Perfect Equilibrium in a Bargaining Model
Author(s): Ariel Rubinstein
Source:
Econometrica,
Vol. 50, No. 1 (Jan., 1982), pp. 97-109
Published by: The Econometric Society
Stable URL: http://www.jstor.org/stable/1912531
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Econometrica, Vol.
50, No.
1
(January, 1982)
PERFECT
EQUILIBRIUM IN
A
BARGAINING
MODEL
BY
ARIEL RUBINSTEIN1
Two
players have to
reach
an
agreement
on the
partition of
a pie
of size 1. Each
has
to
make
in
turn,
a
proposal as to how
it
should be
divided.
After one
player
has made an
offer, the
other must decide
either
to
accept it, or
to reject it and continue
the
bargaining.
Several
properties
which
the
players' preferences
possess
are
assumed. The
Perfect
Equilib-
rium
Partitions
(P.E.P.) are
characterized
in all the
models
satisfying
these
assumptions.
Specially,
it
is
proved
that
when
every player bears
a
fixed
bargaining cost
for
each
period
(cl
and
c2),
then:
(i)
if
cl
< c2
the
only P.E.P.
gives
all the
pie to
1; (ii)
if
cl
> c2
the
only
P.E.P.
gives
to
I
only c2.
In
the case where
each
player
has
a fixed
discounting factor
(83
and 82) the
only P.E.P.
iS
(1
-
82)7(1
-
8182)-
1. INTRODUCTION
WHEN
I
REFER
IN
THIS PAPER
to the
Bargaining Problem I
mean
the
following
situation
and
question:
Two
individuals have
before
them
several
possible
contractual
agreements.
Both have
interests in
reaching
agreement
but their
interests are not
entirely
identical.
What "will be"
the
agreed
contract,
assuming
that
both
parties
behave
rationally?
I
begin
with
this clarification
because I
would like
to
prevent
the common
confusion of the
above problem
with two other
problems that
may be
asked
about the
bargaining
situation,
namely: (i)
the
positive
question
-
what is the
agreement reached in
practice; (ii)
the
normative
question
-
what is
the
just
agreement.
Edgeworth
[4]
presented this
problem
one hundred
years
ago,
considering
it
the most
fundamental
problem
in
Economics. Since then
it seems
to
have
been
the
source
of
considerable
frustration for
Economic theorists.
Economists often
talk in
the
following
vein
(beginning
of
Cross
[3]):
"Economists
traditionally
have had
very
little to
say
about
pure
bargaining
situations
in
which the outcome is
clearly
dependent
upon interactions
among
only
a few
individuals"
(p.
67).
The
"very
little"
referred
to above is
that the
agreed
contract
is
individual-
rational
and
is
Pareto
otpimal;
i.e. it
is
no worse
than
disagreement, and
there
is
no
agreement
which
both would
prefer.
However,
which
of the
(usually
numer-
ous)
contracts
satisfying
these
conditions will
be
agreed?
Economists tend to
answer
vaguely by
saying
that this
depends
on
the
"bargaining
ability"
of the
parties.
'This
research
was
supported by
the
U.K. Social Sciences Research Council
in
connection
with
the
project:
"Incentives,
Consumer
Uncertainty,
and
Public
Policy",
and
by
Rothschild Foundation.
It
was undertaken
while
I was a
research
fellow
at
Nuffield
College,
Oxford.
I
would like
to
thank J.
Mirrlees and Y. Shiloni for their
helpful
comments.
I owe
special
thanks to Ken Binmore
for his
encouragement
and remarks.
97
98 ARIEL
RUBINSTEIN
Many attempts
have been made
in order to
get
to a clear cut answer to
the
bargaining problem.
Two
approaches
may
be
distinguished
in
the
published
literature. The first
is
the strategic
approach.
The
players' negotiating
maneuvers
are moves
in a
noncooperative game
and the
rationality assumption
is
expressed
by investigation
of the Nash
equilibria.
The second
approach
is
the axiomatic
method.
"One states as axioms several properties
that it
would seem natural for the solution to
have
and then one discovers
that the axioms
actually
determine
the solution
uniquely"
[11,
p.
129].
(For
a
survey
of the axiomatic
models of
bargaining,
see Roth
[13].)
The
purpose
of
this approach
is
to
bypass
the difficulties inherent in the
strategic
approach.
