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Institute of
OPTIMISM, DEADLINE EFFECT,
AND
STOCHASTIC DEADLINES
Muhamet Yildiz
MIT
Working Paper 04-36
August 2004
RoomE52-251
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Cambridge,
02142
MA
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MASSACHUSETTS INSTITUTE
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i
Optimism, Deadline
Effect,
and Stochastic
Deadlines*
Muhamet
Yildiz
MIT
August, 2004
Abstract
Under a firm deadline, agreement
until the deadline.
I
in bargaining
propose a rationale
is
often delayed
for this deadline effect that
naturally comes from the parties' optimism about their bargaining
power.
is
I
then show that the deadline effect disappears
stochastic and the offers are
made
Key words: optimism, deadline
if
the deadline
arbitrarily frequently.
effect,
bargaining
JEL Numbers: C72, C73.
Introduction
1
When
last
there
is
a firm deadline, the agreement
minute before the deadline.
is
often delayed until the very
This "deadline
effect" is
served in a wide range of laboratory experiments as well as
ations (See
e.g.,
commonly
ob-
real-life negoti-
Roth, Murnighan, and Schoumaker (1988)). For example,
*This paper stems from
my
dissertation submitted to
Graduate School of Business,
Stanford University, and some of the ideas here are also expressed in an earlier working
paper
(Yildiz, 2001).
I
thank
my
advisor Robert Wilson.
and John Ktnnan.
1
I
also
thank Thomass Helmann
the unions and employers reach "eleventh-hour agreements" just before the
unions' deadline to strike. Litigants often reach a "settlement on the cour-
thouse steps." In these and
many
other real-life negotiations the parties re-
portedly hold overly optimistic views about their relative bargaining power
(Babcock and Loewenstein (1997)). Such optimism has long been recognized
as a major factor in bargaining delays (see,
I
propose a rationale for the deadline
optimism about their
parties'
The
Hicks (1937)). In this paper,
effect that naturally
relative bargaining power.
the deadline effect disappears
relative bargaining
e.g.,
if
power may
the deadline
is
I
stochastic
comes from the
then show that
and the
parties'
shift quickly.
rationale for the deadline effect
as follows. Consider
is
and B, negotiating under a firm deadline
d*,
which
two
parties,
A
the last date at which
is
they can strike a deal. After a deadline, they can no longer negotiate, and
each party receives his disagreement payoff, which
is,
at d*
,
the cost of delay
high. Hence, at d*
is
,
is
presumably low. That
a wide range of agreements
are possible as outcomes of bargaining. For example,
if
party
A
has a strong
bargaining position that allows him to set the terms of trade, leaving the
other party no other option than that of accepting or rejecting these terms,
then they reach an agreement in which
gains from trade
if
B
is
and
leaves
B
A
B
indifferent to disagreement.
bargaining power, they
may
extracts almost
all
of the large
almost indifferent to disagreement. Similarly,
in such a strong position, then
and leave
A
strike
extracts
all
the gains from trade
Depending on the
any deal that
is
in
parties' relative
between these two
extreme points. Therefore, a party's share in an agreement at d*
affected
by
his relative bargaining power. Hence, at
negotiation, each party
his share in a possible
may have
any
is
greatly
earlier date in the
very high or very low expectations about
agreement at
d*
,
about bis relative bargaining power at
depending on his
d*
.
level of
optimism
In case of excessive optimism,
these expectations will be so high that, unless waiting until the deadline
too costly, there will be no agreement at the
moment
parties' expectations. Therefore, the players wait
is
that could satisfy both
and reach an eleventh- hour
agreement just before the deadline. The cost
may be
gain from trade
lost
due to the deadline
Now consider an environment
stochastic. Formally,
may be
in
very high; half of the
effect.
which the deadline
assume that the deadline
is
is
not fixed but rather
a random variable with a
continuous cumulative distribution function. For example, consider a market
may
that does not necessarily close sharply at 5:00pm but
in
close at
any time
between 4:50pm and 5:10pm. For another example, consider a labor con-
tract that does not expire at a fixed date but expires
event happens,
e.g.,
when the
when a
certain
random
inflation rate exceeds a certain threshold.
commits
yet another example, consider a union that
For
to a strike that will
be triggered by such a random event, rather than automatically starting at
a fixed date. Assume that the factors which determine
may change
power
quickly, so that
parties' bargaining
whether a party has a strong bargaining
position today does not have an impact on the probability of that party hav-
ing a strong position tomorrow.
