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DEWEY
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Massachusetts Institute of Technology
Department of Economics
Working Paper Series
SEQUENTIAL BARGAINING WITHOUT
A COMAAON PRIOR
ON THE RECOGNITION PROCESS
Muhamet
Yildiz,
MIT
Working Paper 01 -43
June 2001
Room
E52-251
50 Memorial Drive
Cambridge, MA 02142
This paper can be downloaded witiiout cliarge from tlie
Social Science Research Network Paper Collection at
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id=xxxxx
^
Sequential Bargaining witliout a
Common
Prior on the
Recognition Process
Muhamet
Yildiz^
MIT
June
^This paper
my advisor
Feinberg,
Eric
Yale.
based on
my
dissertation,
for his
Steen,
I
in
and
submitted to Stanford Graduate School of Business.
especially
like to
I
also
I
thank
thank Chaya Bhuvaneswar, Yossi
Kreps, Paul Milgrom, Piero La Mura, Steve Tadehs,
Daron Acemoglu and Glenn
seminars at BU, Chicago, Harvard,
would
2001
guidance and continuous help.
Thomas HeUmann, John Kennan, David
Van den
was presented
and
is
Robert Wilson
2,
Ellison for helpful
MEDS, MIT,
comments. This paper
Princeton, Rochester, Stanford,
USC,
thank the members of these seminars, especially Drew Fudenberg, Bentley
MacLeod, and Roger Myerson,
for their inspiring
comments.
Abstract
We
analyze a sequential bargaining model, where players are allowed to hold different beliefs
about which player
will
make an
offer
and when. Excessive optimism about making
the future can cause a delay in agreement. Despite this, the main result states that,
will
remain
sufficiently optimistic for
agree immediately.
offers in
if
players
a sufficiently long future, then in equilibrium they
will
Introduction
1
Considering a
strict
procedure that determines which player
(1972) and Rubinstein (1982) show that, when the delay
will reach
players
will
make an
is costly, in
an agreement immediately. Nevertheless, when there are no such
may
and when, Stahl
offer
equilibrium, two players
strict
procedures,
hold any beliefs about the negotiation process, thereby holding any beliefs about
what each player
about what he
will get in case of
will get in
In this paper,
we
a delay. In particular, each player
case of a delay that they
may
may be
so optimistic
not reach an agreement at the beginning.
analyze the problem of reaching an agreement in a model that
will
extends the Rubinstein-Stahl framework, where we will allow players to hold their own (possibly
optimistic) beliefs about the recognition process determining which player will
when.
We
take the lack of a
common
make an
offer
prior as the only source of the differences in beliefs.'
In our model, the recognition process
the only source of bargaining power. In equilibrium,
is
the recognized player at a given date extracts a non-informational rent, as he makes an offer
that can be rejected only by destroying
power: a player's continuation value
when he
is
is
some
of the pie. These rents constitute the bargaining
the present value of the rents he expects to extract
recognized in the future. Therefore, our analysis will help us to explore the broader
question of when two rational individuals can reach an agreement even
beliefs
about which parties
will bring
which advantages to the table
In our model, we have optimism about a date
the other player assesses.
pI
is
We
the probability player
i
measure the
level of
Excessive optimism can cause a delay.
To
players are trying to divide a dollar, which
afterwards.
will
It is also
common knowledge
be recognized (and hence
Since the dollar
is
will
worth
is
=
only
have only one
if
make an
worth zero afterwards, at
each gets at least
dollar,
common knowledge
beUefs, as
by lack of
i
=
common knowledge
plausibility of the
1,
is
=
hy
0, 8
E
t
yt
is
=
higher than what
+ p^ —
pj
will
is
6.
i
where
t.
where two risk-neutral
(1/2, 1) at
i
=
willing to accept
Anticipating
worth
1,
and zero
1,
that he
1
independent of recognition at
1
minimum
beginning that players
(as in
—
each player
dollar.
requiring a
common knowledge and
t
recognized at
1 at t
offer) at i
this, at
t
=
t
=
0.
any division,
0,
each player
Thus, they can agree on a division
total
=
0.
amount
of 26
>
1.
Since they
Therefore, in equilibrium,
not reach an agreement before
i
=
it is
1.
the common-prior assumption holds, then they must hold the
shown by Aumann (1976). Hence, the
prior (as in Hicks (1932),
i is
they cannot reach an agreement at
at the
'If players' beliefs are
same
6,
offer) at
that each player beheves with probability
expects to take the whole dollar next day, which
f
they keep bargaining.
see this, consider the case
hence the recognized player takes the whole
at
make an
optimism about
assigns to the event that
if
they hold incompatible
whenever each player thinks that the
t
probability that he will be recognized (and hence will
if
differences
between players' behefs may be caused either
Myerson (1979) and Kennan and Wilson, 1993),
or
by lack of a
common
Landes (1971), Gould (1973), Posner (1974), and Farber and Bazerman, 1989). For
common
prior assumption, see
Kreps (1990),
Aumann
(1998) and Gul (1998).
main
In contrast with this Example, our
if
result
is
an Agreement Theorem.
It
states that,
the players will remain sufficiently optimistic for a sufficiently long while, in equilibrium,
they will reach an agreement immediately. That
the excessive optimism alone cannot be a
is,
The reason
is
simple:
future will not settle for a small share, even
if
his
reason for a delay in agreement.
less if
when
they wait. Hence,
A
player
is
optimistic about the
opponent beheves that he would get even
opponent recognizes that he
his
who
will
remain very optimistic for
a very long while, she will lower her expectations. In equilibrium, their expectations will be so
low that they
will
agree immediately.
These two seemingly conflicting
players will delay the agreement
the near future.
Particularly,
if
But the
if
size of this rent
As
intuition.
in our
depends upon players' expectations
about the rent at
their expectations
common
results share a
Example,
they are very optimistic about getting a very high rent in
t
+1
are high, the rent at
t
the future.
in
will
be small.
If
the players will remain sufficiently optimistic for a sufficiently long while, in equilibrium, the
rents in the near future will be so small that each player will prefer to agree
on
opponents
his
terms rather than waiting and getting these rents. Hence our Agreement Theorem.
As our Example
indicates,
some
are excessively optimistic for
+
until t*. Here, t*
Agreement Theorem that the optimism stays
this example, in Subsection 3.4
t* in
we show
that,
players
if
the near future, while they are excessively pessimistic
very high while yt'+i
1 (i.e., ii yt' is
from trade
crucial for our
Generahzing
high for a long while.
for t*
it is
is
very low), then the agreement
may be
delayed
need not be very small: the delay can be so long that almost half the gains
is lost.^
In order to cause such a long delay, the change in the level of optimism must also be
abrupt. For the special case of transferrable utility and independently-distributed recognition
process.
Theorem
2 states that they will reach
an agreement immediately, so long as the
optimism does not drop substantially at any given date
low at each
t)
Theorem
,
relaxing the hypothesis of our Agreement
2 identifies the
Stahl framework.
optimism
These
is
It is
results also
make any
offer.
by Subsection
other hand.
This
is
Theorem
—
yt+i
is
sufficiently
in this case.
assumption behind the immediate agreement
result in the Rubinstein-
that the level of optimism stays constant - not necessarily that the
shed some
al,
light
1988) and in
settle just before the deadline.
t*,
so long as yt
absent.
experiments (Roth et
to
(i.e.,
level of
Hence,
if
Above,
on the deadline
some real-world
t*
may be a
the theoretical
negotiations, where players
commonly
hope
the players are very optimistic about making the final offer at
the deadline, as
2 suggests a further test for our theory:
bound
observed in some bargaining
deadline, after which players do not
3.4, players will settle just before
Theorem
effect
of the length of deterministic delays.
agreement depends upon the history,
in
when
When
it
is
observed.
the deadhne
is
On
the
random
people learn from the past, the
which case the length of the delay can be unbounded (Example
1).
and players can make
By
must disappear.
offers very fast, the deadline effect
That
implicitly requires that players' beliefs are sufficiently firm.
beliefs drastically as they observe
who
such change of beliefs "learning.")
may
player, say j,
believe that,
if
If
be higher than what
would be
it
gets a chance to
a player
i
is
make an
known
they do not update their
is,
and when. (We
offer
will realize that
i
he
is
expectation about the future
j's terms. In that case, j's
if z's
will call
to update his beliefs fast, the other
they disagree now, in a short while
wrong, and thus be willing to agree on
will
Agreement Theorem
requiring the level of optimism to stay high for a long while, our
behefs were very firm. But
does not find
i
it
very
hkely that he will change his mind; he might even expect that j will change her mind. This
may
cause a disagreement.
Furthermore, in a typical learning process, agents learn faster at the beginning, in which
case the disagreement will be at the beginning of the game, and hence
we
will
observe a delay in
agreement on the path of equilibrium. Considering a canonical model of learning and confining
ourself to the transferrable-utility case,
some date
such that players
t**
reach an agreement thereafter.
not firm,
t**
we show
for the infinite-horizon case that there exists
will
never reach an agreement before
We
further demonstrate that,
when
can be large, generating long delays in equihbrium.
the beginning of the
game
that they will disagree until
t**,
and they
will
always
players' initial beliefs are
It is
common knowledge
at
distinguishing our theorem from
t**,
usual delay results in bargaining with private information.
The
outline of the paper
we analyze the
is
as follows. In the next section,
special case of transferrable utility
we
lay out or model. In Section 3,
and independently-distributed recognition
There, we use only simple algebra to derive our main
process.
equilibria in Section 4,
and present our Agreement Theorem
in
We
results.
Section
5.
describe the
Section 6 contains
the analysis of our canonical model of learning. After discussing the impact of risk-aversion
on reaching an agreement
in Section 7,
we conclude
in Section
8.
Model
2
In this section
N
space,
We
we
will lay out
We
will write K*^ for
T=
{t
e N\t <t} tohe our time space
be our set of players, and a compact and convex set
utihty pairs.
that
is
for
some
U cM?
U contains
than
0.
We
will further
<
i
and
=
{1, 2} to
feasible
expected
oo, a set
to be the set of
Throughout the paper, we assume that
strictly greater
any /c-dimensional Euclidean
non-negative integers.
for the set of
take a grid
our model.
all
A'^
at least another
member
assume the following the following regularity
condition.
Assumption
exists
some
v^
FDA
>
u^
Given any (u^,u^) G U, any distinct
such that (w^u^) G U.
i,j
€
A'^,
and any
v'
G
[0,tx^),
there
.
That
U
the frontier of
is,
decreasing on the non-negative quadrant.
is
This condition
is
satisfied under the free-disposal assumption with locally insatiable preferences.
We
will
a player
i
analyze the following game, denoted by G^[5,p]. At each
£ N;
offers
i
=
an alternative u
(i/\u^) G U;
then the game ends yielding a payoff vector
the
game proceeds
vector
(0, 0).
We
to date
write p
=
t
+
t
= {6^u^,S^u'^)
= i — 1, when
for
some
the
game ends
6
S
(0, 1);
offer,
otherwise,
yielding payoff
{pt}t^T ^°^ ^^^ publicly observed stochastic process that recognizes
the players (so that they can
make
=
offers), Pj
(/Oq(w),Pi(cj),.
history of the recognized players at the beginning of date
conditional beliefs at any history p^
6
we have two
t,
.
.
,pj_j(w)) 6
and P^
A''*
i'lPs) for
for a generic
an agent
i's
Players recall everything happened in the past; and
TV*.
everything described in this paragraph
Notice that
recognizes
the other player accepts the
8''u
except for
1,
if
eT, Nature
t
common
is
knowledge.
one for each player;
sets of beliefs,
this
is
our only departure from
the Rubinstein-Stahl framework.^
Given any
e
i
N
and any history p^ E N^, we write
=
pliPs)
for the probability player
By
a belief structure,
We
will also
that no player
Pt (Ps)
he
1-
E
t
is
Now,
[6,p] for
iff
pI (Ps)
pI (Ps)
+P?
(Ps)
-|-
<
APs)
A''^
going to be recognized at date
with
s
<
t>s.
of such probability assessments.
{pj (Ps)}
game where
the
G
is
the belief structure
t.
Since
is p.
we excluded the contingency
+
recognized, players' probability assessments agree at {t,Ps) iS p] [p^)
if
p\ (ps)
+ Pf
(ps)
t is
Pf (ps)
We
1-
>
>
1,
is
not bad; so
Likewise,
1.
then each player thinks that the probabihty that
higher than what the other player assesses.
in the Introduction, being recognized
at Ps
=
p
full list
T and any history p^
going to be recognized at
is
=
assigns to the event that he
we mean any
sometimes write G*
Take any date
—
i
P'{Pt
we say
As explained
that players are optimistic for
we say that players
are pessimistic for
t
at p^
t
iff
write
yt(pJ=Pf (Ps)+p?(pJ-i
for the level of
optimism
for
t.
Finally,
common-prior assumption
in our context corresponds
to
yt
When CPA
holds, by No-trade
incentive to bet on p.
With independently
Since
we
distributed
(1993), originally developed by Perry
when
to
make an
offer.