We make
assumptions
about the solution without
specifying
the
bargaining
process
itself. Notice that
in order to be relevant
to
our
problem, these
axioms
may only either
restrict the domain
of the solution or
be obtained
from
the
assumption
of
rationality. Thus,
for
example,
Nash's
symmetry
axiom can
be
considered
as an
assumption
that
all the differences
between
the
players
can be
expressed
in
the set
of
utility pairs
arising
from the
possible
contracts and that
there is
no
other
relevant element that
distinguishes
between
them.
But,
the
key
axiom
in
most axiomatizations
-
the
"Independence
of Irrelevant Alternatives"
has not
received
a
proper
defense and in fact
it
is more
suited
to the
normative
question (see
Luce and
Raiffa
[9]
and Binmore
[2]).
It
was Nash himself
who
felt the
need to
complement
the axiomatic approach
(see [10]) with
a
non-cooperative
game. (For
a
wider discussion, see Binmore
[2].)
In his
second paper
on
the solution
that he proposed [11], Nash proved
that the
solution is
the limit of a
sequence
of
equilibria
of
bargaining games.
These
models, however,
are
highly stylized
and
artificial.
Among
the later
works,
I
mention
here
three,
wherein the
bargaining
is
represented by
a
multi-stage
game.
Stahl
[19,20]
and Krelle
[7]
assume the existence of
a
known
finite number
of
bargaining periods
and
their
solutions are based
on
dynamic programming.
Rice
[12]
uses the notion
of a
differential game. The bargaining period is
identified
with
an
interval, equilibrium
strategies
are
the limits of
"step-wise"
strategies,
and
the
lengths
of
those
steps
tend
to
zero.
In
this
paper
I will
adopt
the
strategic approach.
I
will consider
the
following
bargaining
situation: two
players
have
to reach an
agreement
on
the
partition
of
a
pie
of size
1. Each has to make
in
turn,
a
proposal
as
to how it
should be
divided. After
one
party
has
made
such
an
offer,
the other must decide either to
accept
it or to
reject
it and continue with
the
bargaining.
The
players'
preference
relations
are defined
on the set of ordered
pairs
of the
type (x,
t) (where
0
'
x
'-1
and t is a
nonnegative
integer).
The
pair (x, t)
is
interpreted
as
"1
receives
x
and
2
receives
1
-
x at time
t."
This
paper
is limited to the
investigation
of
a
family
of
models
in which the
preferences satisfy:
(A- 1) 'pie'
is
desirable,
PERFECT EQUILIBRIUM
99
(A-2) 'time' is valuable,
(A-3) continuity,
(A-4) stationarity (the preference
of (x, t) over (y, t + 1) is
independent of t),
(A-5) the larger the portion
the
more
'compensation' a player
needs for a delay
of one period
to be immaterial to him.
The
two elements
in which the parties may differ are the
negotiating order
(who
has "first
turn")
and the
preferences.
Two sub-families
of models
to which I will
refer,
are:
(i) Fixed bargaining cost:
i's preference
is derived from the function y -
ci
t, i.e.
every player
bears a fixed
cost for
each
period.
(ii) Fixed discounting factor:
i's preference is derived from
the
functioin
y *
6,',
i.e.
every player has
a
fixed
discounting factor.
So
my
first
step
has been
to
restrict the
bargaining
situation to be considered.
Secondly,
I
will
give
a severe interpretation
to the
rationality
requirement by
investigating perfect equilibria
(see Selten [17, 18]).
A
perfect
equilibrium is one
where not only the strategies
chosen at the beginning of
the game form an
equilibrium,
but also the
strategies planned
after all
possible
histories (in every
subgame).
Quite surprisingly2
this leads to
the isolation
of a
single solution
for most of the
cases examined
here.
For
example,
in the
fixed bargaining
cost model, it turns
out that
if
cl > c2, 1
receives
c2 only.
If
cl < c2, 1
receives all the
pie.
If
cl
=
C2,
any partition
of the
pie
from which
1
receives at least
cl
is a
perfect equilibrium
partition (P.E.P.).
In other
words,
a weaker
player gets
almost 'nothing';
he can
at most
get
the loss
which his
opponent
incurs
during
one
bargaining
round.
In
the fixed
discounting
factor model there
is
one
P.E.P.,
1
obtaining
(1-
82)(1
-
8182).
This solution is
continuous,
monotonic in the
discounting factors,
and
gives
relative
advantage
to
the
player
who starts the
bargaining.
The
work
closest to that
appearing here,
is that of
Ingolf
Stahl3
[19,20].
He
investigates
a similar
bargaining
situation but
which has a finite and known
negotiating
time
horizon,
and in which the
pie
can be
only partitioned
discretely.