Assume
opportunities to strike a deal.
In such an environment, 1
have frequent
also that the parties
I
show that the
deadline effect disappears, and the parties reach an agreement immediately.
An
intuition for this result
the parties find
it
very
is
Consider a date
as follows.
likely that there will
explained above, the perceived cost of delay
is
deadline
is
at which
be no more negotiations. As
high at
t*
,
and the parties may
be highly optimistic about their shares in an agreement at
when the
t*
t*
.
Nevertheless,
stochastic, the ex-ante probability that the parties will
face the deadline before such a date t*
must be very
high.
2
not willing to wait until
sufficiently earlier date, the parties are
Hence, at a
t*
to realize
their expectations because that incurs a very high risk of receiving the
1
that
2
is
In the formal model,
is
I
also
make an assumption
low
to rule out any anticipated deadline
implicit in players' beliefs.
This
is
because the ex-ante probability that the deadline arrives before the date
at least as high as the conditional probability of that event at t*
be very high. But the former probability
the the deadline arrives before the date
is
,
which
is
t*
+1
assumed to
approximately the ex-ante probability that
£*, for
the dates
t*
and
very close in real-time and the cumulative distribution function
f+
is
1
are
assumed to be
continuous.
payoff of disagreement.
On the other hand,
the parties expect to have
if
many
opportunities to strike a deal in the future and parties are not pessimistic
about their bargaining power in the future, then their expectations about the
future are high. Hence the perceived cost of delay
limits the scope of the individually rational trade.
in a possible agreement are not affected
is
That
relatively small.
Then, the parties' payoffs
much by
their bargaining power.
Therefore, the parties' optimism about their bargaining power induces only
low
levels of
optimism about their shares. 3
There are arbitrarily close stochastic deadlines to any deterministic dead-
The above
line.
results describe quite different equilibrium behavior
such two similar environments.
mism
under
While a deterministic deadline and
opti-
naturally lead to delaying of agreement until the deadline, under any
nearby stochastic deadline, there
environment
From
is
will
be immediate agreement whenever the
sufficiently variable.
a theoretical point of view, the above two results extend the
cri-
tique in Yildiz (2003) of the literature that explains the bargaining delays
by optimism, using informal arguments that
fit
to a two period model,
excessive optimism leads to a delay. In Yildiz (2003),
mism
The
is sufficiently
first
persistent, then there will
result in this
showed that
I
opti-
be an immediate agreement.
paper shows that optimism naturally leads to the
—just as
deadline effect under a deterministic deadline!
it
leads to a delay in
a two period model. The second result shows that, nevertheless,
slight uncertainty
if
when
about the deadline so that
be an immediate agreement
it is
if
there
is
stochastic, then there will
in the continuous-time limit, the limit case that
attracted most of the attention in bargaining literature.
In the continuous-time limit, the players' relative bargaining power
allowed to shift back and forth instantaneously.
In practice, however, the
factors that determine players' relative bargaining power
suddenly and that often. Hence, optimism
3
Although
this intuition
is
may
do not change that
lead to the deadline effect
valid also for multilateral bargaining,
some delay when there are many
parties (Ali, 2002).
is
it
may
not prevent
even under a stochastic deadline. Therefore, from a practical point of view,
one must interpret the
last result as follows:
the deadline effect disappears
when there is sufficient uncertainty about the deadline relative to how quickly
the players get a chance to strike a deal and
how
quickly their relative bar-
gaining power changes.
