s and let the arbitrator's (von
=
function be va (y,s)
-(y-s) 2 .
arbitrator, the parties are
preferences that are independent of the
Neumann-
assumed
state:
to
be risk-neutral and to have
v e (y,s) = -y and v u (y,s)=y. As
in the
previous section, the employer simply seeks to minimize the arbitrator's expected
settlement, while the union seeks to
what a
maximize
Thus, the parties are well informed about
it.
settlement would be, but this information does not affect their preferences.
fair
These assumptions about preferences keep the analysis simple.
however,
'
It
may seem
odd,
that the state s appears in the arbitrator's utility function but not in the parties'
utility functions.
Two justifications for this
here seems unlikely to disappear
when
can be given.
First, the intuition
the parties' preferences
do depend on
developed
the state.
And
second, the assumptions could be correct as they stand. Let the value of the enterprise be
when
V(s)
Then
the state
is s, let
the settlement y be the
the union's utility function
is
3b.
Info— adoLet
s
s_
=
is
let
employment be
v e (y,s) = V(s)
-
fixed.
y (which
arbitrator's ideal settlement z(s)=s reflects a
of V(s).
be Normally distributed with mean
s -r
and
Structure
knowledge of the
(7)
fair division
bill,
v u (y,s)=y, the employer's
changes nothing in the analysis), and the
judgment about the
wage
state
£,,
m and precision h.
be summarized by the noisy signal
Also,
let the parties'
1
where Ep
Normally
is
common
share a
mean and
precision h
Notice that the parties
perception of the state of the employment relationship. This
what follows, and
in
distributed with zero
discussed in Section
is
7.
is
important
Finally, let the results of the arbitrator's
private investigation be represented by the noisy signal
i
i
(8)
sa
=
where E a
is
variables
s,
s
+ EB
,
Normally
Ep,
distributed with zero
and £ a are assumed
to
mean and
precision h a
that hp
> h a but
,
To summarize,
the arbitrator observes
parameters
this is not
three
random
is
the arbitrator,
one could
necessary for what follows.
the information structure
sa,
The
be independent of each other. In keeping with the
idea that the parties are better informed about the state than
assume
.
no one observes
s,
is
as follows: the parties both observe s
and everyone observes m,
h,
h p and h 2
,
.
p
,
The
m and h that specify the prior distribution of s can be interpreted as the publicly
observable facts of the case.
Because of the independence of
learning
in the
s, Ep,
model (DeGroot, 1970, Chapter
model.
Some
9) characterize the Bayesian updating that occurs
s
given
(where
s
s
precision H^, where
(10)
M. (Si) =
:
hm
Hj = h + h
.
:
-rh;S:
_
n
1a
,
of the results of this updating are recorded here for future use.
conditional distribution of
(9;
and E a the simple formulae of the Normal
n
i*
QJjH
.
,
:
£ {p.a})
is
The
Normal with mean M(s,) and
1
Similarly, the conditional distribution of
given Sp and
s
s a is
Normal with mean
M pa
2
(s ,s a )
p
and precision Hp a where
,
hm +
(11)
M pa
(12)
Hpa = h+h p +h a
, „
x
,
(s
p
,s a
)=
h_s r
+
h.s.
a8
and
^+ha
h + h
.
These definitions provide a convenient shorthand
making
4.
in the
Learning
in
for describing the beliefs
and decision-
models below.
Final-Offer Arbitration
This section analyzes a model of final-offer arbitration in the economic environment
specified in Section
3.
The
analysis focuses on a separating equilibrium:
perfectly infers the parties' private information (Sp)
information and the signal
(s a ) to
compute
from
their offers
The
arbitrator
and then uses
a posterior belief about the state (s)
this
and to
choose the expected-utiiiry-maximizing offer as the settlement. Tne parties understand that
the arbitrator
-will
draw an inference from
misleading the arbitrator
when choosing
the offers and so consider the payoff
their offers, but find
it
from
optimal not to submit
misleading offers.
The timing of the game
observes
arbitrator
s2
.
is
and the arbitrator
as follows. First, the parties observe s^
Second, the parties simultaneously submit offers, y e and y u
chooses the offer that maximizes
Es [v a (y,s)
I
sa ,y e ,y u ],
.
Third, the
where the conditioning on
13
y e and y u denotes the information about s p
Note
offers.
that the arbitrator
rather, the arbitrator's choice
is
that the arbitrator extracts
not allowed to
must
commit
from the
to a decision rule in
parties'
advance;
satisfy sequential rationality at the third stage
of the
game.
Before describing the separating equilibrium
subsequent analysis to show
that
in detail,
some learning must occur
it
helps set the tone for the
cannot be equilibrium behavior for the arbitrator to leam nothing from the
To
see
this,
suppose the arbitrator thought
offers.
Then
(13)
y a(
Given
s
p
,
precision
(14)
, .
-s
is, it
parties' offers.
no information could be extracted from the
the arbitrator's ideal settlement, y a (sa),
would be
M a (s a
)
from
(9),
or
^^
o=
the parties' belief about y a (s a )
is
Normally distributed with mean m' and
where
h',
ha
m' =
.,
that
That
in equilibrium.
t
n
^
ii
•
iia^n
^?^
h
-r
na
it- Mil
-r
nD
^
T jjoU
- ii-j
(Note that the dependence of m' en Sp has been suppressed in the notation.) The logic
behind Fafbefs equilibrium
parzes' offers then
s-,
is
now
applies directly: if the arbitrator learns nothing
relevant only because
arbitrator's ideal settlement.