=
(Vf
Theorem
of
(CPA)
e T)
Milgrom and Stokey
(1982), players have
where
CPA
fails (i.e.,
also isomorphic to an extension of the
model
of Sakovics
are particularly interested in the case
p,
our model
is
no
and Reny (1993), a bargaining model where players endogenously choose
.
y
7^
0),
our players do have an incentive to bet on
so merely by our
p;
and they are precluded from doing
imbedded assumption that the side-bets are not
feasible (e.g., they are not
enforceable).'^
Notation
We
with
and a
i
j^ j,
belief structure
defined on a large space
whose values
at
any
T
and Pg are
s
we
In this section
G
histories p,
and
pi (p^)
An
each
for
t
Q.T, p] and
Assumption
TU U =
Assumption
NL
are stochastic processes,
Intuitive Exposition
p
is
{u e
utility
[0,
l]^\u^
two histories p^ and
p'^,
game
G^[6,p\ under the following
and no learning from previous recognitions.
+ u'^ <
1}
independently distributed under both probability distributions P^ and
Note that, according to Assumption NL, p
with s,s'
<
t.
Under these two assumptions, we
ment Theorem, stating that players
will present
yt
confine ourself to the simple bargaining problem of dividing a dollar, and
two assumptions of transferrable
sufficiently long
and y >
0.
Hence,
is
pi
deterministic,
G
[0, 1]
and
yt
i.e.,
pi (p^)
e [—1,
will present
any
pi (p'^,) at
1].
will first construct the equilibrium,
will reach
=
and prove the Agree-
an agreement immediately, so long as the game
Then, allowing y to take both positive and negative values, we
an example with a long delay, and discuss the general properties of delay
model. Finally, we
N
yt (Ps), respectively.
analyze the subgame-perfect equilibria of a fimte-horizon
is
p^ e TV*, players i,j,k 6
A''*,
Note that p and y are functions
p as generic members.
x {Us^t^^) and,
Dividing a Dollar —
3
£T,
will designate dates t,s
our
in
another version of Agreement Theorem that requires only that
y does not drop suddenly.
Continuation values in Equilibrium
3.1
For any given finite-horizon game G*[5,p], we
any subgame perfect equilibrium
Let us write V^
St
=
Vf^
"Even
-I-
if
V^
is
first
payoff equivalent to the one
for the continuation value of player
for the "perceived size of the pie." If
the side-bets on p were feasible, agents could
the other hand,
if
construct a subgame-perfect equilibrium;
players do and that yields p( measurable with respect to
immediate-agreement theorem similar to the one
p by some contracts on B.
we
at the
construct.
beginning of date
and
t,
they have not reached an agreement before
make a
there exists another contractible process
i
B
side-bet on Pj void by agreeing prior to
t.
On
that remains publicly observable regardless what
a (Bo.
in Yildiz (1998).
,
Bt), then
we would simply have a
general
For they can simply emulate the side-bets on
.
each player
t,
we
will get 0; hence,
set
= V^ =
V^'
Taking
t
V
we determine
this as final values,
Si
and
=
0.
(1)
We
e T. There are two cases we need to address separately.
If
they do not agree at
will
then agree on a
+u^ >
that u^
will
each
t,
5V'(l|_j
t
=
u
utility pair
+ <5Vj^j =
not reach an agreement at
of date
will get V^\i at
i
[u^ ,u^^ only
6St+i
>
Note that
their beliefs
Now we
=
=
SV^_^_^
at each
is
i
1.
They
t.
E N, which requires
not feasible. That
i
is,
they
at the beginning
the recognized player
<
6St+i
{6St+i
t is
Any
1.
value of player
i
~ ^^+1
*'^'^^ ^
^° himself;
at the beginning of date
V;
=
=
+
1
adding equation
(4)
up
—
for players,
St
Briefly, the processes
V^,
=
l
V'^,
perfect equilibrium. Furthermore,
player j accepts an offer
Note that,
with
is left
iff it
gives
him
at least
1 — SVi^^, which
Thus, in equilibrium, player
i
offer is accepted. Therefore,
is
now
offers to give
continuation
z
+ (l-pD'5VtVi
+ 6V^^,
{6St+i<l).
at
i is
(4)
the present value ^V^+j of his continuation
when he makes an
offer.
By
we obtain
+ yt- SytSt+i
{6St+i
<
!)•
and 5 defined by the equations
V^ and
S
V"^ are
(5)
(1-5) describes a
subgame-
the only continuation values consistent with
prescribes whether they agree at any subgame-perfect
in equilibrium,
aggregate data y, 6 (and 5).
he
SSt+i) he expects to extract
any subgame-perfect equilibrium; and
equilibrium.
(3)
t is:
pi{l-6St+i)
plus the rent pl{l
1)
immaterial to the problem.
and the
pj(l-'5^4i)
the continuation value of player
i
>
offers to give SV^^-^ to j,
i
(2)
N, we obtain
about recognition at
consider the case 6St+i
*° j ^^"^ *°
value at
>
equivalent to ^V/^j at
{6St+i>l).
6V^^,
at least as high as his continuation value SV^^^.
is,
u^
is
>
consider the case SSt+i
first
showing that such u
1,
sides of the equation over
St
That
if
which
1,
us take any
will be:
Adding up both
^^+1
+
this, let
Therefore, the continuation value of each
t.
V;
SV^_^_^. If
t
To do
5, recursively.
agreement at any date
is
fully
determined by the
Agreement and Disagreement Regimes
3.2
Our equilibrium has two regimes. The
>
terized by inequality St+i
back
t is
irrelevant;
at
t,
we
one, which
In this regime,
1/5.
players do not reach an agreement
recognized at
first
call
disagreement regime,
is
charac-
they have not yet reached an agreement,
if
Hence, their beliefs about which player will be
either.
and the perceived
size of the "pie" shrinks geometrically as
we go
in time.
If i is
the
date they reach an agreement after a period of disagreement regimes,^ the
first
perceived size of the pie at any
be
in that period will
t
S^
Hence, the length of such a period
will
=
6'-'S:.
(6)
be
log Si
L{Si,6)
(7)
log(l/<5)
where operator
[•]
finds the smallest integer that
Since S^ can be at most
2,
is
greater than or equal to the argument
the length of a disagreement period can be at most
log 2
1(6)
(8)
log(l/<5)
Hence, given any
6,
the lengths of disagreement periods are bounded uniformly. In efficiency
1/2
<
and characterized by inequality St+i
<
terms, the length of delay can be at most as large as almost half the pie
The second regime
1/6. In this regime,
if
is
called the agreement regime,
is
i.e.,
they have not reached an agreement yet, players immediately agree on
a division that gives the recognized player (say
this
is lost,
what he would get
if
i)
I
—
^^t+i' which
is
the other player j were recognized. That
higher than
is,
6V^_^_.^,
and
the recognized player
extracts a rent
Rt
The discrepancy among
= l-
<5V/+i
-
players' beliefs on
crepancy between the perceived
<5V;Vi
=
1
-
6St+i
which player
size St of the pie
and
its
.
will get this rent causes the dis-
actual size of
1.
In fact,
we can
re-write equation (5) as
St
so that the discrepancy
is
5t
=
l
1
=
—
aligned with excessive optimism
+ yt{l- SSt+i) = 1 + ytRt
ytRt-
may
Since they agree at
t
—
1,
there
is
(9)
Therefore, a high rent for the recognized player
prevent players from reaching an agreement in some
previous dates.
^
'
such a date
for each
disagreement period.
at
If yt
<
-
as well. Thus, iiyt
i
1,
and we have an agreement regime at
<0
at each
t
£T,
t,
then by
then we
(9),
we have an agreement regime
have agreement regimes throughout
will
the game. In fact, this case can be imbedded in the Rubinstein-Stahl framework by recognizing
each player
with probability
i
now analyze the
will
pi,
>
case of y
and by recognizing nobody with probability
— pj —Pt- We
1
0.
Perpetual Optimism
3.3
Confining ourselves to long but
finite
games, in this section, we will show that players
reach an agreement immediately, so long as players are optimistic throughout the game,
yt>0
We start with the following Lemma,
the pie at
Lemma
<
1
+
1 is
will
i.e.,
(PO)
Vi G T.
< t whenever
giving us an agreement regime at each date s
perceived to be of the "right size."
Assume
TU
and NL. Given any
t
with
>
yt
0,
6
St+i
if
then St G
[l,l/i5],
[l,2-,5]c[l,l/.5].
Proof. Assume that St+i G
by
(9),
we have
above. Since
2
—
<
6
yt
St
€
=
I
[0, 1],
+
[1,1/^].
Then, we have an agreement regime at
ytRt G
[0, 1
—
6]
—
and therefore
St
= 1).
(PO), Lemma
long games. To see this,
first
1
< {1 — S) /6, we have 5f_i = 1 + yi-i <
Lemma 1, we can conclude via mathematical
Now we
consider the case
when
yi^i
and hence a disagreement regime
game
i
is
>
>
at i
{1
—
—
t
ytRt G [1,2
—
Note that
6].
= i — 2—
L{St-i,6).
By
5(_,_i
=
6Si_^2
>
three periods of constant regimes:
t
we have
1' i-^-' Si_^i
<t. To sum up,
We
< L (6)
disagreement regimes at {i— L{St-i,6)
exhibited
In either case,
proving our
first
ending at
1,
.
.
.
,
When
In that case, using
t
<
t
—
i
—
2.
1,
Now, assuming
with an agreement regime before i—
<
S^_^_i
G
[1,
[1,
1/6 and
1/6].
t
>
{0,
.
>
The
1/6.
again, using
1/6] at each
in this case, for
i — 2}, and
5j_,_2
Once
have agreement regimes at
—
1/^].
1-
we have 5f_i = 1 + yi-i >
we have already shown that there is a
we can conclude via mathematical induction that St+i G
have an agreement regime at each
[1,
+yi-i >
1
6) /6. In this case,
In fact,
2.
definition,
=
hence 5f_i
0,
&T.
sufficiently long, consider the last date
inequahty also implies that
is
+
bounded from
is
6]
induction that 5^ G [1,1/5] at each
period of disagreement regimes with length L{St-i,6)
(This
I
1/6 so that Si-i G
showing that we have an agreement regime at each
namely
=
—
[0, 1
gives us immediate agreement for sufficiently
note that, by (PO), yj_i
j/t_i
that the
6St+i €
and hence
1/6 (with equality only at 5
Under perpetual optimism
1/6,
=
I
ytRt, where the rent Rt
t,
t
2,
latter
Lemma
1,
<t, showing that we
L{St^i,6)
+ 2, we have
—2—
L{St-i,6)},
.
.
,i
again an agreement regime at
t
—
1.
in Figure 1 for the extreme case when yt = pj = p^ = 1 at each
£T.)
—
—
2—
2—
<
<
we have an agreement regime at each t
i
L{6)
i
L{Si^i,6),
Agreement Theorem:
t
1
The Perceived Size
1
1
1
of the
Pie
1
1.8
-
1.6
1.4
1.2
1/5
y
/
'
]
-
0.8
0.6
^
Agreemeni negimes
"*
Disagreement'
Regimes
;
0.4
0.2
1
11
10
Figure
t
£T.
1;
[i
The perceived
=
30, 6
=
O.9.]
size of the pie in the
15
1
-H
20
extreme case where
Note that we have an agreement regime at
25
yt
=
i
—1,
30
p}
=
too.
pf
=
1
at each
Theorem 1 Assume
with
< t- L{6) -2.
TU, NL, and PO.
Then, we have an agreement regime at each
t
£
T
t
Excluding the end of the game, our Theorem states that,
if
players are optimistic throughout
Example we have
the game, they will reach an agreement immediately, suggesting that the
may be
given in the Introduction
Example
Actually, our
rather singular.
illustrated that
sequential bargaining allocates a non-negative rent for the recognized player at any date
can be very
this rent
rent,
may
they
large.
If
choose to wait at
Our Theorem goes one
t;
and
players are also extremely optimistic about getting this large
t
—
1.
step beyond this intuition. At such an agreement regime
players
t, if
are also sufficiently optimistic for subsequent times, the rent for the recognized player will be
we estabhshed
small, a fact that
Lemma
in
Our Theorem
1.
states that, in equilibrium, this
rent will be so small that the players will reach an agreement at
optimistic about getting
Theorem
1
even
if
they are very
it.
establishes that, in case of perpetual optimism,
game except
out the
like in
we have agreement regimes through-
a bounded period just before the end. Naturally, as
for
of disagreement regimes disappears,
—
i
oo, this period
>
and thus we have agreement regimes throughout the game
the Rubinstein-Stahl framework.
Take any
£
8
(0, 1)
and
G*, write S[t\
tively.
1
Perpetual Optimism — Infinite Horizon
3.3.1
-
—
i
Given any
solving (5)
on
infinite
behef structure
for the size of the pie
i[t\
t
and any
<
t[i\,
and the
p.
last
For any sufficiently long truncation
date before any disagreement, respec-
we have an agreement regime
this interval,
at each s with
t
<
s
<
t[i\;
hence,
we obtain
t[t]-i
St[t]= 5]7rf(l
where
is
ttj
and since
YVjZ^
<
S^<n
{Syj) when
<
2, St[t\
t
<
s
and
1
+ y,) + 7rf5,-[i],
when
t
=
As
s.
?
