Durable Bargaining Power and Stochastic
Deadlines
Alp Simsek and Muhamet Yildiz
MIT Economics Department
January 4, 2008
Abstract
Observing that standard assumptions imply extreme discontinuity of bargaining power
in continuous time limit, we develop a model of continuous bargaining power in continuous time. We formally introduce durability of bargaining power and show that it
plays a central role, especially when the parties are optimistic. In particular, when
bargaining power becomes more durable after a date, optimistic parties are enticed
to wait until that date, leading to a period of disagreement. We show that when
players are optimistic and bargaining power is durable around a period during which
a (possibly stochastic) deadline is likely to arrive, the parties are enticed to delay
the agreement until the deadline, as widely observed in experiments and real-world
negotiations. We show that this deadline effect is stronger when the bargaining power
is more durable around the likely time of deadline, or when the uncertainty about the
deadline is smaller, or the parties are more optimistic.
1
Introduction
In sequential bargaining, the bargaining power is modeled by the process that determines who makes an offer and when. We observe that the usual assumptions on this
process tantamount to assuming that the bargaining power is extremely transient
and discontinuous in the continuous-time limit. Moreover, many difficulties in the
continuous-time limit seem to arise from such an extreme and unrealistic discontinuity assumption. In this paper, we develop a continuous-time model of bargaining
power in which the bargaining power is piecewise continuous and analyze the effects
of durability of bargaining power, optimism, and stochastic deadlines. We show that
with optimism, durability of bargaining power plays a central role. As an application, we provide a rationale for the deadline effect, arguably one of the most robust
empirical regularities in bargaining.
More precisely, we consider two agents who want to divide a dollar. We define an
agent’s bargaining power as the amount of share the agent gets from the surplus, in
addition to her continuation value. (Here the surplus is the total cost of not reaching
an agreement at the time.) In sequential bargaining, the proposer gets the entire
surplus, leaving the other party indifferent between agreeing and disagreeing. Hence,
the proposer has all the bargaining power (for that time), which is sometimes called
temporal monopoly power. While this could be a reasonable way to model bargaining
power, the assumptions we usually make are appropriate for literal interpretation of
bargaining procedure, but not for real-life bargaining power. In particular, under the
usual assumptions, bargaining power becomes extremely discontinuous and transient.
For example, Rubinstein’s (1982) alternating-offer bargaining assumes that bargaining
power shifts from one end to the other frequently, so much so that the process is
not defined in the continuous time limit. Similarly, random proposal model with
independence (Binmore, 1987) assumes that bargaining power is so transient that the
bargaining power now has no impact on the bargaining power a moment later, leading
1
to highly discontinuous sample paths.
In this paper, we take bargaining power exogenously given as a stochastic process
defined on continuous time. Allowing bargaining power of an agent to take any value
between 0 and 1 (as opposed to just 0 or 1), we focus on the case that the sample
paths of this process are piecewise continuous almost surely. We allow parties to
have differing, possibly optimistic, beliefs on the future bargaining powers, but we
assume they know what each agent’s current bargaining power is. Applying backward
induction under the given process, we determine, for each instance, whether there
will be an agreement and what the agreement will be.1 We formally define and
quantify the durability of the bargaining power. It is roughly defined as how slowly
the bargaining power changes as time passes.
We illustrate that durability of bargaining power plays a profound role when the
parties are optimistic. Imagine that bargaining power will be very durable starting
a particular date t∗ . For example, at t∗ , there will be a general election that will
determine who will be in the government for the next several years; e.g. we may
have a Democratic president with a Democratic majority in both houses, or a form
of split government, or Republicans controlling all branches, etc., each leading to a
different bargaining power for each party. Now at t∗ , knowing that their bargaining
powers in a considerable future will be similar to their current bargaining powers,
the agents will hold very similar beliefs about their future bargaining powers, and
they will reach an agreement, each getting a share that is in accordance with her
bargaining power at t∗ . Now consider an earlier time t̂ at which the parties have a
considerable uncertainty about the bargaining power after t∗ . For example, a year
before the election, the parties may expect that there may be several different events,
such as a terrorist attack or a political scandal, that will affect the election results.
(Notice that this assumes that the bargaining power is somewhat less durable before
1
While our method can be modeled as a subgame-perfect equilibrium of a game (see Proposition
1), we consider our method of solution more straightforward and we do not necessarily refer to the
game.