A
firm deadline
is
often imposed
upon
them from dragging out the negotiations
negotiators in order to prevent
indefinitely. Ironically,
such dead-
themselves sometimes entice parties to delay the agreement.
lines
happens because of the
such a deadline
effect
that will happen at a
then one
parties' optimism,
by imposing a deadline that
random time and
is
may be
is
beyond the
If this
able to avoid
triggered by an event
parties' control.
The
uncertainty about the timing of the deadline must be high enough so that,
during the period at which such an event happens, the parties will have
many
opportunities to strike a deal and their relative bargaining power will
potentially change
many
times.
Spier (1992) shows that, in a pre-trial negotiation with incomplete in-
formation, the settlement probability will be a U-shaped function of time,
consistent with the deadline effect.
model that explains the deadline
Ma
effect.
and Manove (1993)
also develop a
In that model, the delay
and a party can wait as much as he wants before making an
opponent has to wait for his
fer waits until the deadline
In their model, there
is
made and
it is
is
offer.
Then, the party who
and makes a
his
make an
of-
it offer.
between the times an
offer
may not go through. This
model may not disappear when the
accepted. Hence, a late offer
Indeed, Roth, Murnighan, and Schoumaker
deadline becomes stochastic.
is
to
costles
and
offer,
minute take-it-or-leave
also a stochastic delay
suggests that the deadline effect in that
(1
last
is
is
988) informally discuss a possible explanation based on the idea that there
no cost of delay except
about the deadline. 4
''Roth
for a cost at the
When
end due to a
the deadline becomes stochastic, the deadline
and Ockenfels (2002) consider a similar model
by the possibility that the
slight uncertainty
last offer
may
in
which the delay
is
motivated
not go through. They use this model to explain
effect disappears in
some models while remains
intact in
some
other.
Hence
one can use stochastic deadlines to test these models.
Model
2
There are two
among themselves
dollar
N=
6
risk averse agents, i,j
who want
{1,2},
before a deadline; the set of
all
to divide a
feasible expected
place on a
U — {(u ,!?) G [0, lplu + u 2 < 1}. The negotiation takes
grid T = {0,1,2,...} of index-times t that approximates the
continuum
[0,
1
utility pairs is
r
(t,
=
k)
Each index
oo) of real times r.
where
t/k,
The
grid.
1
a large integer that measures the fineness of the
is
A;
corresponds to a real time
t
players' time preferences
and the deadline are given by the
time and do not depend on the grid. Each player's
r
is
e
_rT
x where r
discount rate
I
is
>
(£,
d
if
=
<
k)
2
,
—
At t
0.
.
if
and only
d,
The
with probability
d*
)
d,
if
deadline
1 for
F
the negotiation automatically ends and
is
some
is
d*
said to be deterministic
S
[0,
oo).
6 U;
if
game.
Given any
each player
i
t
,
<5*u );
6
iV;
otherwise, the
is
is
i
offers
Notice that,
if
At each
t
with
an alternative v
=
game ends yielding
game proceeds
—
real
time
to date
t
+ 1,
F (t (t,k)
before the recognition at
r'
\d
>
t'))p\ to the event that
observed in p-Bay auctions but not
firm; in
said
is
which case the game ends yielding payoff vector
and any history with
the deadline effect
10 minutes.
2
1
assigns probability (1
e-Bay, the deadline
Amazon can be
i
the other player accepts the offer, then the
a payoff vector S u = ((5'n
unless r (t + 1, k) > d, in
(0, 0).
The deadline
and
if
continuous.
Nature recognizes a player
t
why
rl k
will analyze the following perfect-information
(w 1 u
t,
e~
model the deadline as a positive random variable d with cumulative
to be stochastic
r
=
denoted by 5
each agent gets payoff of
I
from getting x at
measures the real-time impatience. The index-time
distribution function F.
only
utility
real
Amazon, auction may not end
if
in
Amazon
auctions.