—
>
oo,
i[t\
—
>
oo,
hence
tt^
'
—y
0;
converges to
oo
St[<^]=J2nt{l +
(10)
ys).
s=t
By Theorem
1,
St[t\
e
[1,1/5] for every sufficiently large i so that St [oo]
G N. That
is,
for i
=
oo,
f
player
i
we have agreement regimes throughout:
gives the other player, say j, ^Vj-^^Joo]
G
[0,6],
keeping
himself - a behavior that replicates the equihbrium behavior
work, characterized by
CPA. In
fact,
when CPA
10
holds
(i.e.,
y
1
—
at
6
any
t,
5VJ-!^j[oo]
[1,1/5] at each
the recognized
G
[1
—
8,1] for
in the Rubinstein-Stahl frame-
=
0), V^'[oo]
can take any value
in
thus any such division
[0, 1],
Rubinstein-Stahl framework)
To
=
see this, let yi
is
CPA; and
consistent with
therefore
CPA
(and thereby the
not refutable in the case of perpetual optimism.
is
y 6 [0,1] at each
i
Now, by inserting y
G N.
each y^ in (10),
for
- 8St+i [oo] =
one can
j^§^, and hence Rt [oo] = 1
Moreover, by Proposition 2 of Section 4, we have V} [oo] = Y!^=t ^^"Vs-^s [oo]- Thus,
easily
compute that
Since p\ G
^E^t^'"Vs-
therefore the range of V^
[oo] is
offers
6 T, the range
•
is
+
far
holds
5)
E^t ^'"Vs
at y
(i.e.,
=
=
=
and
[y,l],
is
R.
V^ [oo]
0), this interval is
CPA.
=
it
1
(i.e.,
=
pj
+
consists of 1/(1
That
6).
make
believing that he will
pj
leaving the other player 6/(1
6),
-
of (1
when everyone keeps
no matter what happened so
player offers to take 1/(1
When CPA
and at y
increases, the interval shrinks;
remaining
in
is,
all
the
= y = 1 Vi G T), the recognized
+ 6)\ and the offer is accepted.
the equilibrium outcome under the Alternating-Offer Procedure, analyzed by Rubinstein
is
Note that Rubinstein's well-known division
(1982).
by any
Long Delay
Theorem
mistic.
1
Now, allowing
this
in
is
we
players be pessimistic sometimes,
will create
example
is
simple.
When
1/2^
i.e.,
opti-
a simple example where
game
delayed for a long while, no matter how long the
have an upper bound
<
not ruled out
games where players remain
The
is.
rationale
each player believes that he will not be able to make an
will act as
if
the negotiation stops there, imitating a short game.
Since the length of a period of disagreement regimes cannot be longer than
^^( )+i
is
Agreement
a given date, they
offer after
the only division that
establishes immediate agreement for the long
the agreement
behind
is
optimism.
level of
A
3.4
j^
jf^-, yij^
consistent with
the equilibrium of the extreme case
This
=
at each 5
[y, 1]
any outcome
entire [0,1], thus
As y
St [oo]
for the delay in
L [6), we already
agreement. Note that this upper bound
almost half the benefit
Nevertheless,
is lost.
it is
is
rather long:
not loose, as our example
reveals:
Consider any belief structure p with
any date with
<
i*
< min{L
At
{8) ,t}.
have any other chance to make an
yt-
offer,
t*
=
1
and
and
recognized player can extract a huge rent: we have
extreme pessimism
for the future, their
= —1
at each
is
any
willing to accept
iilj
=
1
at each
t
>
going to make an offer at
is
optimism
>
t
thereafter, thinking that
each player
an auspicious moment, each thinking that he
yj
for t* leads
them
t*
.
t*.
t*
he
,
where
will
t*
no longer
offer.
Hence, the
They
consider
The
pie at t*
is
=2>
to perceive the pie at t*
1/6.
perceived to be so large that they will never reach an agreement before
St+i
=
6^'-'-^Sr
=26^'-''^ >l/6,
11
(0
<
i
t*
Amplified by the
very large:
Sf = l+yt'Rf
is
<
i*)
t*:
where the
That
is,
when
all
inequahty
last
during {0,
.
.
is
,t*
.
due to the definition of L
—
[6),
which
without agreeing, players wait
1},
defined by 26
is
^
for the auspicious
S
'
(1, 1/6].
moment
t*,
the offers are going to be accepted.
This example illustrates some key properties of delays caused by lack of a
Firstly,
the delay in our
case yi
=
model
Now, giving the
1.
is
To
not Pareto-optimal in general.
dollar to the player recognized at
i
=
expected
utility levels
under these two schemes are 6 and
6*
,
prior.
see this, consider the
1
equilibrium outcome, which gives the dollar to the player recognized at
common
Pareto- dominates the
t*.
Note that a
respectively. [This also
player's
shows that
the equilibrium outcome in the example presented in the Introduction was Pareto-optimal.]
Second, at the beginning of our game
an agreement before
t*;
whereas the delay
common knowledge
it is
is
that players will not reach
only a possibihty in usual bargaining models with
private information, as the types with the least advantageous information reach an agreement
immediately in any separating equilibrium
Third, there
division at
t
is
=
among our
a consensus
that gives each player
agree that the size of the pie at
would get more than
giving 6
+
—
(1
t
=
1 is
their equilibrium
Finally, even
though there
=
i
is
players that the delay
more than what he
only
—
(1
if
gets in equilibrium.
6)/2 to his opponent. There
at the beginning
is
costly,
1, it
would give him
and would be rejected by player
A
at least 5*
reflect
is,
a
they
is
f
=
by
no consensus,
,
one cannot find a mecha-
on a procedure that recognizes
if
such a mechanism were
thus would give player 2 at most
1
—
6*,
2.
period of low y does not necessarily
simply
That
is
and would be accepted by each player to
"objective" probabilities throughout the negotiation. For,
accepted by player
that there
dominates the equilibrium outcome.
replace the current situation. In particular, they cannot agree
them with
and
they divided the dollar at
a consensus that the delay
nism that would yield an agreement
is costly
and each player further thinks that they both
6;
consumption
6)/2 to himself and leaving
however, on which division at
models.
in these
mean
that they are unreasonably pessimistic;
it
may
the fact that agents are not going to negotiate during this period, as happens
in bargaining with a deadline or several "bargaining sessions."
illustrates that, if the
end of the session or the deadline
imtil the deadline to settle. This deadline effect
is
is
In that case, our analysis
sufficiently close, agents will wait
observed in laboratory experiments and the
real- world negotiations.
3.5
A
sudden
loss of
In the previous section,
incentive to wait.
regimes,
it
optimism
we
Now we
illustrated that a
will establish that,
must be followed by a sudden
Towards
sudden
this goal, our first
lemma
loss of
optimism may give players an
whenever there
is
a period of disagreement
optimism.
states that
12
loss of
if
a disagreement regime proceeds an agree-
ment regime, there must be a substantial drop
Lemma
2 Given any G^[S,p] and any
<
t
i
in y.
—
I,
>
St
if
and St+i <
^
then
-j,
y^
—
j/^^j
>
{l-b)/6.
Take any
Proof.
agreement regime at
Since
St+i
1
<
—
>
SSt+i
i
+
t
—
St
=
l
we then have
yt
>
and St+\ <
\
Since St+i
|.
<
|,
we have an
+ yt{l-8St+i)>^.
Since St
0.
we have 5f+2 <
=
rendering St+i
1,
>
with St
1
and hence
In that case,
1-
regime at
t,
0,
<
t
1/S and
1/^, for otherwise
>
8St+2
>
(11)
1-
yt
>
Lemma
by
0,
1,
we have
we would have a disagreement
Hence, we have an agreement regime at
t
+
1,
and thus
= l+yt+i(l-<55,+2)<l,
5t+i
yielding yt+i
<
0.
By combining
1
-^<St =
Since ytyt+iSt+2
<
0, this
and
(11)
+ yt~Syt[l + yt) +
l
+
8ytA > 8ytA
gives us
A = yt
and writing
(12),
+
SytA
(12)
—
yt+i,
we obtain
6^ytyt+iSt+2-
6'^ytyt+\St+2
>
-
^
(13)
I
-
yt
+
6yt{l
+
yt)-
Hence,
Once can check
taking the value of
By Lemma
hand
that the expression on the right
2, if
2w|
(^
—
l)
—
—
(^
l),
which
is
we have a disagreement regime
is
minimized
greater than \/8
at
i
—
1
while
-
at y*
= -j^Jj —
1,
1.
we have an agreement regime
yt — yt+i < {1—6) /6 and we
— ^) /^- Put it differently, if
have an agreement regime at i < i — 1, we must also have an agreement regime at t — 1. We
already know that we have an agreement regime at some i with t — 2 — L{6) < i < t — I.
Hence, if y is smooth enough so that yt — yt+i < (1 — 5) /5 at each i G T, we would have an
agreement regime at each t < i:
at
t,
we must have
Theorem
2
yt
—
side
yt+i
>
(1
Assume TU, NL, and
an agreement regime
at each
t
£T
that yt
with
t
—
yt+\
<i —
Excluding the end of the game, Theorem
1
<
L{6)
—
(1
—
6)
/6 at each
t
£ T. Then, we have
2.
stated that,
if
players are optimistic throughout
the game, they will reach an agreement immediately. Proposition 2 establishes that, actually,
we do not need optimism stay
high;
all
we need
13
is
that
it
does not drop too
fast.
In fact, by
requiring perpetual optimism,
yt
—
<
yt+i
Of
1-
Theorem
bound
course, this
is
1 also
bounded the drop
much
looser than that of
We may
results are logically independent.
the game, precisely because after the
smooth enough, there
is
We
game ends
yt is identically
some prehminary
Theorem
2 also implies that
we
In this section,
we
will
a
the transition to
if
either.
Beforehand, we
5.
results.
will describe the
Ccise
subgame-perfect equilibria of our game.
be payoff equivalent to each other. In the
horizon case, all the equilibria will
case,
then
may have
be no disagreement regime at the end of the game,
Equilibrium and the Rents — General
4
[0,1],
and these two
2;
—1, and thus we
our Agreement Theorem for the general case in Section
will present
present
will
Theorem
6
have a disagreement regime just before the end of
substantial drop in y at the end of the game.
—1
in y: iiyt,yt+i
In the finiteinfinite
horizon
only consider the limit of the equilibria of the finite-horizon truncations.
In
and these
equilibrium, at each date, the recognized player extracts a non- informational rent,
rents determine players' equilibrium-payoffs.
Towards describing the subgame perfect
(7* [6,p],
and write
The
history P(.
V/
[t\
[t\
Vf^
(pj
for the
V
stochastic process
iPt)
=
P'
=
[p,
i\Pt)
equilibria, let us take
any finite-horizon game
equihbrium continuation- value of a player
[t\
is
at any
i
t
with
defined by the recursive equation
m\bVt+i
{p„
[t\
i))
+ P'
=
{p,
j\p^) 6V,\,
[t\
(15)
{p„j)
and the boundary condition
Vi[i\=0,
where m'
each
i;
:
G K^
K^
— R+
for
i 7^ j.
is
= max
defined through m^(u)
Note that
rn}{v)
is
if
i.e.,
when v
U.)
will get the
maximum
=
i|p(), player i
i
will get his continuation value in
fact that players automatically get
The equihbrium
Given any
t
to give ^V/^_j
strategy
with
[F|
pj,
s,
[t]
if i
(P(,t) to j
at
can enjoy
>
if
v^}\ at
he needs
gets m^{v), j gets
i
wiU be recognized,
(/9(
= j|pt),
[i]
vK Hence,
in
(15)
which case he
{Pf,i) to player j in
which
j will be recognized, in
the case of a delay. Condition (16) simply states the
t.
of a player
is
i
G U, u^
too costly to convince the other
payoff given that he needs to give at least 5V(-^j
order to reach agreement, and with probabihty P'
case
payoff player
if it is
By Assumption FDA, when
expresses that, with probability P^ {p^
{u'| (u^,u^)
the payoff vector v would be realized in case of
a disagreement. (Of course, he wiU choose to disagree
player,
U
[{v^}
maximum
the
the consent of player j for an agreement and
(16)
i
[^
j) will
recognized at
and m}{6Vt+\
14
[i]
t,
be as
and
{Pt^'^))
if
to
follows:
6Vt+\^{p^,i) G U,
i;
if i
is
offer
recognized, but
^Vj+i
[t\
^ U, then
{Pf,i)
accept his offer
iff
offer to give
he gives
i
at least SV^^^
Given any infinite-horizon game G°°
and m'
to j
[t\
cation G* [5,p], which agrees with G°° [6,p] until
if
z;
j
is
recognized,
{pt,j).
each
[6,p], for
(0) to
G N, consider the finite-horizon trun-
t
and ends
i,
Defining
there.
V
as above,
[t\
assume that
asf-cx)
Vt[t\{p,)^Vt[^]ip,)
some
for
t
=
oo.
V [oo]. We
The
define the strategy profile s [oo]
when
following Proposition states that,
the unique subgame perfect equilibrium;
when
i
=
(Vp(,Vi)
(sj [oo] ,S2 [oo]) as
is finite, s
<^= oo, s
[oo] is
but not necessarily unique. The proof of this Proposition
be found
[f]
=
(si
Given any G°°
1
any finite-horizon game G* [6,
p]
with (17), s
{6,p\
s
,
is
[i]
any subgame-perfect equilibrium, and
S2
[i]) is
essentially
a subgame-perfect equilibrium,
is fairly
and can
straight- forward,
[oo]
is
at
any date
if
8Vt+\{pi,i)
^ U and
In the rest of the paper,
we
i
is
Given
a subgame-perfect equilibrium.