2
t∗ , which is natural in the example of an election.) Now, at t̂, the agents’ belief about
the bargaining power at t∗ may differ dramatically. Since their share in the actual
agreement at t∗ is closely related to their bargaining power at t∗ , this means that
their beliefs about each agent’s share in an agreement at t∗ may differ dramatically.
In particular, if the agents are optimistic about their bargaining powers at t∗ , they
will be optimistic about those shares. Such optimism may render an agreement at
t̂ impossible because agreement has to give each agent at least as much as what
she expects to get from waiting until t∗ . More generally, we show that if players
are optimistic about their bargaining powers at t∗ (which requires that bargaining
power is somewhat less durable before t∗ ), and the bargaining power will become
more durable after t∗ , then there will be a period of inactivity before t∗ in which
there will not be any agreement, and there will be an agreement afterwards with high
probability. (In the election example, this suggests that there is a force that leads the
congress to pass fewer laws before the elections.)
Now imagine that the durability of bargaining power does not necessarily change
much, but there is a deadline, after which the parties cannot agree. Suppose that the
time of the deadline is stochastic. For example, it is uncertain when an arbitrator
will make a decision, after which bargaining is mute. Suppose also that it is very
likely that the deadline will arrive within a small interval of time that starts at
t∗ . In that case, the effective discount factors in that interval will be very small,
depending on the conditional probability of arrival at that moment. This is the same
as stretching the time in that interval, stretching more when the arrival is more likely.
If the bargaining power is somewhat durable in that interval (e.g. it has Lipschitzcontinuous sample paths with high probability), this stretching of time amounts to
making bargaining power more durable (e.g. making the sample paths Lipschitzcontinuous with a smaller coefficient). In that case, the argument in the previous
paragraph tells us that this will lead to a period of inactivity before t∗ , followed by an
agreement with high probability. In fact, under the assumption that bargaining power
3
is durable around t∗ , we show that if it is sufficiently likely that the deadline arrives
within a sufficiently small neighborhood of t∗ and the parties are sufficiently optimistic
about their bargaining power at t∗ , then they will not settle until sometime around
t∗ , and they will reach an agreement "just before the deadline" with high probability.
(With some small probability they will fail to agree before the deadline.)
This behavior is the “deadline effect” that is commonly observed in a wide range
of laboratory experiments as well as real-life negotiations (see e.g., Roth, Murnighan,
and Schoumaker (1988)), where the agreement is often delayed until the very last
minute before the deadline. For example, the unions and employers reach “eleventhhour agreements” just before the unions’ deadline to strike. Litigants often reach a
“settlement on the courthouse steps.” The above result provides a rationale for the
deadline effect, based on optimism, which is reportedly common in real life negotiations (see Babcock and Loewenstein (1997)).
More importantly, we find explicit conditions under which the deadline effect occurs. In a canonical, stylized model (and more generally as a sufficient condition), we
show that whether there will be a deadline effect depends on the ratio of arrival rate
of deadline to the "inverse rate of durability". When this ratio exceeds a cutoff value,
there is a deadline effect and otherwise there is not. That is, keeping everything else
constant, if we make the deadline arrive faster, or make the bargaining power more
durable around the deadline, then the deadline effect will be stronger. Moreover, the
cutoff is a decreasing function of the optimism level about the bargaining power at t∗ ,
i.e. optimism makes the deadline effect stronger. Interestingly, our analysis also establishes that there is no deadline effect under the usual independence assumption for
the bargaining power when the deadline arrival has a continuous distribution. That
is, even in cases in which there is a deadline effect with a deterministic deadline, the
deadline effect disappears in the continuous time limit when the arrival process has a
continuous distribution, no matter how close this distribution approximates the deterministic deadline. The extreme independence assumption implies that the bargaining
4
power is extremely non-durable, which further implies that the potential agreement
payoffs are not related to the instantaneous bargaining power.(as long as the deadline
arrival is not completely deterministic). Hence, initial levels of optimism about the
bargaining power does not translate to optimism about agreement payoffs, enticing
the agents to agree earlier despite their optimistic beliefs about the bargaining power
around the deadline. This result suggests that some durability for the bargaining
power is necessary to have a robust deadline effect.