Tne equilibrium
it
changes the
parries' beliefs
from the
about the
offers are therefore analogous to (5)
and
(6):
1
=m' +
(16)
y u (s p )
(17)
y e (s p ) =
4
"^
Vir
m'--\j^T
Contrary to the hypothesis, the arbitrator can learn something from these offers. The
m', and (9) and (14) determine
average of the offers
is
(point) estimate of s
computed
p
arbitrator's ideal settlement
(18)
The
y a (y c ,y u ,sa) =
1
learns nothing
The
from
m'. Let s (m') denote the
p
way. Sequential rationality then requires that the
(s (m'), s
p
a)
from
(11), or
(13) establishes the following result.
exist a perfect
Bayesian equilibrium
in
which
the arbitrator
parties' offers.
crucial difference
parties' offers
this
from the
from
+ h r s p (m')+ h a s a
h + h„ + h,a
p
There does not
.
Mpa
p
hm
fact that (18) differs
Proposition
be
in this
s
between (13) and
(18), of course, is that in the latter the
help the arbitrator determine the ideal settlement. In an equilibrium featuring
kind of learning, each parry's offer must balance not only the probability of and the
gain from submitting the offer chosen as the settlement, as in Farber's model, but also the
effect the offer has
on
the arbitrator's inference about the ideal senlement.
section establishes that a separating perfect Bayesian equilibrium exists in
parties' offers perfectly reveal Sp to the arbitrator.
The
rest
of this
which the
1
Suppose
that the arbitrator believes that y, the average of the parties' offers,
perfectly reveals
Sp.
point estimate Sp
=
That
is,
for any pair of offers y e and y u the arbitrator
(Other separating equilibria
Sp(y).
The
function of ye and y u reveals Sp to the arbitrator.)
h
(i9)
y>(y*.y^-
V £?X'
h
may
exist in
then
S
'
settlement, y a (y e ,y u ,Sa),
less than the average of the offers, y.
to
which some other
+
arbitrator
simple
computes the
arbitrator's ideal settlement is
and (assuming y c < y u ) the
It is
is
show
chooses y e as the settlement
that if the equilibrium value of
y
is to
if
and only
if
the ideal
reveal Sp in this
way
then
the arbitrator's inference rule must be
(20,
To
Sp (v)
.
£ - My
see this, use (19) and
-
hm
(1 1) to
express the event that the arbitrator chooses y e as
Sa<S(y), where
(-1J
i>0)
=
r
'
.
>
"a
A derivation analogous :o that leading to
(22)
l-?[S(y)]
5
=
(y B
-
y e) fTS(yj]S'(y)
(3)
and
(4)
and
then yields the first-order conditions
1
=
F[S(y)]
(23)
where F is now the
-y e )f[S(y)]S
(y u
As
and
in (5)
for s
p
in
Mp
(6),
(s
p)
,
(y),
distribution of s a conditional
given by (9) and precision
on sp which
,
is
Normal with mean
M p (sp
i
these first-order conditions imply that S(y)
and solving for s p (y) yields
The inference
=
Mp
(s
p ).
Substituting
choose y c
analogous to the decision rule used
if
s (y)
p
(20), the desired result
rule s (y) in (20) implies that (21) simplifies to S(y)
p
arbitrator's decision rule is to
and only
in Farber's
if s a
<
y.
model (where
=
y: the
Note
that this is directly
there
is
no learning). Not
surprisingly, therefore, the equilibrium strategies for the parties that solve the first-order
irtions (22)
and (23) are
Q5)
y u (sp ) = Mp(Sp)
(26)
y.fsp)
=.Mp(s P )
Tne average of the
+
directly analogous to (5)
offers is
Mp
(s
p ),
(6):
and
-y T^r
--\/w
and
•
which
yields a
more convenient expression of the
inference rule (20): in equilibrium, the arbitrator infers s« from the average of the parties'
offers, uses
this ideal.
it
in (19) to
6
compute
the ideal settlement,
and then chooses the offer closer to
)
17
The summary of this argument
Proposition 2
The
.
is
the
main
result of this section.
parties' offer strategies given in (25)
and (26) and the
arbitrator's
decision strategy based on the ideal settlement (19) and the inference rule (20) constitute a
separating perfect Bayesian equilibrium of the final-offer arbitration game. In this
equilibrium, the arbitrator's ideal settlement can be written
ay+
ya =
where
sa ,
hm +h r s r
-
h
+ hp +
The key
offers
(1-a)
from the
h
;
feature of the arbitrator's equilibrium strategy
parties. In equilibrium, the panics' offers differ
arbitrator perfectly infers the parties' private information
Any
pair of offers that differs by this prescribed
that differs
is its
by any other amount
is
amount
treatment of unexpected
by \2~/H'
,
and the
from the average of the
is
offers.
on the equilibrium path; a pair
off the equilibrium path.
Tne
arbitrator's inference rule
ignores this distinction, however: (20) oroduces the ooint estimate of Sn both on and off the
equilibrium
of Sp
path".
at the rate (h
the parties
Thus, any change in either party's offer changes the arbitrator's estimate
-
hp) / 2h n
must balance
this
.