<
oo, at
unth pj, the following are true: the vector of
t
is Vj (pj); if
6Vt+i{pt,i)
is
in the interior
recognized, then they do not reach
the equilibrium
will confine ourself to
ofU, and
and m^{6Vt+i
recognized, then they reach an agreement that gives 5V^_^_^{p^,i) to j
and
,
a subgame-perfect equilibrium. Moreover, for i
continuation values at the beginning oft
to i;
[t\
above, by taking
in Yildiz (2000).
Proposition
is
(17)
s;
[t\
an agreement
i
{pt,i))
at
t.
we suppress the terms
in square brackets.
In equilibrium, the recognized player extract a rent; and these rents constitutes the bar-
gaining power.
To
see this, observe that, given any pj
when he makes an
his payoff at {Pt,i),
m'(5Vt+i(pt,i))-5ViVi(p„j)
Here, the last term
players' beliefs
equihbrium.
is
=
and
offer,
[vn}{8Vt+^{p„i))
informational; that
is,
first
is
the difference between
when he does
6Vl^,{p„i)\-^[6V^%^{p,,i)
=
t-\- \\
V^ip,)
Pi 7^
respectively.
i,
is
is
t
affects
in turn affect their continuation values in
m\dVt+^{p^,i))
- dVi^M^i),
and we
call
it
(18)
offer that
can be rejected only by
the (non-informational) rent for
Substituting (18) in Equation (15) and carrying out the necessary algebra,
l{p,=i}
not,
6V^\,{p„j)\.
the identity of the recognized player at
caused by his opportunity to make an
delaying the agreement until
where
-
term,
B\{p,)
on the other hand,
i,
his payoff at {Pf,j),
-
about the future recognitions, which
The
and any player
at
t.
we obtain
= E%,-,yRl\p,] + 8E'[V^^,\p,],
the indicator function of {p^
Equation
i
=
i},
taking values
(19) states that the value of the
15
(19)
1
game
and
when
for a player
i
pt
=
at
any date
i
and
t is
his discounted
expected value of the game at
being a recognized player at
i
+1
plus the rent he expects to extract from
game
Since the value of the
t.
at i
is
identically nil, this gives
us the following Proposition, stating that, in equilibrium, the continuation value of a player
is
the present value of
all
the rents he expects to extract in the future. In an agreement, the
recognized player gives the other player his continuation value, keeping the rest of the pie for
himself. Therefore, a player's share in an agreement
he expects to extract
the discounted
is itself
he
in the future, plus the current rent in case
Proposition 2 Given any
i
G
A'^,
t
Q
T
and any
pi,
is
sum
of the rents
the recognized player.
we have
i-l
V^Pt)
= J2^'~'^'
[l{p.=,}^slPt]
(20)
•
s=t
Using
and the law of iterated expectations, one can
(16), (19),
induction that (20) holds for any
(2000) shows that
it
finite
i.
Using the fact that
he expects to extract
that he
is
more
mathematical
R
is
is
the present value of the rents
uniformly bounded, Yildiz
holds for the infinite-horizon case as well.
Proposition 2 states that the continuation value of a player
well as the history,
easily check via
in the future.
it is
Since the rents themselves depend on players' beliefs as
not clear that a player would not lose
likely to
example where a player
make
loses as
offers in the future.
if
each player comes to believe
In fact, in
he finds himself more
likely to
A
Appendix
make an
we present an
offer at
i
=
1,
while
everything else remains unchanged. Nevertheless, there we also show that a player will not lose
when each
player comes to believe that he
as the recognition process
the cases that
we mainly
is
is
more
affiliated, e.g., it is
make
likely to
offers in the future, so long
exchangeable or independently distributed -
focus. In that case, our definition of
optimism
will
be unambiguously
valid.
According to
we
will say that
s,
players reach an agreement at any {pt,i)
we have an agreement regime
SVt+i{pt,i)
G U. In that
case,
and a disagreement regime, otherwise.
Agreement Theorem
5
Assuming that players do not change
an
if
at {pf,i),
iff
offer
and when (NL),
their beliefs as they observe
for the case of transferrable utility
who
gets a chance to
(TU), Theorem
1
make
estabhshed that,
the level of optimism y stays non-negative for a sufficiently long future, players will reach
an agreement immediately. In this section, we
extension will state that,
if
will present
an extension of
this result.
the level of optimism y stays sufficiently high for a sufficiently
long future, players will reach an agreement immediately. While Assumption
entirely,
it
will
Our
be clear that a weaker form of
NL
is
dropped
remains embedded in the assumption that y
stays sufficiently high for a suiBciently long future.
16
TU
.
We
Theorem
y* e
yt
>
(
present the result for the finite horizon case.
first
— 1,
3 Given any
such
1)
y* it each
Theorem
Intuitively,
t
and any non-negative integer
(0, 1)
for every finite-horizon
that,
game
players
Theorem
know
>
t
+
t*
there exists
+
L{6)
some
3 and with
t*
TU
a weaker form, by dropping Assumptions
and NL.
that their opponents will remain sufficiently optimistic, they will
down
adjust their expectations
1 in
<
t
G N,
t*
G*[6,p] vjith
Q.T, we have an agreement regime at every
3 extends
if
G
6
so that they will reach an agreement
immediately, except for
the end of the game, where we may have a period of disagreement regimes due to a large
last-mover-advantage. Under Assumptions
do not
3
we
lose their
optimism, hence y*
will establish that y*
The proof
of this
therefore relegated to
show
Theorem
was
and TU, Theorem
sufficient for
The proof
Appendix B.
required only that players
immediate agreement. In Example
to be strictly positive with risk-averse players.
Theorem
similar to that of
is
1
but
1,
and
less transparent,
consists of the following steps.
we
Firstly,
NL, the lengths of disagreement periods are uniformly bounded. Then, we
that, under
show that our
may need
=
NL
result holds
with
strict inequalities for
the case
when y
identically
is
Finally,
I.
invoking a continuity property of equilibrium payoffs with respect to the belief structures,
we
obtain our result.
We
extend Theorem 3 to the infinite-horizon case as follows.
Theorem
4 Given any
6
6
and any non-negative integer
(0, 1)
Vt [i,p] -^ Vt [oo,p]
uniformly over belief structures
exist
some
i
G
N
and y* G ( — 1,
(Vt
<
such
have an agreement regime at every
t*)
(21)
t
that, given
any G'^[6,p] with
yt
>
under
3 to infinite-horizon games, stating that,
(21), the infinite
t
<
t,
we
<t*.
stays sufficiently high for a sufficiently long future, players will reach
Intuitively,
Then, there
y* at each
Assuming that equilibrium continuation-values converge uniformly over
Theorem 4 extends Theorem
that
where i and p indicates the underlying game.
p,
1)
asi-*oo
G N, assume
t*
if
belief structures.
the level of optimism
an agreement immediately.
horizon can be approximated by a
finite horizon,
whence
a version of Theorem 3 (with strict inequalities) would give us immediate agreement for the
infinite
horizon case.
Here, a finite period of a sufficiently high level of optimism suffices for an immediate
agreement.
By
requiring the optimism stay high only for a finite time,
to merge, a plausible property exhibited by
employ
many
in Section 6.
17
we
allow players' beliefs
learning models, such as the one
we
will
While Assumption
Theorems 3 and
i,
as p\
+
yt
I
dropped
is
any two histories p^ and
for
NL
a weaker form of
entirely,
remains embedded in
our Theorem also requires that p]
further requires that
(pj —
|pj
bounding how much can be learned before
p[,
>
y* for each
<
p\ ip't)\
1
—
y*
t.
Discussion
5.1
If
=
TU
By requiring that yt >y*,
— pI > yt- Since pj < 1, this
4.
a player
opponent j
believes that his
i
very optimistic and will remain
is
so,
his expectations. For such an optimistic opponent will not settle for httle, even
the value of continuing on bargaining for j
is
If
If
will
be so low that they
an agreement immediately.
the level of optimism stays very high for a long while,
much with
conditional beliefs do not vary
respect to
Bayesian, this means that they adhere to their
very firm.
will lower
thinks that
the level of optimism will remain
then players' expectation
sufficiently high for a sufficiently long future,
will reach
even lower.
he
if i
On
the other hand,
if
it
must be the case that
what happens so
initial beUefs,
and thus
far.
players'
Since they are
their prior beliefs are
players' prior beliefs reflect a high level of
optimism
for a long
future and their prior beliefs are firm, then the level of optimism will remain high. Therefore,
we can
restate our
theorem as
follows.
If players
'
prior beliefs reflect a sufficiently high level
of optimism for a sufficiently long immediate-future, and if these beliefs are sufficiently firm,
then they will reach an agreement immediately.^
Our theorem asks
for three conditions for
a sufficiently long future.
about making an
opponent
will
If
offer at the
the
game
is
an immediate agreement.
to end very soon, and
if
players are very optimistic
end of the game, each may choose to wait
know that she will get
if
until the end,
they disagree - as we demonstrated
if
as they
may do
Finally,
it
in a short
if
the
game ends
there,
is
it
is
may
asks players' initial beliefs be sufficiently firm, that
the
game
offer after
is
a
disagree at the beginning,
is,
they should not update
There are three reasons
for this requirement.
essential for our theorems that the length of a period of disagreement regimes
uniformly bounded under NL, as there
game.
hence they
his
game.^
their beliefs drastically with a limited observation.
Firstly,
if
each player thinks that he will never be able to make an
given date, they will act as
when
in the Introduction.
Second, they must remain sufficiently optimistic for a long while. For, even
to continue indefinitely,
must be
Firstly, there
When
may be a
disagreement period at the end of the
players update their behefs as they observe which players are recognized
and
when, whether they reach an agreement at a given date depends on which players have been
For formal statements, see Yildiz (2000).
Since players will wait for the end only
when they
are optimistic about the end,
the level of optimism does not drop suddenly and drastically, as suggested by
open question whether we can (qualitatively) extend Theorem
stronger theorem.
18
2
we may only need that
Theorem
beyond Assumptions
TU
2.
It
remains an
and NL, obtaining a
The
recognized and when.
drastically, the length of
when
following example demonstrates that,
may
such a disagreement period
the beliefs are updated
be unbounded, causing (possibly
long) delays.
Example
we have
Pgj^i
Consider the following
1
=p
(pj
p]
at each s
[ijj]
at each
<
—
i
On
1.
Pt
we
~
{Po
i'^)
i.e.,
=
we have P^{pt
=
iP(-i (^)) "^^^^ s*
I
that
alternating history. In that case, given any
<
t
Consider any alternating history
then he
Hence,
i
Pt iPt)
—P
will be revealed to
will offer to take I
each k E
/^'~
N
—
where {pt,j)
is
=
= y^i
these values into (22),
>
1,
p^^j (w)}
>
p
at each
Letting i
0.
=
oo,
never reach an agreement at an
game each player
believes
t.
—
i
is
recognized at
2,
we
is
recognized at
=
p), leaving S {I
— p)
= V^
{p,6)
1
-p
for j
t
^
who
to j,
I.
If
(Note that
z.
will
have
R=l-6.)
will accept the offer.
Since
The subgames starting
at pj
(22)
and
{Pt,j,i)
o,re
game
and V/
(p,)
is
symmetric with respect
=
V;Vi (Pt,j)
to the players
= V^ {p,6).
Substituting
we obtain
1
1
6{\-p)
l-6^p{l-p)
dp
1
one can check from. (23)
close to 1 such that
will
=
Ps' {^)\Pt)
Vt{pt,j,i), hence, at any alternating history, payoffs depend only
(pt,i)
V^(p,6) \
VP{p,6) j
—
=
=
In that case, by Proposition
i
player recognized. Furthermore, our
last
so that V^ (pt)
^
has been recognized twice in a
P^iPt
l\pg {ui)
and V,\, {p„i)
again an alternating history.
identical so that Vt{pt)
Letting ^
where
1/2,
{ui)
= p{\~6{\-p)) + {l-p)8Vl^,{p„3)
= p6V^^,{p„j) + {l-p)6{\-p),
y/(pj
.
—
.
>
where p^
we now have
,
VI [p^)
•'
6 {I
p^,
and for some p
,
,Pf_i (w))
an agreement before
be blessed.
= Es>t '5^"'p(l -5)=p
Vt+i [Pt^T)
on the
—
,
.
at the beginning of the
t,
that with positive probability they will not reach
t,
=
Ps' {^)\Pt)
niin {s
.
"blessed" so that he will be recognized
is
i
i
N
6
i
(u)
(/Jq
demonstrate for high values of 6 that players
will
too,
=
the other hand, if a player
row before the other agent, each player thinks
with probability p,
For each
belief structure:
''alternating" history pj
V^
(p, 5)
+ V^ (p,
for every p
that,
<5)
>
1/5,
>
when we
**<'
-*"-/».
(23)
1/2 there exists some 6 sufficiently
will
have a disagreement regime at
each alternating history.