A firm deadline is often imposed upon negotiators in order to prevent them from
dragging out the negotiations indefinitely. Ironically, such deadlines themselves sometimes entice parties to delay the agreement. The above comparative statics show that,
if this happens because of the parties’ optimism, then one may be able to avoid such a
deadline effect by making the time of deadline more uncertain, or putting the deadline
in a period at which the bargaining power is less durable. For example, our results
suggest that if one wants to avoid the deadline effect, he should put the (stochastic) deadline before an election, rather than after the election. Before the election,
the bargaining power is less durable because of the upcoming elections and initial
optimism about the bargaining power for that period is relatively low. After the election, the bargaining power becomes more durable and the initial optimism about the
bargaining power is high. Both of these features make the deadline effect stronger.
The outline of the paper is as follows. In the next section, we lay out our model
and characterize the unique solution. In Section 3, we analyze a canonical stylized
model in which the bargaining power changes only after certain events that arrive with
Poisson distribution and the deadlines are exponential. In this model, the inverse rate
of durability is the arrival rate of the Poisson process, i.e., the rates of durability and
deadlines are constant. Consequently, we are able to find an explicit solution and
characterize the cases with delay explicitly in terms of the ratio of these rates. In
Section 4, we establish that the solution is discontinuous under independence. Section
5 is the heart of the paper. There, we formally introduce our notions of durability and
5
present our main results for the general case. In Section 6, we present the continuous
time solution under the common prior assumption. The solution is very simple and
intuitive, suggesting that the problems in continuous time limit in usual models may
be an artifact of unrealistic discontinuity assumptions. We present the proofs in an
appendix.
2
Model
Two risk-neutral agents, i, j ∈ {1, 2}, want to divide a dollar among themselves before
a deadline. (The possible divisions are of the form (x, 1 − x) for x ∈ [0, 1].) The set
of times t is a continuum T = [0, ∞). The agents discount the future payoffs, so
that the payoff of getting x at t for an agent is e−rt x for some constant discount rate
r. The payoffs are 0 if agents never agree. The agents can strike a deal only on a
grid T ∗ = {0, 1/n, 2/n, . . .}, where n is a large integer. The (instantaneous, relative)
bargaining power of an agent is defined as the portion of the surplus, or the gain from
agreement, the agent gets in addition to his continuation value from delay. Here, an
agent i’s continuation value from delay at t is defined as the amount x̄i such that i is
indifferent between getting x̄i at t and waiting until t + 1/n for another opportunity
to strike a deal. The surplus is defined as 1 − x̄1 − x̄2 . Hence, assuming that the
surplus is positive, the instantaneous bargaining power of an agent i is
π it =
xi − x̄i
1 − x̄1 − x̄2
(1)
where xi is his share. In this paper, we take the agents’ bargaining powers as the
primitives of the model and compute the agents’ continuation values, which in turn
determine whether there will be agreement at a given time and how the dollar is
divided when there is an agreement.2
2
By taking the bargaining powers as given, we implicitly assume that the bargaining power at
a given time does not depend on the players’ continuation values, a standard assumption in the
6
We allow the agents’ bargaining powers to be stochastic and the agents to have
subjective beliefs about how their bargaining power will evolve. We consider a state
space Ω, endowed with a σ-algebra Σ and a filtration (Ft )t∈T . The instantaneous
bargaining power of agent 1 is given by an adapted stochastic process π 1t : Ω → [0, 1],
t ∈ T , such that the mapping (ω, t) 7→ π 1t (ω) is measurable. We define the process
(π 2t )t∈T by π2t = 1 − π1t . If the true state of the world is ω, then the bargaining
power of agent i at t is π it (ω). We endow (Ω, Σ) with two probability distributions P 1
and P 2 , representing the beliefs of agents 1 and 2, respectively. We write Eti for the
conditional expectation with respect to (Ω, Ft , P i ). We suppress the dependence on ω
in our notation when it does not lead to a confusion. Note that with this specification,
at any t, the agents know and agree on the instantaneous bargaining power at t but
may hold subjective beliefs about how the bargaining power will evolve.
The common-prior assumption (CPA) is
P 1 = P 2.
(CPA)
We define the optimism at time t about time s ≥ t (as a function of ω) as
£ ¤
£ ¤
yt,s = Et1 π 1s + Et2 π 2s − 1.