In
there was a bid in the last
there are uncertainty about the entry, then the deadline in
considered stochastic. Incidentally, the results here can also be extended
to auction environments.
6
i
be recognized at date
will
about
t,
where
on r
this event conditional
(t,
E
p\
<
k)
his probability assessment
[0, 1] is
d.
Everything above
is
common
knowledge.
This model
it
is
similar to that of Yildiz (2003)
allows a stochastic deadline.
dom bargaining breakdowns,
(1987), Merlo
and Wilson
Its
only difference
the discounting
is
that
due to the ran-
is
the Rubinstein-Stahl framework
(e.g.,
Binmore
(1995)) also allows stochastic deadlines. In that
framework, the discount rate
is
When
.
is
usually taken as constant over time, and
it
thereby assumed that the deadline has an exponential distribution. This
paper does not impose any such restriction on deadlines because that would
rule out the critical class of stochastic deadlines that
istic deadlines.
however,
is
Its
that
it
approximate determin-
main departure from the Rubinstein-Stahl framework,
allows the players to hold subjective beliefs about which
player will be recognized
and when, and these
beliefs possibly differ
from
each other.
A
player's bargaining
power
is
determined by the recognition process; a
the present value of the temporal monopoly
player's equilibrium payoff
is
rents he expects to extract
when he
is
recognized (Yildiz, 2003, 2004).
The
differences in players' beliefs about the recognition process reflect their opti-
mism
date
or pessimism about their bargaining power. The level of optimism for
t is
measured by
=
Vt
The
= p\ + Pt ~
1-
players' probability assessments for date
t
agree
players are weakly optimistic (resp., pessimistic) for
?/t
<
0).
The main
focus in this paper will be
t
when y =
when y >
do not update
player
is
is
0.
The
(resp.,
t
on the case that the players
remain (weakly) optimistic throughout the game, although
Notice also that p\
t
it is
not assumed.
a constant. This reflects the assumption that players
their beliefs about the future events as they observe
recognized at a given date. This independence assumption
in order to rule out the delays that are
unrelated to the deadline
effect.
due to learning
is
which
made
(Yildiz (2004)),
Equilibrium
3
One can
iteratively eliminate all conditionally
dominated strategies to reach
an essentially unique subgame-perfect equilibrium. The equilibrium
is
unique for
all
own
In this section,
I
a
trivial
(The continuation values are
as the present value of continuation payoffs at
expectations.)
may be
subgame-perfect equilibria; there
multiplicity of equilibria in knife-edge cases.
computed
unique
any player at the beginning of
in the sense that the continuation value of
any date
is
will derive
using the player's
t,
a difference equation that
determines these continuation values and describe the equilibrium behavior
and the conditions that determines whether players reach an agreement
at
any given date.
Let VJ be the continuation value of a player
t
<
conditional on that r (t,k)
when r (t,
k)
>
any date
the players share a
(t,
k)
<
the surplus or the perceived
d,
t is
St
If
at the beginning of a date
the continuation values are identically zero
Conditional on r
d.
size of the pie at
d;
i
common
=
v?
+ v2
t
.
set of beliefs, St is identically 1.
may be higher
players' subjective beliefs are allowed to differ in this paper, St
or lower than
For each
1.
qt
write also
t,
=l-F(T(t + l,k)\d>T(t,k))
for the conditional probability that the deadline
t
+
1
given that the deadline
then payoffs at
t
+
1
Since the
by the
is
not reached at
is
t.
not reached at index time
At date
players discount
t
effective discount factor of Sqt
The discounted
-
value 8qt St+i of the next period surplus determines whether players reach an
agreement
at
t
in equilibrium
if
they have not yet agreed.
Consider the case that 5qt St +i
>
1.
Since each player
counted payoff of Sqt Vt +1 from delaying agreement beyond
i
expects a dis-
t,
an agreement
l
at
t
must give
5<ltV£-i
=
at least 5qt
Sq t St +i
>
1.
V +l
z
t
to each player
i,
requiring a
Hence the players cannot agree
at
sum
t
of 5qt V*+1
+
in equilibrium.