Second, when players update their behefs drastically, we will have another form of the
deadhne
effect, as illustrated in
Example
{P(),Pi,
the following example.
2 Assuming TU, take any
,Pt')
0.1^6
t*
with
stochastically independent
<
i*
< min{L
(6) ,t].
Consider the case where
under both P^ and P^, P^{pf
19
=
i)
=
1
for each
)
i
G
N
,
and
that he will
=
pj
no longer
at each p^.
^t'
is
{Pf )
=
1
each
p^. at
t
>
be able to
t*
.
player
If a
make any
i
offer,
recognized at
is
=
In that case, we will have m^ {8Vt+i {pf,i))
.
^y (15). Now,
Sf [pt, ) =
Vjl (p(.
As we showed
independently distributed.
not reach an agreement before
t*
)
+ 14?
(pj.
)
=
,
the other player j will
=
and
1
1
—
2,
we have
8V/_^_^
2 at each pj.
V^._^_^{pf..,i)
{pi.,i)
,
and
know
=
(pg, pj
=
yt
=
yielding
1,
,
.
.
.
,
under these conditions, they
in subsection 3.4,
[Note that y^
.
t*
hence by Proposition
pj.
viill
thereafter, consistent with
perpetual optimism.]
Example
beliefs
2 also suggests the third
When
be firm:
servation, his
a player
is
and the most important rationale
known
to update his beliefs substantially with limited ob-
opponent may be willing to wait and
convinced that she
is
right.
illustrate
Assume
s as
i
more
likely to
s.
Now, player
she will be recognized at
very likely to be recognized at
not be willing to give
t
if
i
i
thinks that,
if
i.
t
offer.
t,
and
is
will
s,
they disagree until
be willing to agree on
player
will
i
for
j, will
i's
large, or the required
is
s, it is
terms. Being very
i
is
by
an agreement at the beginning. In
j,
and
if s is
2.
s,
and
fact, if
close enough,
Oiu:
so slow that either the impact of convincing j
time to convince him
very
be convinced
about convincing
continuation value at s increases substantially as she convinces
assumes that the rate of learning
find
not settle at the beginning unless
also very optimistic
what she wants
and that they
then they would choose to wait until s - as we demonstrated in Example
On
their
In that case, they must
and hence her opponent, namely
s,
she gets a high share. But her opponent
z's
be recognized at
a strong evidence of being recognized at
be very optimistic about
may
no longer be able to make any
will
that players are very optimistic about
optimistic about convincing her opponent by
thus
when
is
being recognized at
i is
,
p,
s.
that
t*
they know that she
finds a player
recognized at some
likely that
observe, hoping that he will be
our rationale in a greater detail, consider the canonical case of affiliated
where each player
also
him
let
For instance, in our example, players wait until
opponents are expected to be convinced that they
To
for requiring players'
Theorem
is
not so
too long to wait.
the other hand, in canonical learning models, unless their beliefs are sufficiently firm,
players update their beliefs substantially at the beginning,
whence we have disagreement
regimes at the beginning of the game, and thus a delay in reaching an agreement on the
path of equilibrium. Analyzing such a canonical learning-model,
show that agreement can be delayed, and
6
A
Without
effects,
stylized
restricting
we
will
by Assumption
now
TU
will derive
model with
le2irning
how
update their
fast players
explore
when
a
strict
beliefs,
in
our next section we
bound on the settlement
will
date.
but assuming away the deadline
players reach an agreement in a canonical case, characterized
and Assumption
XB
below, which
20
we
will
maintain throughout this section.
Under these two assumptions, we
show
will
for
f
=
oo that there exists a predetermined
date such that players will never reach an agreement before that date and will always agree
Throughout
thereafter.
any
valid for
sufficiently large
XB
Assumption
where player
we
this section,
i;
N
Given any t,s,r G
<
with r
t,
for
N
some kj,h &
with
<
k^
N,
<
t
each
^'^- = '"'' =
for
T =
=
i
i.e.,
(Our
oo.
results are also
see Yildiz, 2000).
recognized r times out of
1 is
take
will
and any history p[ of recognitions
s,
& N,
i
have
v/e
T^
P^'
h.
This assumption holds when each player believes that p
is
exchangeable,
identically
i.e., it is
and independently distributed with some unknown parameter ^ measuring the probability
=
{Pt
and p
1},
is
distributed with a beta distribution such that as
p and observed h
distribution on [0,1] for
player
was recognized
1
believes that his
own h
While h measures the
measure the
Note that, since
depend on
s.
at each pj
e
beliefs
A^'
s
and
level of conviction in players' prior beliefs,
believes that p
s
>
Therefore,
it
by
measures the discrepancy
of date
at
t,
that
We
t,
Two
t.
do not depend on
we denote
i
an interim stage where
Of
course, each player
will write
K/
had a uniform
[h
+
K=
2) will
k\
—
k2-
be shown to
identically distributed, his beliefs
is
y^^j
about p^ does not
we obtain that
properties of y are important for us. Firstly, since players'
s,
the discrepancy in beliefs about s does not depend on
i.e.,
ys{Pt)
=
VtiPt) for each s
- indexed only by the date the behefs are held. Note that
in players' behefs
about any recognition
>
s,
t.
j/(f)
in future at the beginning
while yt denotes the entire process of the discrepancy in behefs about the recognition
whose value
at the beginning of
t is
yuy
When
assume that
Second,
ki
>
k2
(i.e.,
K
>
0),
yt is deterministic, i.e., yt {Pg)
due to our assumption that h
- even though
V
is
is
same
for
rendering
=
<
ki
we have agreement regimes throughout the game
will
each
optimism.
depends only on the date the behefs are held,
It
We
are relevant for this bargaining.
trials
Substituting (24) in definition of y,
about
either.
i
(pre- bargaining) trials at
times (see Fudenberg and Levine, 1998).
k^
initial level of
if
of
k2,
we have
j/(j)
<
throughout so
as in the Rubinstein-Stahl framework.
>
y(()
yt (p's) for
throughout.
each Ps,
p's
both players. This renders
not deterministic. (Recall that St
=
V/
+
VJ^
€ N^ with
s
<
t.
This
is
R and S also deterministic
and B\{pt)
=
max{0,
1
—
6St+i{Pt,i)}.)
Lemma
3 Under Assumptions
deterministic,
and hence R^
TU
= B^ =
and XB, given any
t
Rt for some Rt & K.
21
E
T=N
and
i
£ N, St and R\ are
When
5(+i
deterministic, Rl
is
Lemma.
deterministic, giving us our
regime at
i
=
deterministic and B\
is
B^. In that case, Si will also be
Since St+\ and 6 determine whether
G T, the event that players reach an agreement
at
t
is
we have an agreement
also deterministic.
Writing
Kt
= Y.^'-'Rs
(26)
s=t
Lemma
The
4 Assume
= pI
is
Using
(28),
each
(pj) for
stochastic, so
On
is Vj'.
=
p\{Pt)Kt
StiPt)
=
_(l
>t. Adding up
we have At
By
(28),
A
=
Rt
we have an agreement regime
the right-hand side, as
the left-hand side,
A
t
is
t,
we
t
>
to
=
jzrs-^
-^
At<—
^(i
,
y(^t)
t,
—
have At
=
fact
(28). Since pj
is
St-
any
t
These difference
SAt+i.
i
—
=
1
G
T
iff
D{t).
(30)
1,
which
is less
D
(t)
|
—
1,
yielding
D (s)
Vs
>
t}
j/(j)
<
approaches to 1/6.
than 1/S. Thus, inequality
yielding an agreement regime at
we have
2,
i
we obtain
and the
(29)
converges to zero so that
uniformly bounded by
rendering an agreement regime at
3,
1.
any
gets larger,
—h—
Lemma
difference equation for A. Consider
clearly
at
(30) holds for every sufficiently large
whenever
2,
and Aj are deterministic, and so
j/(t)
+ y(t))
On
On
(28)
= !-%+!) At+i-
uniformly bounded by
is
(27)
= 1 — 6St+i, hence (28) yields Rt = 1 — 6{1 +y(t+i))^t+i+ ^Aj+i. Combining these two equations, we obtain
have a disagreement regime at
equations imply that
& N, we have
+ y(t))At.
and obtain a
(26),
At
When we
i
(27) over the players,
the other hand, both
we can decouple
(26),
terms of A:
in
a simple application of Proposition
is
s
Vt\pt)
with an agreement regime. Now, Rt
Moreover, by
and S
and XB. Given any p^ e N^ and any
statement, (27),
first
that pI (pj)
TU
V
we express
for the present value of all future rents,
i
—
D (t) >
1.
1,
For instance,
and thereby
1.
Let us write
PA =
for the interval of perpetual
regime at each
t**
<
s
>
t.
Since
{te T\As <
agreement, consisting of the dates
to
e PA,
PA C N is non-empty,
to.
22
t
such that we have an agreement
and thus possesses some minimum
We
ment
now
will
derive a stricter upper-bound for
B{t)=
^
+ ^y(t)
1
Lemma
each
we
(29) repeatedly,
=
5 Let i
e
oo, 6
obtain the following bounds for Af.
first
TU
and assume
(0,1),
B {i) <
and XB. Then,
< B {t +
h.t
1)
at
>t**.
t
We
large
the date the period of perpetual agree-
,
Writing
starts.
and using
t**
plot our
t,
B {t)
bounds
and
As shown
for A.
in Figure 2.
B {t +
B{t) converges to
as
1
—
i
>
Hence,
oo.
for sufficiently
are arbitrarily close to each other, providing very precise bounds
1)
our bounds are valid only when
in the Figure,
>
t
t**,
because the recursive
equation (29) holds only at agreement regimes.
Let us compare our bounds with D, the process that determines whether we have an
B {t) < D {t)
agreement at any given date. Clearly,
at each
t
so that our lower
agreement regime throughout. In comparing our upper bound B{t
Since
y^t)
-
y(,+i)
=
(i_^;,_^2J(t+fe+3)
'
B{t
*^*^ implies that
t>tcut
+
1)
D [t)
<
l)
+
bound
stays in
with D{t), we write
iff
1
_viiM.
=
h-2.
(32)
D {t)
holds, yielding an agreement
^
when
Consequently,
regime at
i
—
t
>
tcut,
< B{t +
inequality At
This gives us our bound for
1.^
<
\)
t**:
0<<** <max{0, [Wj}-
We
small,^
As
exhibit
some values of
t**
and
h
tcut ior
we have agreement regimes throughout,
d increases,
some disagreement becomes
=
K
=
I
(33)
aX Table
When
1.
in particular at the
6
is
sufficiently
beginning of the game.
possible; in fact, the perpetual
agreement may be
delayed for a long while.
It
turns out that there cannot be any agreement regime before
f**,
the date the period of
perpetual agreement starts. This gives us our main Theorem in this section.
(1
—
Note that
this
S) /6 for
each s
is
precisely the region
>
an agreement regime at
t.
t
'For instance, when 6
This
—
<
1
.
is
where
y^t)
—
2/(t+i)
analogous to Theorem
whenever
y,
j'^2_
-1-1
—
ys+i
,
<
(1
2,
—
we have
^
(^
6) /i5 for
[tcutj
23
~
^) /^t
where we also have
which implies under
<
each s
>
so that
NL
t.
t"
=
0.
and
TU
j/(s)
that
—
we
y^s +
will
l)
<
have
1
1
0.95
1
—
1
-
B
-_
^g^f^
-
>x'
-
^'
0.9
D
J,
-^ A
-^ A
/7/,'
0.85
B(t+1)
*
i
"
0.8
1
0.75
1
r
5
Figure
only on
2:
PA. Note that
tcut
=
15
10
cut
Functions D, B, B{t
t
+
1),
1
1
20
25
and A. A* denotes the solution to
6.4624.
[6
=
0.99,
h
6'"
0.95
1
1
Table
1:
tcut
-
1
0.99
6
6
0.9415
0.999
28
27
0.9733
10-*^
996
996
0.9990
10-9
31619
31618
1
and
(29),
K =l,t = 500.]
t**
[tcutl
<5
=
30
t** vs. 6.
{h
24
=
K=
I, t is
very large.)
which
is
vahd
.
Theorem
such
5 Let i
agreement
oo, b
>
t**
t
G
TU
and assume
(0,1),
By
and XB. Then,
players reach an agreement immediately
,
and they do not reach an agreement
yet,
Proof.
we
each
that, at
=
show that we have a disagreement regime
will only
vacuously true,
so,
we assume that
disagreement regime at
—
t**
<
t
tcut,
we
at each
at each
In that case, i**
0.
by definition of
1
a disagreement regime at any
>
t**
Now we
t**.
will
they have not reached an
—
t**.
tcut
by
show
will
that,
Rt
=
0,
and hence Aj
=
>
Hence, St
is
By
(28), this yields
we have
>
St
St+i
>
+ y^t+i) <
^(1
+ 2/(1)),
i-e-,
-
whenever
>
and hence a disagreement regime
at
and
t
<
tcut,
hence
i/(f)
y(t)
y(t)
-
—
1.
t
For a very canonical case, Theorem 5 states that, unless players'
that our Agreement
typically, at the
Theorems
is
>
— 1, showing
— 1.
1/6.