(2)
Here, yt,s is the amount by which an agent over-estimates her own bargaining power
with respect to the other agent, i.e. yt,s = Eti [π is ] − Etj [π is ] for i 6= j. Under the
common prior assumption, yt,s = 0 for all t and s ≥ t. In general, when t < s, yt,s
can take any value in [−1, 1]. We simplify our notation by using
yt = yt,t+1/n .
We model the deadline as a positive random variable d with cumulative distribubargaining literature.
7
tion function F . For simplicity, we assume that d is stochastically independent from
Ft , t ∈ T . At t = d, the negotiation automatically ends, and each agent gets payoff
of 0 at any t > d if they have not agreed at d or before. The deadline is said to be
deterministic if and only if d = d∗ with probability 1 for some d∗ ∈ [0, ∞], where
d∗ = ∞ corresponds to the case that there is no deadline. The deadline is said to be
stochastic with continuous distribution if and only if F is continuous.
For t ≤ s,
1−F (s)
1−F (t)
is the conditional probability that the deadline will not arrive
before s given it has not arrived by t. We define the effective discount factor between
t and s as
δ t,s = e−r(s−t)
1 − F (s)
.
1 − F (t)
For notational convenience, we let
δ t = δ t,t+1/n .
Agents discount the t + 1/n payoffs by the effective discount factor δt .
Deadlines as Transformations of Time In our model, time is a relative notion
that affects the behavior only through the effective discount factor δ t,s . Hence, any stochastic deadline can be represented as a transformation τ of time by e−rt (1 − F (t)) =
e−rτ (t) , or
τ (t) = t −
1
log (1 − F (t)) .
r
Then, a high arrival rate at a given t can be replicated by stretching the time around
t:
τ 0 (t) = 1 +
1 f (t)
.
r 1 − F (t)
If π it is continuous at t, such a transformation of time is equivalent to making bargaining power change more slowly, or more "durable." Therefore, in our analysis deadlines
and variations in durability of the bargaining power will play similar roles.
8
We let Vti denote the continuation value of agent i at t, as a function of ω. By
individual rationality, Vti is restricted to be in [0, 1]. For any t = k/n ≥ 0, we let
£ 1 ¤
£ 2 ¤
St+1/n ≡ Et1 Vt+1/n
+ Et2 Vt+1/n
denote the size of the pie at t + 1/n as perceived by the agents at t. Under the
common-prior assumption, St+1/n = 1 for each t ≥ 0. It can take any value in [0, 2]
in general. If
δ t St+1/n > 1,
there cannot be an agreement that would satisfy both agents’ expectations, as the
sum of their continuation values from delay exceeds 1. Assuming that the agents are
individually rational, we reckon that there will not be an agreement at such t. In that
case, the continuation value of i at t will be
i
Vti = δ t Vt+1/n
.
Now consider the case that δ t St+1/n ≤ 1. In that case, the surplus is 1 − δ t St+1/n , and
by (1), we conclude that the agents agree on the division (Vt1 , Vt2 ) with
¡
¢
£ i
¤
;
Vti = π it 1 − δ t St+1/n + δ t Eti Vt+1/n
an agent’s continuation value will be simply his share in the agreement. The last two
difference equations can be combined as
©
ª
£ i
¤
.
Vti = π it max 1 − δ t St+1/n , 0 + δ t Eti Vt+1/n
(3)
The continuation values are determined by the difference equation (3), the boundary
condition that Vti = 0 for t > d, and the condition that Vti ∈ [0, 1].
In this model, the agents’ bargaining powers are taken as primitives, and the
9
agents’ behavior is derived using backward induction and individual rationality, assuming that, at a given date, the agents will divide the dollar according to the agents’
instantaneous bargaining powers–when there is an individually rational division. As
in Nash (1950), we take the instantaneous bargaining at a given date as a black
box, hence the analysis in this paper does not necessarily refer to a game. Nevertheless, it can be formalized as a game. As an example, consider the following
perfect-information game, Γ. At each t ≤ d, Nature chooses a publicly observable
number π1t ∈ [0, 1], and we set π 2t = 1 − π 1t . If t = k/n ≤ d for some k, then one of
the agents is recognized where the probability of recognition for agent i is π it . The
recognized agent offers a division (x1 , x2 ), with x1 +x2 = 1. If the other agent accepts
the offer, then the game ends yielding a payoff vector δ t x = (δ t x1 , δt x2 ); otherwise,
the game continues. If t > d, the game ends with payoff vector (0, 0). The agents’
beliefs about π it and d are as described above. Here, we introduce an additional public
signal π1t to the model of Yildiz (2003), a signal that represents the "true" probability
of agent 1 making an offer at t.