This case
i
is
at the beginning of
and the surplus
Now
The
called the disagreement regime.
continuation value of player
t is
is
St
=
consider the case 5qt S +i
<
t
5qt St+ i.
1.
(1)
Now
an agreement
at
t is
possible but also necessary for equilibrium whenever the inequality
This case
is
called the agreement regime.
other player j
i
offers j this
is
willing to agree to
When
a player
%
— 5q V^+1
rest, 1
t
,
is strict.
recognized, the
is
any division that gives j at
amount and keeps the
not only
least Sqt
for himself,
VJ+1
t
;
an amount
more than 5q V?+1 his continuation value from delay. (The latter
amount would be his share if he were not recognized.) The continuation
that
is
t
value of player
V;
i
,
at the beginning of
= P\{1- 5q VJ+l +
(1
)
t
- pi)
t is
now
5qt Vi+l
= p\ (1 - 5q St+l + 5q
t
In that case, the surplus at the beginning of
S =
t
4
l
.
t is
+ y {l-5q St+ i).
t
(2)
t
I will
show that optimism and a deterministic deadline
near future lead players to wait until the last date to
effect
disappears and there
stochastic
and the
Proposition
e
VJ+1
Deadline Effect
In this section,
t*
t
)
=
-rd'
but this deadline
immediate agreement whenever the deadline
is
offers are
settle,
made
frequently
(i.e.,
k
is
1 Let there be a deterministic deadline at
some
d*
> 0, and let
Assume that
(t,
k)
<
(1 -f-^„)
>
1.
Then, in equilibrium, the players disagree at each
d*} be the last date before the deadline.
at t*.
is
large).
max{i|r
and reach an agreement
in the
t
<
t*
Proof.
regime at
each
<
t
t*
and S =
since q =
,
t*,
and there
will
£**"*&.
5
>
r
t»
1
t
1,
0,
+
=
5qt *St *+i
yt *
(1
—
S >
e~
t*
Td '
(1
yt .)
>
=
t
1
t
*
t
l
1
each
1 for
<
t
t*
by the hypothesis.
a simple generalization of Example
1 is
an agreement
is
Moreover, for
+y
5*
S = 5S +i =
^Sf will hold,
at t, so long as 5 ~ Sf > 1- But
the equality
+
Hence, there
0.
5qt *Sf+i)
be a disagreement regime
Proposition
is
—
Firstly, since qt *
of Yildiz (2003). It
based on the basic observation that the recognized player just before the
deadline has a great bargaining power, as he makes a take-it-or-leave-it
If
the deadline
not too far away and the players are sufficiently optimistic
is
about being recognized and dictating a very favorable settlement at
it is
hypothesized in Proposition
before
t*
agreement until the
This behavior
is
last
The next
many
when
will delay
the
and
is
commonly observed
in real-
under which the deadline
effect dis-
the deadline becomes stochastic.
e
>
the deadline be stochastic
0, there exists
in equilibrium before real-time
(i.e.,
let
F
be continuous).
k such that the players reach an agreement
for each k
e
ing two conditions are satisfied:
(i)
yt
>
>
k whenever either of the follow-
for each
t,
or
(ii)
yt
= Y (r (t, k))
k for some continuous function Y.
is,
if
the deadline
is
agreement in continuous-time
tions
be no feasible agreement
experiments.
result provides conditions
Then, given any
That
will
then predicts that the players
called the deadline effect
Proposition 2 Let
all t,
then there
—as
date before the deadline and settle at the last date.
world negotiations and in
for
1,
t*
that satisfies both players' (highly optimistic) expectations. Under
this hypothesis, Proposition 1
appears
offer.
above
is satisfied.
stochastic, then there will
limit, so long as either
be an immediate
one of the two condi-
Such conditions are not superfluous because the play-
ers' beliefs (particularly their
pessimism)
may reflect an
anticipated deadline,
and thus considerations about a deterministic deadline may be disguised as
pessimistic beliefs.
pessimistic about
To
t),
see this, note that
when
yt
<
the analysis remains unchanged
10
(i.e.,
if
when
players are
we instead assumed
commonly
that the players
with probability
believed that, at
and no player
p\
each player
t,
recognized with probability
is
recognized
is
i
—yt
.