We
have
+ y{t+\)
^y{t+i)
>
1/6,
l
t
t**
6At+i-
St+i whenever
always true: y{t+i)
<
with a disagreement regime so that St+i
i
this
0,
whenever we have
have a disagreement regime at
s
=
and we have a
(33);
tcut
t
ra\nPA, hence
If t**
<
some
this end, take
=
t**
<
t
<
1
>
t
by mathematical induction that we have a disagreement regime at each
To
T
G
before t**
we have an agreement regime
definition,
if
there exists at**
-
•|y((+i)
y(t+\)
>
^. But this
^- Therefore,
>
initial beliefs are
so firm
apply, reaching an agreement will be delayed for a while. For,
beginning of a learning process players are more open to new information, in
the sense that they update their beliefs substantially as they observe which player gets a chance
to
make an
proceed
in
off'er.
Knowing
this,
each player waits, believing that the events are very
likely to
such a way that his opponent will change his mind. As time passes, they become
experienced. In this way, two things occur simultaneously, both facihtating agreement. Firstly,
having similar experiences, the discrepancy in their beliefs diminishes. Secondly, they become
so closed
minded that
wilUng to agree
,
when they
share the pie at
opponents
hope to convince them and thus become more
lose their
a while, they reach an agreement.
in their terms. Therefore, after
At the beginning
until t**
their
of our
game it
will reach
t** will
Since they disagree about
is
common knowledge
that they will not reach an agreement
an agreement no matter what happens by then.
depend on how many times each player
how many
times each player
is
will
How
they
will
have been recognized.
hkely to be recognized by
t**,
there
is
no consensus among our players on how they can better each of them by agreeing on a decision
at the beginning. Therefore,
that there
How
is
they wait imtil
t**
even though there
a consensus among them
an agreement at the beginning that would leave each player better
long can they delay the agreement?
tcut{h,K,6).
is
Now, given any
K
for sufficiently large values of h.
and
6,
as
/i
We showed that the delay
— oo, tcut{h,K,6) — — oo,
>
There are two factors behind
25
>
off.
can be at most
so that t**
this limit. Firstly, as
/i
—
=
>
oo,
become very
players' initial beliefs
limit,
thus by
Theorem
1,
firm,
approximating an independently distributed p
they reach an agreement immediately. But more importantly, the
discrepancy in beliefs also disappears since
would be more interesting then to
It
optimism
level of
h —^
as
Now,
oo.
always
is
tcut
=
set
y(()
yo- In that case, for
^
Finally, as 5
agreement
in
=
y(o)
i
>
0,
\/ah~+b — h — 5/2 where a
K,
1, t^ut {h,
we
=
-^ oo.
The
converging to
some yo G
for
have
will
y^S/
—
(1
in
=
and
as
/z
—
»
oo.
so that the initial
(0, 1]
y(j)
5)
due to the increasing
for large values of h, consistent
6)
+ 2),
t^/{h
and approaches to — oo
point, however, tcut starts decreasing,
be no delay
<
K = yo{h + 2)
increases, initially, tcut also increases, possibly
will
in the
+
yo/{-Ar[
b
level of
=
2a
+
1)
^* yo
1/4.
As h
optimism. After a
the limit. Once again, there
with our Agreement Theorems.
intuition for this result
is
clear.
Very patient players
can wait very long to find out whose beliefs are more accurate.
We will now measure how long real-time delay we can have in the continuous-time hmit. We
measure index-time t in terms of real time by r {t, n) = t/n, where
> measures the fineness
ra
Now, discount
of the grid.
exp(— rr
Clearly,
in real time
If
is
—
>
=
and thus t
(5*,
exp {—r/n) where r >
=
{t,n)
the real-time impatience.
is
Mog(6~'). Therefore, the maximum delay
K do not depend on n, we can measure the real-time delay in the continuous time
h and
1,
=
is
(n)
('5-«-t('^,/c.5)^
^ log
hmit by hm5_i
5
(<,n))
rate
<5
5""''="*'
'
^
'
and K, the delay
One can check from
^og (s~^'="^-^'^'^^^n.
^
^
1
so that there
is
no real-time delay
(32) that, given
in the limit.
any h and K, as
That
is,
given any h
agreement can be bounded to be arbitrarily short by recognizing players
in
sufficiently fast.
It is
h and
crucial for this result that
K
are taken to be independent of
n
discrepancy in beliefs vanishes arbitrarily fast as
at
any
real
time r
is
K/ (rn
-\-
h+2), converging
gets sufliciently large,
to
as
n goes
In updating his beliefs, a player might take into account
how
i.e.,
n
so that the
the discrepancy
to oo.
fast
the players are recognized.
In particular, one might attribute the recognition of a player at a given time to the player's
innate abilities
measure the
little if
players are to be recognized very frequently.
effect of the frequent recognition purely,
one
may want
to adjust h
ingly so that the discrepancy in beliefs at a given real-time does not
of the grid
much. To do
this, let
in behefs at
a given real time t
values of n.
As n —
of
»
optimism stays at
oo, 6{n)
j/o
>
oo, however,
>
and
K accord-
depend on the fineness n
= n and K = yo{h + 2) so that the discrepancy
yo/{^^T + 1), which is approximately yo/{T + 1) for large
us take h
1,
and h and /C go
unchanged.
taken separately, in both cases
When n —
is
—
Therefore, in order to
6'*="'
we compute
We
to oo in such a
way that the
initial level
have already checked that, when these two limits are
goes to
1,
rendering immediate agreement in the limit.
that
lim T{tcut{h{n),K{n),8{n)),n)
26
= \/yJr -
1
we can have very long
so that, in the continuous time limit,
both patient (with low
and
r)
if
the players are
optimistic.
Impact of Risk- Aversion
7
In this section, reinstating
Assumption NL, we wish to demonstrate the impact of risk aversion
We
on the problem of reaching an agreement.
beginning of the game even though y
of
real-time delays
Theorem
TU
>
present a case where players disagree at the
and the game can be
3 needs to be strictly positive,
and thus Theorem
1
arbitrarily long.
That
is,
y*
could not be extended beyond
in its full strength.
Example
3
Assummg NL,
take
U =
{u e [0,l]'^\{u^f
+
[u'^)^
<
l},^^ i
>
e (0.907,1),
3, 6
and
=
so that y
(0,
1,0,0,
.
.
Now, by Proposition
(vl —S^,6) and
,0).
.
V3
2,
(1,1,1/2,0,0,... ,0)
p2
=
(0,1,1/2,1,1,... ,1),
We will have a disagreement regime at
= (0, 1). Hence, att = 2, if recognized,
,^^2^),
The
offers will be accepted.
-^
N
\( I
1,^
= l(x/rr7^..)+i(o,
At the beginning of date
— {6V2)
=
(0,1), respectively.
v.
1
p'
which
t
=
1,
player
Vl
(\/\-b^
=
0.
players
1
and 2
will offer
and
will offer
Thus,
l
+ 5\
certain that he will be recognized,
1 is
will be accepted.
,
t
Thus, his continuation value will be
.2/1+5^'
^l-6^^]
.2_./.
= ^l-{6ViY
=
(34)
.
Likewise,
V,'
We
=
y/i-mf = ^i-S^'-^
4
have a disagreement regime att
the case
when2-^
+ ^) > ^.
thus f{S) >
(1
/'(I)
= —1/2 <
that,
for any 6 G (0.907,
0,
"This would be the case
1),
if
=
whenever
we have f{5)
+ [Vi) > 1/6 By (34) and (35), this is
= 2- ^ (1 + 5) -•^, we observe that f{l) = 0,
Oif {V^)
Writing f {6)
>
6 is sufficiently close to
0,
(35)
.
1.
In fact, one can check
hence we have a disagreement regime at
players were dividing a dollar and each had utility function x 1— ^/x.
>
27
t
=
0.
In a
game with perpetual optimism, each
player adjusts his expectations down, knowing
that his opponent has very optimistic beliefs - even though he believes that his opponent
wrong. With transferable
utility this
adjustment was enough for agreement.
When
is
players are
however, uncertainty about which player will get the rent lowers their continuation
risk- averse,
values at the beginning of any date with an agreement regime, increasing weakly the rent for
the player recognized at the previous date. If in addition players are highly optimistic for this
date, such an increase
before - as our
may be
high enough for preventing them from agreeing at the date
Example demonstrates.
Conclusion
8
As suggested by Edgeworth
results in
agreement
(1881), with complete information, in equilibrium, bargaining
an optimal outcome.
is
Given that the delays are
Yet, the agreement
reached immediately.
prominently proposed explanation
for the delay
we analyzed the problem
In this paper,
is
costly, this also implies that the
delayed as a rule in real
parties' excessive
is
life.
A
optimism.
of reaching an agreement in a
model where players
are possibly optimistic about the recognition process, the ultimate source of bargaining power
in sequential bargaining
in
with no outside option. In order to see the pure impact of optimism,
our analysis we relaxed only the
common
prior assumption of complete information,
and
adhered to the equilibrium.
We
showed that the excessive optimism alone cannot be a reason
players are sufficiently optimistic for a long while, recognizing that his
When
for the delay.
opponent
will
remain
optimistic for a long while, each player will lower his expectations about the future so that they
will
reach an agreement immediately. In other words, presence of perpetual optimism includes
also the
bad news
for
each player that his opponent wiU remain optimistic, inducing a form of
pessimism that moderates
Corroborating
this,
his
optimism to the extent that they
we further showed
settle at the beginning.
in a special case that
agreement so long as there are no sudden jumps
in the level of
we
will
have an immediate
optimism, providing a
ratio-
nale for the immediate agreement result in complete-information models, where the level of
optimism
is
constantly
nil.
It is
not the absence or the presence of optimism that determines
whether we have an immediate agreement; that
is
determined by the change in the
level of
optimism.
Relaxing the
common
prior
assumption enriches our understanding further. Taking
ences in information to be the only source of differences in beliefs, the
common prior assumption
allows only one notion of convincing, namely communicating the privately
in a credible way.
While the delays are used
sense, another notion of convincing
for delay.
When
becomes
as a
means
salient in
known information
to convince the other party in this
our model, providing a distinct rationale
players' beliefs are substantially different
28
differ-
from each other, a player may be
J
opponent
willing to wait so that his
of the truth, which
A
show
will
becomes more
make an
U =
4 Take
6
(^u^,u^)
<
Player 2
\.
be recognized at
any
p^
=
=
t
and
1.
(pg (w) ,pi
{uj) ,..
he
if
=
1)
P^
0,
=
t
t
>
2,
2|p,)
=
|
with
(oj))
=
p2(p.
(po
=
=
2)
beginning of date
player
1
1
and P^
hence he
1 is
—
^t
1,
(Pt)
—
^i+i {Pt+i)
will offer [6/ {2
1,
we
is
by Proposition
—
6) ,1
—
6/ {2
—
(pj)
6)),
6
oo, 6
=1) =
1
beliefs
's
tt,
— v^,
(3
and P^
1),
and consider
about pj
[p^
=
1)
Under P^
.
=
the
,
p
1/2 at each
will
never be recognized again. That
given
is,
>t, we have
s
^/2
^/^iM =
U
othenmse;
show
i.
that player 1
Hence, when
continuation value at the
's
gets higher, player 2 will offer
it
to be accepted.
is
=
(The continuation values
Now, Player 2
1.
Vj'^j
2,
Thus,
be accepted.
to
this
have the Rubinstein- Stahl framework with identical players,
unll
^^ then have V^
'
he
I,
recognized at date
is
hence,
offer;
which
offers (1,0).
he
n
an example showing that
except that player 2 believes that he will not
1,
6/2,6/2), which will be accepted.
Take any Pf where player 2
never make an
=
recognized att
will offer (1
(pj
= 0. We will
= P^ (pj = 1).
2|po)
a lower share at date 0; and the offer
1
If player
1
=
{pi
decreasing with
is
=
i
f,
for any
I
and P^
1
parametrized by player
p,
=
recognized at
is
,Pi_i
.
+u^ <
[0.1]" 1"^
equilibrium payoff increases as he
will first present
affiliated.
similar to those of player
beliefs are
's
not
is
independently distributed with P^ [pQ
>
We
the future.
offer in
following family of recognition processes
s
payoffs with respect to beliefs
for affiliated recognition processes that a player's
likely to
not true when the recognition process
Example
IS
our player thinks.
like
Appendix - Monotonicity of
Here we
is
presumably
is
observe more facts and have a better understanding
will
(P(+i)
=
by (20), V^ (pj
1/(2
which
—
i5).
is to be
=
'
2
^
at
"*"
t,
'
2
Therefore, if player 2
accepted.
t
=
2 are 1/2.
always think that he will
will
Hence
0.
at
if recognized,
player
^^(+1 (P<+i)
Since
is
recognized at date
The continuation value of player
1
at the beginning of date 1 is then
V/
Since 5 6 (3
—
Vl +V\ <\,
\/5, 1),
6/ {2
—
6)
>
I
=
—
TT
•
(1
-
6/2)
6/2, hence
thus they reach an agreement at
t
=
+
V^
(1
some non-negative
rent. In
the bad news that his opponent
if
is
our example,
tt)
•
^
2-6'
decreasing with
is
tt.
(Note that V^
t
1 is
overall
the recognition at date
not excessively pessimistic, hence
bad news
now
it is
for player 1. If player 1
in the dollar.