Stahl studies
cases for which there exists a
single
P.E.P.
which is
independent
of
who
has
the first move.
The discussed
bargaining
model may be modified in numerous
ways, many
being only
technical
modifications.
However I would like
to
point
out one
type
of modification
which I
believe to be
extremely interesting.
A
critical
assumption
in
the model
is
that each
player
has
complete
information about the
preference
of
the other.
Assume on the other hand that
1
and
2
both
know
that
1
has a fixed
bargaining
cost.
They
both
know
that
2
has also a fixed
bargaining
cost,
but
only
2Especially
considering
that the
perfect
equilibrium
concept
has been "disappointing"
when
applied to the supergames,
see Aumann
and Shapley
[1],
Kurz
[8],
and Rubinstein [14,15,16].
3I
would like to
thank Professor
R. Selten for referring
me to Stahl's work,
after reading the first
version of this paper.
100
ARIEL
RUBINSTEIN
2
knows
its
actual
value. In
such
a
situation some
new
aspects
appear.
1
will
try
to conclude
from
2's behavior what
the true
bargaining
cost
is,
and 2
may try
to
cheat
1
by
leading
him
to believe
that
he, 2,
is
"stronger"
than he
actually
is.
In
such
a situation
one can
expect
that the
bargaining
will continue for more
than
one
period.
I
hope
to
deal with this
situation in another
paper.
2.
THE
BARGAINING MODEL
Two
players,
1
and
2,
are
bargaining
on
the
partition
of
a
pie.
The
pie
will
be
partitioned
only
after
the
players
reach an
agreement. Each
player,
in
turn offers
a
partition
and his
opponent
may agree
to the
offer "Y"
or reject it
"N".
Acceptance of the offer ends the
bargaining. After
rejection,
the
rejecting player
then
has
to make a counter offer
and
so on.
There
are no
rules which
bind the
players
to
any
previous
offers
they
have
made.
Formally, let
S
=
[0,
1].
A
partition
of
the pie
is
identified with a
number s in
the
unit interval
by interpreting
s as the
proportion
of
the
pie
that
1
receives.
Let
si
be
the
portion
of
the
pie
that
player
i receives in the
partition
s: that is s1
=
s
and
s2
=
1
-
S.
Let F be the set of all
sequences
of functions
f
=
{ft
}
t I
where
f1
E
S,
for t
oddft: S'-t
S,
and for t
evenft: St'>{Y,N}.
(St
is
the
set
of all
sequences
of
length
t of elements in
S.)
F
is
the set
of
all
strategies
of the
player
who
starts
the
bargaining.
Similarly
let G
be the set of
all
strategies
of
the
player
who
in
the
first move
has
to respond to
the other player's
offer; that is,
G is the set of
all
sequences
of func4lions
g
= { gt
such
that,
for t odd
gt: S'--
{
Y,
N
}
and
for
t even
gt: St-'->
S.
The
following concepts
are
easily defined
rigorously.
Let u(f, g) be
the
sequence
of
offers in which
1
starts the
bargaining
and
adopts f
E
F, and
2
adopts g
E
G.
Let
T(f, g) be
the
length
of
a(f, g) (may be
xo).
Let D(f, g)
be
the last
element of
a(f, g) (if
there is
such an
element).
D(f, g) is called
the
partition
induced
by (f, g).
The
outcome function of
the game
is defined by
(D(f, g),
T(f, g)), T(f,
g)
<
oo,
P(f, g)=
(0,
x),
T(f,
g) =
xc.
Thus,
the
outcome
(s, t)
is
interpreted
as
the
reaching
of
agreement s
in
period
t,
and the
symbol
(0, oo)
indicates a
perpetual
disagreement.
For the
analysis
of the
game
we will
have to consider
the
case in which
the
order of
bargaining
is
revised and
player
2
is the
first to
move.
In
this
case a
strategy
for
player
2
is
an
element
of
F and a
strategy
for
player
1
is an
element
of
G.
Let
us define
a(g, f),
T(g, f), D(g, f)
and
P(g, f)
similarly
to the above
for
the case where
player
2
starts the
bargaining
and
adopts f
E
F
and
player
1
adopts g
E
G.
The
last
component
of
the model is the
preference
of
the
players
on the
set
of
outcomes.
I
assume
that
player
i has a
preference relation
(complete,
reflexive,
and
transitive)
z,
on the set of
S
x
N U
{(0,
xc)),
where N
is the set
of
natural
numbers.