Hence,
a deterministic deadline at a date d* can be implicitly incorporated in players'
by assuming that yt
beliefs
= —1
each
for
>
with r(t,k)
t
d*
Condition
.
(ii)
by
rules out such possible incorporation of deterministic deadlines in beliefs
jumps
ruling out sudden
one only needs to rule out sudden drops in the
below. Condition
out,
random
variable
all
Proof of Proposition
agreement regime.
2,
is
t
(tk,
k)
€
for the proof
consistent with empirical evidence for
is
I will
For each
2.
show that
limfe^,*,
log (2) /r] for each k,
[0,
be the
k, let tk
r
k)
(tk,
=
and hence
the only limit point of the sequence r
Since
0.
date with an
first
Stk
it suffices
is
to
tk
5-q
the inequality
>
By
0.
=
is
5
show that
.
Consider
(1),
Stk =
tk
(
bounded
Let r* be any limit point
(tk, k).
of this sequence, and consider a subsequence that converges to r*
any k with
by the
optimism (Babcock and Loewenstein (1997)).
players' persistent
by
optimism
considerations about deadlines are captured
This condition
d.
level of
which states that the players are optimistic through-
(i),
makes sure that
with this intuition,
in level of optimism. Consistent
J!
ft
t
5
*(l-F(r (t k
,
k)))
Stk >
1;
(3)
J
due to the
fact that there
is
a disagreement regime at
0.
Hence,
=
—
A
(l-F(T(t
k
1
will
show that 1/A k converges
e-
to
rT "
k ,k)))Stk
th
lim 5
fc
and
therefore, r*
=
0.
Towards
St k
as there
is
5
tk
<
1.
(4)
{)
proving that
1,
=
<
-
—>oo
=
1,
this goal, note first that
=l + yt
an agreement regime at
k
(1
tk,
-
5qtk Stk+1 )
which
11
also implies that 5qt k Stk +\
(5)
<
1.
I
1.
consider the two conditions separately. First assume
[If
= 1 + Vt
S*»+i
>
5qtk+1 Stk +2
Stk+1 = 5qtk+l Stk+2 > 1- If 5qtk+1 Stk+2 <
- 5qtk+1 Stk + 2 ) > 1, as ytk+1 > 0.] Hence, by (5),
then
1,
+\ (1
k
Stk +i >
Then,
(i).
Stk <2-5qtk
1,
then
(6)
.
Thus,
< (l-F(T(t k ,k)))(2-5qtk )
^4fc
= 2(l-F(r(t
Since
F
Ak >
Now
converges to
by
1
<
assume
>
Vt k +i
to
be
by
0,
(ii)
then
0,
and similarly
=
that Y{t*)
are covered,
By
bound
Ak
for
1.
=Y (r
yt k +i
1,
(ifc,
is
Ak <
—
t,
be delays
shows.
=
1, fc)
r*, the
—
now
1
it suffices
in the limit.
If
when
is
1.
0.
But,
1,
&))
<
if
this case occurs infinitely
Y (r*) >
by continuity of F,
imphes that
A k < Stk <
1
+
y (r*) <
y tk -+
1-
0,
AH
showing
the cases
complete.
there
is
no deadline, condition
—
(i)
guaranties im-
qtS as the discount rate for
model with time varying discount
One
cannot, however,
may
rates, as Proposition 1
in other sequential bargaining models, equilibriumi
5=1, and
1).
1.
>
clear that
not greater than
=
part of the proposition from Yildiz (2003) because there
in such a
As
ous near
first
+
(tfe
and thus 1/A&
obtaining a bargaining model with no deadline.
deduce the
-M,fc)))-.