29
1
comes
also with
common knowledge
is
0, in
that,
dominant; and being
comes to believe that he
to receive such bad news, he will have a weaker bargaining position at date
a smaller share
6/2, hence
contains a good news for a player that he will
for player 1,
they wait, he will not be able to get more than half the dollar. This bad news
recognized at date
=
0.)
Proposition 2 stated that being recognized at some
extract
-
is
more
which case he
likely
will get
Now we
focus on a canonical class of recognition processes, where
recognition-process p
is
said to be (weakly) affiliated
P'iPs =i\Pto =hPu,Pt^,--- ,PtJ > P"
each
for
if
p
is
sequence {to,ti,
finite
.
.
among
affiliation
members
the
all
=
[Ps
T-\Pto
^i,Pu,Pt^,--- ,Pt„)
,i„) of non-negative integers, for each
.
independently distributed or exchangeable,
weakly
is
it
of a process, while
A
these news are good.
all
iff
N
€
i
and
Note
affiliated.
s
usually defined between two
it is
=
say Pj and p^g, which corresponds to the sequence with to
*i
=
•
=
-
•
If a
*«•
G N. Note that,
we
also that
random
process p
define the
variables,
is affiliated,
then we have
=
F' {p,
at
any two
> P'
i\i,pt)
histories {i,Pt)
and
=
{p,
and P^
i\j,p^)
{p,
=
< P^
j\i,p,)
{j,Pt) that start with distinct players
i
=
j\j,p,)
(36)
j at date 0,
but continues
{p,
and
with identical players.
Proposition 3 Given any two games G*
P'
at
= i\Pt) >
i'Ps
P'
each s and p^ for some distinct players
V'
where the terms in
That
is,
opponent
when the
(if
Apt) and P' {p,
and
i
> V'
[p]
with
[6,p]
[p]
j, if
p
recognition process
i
is
at least as likely to
A
anything) loses.
and V^
< V^
[p]
is
[p]
Vj
[p]
our Proposition holds for
any two games G*
hi,o
=
(k,
which are of length
(37)
i
&
=
1
and each
Recall that, by (15),
=
[p]
games
1.
change
i
(if
in
such a way that each
anything) gains and his
is
by their modelling assumptions.
affiliated
of
limit.
mathematical induction on the length
V^
[p]
=
0,
and our Proposition
some length
i
and
Conditional on that
p^.
^ [5,/9]
[p]
t.
Firstly, for
is trivially satisfied.
(n,P^,P^). Setting
for every
[6,p\ satisfying the hypotheses of the Proposition.
jPt-i)
Gl^^
V^
>
(38)
,
offer as before, player
N xU x {Reject}, consider subgames G{,
and process {pi,P2,
subgames
t
Vj
all
=t —
(38) holds for
at each
[p]
similar proposition for the Rubinstein-Stahl framework can be found
will use
and G*
[5,p]
ui, Reject)
(37)
jift)
then
affiliated, if players' beliefs
make an
remain to hold as we take the
=
=
defined as limits of the continuation values in finite-horizon games; and the
Assuming that f 6 N, we
simply have V^
[p,
to prove the Proposition for finite-horizon case. For, the continuation values in
It suffices
an infinite-horizon game
inequalities will
< P^
j\p^)
is affiliated,
Merlo and Wilson (1995), where the recognition process
Proof.
=
indicates the underlying recognition process.
[]
player finds a player
in
=
{Ps
and G*
[6,p]
is affiliated (for
and
iPt)
>
Gj,j
VI
[p]
^
[6, p]
is
recognized at
p
is
affihated). Hence,
[S,p], i.e.,
[p\
(pj < K/
Thus, we only need to show that Vq
we have V^
\p]
=
P" (Po
i)
m'{SV,
30
Gf,
0,
we
=t+
\,
take
Given any history
[S, p]
starting at hi,o,
by our induction assumption,
we have
(pj and Vi
=
and
=
these subgames trivially satisfy
A;
0,
t
t
Assume that
[p] {i))
+
\p\
[p]
(39)
ip,)
>
P' (po
Vq
[p]
and Vq
= j) 6V{
[p]
(j)
\p]
<
Vq
and V^
[p]
[p\.
=
=
P' (po
Tn'(6Vi
i)
+
[p] (?;))
=
P' (po
=
yo[~p]-Vo{p\
j)
P'(Po
5K/
=
Hence, we can decompose
[p] (j).
V'J [p]
- V^
\p]
as
i)[m'{SV,[p](i))-m'{6V,{p]{i))]
(40)
_
+P'{Po = j)[SV^\p]U)-6Vnp]U)]
+
We
show that
will
with
all
=
[P= (po
the terms in
- P^
z)
(po
=
are non-negative.
[J's
[m'{5V,
i)]
(41)
- 6V^
[p] {i})
note that m'
Firstly,
[p]
(j)]
(42)
.
non-decreasing
(v) is
v^
Hence, by (39), m^{6Vi \p\ [i)) — m^{6Vi [p] (i)) > 0. By (39), we
—
>
thus the terms in lines (40) and (41) are non-negative. To show
0;
6Vl
[p] {j)
[p] (j)
—
>
m^(6Vi [p] [i))
0, we need to compare the subgames G* [6,p] and G' [S,p] starting
6Vi [p] {j)
and non-increasing with
v'
.
also have 6Vi
that
and
at histories (i,w, Reject)
have a game G'~'
['^'Z''^]
p'{p';
at each p,
P'
{Pl
=
and
£ N. {p and
I
> P'
^\Pt)
so are p' and p'
Vj' [p] (i)
=
V^
decreasing in
where the
term
=
p'^
^\Pt)
> V^
[p']
and P^
[p']
=
VI
< V^
[p];
in
=
p'{p^
=
{pi
< PJ
j\pt)
Similarly,
u-',
(v)
it
>
=
{p^
we have V^
[p] (i)
m^{6Vi
[p] (z))
Therefore, Tn^{SVi
[p] (i))
- Vq
[p]
>
[p]
is
affiliated,
j\pt) at each
we have V/
follows that
v.
showing that Vq
we
p,.
> V^
affiliated,
is
[p']; in particular,
< F/ [p] (j). Since m' is non> m^{SVi [p] (j)) > 5Vi [p] (j),
— 6Vi [p] {j) > 0, and thus the
and thus
0,
is
[p']
by (36), we have
Since p
V
> V-
[p].
Similarly,
will present these proofs
under the
[p]
complete.
Appendix— Proofs
will present the
titles of
the sections they belong to.
[],
e.g.,
B.l
—
we omitted
proofs that
We
K[i,p], V[p], etc.
maxt,j^p^ \pI (pj)
ql (Ps)|,
When
We
in the text.
needed,
is
it
K
ifTpJ (p^)
>
A
for
for
the underlying parameters
the functions,
i.e.,
|p
—
g|
=
each {t,i,p^).
Agreement Theorem
size of the pie.
bounded.
Then, we
Then, we
will
will
show
that, under
show that our
first
develop our notion of measuring the
NL, the lengths of disagreement periods are uniformly
result holds
with
strict inequalities for the case
when y
is
Finally, invoking a continuity property of equilibrium payofTs with respect to the belief
identically
1.
structures,
we wiU obtain a stronger
Measuring the
>0
will write
supremum metric
will also use the
and we write p > A €
we
Now we will prove Theorems 3 and 4. Towards this goal, we will
V
[6,p],
i\k,p,)
and by induction hypothesis, the proof
Here we
in
=
\p] ij).
due to m'
in line (42) is non-negative,
[p]
i\p,)
can be defined on different spaces.) Since p
and non-increasing
ti'
=
Therefore, by our induction assumption,
.
last inequality is
we have V^
B
{pi
Isomorphic to each such subgame GJ.
(j,u, Reject), respectively.
where
relative to U,
size of
version of
the pie
Define
Theorem
E
:
3,
R\ —
which
IR+,
will also
measuring the
by setting
IU,II
max{A||t;|||AvG
31
imply Theorem
[/,AeR+}
size of
4.
each payoff vector
=
v G IR^\{0} and setting E{0)
at each
where
0,
continuous, increasing, and homogenous of degree
Moreover,
||-||
any given norm on K^. Thus defined,
is
E(At;)
i.e.,
1,
[/nR^ = {v|E(u) S [0,1]}; and the Pareto-frontier
= E(v\m^(u)) = 1. Given any t, we write
pie at the beginning of date
any
v,
we
is
(m'
(v)
NL
Disagreement under
regimes
we have an agreement regime
t;
m (u) =
will write
,m}
if
t,
where
we have a disagreement regime
=
6Vt+i, yielding 5t
agreement regime after
first
log(^t)
=
is
due to the
Extreme case
Writing 1
equilibrium of any
finite- horizon
When
{Pi,i)
iff
St+i {Pt,i)
<
1/^-
Given
at
interval of disagreement
any given
t,
we have m{6Vt+i) =
= E {Vt) = E {6Vt+i) = 6T, {Vt+i) =
= 6 ~ S^ at each such
5t_i €
[1,
G*[6,
1].
=
p with
is
[y
=
pj
1
at each
by
0,
and
t
i,
we now analyze the
also identically 1.)
we have 14_i
we have an agreement regime at i — 2.
Since Vj
1/5],
L{6).,
< E(m(0)).
for the belief structure
game
-1 =
\og(l/6)
fact that 5;
Consider the end of the game.
1.
we
Hence, the length of such an interval of
t.
log(E(m(0)))
<
1
\og(l/S)
E {m{0)) >
in particular,
(Vj) for the perceived size of the
is
L{Si,6)
where the inequality
any
1;
is
with a disagreement regime; we must thus have St
the date of the
i is
disagreement regimes
=
E
and v £
(v)).
6Vt+i, and hence by equation (15), we have Vt
t
at
T,
Under Assumption NL, the length of an
uniformly bounded. For,
dSt+i- This holds at each
=
St
>
XT,{v) at each A
defined by S(v)
is
have T,{m^{v),v'^)
=
(15),
=
771(0),
and hence Si-i
=
If St-i > l/S, then we have
— 1,. .. ,i — 2}; and we have an agreement
5j_i = S(m(0)). By construction, St_i,^^_i €
disagreement regimes throughout the interval {i — L(6)
regime at
[1,
i
— L{S) —
where L{6)
2,
=
L{St-\,6) as
Together with our next lemma, this
1/6].
will
guarantee that we have an agreement regime at each
t<i-L{6) -2.
Lemma
6 Take any finite-horizon game
Given any
G*[i5, 1].
t
€ T,
if
thus,
by
St+i € [1,1/5], then St £
[1,2-5] C[l,l/A).
Proof. Since St+i <
1/5,
we have agreement regime
at
and
t;
(15), Vt
= m{6Vt+i).
Now,
E (514+1) = 5E {Vt+i) = 6St+i S [5, 1]; therefore, it suffices to show that E(m(u)) € [1,2 — 5] whenever
S (v) € [5, 1]. Given any v € IR+, if E (v) = 1, then m{v) = v, hence E {m{v)) = E (v) = 1 € [1, 2 - 5].
Now take any w € 1R+ with 5 < E (v) < 1. By definition, m(v) > v, hence E (m(v)) > E (v) > 1.
Hence, we only need to show that E {m{v)) < 2 — 5. We accomplish this using a geometric construction.
CONE for the convex huU
C. Since m.{v) € CONE
—
max {E ((2 5)wi) E ((2 — 6)102)} =2 — 6,
See Figure 3 for our notation and graphical exposition. In addition, we write
of
n
and
and
E
is
r2.
[Note that v, wi, Tn{v),
W2 € CONE.]
increasing and convex, this implies
We will show that m{v) <
E (m(w)) <
,
completing the proof.
To prove
B, C,
is
V,
that m{v) < C, we
and m{v) are
all
first
<
B<
v,
<
rn{v), line
In this order, clearly,
linearly ordered.
the center of the rectangle defined by
Hence, we have v
note that, since v
wi, rn{v),
m,{v). Furthermore,
we have
32
and W2,
A<
v.
i.e.,
I
has a positive slop so that A,
we have
B=
v/2
A < B < C.
+ m{v)/2 =
[For otherwise,
Moreover,
B
Wi/2 + W2/2.
we would have v < A < B
Figure
3:
An
illustration for
33
Lemma
6.
,
.
A
so that
CONE
£
would then have
A
On
contradiction.]
A=
and hence
€ 6U,
i.e.,
T,{A)
— X)6w2
\6wi
+
<
But v <
S.
(1
the other hand, since hnes
\\B-A\\_
I,
61,
A
also
and
{2
\\w,-bw,\\
some A €
for
impHes that 6
—
<
T,{v)
<
we have a
T,{A);
we have
_^_
^^g^
•
\\{2-8)w,-w,\\
IIC-BII
we
convex,
is
b)l are parallel to each other,
\l-6\\\w,\\
_
6U
Since
[0, 1].
12-5-lllKII
Hence,
-
\\m{v)
where the
one
is
equality
first
due to
C>B
Since S{_ir^\^ G
-
B\\
the second one
is
due to
\\B
C
-
=
\\C
<
v\\
and since m{v) and
A\\
B>
>
v
Lemma
A, and the last
m{v) < C.
are ordered, this implies that
with mathematical induction,
l/6\, together
[1,
-
\\B
B = v/2 + m{v)/2,
due to
is
(43). Since
=
B\\
gives us the following
6
Lemma.