Tne reason
the arbitrator
is
not misled in equilibrium
opportunity to influence the arbitrator's belief against the two
considerations that determined the equilibrium offers in Farber
navms
a
more aisressivs
offer acceoted
Tnere are two cases
analyzed
is that
in
s
model, the gain from
and the reduced ~robabiiitv
which learning plays no
in this section: (1) the arbitrator
that
it
will be accented.
roie in the private-information
has nothing to learn
(i.e.,
game
h 2 approaches infinity):
1
and
have nothing
(2) the parries
to
communicate
feature of the separating equilibrium analyzed here
Farber's complete-information
As
information game.
about
s
game
h
(i.e.,
is that it
p
becomes relevant only
setdement
(i.e.,
H in (5) and
One
infinity.
is
h
+
h
p
which
,
Similarly, as h
is
p
H in (5) and (6) is hha/(h + ha), which
is
indeed
approaches zero.
perfectly infers not only
s,
model)
does not improve their prior information
As
other limit also bears consideration.
information about
in the private-
of the parties' belief about the
(6) in Farber's
parties' private information
about the arbitrator's ideal settlement, so
p
appealing
as a signal to the parties about the
indeed the limit of H' in (25) and (26) as h a approaches
the limit of H' as h
One
approaches the equilibrium in
becomes unimportant
as the learning
arbitrator's ideal settlement. In this case, the precision
approaches zero, the
approaches zero).
h a approaches infinity, for instance, the arbitrator's uncertainty
disappears, and s
arbitrator's ideal
p
8
s
p
but also
s
from
and
p
approaches infinity, the arbitrator
the average of the parties' offers.
the arbitrator's private signal
arbitrator's ideal settlement,
h
becomes
sa
Given perfect
irrelevant in determining the
this eliminates the parties' uncertainty
about the
arbitrator's
ideal settlement. Recall that as the analogous uncertainty disappears in Farber's complete-
informarion model, the parties' equilibrium offers converge to the
of the arbitrator's ideal settlement. (Formally, as
offers given in (5)
and
(6)
does not approach infinity
K
approaches
(rather,
it
approaches
independent of the value of hp. Thus,
it is
infinity" Sie
equilibrium
,
ha),
so the equilibrium offers given by
model do not converge. This happens because
noted above) (20) and (21) imply that the arbitrator chooses y e
though
the distribution
converge to M.) Interestingly, as h approaches infinity H'
p
(25) and (26) in the private-information
rule even
mean of
sa
if
and only
if s£
<
(as
y,
serves as a tie-breaker in the arbitrator's decision
irrelevant to the arbitrator's inference process.
1
5.
Teaming
in
9
Conventional Arbitration
This section analyzes a model of conventional arbitration
environment specified
in Section 3.
The
in the
economic
continuum of payoff-
analysis focuses on a
equivalent separating equilibria. In each of these equilibria, the arbitrator perfectly infers
the parties' private information (Sp) from their offers and then uses this information and the
signal (s a ) to
compute
a posterior belief about the state
settlement maximizes expected
arbitrator will
utility
draw an inference from
given
this belief.
the offers
(s)
and
The
to
impose whatever
parties understand that the
and so consider the payoff from
misleading the arbitrator when choosing their offers, but find
optimal not to submit
it
misleading offers.
The timing of
the
game
is
Second, the parties simultaneously submit
observes
s
arbitrator
imposes whatever settlement maximizes E s [v a (y,s)
a
.
section, the arbitrator is not allowed to
clear below, however, that in this
commit
rational decision rules associated with the
offers,
I
y e and y u
s ,y ,y ].
a
e u
to a decision rule in
model of conventional
value the opportunity to commit to a decision
below would be an optimal
and the arbitrator
as follows. First, the parties observe s~
rule:
.
Third, the
As
advance.
in the
It
previous
will
arbitration the arbitrator
become
would not
each of the continuum of sequentially
continuum of separating equilibria described
rule if the arbitrator could ccrrrrit.
Tr.s crucial difference
between
this
model of conventional
previous section's model of final-offer arbitration
is that
arbitration
here arbitration
is
and the
modeled
as a
cheap-talk game: the parties' offers are not directly relevant to any player's payoff; they
matter only indirectly through the information they convey.* In this case,
model
J
(as well as faithful to the institutional details
it is
natural in the
of arbitration) to envision Darrv
See Crawford sua Sobel (19S2) for the seminal work on cheap-talk
carries.
i's
offer
20
as a long speech that ends "and that
Final-offer arbitration
is
would not be
why we
parties' offers
is yj."
must be
directly affects payoffs. Similarly, conventional
a cheap-talk
constrained to be the parties'
believe that an appropriate settlement
game because one of the
not a cheap-talk
chosen as the settlement, and so
arbitration
is
game
if
game were
the offers in the arbitration
last offers in a prior
bargaining game, because such offers
could have been accepted and would then have directly affected payoffs. 2
All cheap-talk
talk
game
games share two
distinctive features.