(Recall that linifc^oo^
1,
mediate agreement. Moreover, one can take 5t
date
r
limfc_ 00
It is
(i).
implies that
+
(ifc
<
yt fc +i
(5), this yields
and the proof
Yildiz (2003),
that
>
k))
= Y (r
By
A*
fc
Assume ytk > 0. If we also have
implying the bound in (6), which has just proven
Hence, assume
0.
<
F(r*)
1 yielding
Stk +i >
sufficient.
often, then ytk
—
=
without assuming
Stk <
(5),
1
(£&, fc)
(4), this implies that
to find an upper
Vt k
,fc)))-J(l- JP(r(t
continuous and lim^oo r
is
last expression
Since
fc
is
discontinu-
hence one cannot use a continuity argument. (With a
stochastic deadline, each 5 t
—
>
1
as k
—
12
oo.)
Conclusion
4.1
These deadlines often entice the
Parties often negotiate under a deadline.
parties to wait until the deadline to
that such a deadline effect
may
make a
last
minute agreement.
I
show hat one may avoid
the deadline effect, by making the deadline stochastic,
deadline contingent upon an event that will
so that parties have
many
show
naturally arise from the parties' optimism
about their bargaining power. More interestingly,
order this to succeed there
I
must be
happen
i.e.,
at a
sufficient uncertainty
by making the
random
date.
In
about the deadline
opportunities to strike a deal and their relative
bargaining power has a potential to change during the period in which they
may
face the deadline.
References
[1]
Ali, S.
ases:
[2]
Nageeb M.
Waiting To
(2003): "Multilateral Bargaining with Subjective BiSettle,"
Stanford University, mimeo, 2003.
Babcock, Linda and George Loewenstein (1997): "Explaining Bargaining Impasse:
The Role
of Self-Serving Biases,"
Journal of Economic
Perspectives 11, 109-126.
[3]
Binmore, Kenneth (1987): "Perfect Equilibria in Bargaining Models,"
in
The Economics of Bargaining,
ed.
by K. Binmore and
P.
Dasgupta.
Oxford, U.K.: Basil Blackwell, 77-105.
[4]
Hicks,
John
(1932):
The Theory of Wages.
New York:
Macmillan Pub-
lishing Co.
[5]
Ma, Ching-to Albert and Michael Manove
(1993):
"Bargaining with
Deadlines and Imperfect Player Control," Econometrica 61, 1313-1339.
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[6]
Merlo, Antonio and Charles Wilson (1995):
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of
Sequential Bargaining with Complete Information," Econometrica 63,
371-399.
[7]
Roth, Alvin, Keith Murnighan, and Francoise Schoumaker (1988): "The
Deadline Effect in Bargaining: Some Experimental Evidence," American
Economic Review
[8]
78, 806-823.
Roth, Alvin and Axel Ockenfels (2002): "Last-Minute Bidding and the
Ama-
Rules for Ending Second-Price Auctions: Evidence from eBay and
zon Auctions on the Internet," American Economic Review,
92, 1093-
1103.
[9]
Rubinstein, Ariel (1982): "Perfect Equilibrium in a Bargaining Model,"
Econometrica 50
[10]
Spier,
97-110.
,
Kathryn E.
Review of Economic
[11]
Stahl,
[12]
Yildiz,
I.
"The Dynamics of Pretrial Negotiation," The
(1992):
Studies, 59 (1), 93-108.
(1972): Bargaining Theory (Stockholm School of Economics).
Muhamet
"Sequential Bargaining without a
(2001):
Prior on the Recognition Process,"
MIT
Common
Department of Economics
Working Paper No. 01-43.
[13] Yildiz,
Muhamet
(2003):
"Bargaining without a
Agreement Theorem," Econometrica
[14]
Yildiz,
Muhamet
Economics
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f]
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'
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— An
71, 793-811.
(2004): "Waiting to Persuade,"
119, 223-248.
Common
Quarterly Journal of
Date Due
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