Lemma
This
7 Given any
Theorem
is
G*[i5, 1] urith i
<
oo, at each
t
3 for the extreme case of p
< i— L
=
1
(6)
with
—3, we have St+i €
[1,
On
strict inequalities.
—
2
6]
C
we can
bound p
also
l/S).
the other hand,
continuation values are continuous with respect to the probability assessments p, as our next
states. Since
[1,
to be close to 1 by bounding y from below, this gives us
Lemma
Theorem
3,
again with strict inequalities.
Lemma
8 Given any 6 €
\Vt\p] (Pt)
-
Proof.
Pj.
By
We
continuous. Since
P
^i[p] (Pt)
Lemmas
Theorem
game
whenever
t
<
g|
Q T, and any
e
>
0, there exists
is
continuous in
some A >
such that
A.
=
p.
Take any
t
€
T
and
Vt'[p] (pj
(pj m' {5Vi+i[p] (p(,i)) + (l -pj (Pt)) ^^<+i[p] (Pt>j) at each
and each i £ N. Assume that both p i— Vi_|_i[p] (pj, 1) and p >-^ Vj+i[p] (Pt,2) are
m^
:
M^
—*
M.
and the projection mappings p
must be continuous as
6 Given any 6 €
G*[8,p] with y
t
G N,
-
= pj
7 and 8 gives us
If
i
|p
we have
regime) at each date
Proof.
e
any
use mathematical induction. Clearly, Vj [p]
definition,
belief structure p,
>—
<
Vt[q] (pi)|
(0, 1),
well.
By
Theorem 3 with
(0, 1)
and any
i--» v'^
are
all
continuous,
induction hypothesis this completes the proof.
finite i
>
,
— L (5) —
and v
p] (pt)
strict inequalities:
2,
> y* we have St+i (Pt+i) < {2
<t — L{6) — and each Pj+j
t
>—*
—
there exists
6
+
1/(5)
/2
some y* <
<
1/6
1
such
that,
for every
(and hence an agreement
2>
3
<
0,
then our Theorem
is
vacuously true;
so,
assume that
i
— L (5) — 3 >
0.
= {< S T\t < f — L((5) — 3}, which is finite and non-empty. Given any t € T^ariy, by
Lemma 7, E(Vt+i[l]) <2 — 6 < 1/6; and since E is continuous, there exists some et > such that
S {Vt+i\p] {Pt+i)) <{2-6 + 1/6) /2 whenever \Vt+i\p] (p^+i) - Vt+i[l]\ < £«. But by Lemma 8 there
also exists some Ap^^^ >
such that |yt+i[p] {Pt+i) — ^+i[lll < (t whenever |p— 1| < -^p.+j, i.e.,
and y* = 1 - A < 1. Take
pi (Ps) > 1
^P,+, for each (s, p^, i). Set A = miup^^^, teT,„wv ^p,+, >
any p with y > y*. Now, given any (s, p^,
and any P(_,_i with t € Tgariy, we have pl (p^) >
>2/*
1
= l-A>l- Ap,^, where j / Therefore, at each pj+i with
P^ (Pj + Vs (ps) > Vs (Ps)
t G Teariy, We havc \Vt+,\p]
< eu yielding E {V^+M (p,+i)) < (2 - + 1/6) /2.
Vt+i[l]\
(p,+ i)
Define Teariy
?'),
,
i.
-5
34
Proof.
by
{
(21), for
Theorem
each
<
t
Take any
4)
|S,+i [co,p\
whenever
>
i
iV This
6,
there also exists
(2
-
+
5
+
St+i [Up]
A
B.2
Now we
ii
<
1/^
1/^,
s
>
Vt'
for some
t
[t\
iPt)
=
t
1
with
>
p,_,_i
[t]
P*
{ps
=
i\Pt)
-
(2
6
+
>
1/6) /2
E
Since
0.
[f,p])|
<
continuous,
is
(Vp)
ei
= max {t* + L [6) + 3,maxt<(-
i
with y
G''[<5,p]
< f.
t
-
~^{Vi+i
\J:(Vt+i [^,P])
In that case,
an agreement regime at each
we
We
<
t
t*
.
>
(44)
By Theorem
it}-
we have St+i [i,p] (pt+i) ^
by (44), we will have St+i [oo,p] <
y*,
u
e {0,1) and
6
and
St
prove the following
e N, assume TU,
i
Also, write Aj
i.
=
Rs
[^
The
6.
results in this section are stated for the
and then take the
will first consider the finite-horizon case
will first
[t\
=
[pt)
f
[Pt
=
XB
=
X^^lj 5^~'i?s
(1
+
=
APt)
and that R\
[i].
We
limit.
Lemma.
=
[t\
R'^
= Rs
[t\
Then, given any p( S
[i\
R
G
for each
and any
A''*
i
€
A'^,
y(i)) A[.
Proof. Under our hypothesis, given any pj and
Rs
1/6
such that
such that, for every
each
€i at
{0}.
M
P\ (Pt)
-
Here,
<
N
the proofs omitted in Section
N = (Nu {oo})\
9 For any
=
[f,p]|
=
let ei
e
fj
model with learning
stylized
will present
Lemma
St+i
<
y*
i-C-,
infinite-horizon case.
write
-
and
0,
some
true in particular for i
is
some
=
1/6) /2
>
6
there exists
f*,
Pt
s
>t, we have £' \l^p^-^^R\
{pt)Rs
[f|>
[i]
|pj]
=
Rs
where the penultimate equality
the fact that player's beliefs about different dates are identical.
Hence, by Proposition
E''
[<]
is
2,
[l{p,=i}|P(]
due to
we have
= ^\=t 6'~'E^ [^{P,=^}Rl \Pt] = Pi {Pt)K- This also impHes that St (p,) = y/ (p,) +
(Pt) = pj iPt) Aj + P? iPt) A[ = {pI iPt) + P? iPt)) A[ = (1 + Vd)) Aj.
Lemmas 3 and 9 imply Lemma 4 as an immediate corollary. We now prove Lemma 3.
Proof. {Lemma 3) We will prove the lemma for any t € N. We first consider the case t € N
= Fi^_i
= li which is deterministic. Assume
and use mathematical induction: Firstly, R\_-y
=
= ii^^ = -^3
Then, by Lemma 9, St+i
that R]
e K for each s > < + 1 for some t <
(l +y(t+i)) A(^j G E and hence R\\t\ = R^^ = max{0, 1 — 6St+\ [<]} € K, showing by mathematical
= Rt^ for some i?t (^ € M for each t <
induction that R\^ = iZf
In that case, by Lemma 9,
= (l +2/(f)) Aj € IR for each t <t, proving our Lemma for finite-horizon case. In the infiniteSt
— 14 [oo], and hence St — St [oo] and Rt — Rt [oo]
horizon case, as ^ — oo, we will show that Vj
Vl
Vt^
[<]
[t\
[t\
{Pi)
[i\
[t\
[i]
[^1
t.
[«]
[*]
\t\
t.
[i]
[i]
[i]
at each
To
t,
showing that St
see that Vt
[f]
—
>
[oo]
and Rt
[t]
»
Vt [oo], first observe that A'
agreement and disagreement regimes at
these equalities that A*
is
This also shows that
=
s [oc] is
[i]
(pj
1
—
—
>
|5y((+i)|
Since A|_j
Vt [oo] (pj)
=
6y(t+\)^\+\ and A'
Using the fact that
t.
strongly stable backwards.
pointwise limit A°°. Therefore, Vt
[i]
»
are also deterministic.
[oo]
=
ilt-i
= p\ {pt) A^
an equilibrium. From now on, we
<
6h\^^ when we have
6
<
1,
one can check from
[*]
=
1,
A' thus possesses a
everywhere.
will take
4
=
oo and suppress i in
our expressions.
Towards proving
B{t
+
1)
and C{t)
last inequality,
Lemma
5, let
us define functions B, B, and
= ^^^T+I^T^
^^ ^^^^
*
C
^ ^- ^°^^ *^^t ^(^)
<
^(*)
one can check that
C{t)
-
m
=
V(.-i) (1
35
- ^y(m))
=
^^}
'^(*) ^*
^^^^
by setting B{t)
^
<
,
*•
B{t)
=
^°^ *^^
We
Lemma:
start with the following
Lemma
10 Given any
=
Proof. Since At
€ PA, and
t
1
G R, we have
c
6,
At+i
=
B{t
+ l)-b
At+i
=
C{t
+ l)+c
—
^^
^^
At
=
B{t)+6y^t+i)b,
(45)
At
=
B{t)-6y^t+l)C.
(46)
we have
6y(^t+i}At+i,
r
= -——.
1 + oy^i+i)
A(+i
showing
for
some
1
1
6y^t+i)
1
+
>
=
c
B{t
+
+
+
5y(t+i)&,
6y{t+i)
i
<5j/(t)
B{t)
+
<
and hence we have At
0,
8y(t+i)b
<
B{t). Likewise, whenever At+i
=
Lemma
12 For any
t
€ PA, B{t) < At
Proof. Take any
t
£ PA, and write
Given any
t
we have Aj+i = B{t +
1),
11
e PA,
B{t
if
—
B{t)
+
1)
<
<
5j/(t+i)C
>
+
<5y(t)
< C{t +
At+i
s
>
t.
Then, using
=
A,
C{t)
Lemma
+ e*+2'
10,
C{t
—
+
b for
some
6
<
0,
we have At+i
1),
and hence we
=
C{t
1),
then B{t)
<
At
<
+
1)
+
c
Lemma.
B{t).
C{t).
iit
=
s
we compute that
- C{t +
[At+2i
<
1)
B{t). This gives us the following
1
each
+
1
(46).
Lemma
for
1
+ <5j/(t+i)
y(t+i) 1
^y(t)
Note that, whenever At+i
have At
-
1
(45). Likewise,
y(t+i) 1
showing
<=> At =
b
^*+2fc
-
21)]
^^(j
Y^
^ 2k) - B{t +
2k)]
.
(47)
0<fc<i-l
for
each
t
€ PA, and
I
>
Using
0.
Lemma
10 and mathematical induction on
I,
one can easily check
that (47) holds.
<
Equation (47) implies that At
|— ^2/(f)|
<
1,
as
follows that, as
Z
Z
—
—
»
—
CO, ^t^^'
>
0.
oo, ^*"^^' [At+2i -
multiplication of evenly
that X^o<fc<;-i ^t^
C{t)
many
when
Since
C{t
61+"^'
0.
»
negative numbers.
is
To
see this, note first that, since
—
C{t + 2l)\ < 1 at each t,l, it
> for each k, as it consists of
Since C{t + 2k) — B{t + 2k) is always positive, it follows
further have |Af+2(
+ 21)] —
+ 2A;) — B{t + 2fc)]
exists
€ N such that
[*-^(*
zero. Therefore, there
we
sufficiently large.
is
/
Second,
^*+^''
positive, increasing in
I,
and hence bounded away from
Z'
[At+2i
-
C{t
+ 21)] -
^t"^^*" ["^(^
Y,
o<fc<;-i
36
+ 2't) -
B{t
+
2k)]
<
(48)
whenever
On
/
>
/'.
the other hand, using (47) at
=
A,
t
+
and
1
(46),
one can also obtain
B{t)-6y^,+,)elXf+'[\t+2i+i-C{t
+6y(t+i)
<^lVi^^' [C(t
Y.
+
2l
+
\)]
+ 1 + 2k)-B{t + l +
2k)].
(49)
0<fc</-l
Of
course, by (48), there exists
-6y{t+iAtf^'
[A(+2i+i
some
- C{t +
21
I"
+
N
€
1)]
+
such that
6y^t+,^
^t+f
Y.
^'
[^(*
+
+
2^'
1)
" ^(* +
'2k
+1)]>0
0<k<l-l
(50)
whenever
I
>
whence we have At > B{t). Therefore,
I",
for
any
I
>
Tnax{l' I" }
,
and (49), we have B{t) < At < C{t). u
Lemma 5.
Proof. {Lemma 5) When t ^ PA, our Lemma is vacuously true. Take any t
we have B{t + 1) < A(+i < C{t + 1), hence by Lemma 11, we have B{t) < Aj <
,
inequaUties (48) and
(50) simultaneously hold, hence by (47)
We
can
now
prove
Lemma
12,
Review of Economic Studies
54:
e PA. By
B{t).
U
References
[1]
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in bargaining,"
345-364.
[2]
Aumann,
R. (1976): "Agreeing to disagree," Annals of Statistics
[3]
Aumann,
R. (1995):
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A
Aumann,
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[12]
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Milgrom,
P.
and N. Stokey (1982):
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19]
P.
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strategically
timed
Posner (1974): "An economic approach to legal procedure and judicial administration," Journal
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Reny
Journal of Economic Theory, 59, 50-77.
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Roth, A., K. Murnighan and F. Schoumaker (1988):
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Rubinstein, A. (1982); "Perfect equilibrium in a bargaining model," Econometrica, 50-1, 97-110.
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Sakovics,
J.
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Stahl,
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Yildiz,
M.
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25]
Yildiz,
M.
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I.
mimeo.
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38
Date
Duf"
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