The
first is
that in
any cheap-
there exists a trivial equilibrium in which no communication occurs. In the
conventional arbitration model, for instance, the arbitrator expects there to be no
information in either party's offer and so pays no attention to either offer
settlement, and each party
affect
on the
is
content to choose an offer
arbitrator's settlement.
model of conventional
Thus, there
arbitration there
leams nothing from the
parties' offers.
does
(and imposed) sertiement
independence
not observed in the data,
is
The second
talk
exist
Note
arbitrator's ideal
distinctive feature of
is
this
all
is
at
no analog of Proposition
communicated
1
offer has
a
any
here: in this
an equilibrium in which the arbitrator
well, however, that in this equilibrium the
independent of the parties' offers. Since such
no-communication equilibrium
cheap-talk
games
is that,
class of payoff-equivalent perfect Bayesian equilibria that differ only
costless, tine information
when choosing
random because no
communicated
in
is
suspect.
given a fixed cheap-
game, any perfect Bayesian equilibrium (separating or otherwise)
messages are
.
is
by
associated with a
a translation: since
any given equilibrium also can be
in other navoff-ecuivaiem equilibria that differ onlv in the laneuaee used to
-iJius, in this paper, the parses' offers in arbiration need not be identical to their last offers in a prior
bargaining game. This flexibility arises if (as is required in some contracts) the bargaining record has been
is unavailable to the arbitrator. Also, it will become clear below that the incentive for the parties
to make concessions in arbitrator, can be quite strong, perhaps stronger than the analogous incentive in
sealed and
bargaining. Therefore, even
game may be
if
the bargaining record has not been sealed, the last offers from the bargaining
sufficiently far apart that they
have no bearing on the
parties' choices of offers in arbitration.
2
1
communicate. This accounts for the great multiplicity of payoff-equivalent separating
equilibria
mentioned above.
This section's model of conventional arbitration
distinctive feature: neither party alone can achieve
that together the parties
Bayesian equilibria
in
One scheme
ask each to
disagree. 3
for
And
maximize
which the
arbitrator,
independent of the state of the
it).
a claim about the information
outcomes
and
to shoot
feasible in the arbitration
power
is
from two informed agents
them both
model analyzed
if their
here, however,
limited to imposing a transfer
that are Pareto-inefficient for the
claims
between two
two parties are
second, the arbitrator's behavior must be sequentially rational: given any'
^d any signal (s^, the arbitrator must formulate a belief about the
parties' private information, p.(s
p
I
s a ,yetyu)>
^d act optimally given
In spiie'of the facts that Pareto-inefficient
must be sequentially
crucial feature of the
model
correlated with the signal
from the
that the
~arties.
that drives this result is that the parties' information (s
from the
outcomes
this belief.
outcomes are impossible and
rational, the arbitrator car. extract s
p
o uncerstanc mis result,
Pareto-inefficient
This section demonstrates, however,
parties' offers perfectly reveal Sp to the arbitrator.
First, the arbitrator's
pair of offers (y e ,y u )
i
a third
can achieve perfect communication: there exist separating perfect
Such schemes are not
risk-neutral parties, so
arbitrator
that are
for extracting a single piece of information
make
two reasons.
impossible.
game with
relationship (the employer simply seeks to minimize the arbitrator's expected
settlement, while the union seeks to
is to
a cheap-talk
any communication with the
because each party has preferences over settlements
employment
is
;t
p)
Tne
is
arbitrator's investigation (Sa).
is r.e-prui to
consider tne intermediate case in
wmcn
are imoossibie but sequential rationaiitv is noi recuired. In this
J See Xaiai ar.c Rosenihal
(1979) for z
more
careful (if less dramatic) statement of this result.
22
scheme induces
case, the following
truth-telling:
ask each party for a claim about s p
disagree then compare each party's claim about sp to the signal
who seems
transfer to the party
seems
to
have made the
to
if
less reasonable claim.
they
and enforce a large
have made the more reasonable claim from the party
who
Given any correlation between s p and sa
there exists a sufficiently large transfer such that truth-telling
this
sa
;
is
,
the unique equilibrium of
scheme.
It is
simple to extend the
spirit
of this scheme to the model of conventional
arbitration studied here, in
which the
The argument
fact that in a perfect
on the
relies
game, the
arbitration
freedom ensures
behavior must be sequentially rational.
arbitrator's
Bayesian equilibrium of the conventional
This
arbitrator's belief off the equilibrium path is unrestricted.
that there exists a belief off the equilibrium path that
makes
sequentially
it
rational for the arbitrator to enforce the required transfer if the parties disagree.
There
is
game
arbitration
a close relationship between the incentives created in a conventional
in
which the
arbitrator uses the
created in a final-offer arbitration
which
it is
awarded
to
one
game
:
scheme described above and
the incentives
under
the size of the transfer and the conditions
part}' or the other in the
conventional arbirranon
game
are
analogous to the gap between the offers and the conditions under which the arbitrator
chooses one parry's offer or the other's
in the final-offer arbitration
game. In
proof that there exists a continuum of separating perfect Bayesian equilibria
conventional arbitration
game
fact, the
in the
with learning relies on the construction of the following
continuum of final-offer games without learning.
Recall that in Section 2 the arbitrator's ideal settlement
parties' shared belief
k£
(-»», oo),
was denoted by
z,
and
tine
about z was denoted by F(z). Suppose instead that for some fixed
tie arbitrator's ideal settlement
is
23
HT>
(27)
i,f;
\
z k (s a )-
* hn)Sa I khm
(1 . k)h + hp
Ch
Note
that if the ideal settlement
learn
from the
parties: s a
independent of s p
Given sp the
,
is
given by (27) then there
completely determines the
In this case, the only role for
.
s
parties' shared belief about z k (Sa) is
p
is
nothing the arbitrator needs to
arbitrator's ideal settlement,
is to
help the parties forecast zk(s a ).
Normal with mean Mk(s p ) and precision
Hk, where
(28)
UN
Mrfc..)=
MUSpj-
na\
{ ~ y)
zj,
From
(1
k)hm +
•
.
(1
nk
(5)
-
and
MH
hp s n
and
k)h + h
k)h + h p ] 2
(h + h )(h + h
+ h a)
p
p
-
equilibrium offers in this model of final-offer arbitration
(6), the parties'
witiiout learning are
(30)
yuk(s D ) =Mk(Srj)
'
(3D
yek ( Sp )
=Mk
(s
P)
+ -v/^V
—^k
and
-Vm-at
*~k
\
These equilibrium offers from the
final-offer
game without
learning will
rum
out to be
identical to the equilibrium offers in this section's mo-del of conventional arbitration with
learning!
24
Imagine
that in the
model of conventional
arbitration with learning, the arbitrator
believed that the parties' offer strategies were (30) and (31). The arbitrator could then
invert the offer strategies so as to interpret the offers as implicit claims about the parties'
information. Given the offer y u from the union, for instance, the arbitrator could solve
(30) for s p to arrive at the implicit claim that the parties' private information
(32)
Spuk(y u )
= aJ y u
(33)
o k (x) =
x
solves Mk(Sp)
fl
=x
"
-
y ^r J
k)h
i
for
Sp.
spek(y e )
Note
where
.
(x-m)
Similarly, given the offer y e
could solve (31) for s p to arrive
(34)
= aj|y e +
from the employer, the arbitrator
claim that the parties' private information
at the implicit
1
1
]
that if the parties' offers are in fact
2ri
ecuiiicnum. then
is
y j^
determined by (30) and (31) then these implicit
claims will be identical even though the offers are not.
v,"
is
t*ctt*—
it 1C
^j
*
*"
fo~ n **
*2—n:-— oth~
tr^
If the implicit
r%»"i«»y&
Everything hinges, however, on what the arbitrator believes
T
h**
when
claims agree, as they
^ommo~
"'
2
—
the implicit claims
disagree.
SuDT>ose that 2frer observing the siEnal
arrives at the point estimate s
arbitrator
impose
p
=
Sp(y e ,y u ,Sa).
the ideal settlement
s,
and the offers v. and v„ the
Tnen
Mpa [Sp(y
arbitrator
sequential rationality dictates that the
e .y u .Sa),
sj from
(II), or
2 5
y a (yc.yu,s a ) =
(35)
It
turns out to be
KM
This
Mifv
vu
M
k (y c ,y
last
-i-
h p s p (y e ,y u ,s a ) + h a s a
to
-
-,\
,s
a)
(1
'
k)hm +
(1
-
makes
it
For each k £
(28) yields
h n s r.(ye,Vu,S a )
k)h + h p
possible to state the section's
form. (The proof of the Proposition
.
ht
express the arbitrator's inference rule not as
Mk(y e ,y u .Sa), where
piece of notation
Proposition 3
+
h + h
more convenient
Sp(y e .yu,s a ), but as
(36)
hm
(-°°, °°),
is
given
Appendix
in
main
result in a
compact
2.)
there exists a separating perfect Bayesian equilibrium
of the conventional arbitration game. In each of these equilibria, the arbitrator's ideal (and
imposed) settlement
(31),
and the
is (35).
Mk(y e ,yu ,Sa) =
<
L
k, the arbitrator's ideal
_ -
>2= Ctkyk ^
•
_
K
~
k, the parties' offer strategies axe (30)
arbitrator's inference rule is
r
Given
For each fixed
,.
.
(1-OCk)
>'u
if
Spek(ye)
? Spuk(Vu) and Zk(s a ) >
y
if
Spek(ye )
= sP uk(y u ),
5e
if
Spek(y e )
r Spuk(yu) and Zk(sa) <
serJemeni
khm t
car.
h 2 s*"
kh - h"
d-k";hm - h-s„
(l-k)ij
- h_
(1-kYn 4
hr,
aiUi
be written
,
'
where
y,
y.
and
26
Several observations about this plethora of equilibria are in order. First, as noted
of the equilibria described in Proposition 3 are payoff-equivalent: even though
earlier, all
changes
k
in
affect both the
differences in
k
mean
of and the distance between the parties' offers,
are simply differences in the languages used to communicate. Second,
values of k are simple to interpret: k
=
and k =
1.
When k = 0,
(28)
becomes Mfc(s p ) =
Mp(Sp), so in equilibrium the arbitrator interprets y as the parties' expectation of
When k =
1,
And
itself.
(28)
becomes Mk(s p ) =
s
p
,
two
s
given
s
p
.
so in equilibrium the arbitrator interprets y as s p
third, in the separating equilibrium
with k =
the parties' offers in conventional
arbitration are identical to their offers in the separating equilibrium in final-offer arbitration
described
in
Section 4.
Although the inference rule Mk(ye.yu- S a) stated
knife-edge judgments by the arbitrator
same
spirit as a
(i.e.,
decision rule that seems reasonable in practice:
case. Afterwards,
when
all
The
by
this
have made the less reasonable claim. Note mat
arbitrator gives both
on the
table, the arbitrator asks the
point agree) conducts an investigation,
book
at the
this inference rule
party
imposing a settlement. Thus, for any
described in Proposition 3 would be an optimal rule
if
who seems
has the virtue of
supporting a separating equilibrium and so providing the arbitrator with the
in
has the
to challenge assertions in the other's
the result to the parties' claims, and throws the
amount of information
it
their dispute. If they cannot, then the arbitrator
(thinking that reasonable people should
compares
and
the cards are supposed to be
now resolve
whether they can
depends on
whether Spek(y c ) = s pui;(y u ) or not ).
parties the opportunity to present their cases fully
parties
in the Proposition
k, the
maximum
decision rule
the arbitrator could
commit.
to
27
6.
Relationship of the Learning Models to the Empirical Literature
This section describes the connections between the empirical literature on arbitration
and the learning models developed
much
in
Sections 4 and 5.
As described
of the empirical work on arbitrator behavior assumes the following model of the
arbitrator's ideal settlement:
ya =
As
in the Introduction,
ay
+
,
(l-a)y*(f).
Propositions 2 and 3 emphasize, the arbitrator's equilibrium behavior in this paper takes
exactly this form.
literature's
The paper
thus provides a structural interpretation for the empirical
reduced-form model:
arbitrator's ideal settlement
in both
conventional and final-offer arbitration, the
depends on the
parties' offers
because the offers convey
information, and the importance of the offers (as measured by a) increases as the
importance of
tine arbitrator's
this
independent investigation
The paper
work. The
information increases relative to the publicly observable information and
first
also
fits
nicely with
stylized fact
is
('i.e..
as
two of the
4
h« increases relative io h and h E ).
stylized facts
uncovered by the empirical
that the arbitrator's ideal settlement
is
identical in the
two
forms of arbitration; see Asherieher-Bioorri and Farber-Bazerman. Tne separating
equilibria in the learning
models analyzed here are founded on
of an ideal settlement: under ajv form of arbitration,
the parties' offers then the ideal settlement is
"This structural interpretation
is
if
just
such an invariant notion
the arbitrator perfectly infers
Mr^fs^a) given
a<
1.
p
from
in (11).
incorrect in the conventional arbitraiion equilibria
occur when k < -(h 2 /h), but empirical estimates suggest that
s
ir.
which
cx^
>
1,
which
28
The second
arbitrator's
stylized fact
decision-making
is that
(i.e.,
Bazerman. The learning models
when
a is
in
the parties' offers are
when
higher)
more important
in the
they are closer together; see Farber-
a should
both Sections 4 and 5 predict that
be higher
the offers are closer together, but suggest that this need not be a causal relationship.
Consider Proposition
2, for instance.
Note
that
difference between the offers in (25) and (26)
also increases with h
p
Thus, an increase
.
in h
is
p
a increases with
hp
In equilibrium, the
.
V2rc /H', where H'
is
given by (24) and
results in both an increase in
a and a
decrease in the difference between the offers, thereby producing the observed correlation.
The
causal story behind the correlation
the parties' offers
when
is as
follows.
the parties' information
is
more valuable
in the arbitrator's inference process.
information
more
more
is
precise,
and
known
At
arbitrator puts
to
the
more weight on
be more precise, and hence
same
time,
when
the parties'
precise, the parties' belief about the arbitrator's private signal,
this
s a , is
causes the parties to submit offers that are closer together.
Tne model of final-offer
arbitration
developed
well to one other stylized faci: Ashenfelter and
as the serJement
The
more than
Bloom
in
Section 4 does not correspond
find that the union's offer is selected
half the rime, whereas in Section 4 the parties' offers are
equally likely to be selected. Farber's model of final-offer arbitration without learning
consistent with the Ashenfe::er-3ib6m finding if the union
employer, and so prefers
to
submit
a conservative offer
is
more risk-averse than
a learning
model with
It
would be
interesting to
parties of differing risk aversion could integrate the
Ashenfeiter-3ioo— finding on offer-selection frequencies with the role of the
in the arbitrator's inference nrocess
the
with a large probability of beir.g
chosen rather than an aggressive offer with a small probability.
know whether
is
is
emDhasized here.
parties' offers
29
7.
Extensions
Although
this
paper
complete theory of arbitration, much work remains
models analyzed above and
in building
7a.
to be done, both in extending the
new models.
two extensions of the models analyzed above and
paper's
be required of a
satisfies three of the conditions that will
This section
limited to describing
is
to providing further discussion of the
two most important assumptions.
Comparing Conventional and Final-Offer
Arbitration
This paper analyzes conventional and final-offer arbitration within a single
economic environment. As explained
complete theory of
arbitration.
in the Introduction, this is a prerequisite for a
The obvious next
step
is
a welfare
comparison of these and
other forms of arbitration. Unfortunately, the model analyzed here
make
is
not rich enough to
the comparison interesting. First, because the parties are risk-neutral and have
strictly
opposed preferences, any preference by one party
be exactly reversed for the other parry.
upon the separating equilibrium
based on
full
And
second, the a:biuaioi
in conventional arbitration,
information and the actual settlement
A more
for one
is
form of
cl early
would
cannot improve
where the ideal settlement
is
unconstrained.
interesting comparison might arise if the parties
separating equilibrium exists in conventional arbitration
when
were risk-averse.
If a
the parties are risk-averse
then this will again be the arbitrator's preferred form of arbitration.
might find
arbitration
Tne
parties,
however,
that final-offer arbitration offers better insurance than conventional arbitration
does. If the parties are allowed to choose the form of arbitration then the arbitrator's
preference for conventional arbitration
mav
be overridden.
30
7b.
Modeling Other Forms of
Arbitration
Dow, and Gallagher
Ashenfelter,
arbitration that is practiced in Iowa.
The
(1986) study an interesting variant of final-offer
rules are as follows: first, an arbitrator, acting as a
factfinder, suggests a settlement; then the parties either accept the settlement or continue
their dispute
and submit
offers; finally, if the dispute continues then a
(drawn from the same pool as the
first)
second arbitrator
chooses one of the parties' offers or the factfinder's
proposed settlement.
It is
this version
straightforward to extend the model described in Section 4 in order to analyze
of final-offer arbitration. Suppose the factfinder's investigation yields the
signal Sf
=
the other
random
s
+ Ef, where
Ef is
Normal with zero mean and precision
variables in the model.
Then
model
is as
before, except for
two changes.
yj<Sf)
second arbitrator now has a third choice in the feasible
arbitrator does not
Dow, and Gallagher argue
First,
of arbitral uncertainty that exists in the system"
between
independent of
set
is
(p. 2).
arbitrators also
model of
(9).
The
both parties and the second
And
second, the
of imposed settlements.
that the "extent to
concur in the factfinder's proposal
plausible, the disagreement
is
= Mf(Sf), given by
arbitrator update their beliefs after observing the factfinder's proposal.
Ashenfelter,
and
(as in the conventional-arbitration
Section 5) the factfinder's proposed settlement should be
rest of the
hf
which the second
an obvious measure of the degree
While
this interpretation is certainly
may reflect
the second arbitrator's
learning from the parties' offers.
7c.
Commirmer.: bv the Arbitrator
One of the two important assumptions
commit ex-ante
to a decision rule,
in this
even thoush
this
paper
is
that the arbitrator is unable to
mishi induce a sociallv desirable
3
outcome of the negotiation and
observed practice accurately:
make
their offers.
On
arbitration process. This
arbitrators
assumption seems to represent
to decision rules
before the parties
the other hand, the assumption ignores the important possibility that
an arbitrator might develop a reputation
selection
do not commit
1
in the course of repeated play.
The
arbitrator-
mechanisms described and analyzed by Bloom and Cavanagh (1986a,b)
will play
an important role in the eventual analysis of such dynamic games. Because the complete
dynamic analysis
models
that
commit
is
likely to be difficult, however,
complement
this
paper
to a decision rule ex-ante.
rules described there
The
arbitration.
arbitration
is
is
—
static
models
As noted
an optimal rule
if
in
it
in
would be worthwhile
which the arbitrator
Section
is
to analyze the
allowed
to
each of the continuum of decision
5,
the arbitrator can
commit
nature of the analogous optimal rule under
conventional
in
commitment
in final-offer
an open question.
7d. Arbitration
Between
Parties with
Symmetric Information
The second important assumption
in this
paper
is
that the parties
information about the state of the employment relationship.
It
seems
have symmetric
likely, instead, that
each of the parties will have private information about some aspects of the employment
relationship.
means and
Such an information asymmetry between
the substance of the parties' equilibrium
Perhaps more importantly,
it
the parties
may
influence both the
communication with the
prompts the question of how the parties came
arbitrator.
to
be in
arbitration.
The theory of bargaining under symmetric information, due
Rubinstein (1982). suggests that negotiations should not
symmetric, esneciallv
when
the arbitration
same
fail
when
to
Nash (1950) and
information
that follows failed negotiations
is
imposes
32
costs (such as the arbitrator's fee) on the parties. This does not imply, however, that the
way
prospect of arbitration has no effect. Crawford (1982a), for instance, analyzes the
arbitration
changes the threat point
though the bargainers never
Nash bargaining game:
in a
fail to settle.
effect
models seems
likely to
The analogous
analysis using this
imply that the prospect of arbitration has a smaller
on negotiated settlements when learning
arbitrated settlement to the state
even
Crawford's analysis involves models of
conventional and final-offer arbitration without learning.
paper's learning
arbitration has an effect
is
more important, because learning
links the
and so prevents a disadvantaged bargainer from escaping
an unattractive negotiated settlement through arbitration by an uninformed arbitrator.
Thus, the prospect of arbitration between parties with symmetric information can
have an effect even
if
those developed by
Nash and Rubinstein,
the parties never enter arbitration. Furthermore, in richer
disputes can occur between parties with
symmetric information. Crawford (1982b),
view
that bargaining is a struggle
positions.
If
Attempts
at
for instance, formalizes Schelling's (1956)
between the bargainers
commitment
models than
to
commit themselves
to favorable
are uncertain to succeed, but are irreversible if they do.
only one parry's attempt succeeds then that party wins a favorable negotiated settlement,
but
if the parties
arbitration
are both
committed
may occur. A model
to
incompatible positions then negotiations
fail
and
based on these ideas can produce disputes between parties
with symmetric information, but more work needs to be done in modeling the process of
commitment.
One
intriguing approach to modeling the process of
commitment emphasizes
the
idea that the parties are groups rather than individuals: see "Walton and McKersie (1965).
union ieader, for example, can try
politics
and group dynamics
to incite
may make
and yet control the rank and
the rank
and
file's
reactions
file,
beyond
but internal
the union
A
33
leader's control. In this case, the union leader
may become committed
in the
sense
envisioned by Schelling and Crawford. Alternatively, when the parties are groups a
dispute
may occur
because of asymmetric information between them about features of their
relationship that are of no concern to the arbitrator, including perhaps the career concerns of
individual negotiators. In this case, with regard to the learning in the arbitrator's inference
problem the
parties are appropriately
viewed
as having
symmetric information, even though
asymmetric information on another dimension caused the dispute. These
suggest that
it
may be a
mistake to take too
union as single-agent bargainers.
literally
models
that
possibilities
approximate a firm and a
34
APPENDIX
1
This appendix establishes the global optimality of the offers that satisfy the
order conditions (5) and
(6), (16)
and (17), (25) and
proof is given for the union's offer specified
U(y u ly e) = y e
where y =
F(y")
+ yu
and (30) and (31).
in (5). All the other cases are
Let the union's expected payoff given
(Al)
(26),
in (1)
U'(y u
I
and F
is
a
Normal
It
two
parts.
And the
is
M
^y u
-
-
first
-
simply U'(y u
I
y e) =
0,
y e )f(y).
prove that y u
to
V-/2H from
suffices to prove that
Tnt
is
mean
,
Tne goal of this appendix
I
distribution with
a derivative with respect to y u and
y e ) = [l-F(y)]
U(y u ye *) where y e * =
analogous.
be denoted
M and precision H. In this notation, the first-order condition (3)
(A2)
explicit
[l-F(y)],
(y u +ye)/2 is the average of the offers
where the prime denotes
An
first-
U
=
M + %':t/2H achieves the global maximum of
(6).
quasi-concave in y u
is
establishes thai
U is
second establishes that U' <
when y e = y e x This
.
concave for yu z [y e *
for y u £
(
M+
d, «>
),
,
is
done
M + D], where D >
where
< d < D.
in
0.
(Recall that
union offers below the employer's offer are sub-optimal for the union, and so need no: be
considered here.)
Tne concavity of U on
[ye
*
,
M
-r
Dj follows from evaluating U" on
this interval:
35
U"(y u
(A3)
where y* =
i
(y u
=
e *)
-f(y*)
-
e *)f(y*).
For the Normal distribution with mean
/ 2.
+ye*)
-\ (y u
M and precision H,
-f (y*) = H(y*-M)f(y*), so
U"(y u y e *) =
(A4)
i
which
is
negative for (y u
The negative
(A5)
f(y*){-
H(y*
-
M)
-y e *)(y*
M)2 < rr{% + —) =
-
slope of
[1 -
+ j<y u
1
F(y*)]
U on
<
(
M + d, °°
(A6)
M and precision H.
(y u ->'e*)(y*
which holds for
-
t'v,- -
"£
.
follows from a simple inequality:
-
y*) f(x) over (y*,°°)
Given (A2) and (A5),
~
M) >
M
)
>
,
r?-
M)
f(y*).
Tnis inequaliry follows from integrating (x
mean
D2
-
(—
—
-St)
it
suffices to
when x
show
that
is
Normal wiih
36
APPENDIX
This appendix proves Proposition
3.
2
The proof that
sequentially rational best response to the parties' strategies
inference rule
s (y e ,yu»Sa)
p
Mk(y e ,y u ,Sa)
given
in the
the arbitrator's strategy
is
is
a
simple. Using (33), the
Proposition can be expressed as the inference rule
used in (35). This inference rule
is
clearly correct in equilibrium
and
is
well
defined off the equilibrium path; given this belief, (35) guarantees sequential rationality.
The proof that
the parties' strategies are best responses to each other and to the
arbitrator's strategy is slightly
more involved. The
the conventional-arbitration game,
final-offer
game
that has a
to,
as though the arbitrator
it is
unique Nash equilibrium
payoff Mk(sn), which equals
Consider the payoff
central point is that out of equilibrium in
their equilibrium
say, the
that yields
makes
the parties play a
both parties the expected
expected payoff in conventional arbitration.
employer from submitting an offer different from the
equilibrium offer given by (31). (A deviation by the union
is
By
analyzed symmetrically.)
hypothesis, the union's offer will be given by (30), so any deviation by the employer will
cause Spek^'e)
=^ s
puic (y u ).
This in turn causes the arbitrator to use one of the two out-of-
equilibrium pans of the inference rule
Therefore,
submi: in
tile
tine
problem of finding
Mk(y e? y u ,Sa)
specified in the Proposition.
the optimal misleading claim, Sp^. for the
cor.ver.tionai-arbiratior.
game
is
But the solution
to this
second problem
is
to
equivaien: io the problem of finding the
optimal offer, ye for the employer to submit in the final-offer
,
employer
game analyzed
in the text.
given by (31) and yields a payoff identical to the
employer's equilibrium payoff in the conventional-arbitration game. Thus, no deviation
profitable.
is
37
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