The next result states that the behavior described in this paper corresponds to the
(generically unique) subgame perfect equilibrium of Γ, where the bargaining power is
represented as probability of making an offer (in the present transferable utility case).
The proof is similar to that of Theorem 1 in Yildiz (2003), and hence is omitted.
Proposition 1. There exists a unique solution V to difference equation (3) with the
boundary condition Vt = 0, t ≥ d. In any subgame perfect equilibrium of game Γ, for
each t ∈ T ∗ and each i, after π 1t is revealed but before the proposer at t is recognized,
the continuation value of agent i is Vti . In any subgame perfect equilibrium and at
any t ∈ T ∗ , if δ t St+1/n > 1, then there will be no agreement at t; if δ t St+1/n < 1, the
agents agree on a division that makes the responder indifferent between accepting and
rejecting the offer.
If one restricts πit to be in {0, 1}, the game Γ becomes a sequential bargaining
model similar to that of Yildiz (2003), who extends the usual models of sequential
10
bargaining, such as Rubinstein (1982), Binmore (1987), Merlo and Wilson (1995), by
allowing the agents to have subjective beliefs about the recognition. In view of Proposition 1, this model extends Yildiz (2003) and the above models in several directions.
First, the bargaining power here can take any value in [0, 1], while the sequential bargaining models allocate the entire instantaneous bargaining power to one agent or the
other, by restricting πit to be in {0, 1}. Second, the bargaining power here is defined
as a function of the real time, independent of how frequently the agents come together
to negotiate. This allows us to consider more realistic and better-behaved processes
of bargaining power. For example, assuming alternating offer bargaining or independent recognitions as n → ∞, requires a process of bargaining power that shifts from
one extreme to the other infinitely frequently, while we can focus on processes with
piecewise-continuous sample paths. Third, this model allows a stochastic deadline
with arbitrary distributions. When the discounting is due to the random bargaining
breakdowns, the usual models allow stochastic deadlines, but it is usually assumed
to have exponential distribution, as the discount rate is usually constant. This paper
does not impose any such restriction on deadlines because that would rule out the
critical class of stochastic deadlines that approximate deterministic deadlines. Finally, as Yildiz (2003), it generalizes the usual models by allowing the agents to hold
subjective beliefs about their bargaining power.
Remark 1 (Continuous-time limit). In this paper, to simplify the exposition, we take
real-time differences between any consecutive index times equal. When the sample
paths are piecewise continuous almost surely, this is irrelevant for the continuous-time
limit. In that case, regardless of how one approaches continuous time, as long as one
uses a deterministic grid, the continuous-time limit is the same, and our results remain
intact. This is remarkable because in usual bargaining models there is a discontinuity
at the continuous-time limit, and the limit depends on how we approach continuous
time. For example, in the alternating-offer bargaining, if the real-time delays after the
dates at which Agent 1 makes an offer are r times as long as those after Agent 2 makes
11
offers, then Agent 1 gets r times as much as Agent 2 gets in the continuous-time limit.
The critical difference here is that in those models the process of bargaining power
depends crucially on the grid and is not even well-defined in the continuous-time
limit.
3
A Poisson Model
In this section we demonstrate our main ideas in a tractable and useful model that
embodies certain plausible assumptions. This model is reflective of the situations in
which the bargaining power changes only in occurrence of certain events.
Consider a Poisson process (Nt )t∈T with arrival rate λ. The bargaining power
changes only at the time of an arrival, and at each arrival, the new bargaining power
is drawn from a fixed distribution Gi according to each agent i. That is, if Nt (ω) =
lims↑t Ns (ω), then π 1t (ω) = lims↑t π 1s (ω), and conditional on Nt 6= lims↑t Ns , the
distribution of π 1t is Gi , independent of the history, according to agent i. In this
model, bargaining power becomes less durable as λ increases. In the limit λ = 0,
bargaining power never changes: π it = πis for all t, s. In the other limit λ = ∞,
bargaining power is completely transient: π it and π is are stochastically independent
for all t 6= s.
According to agent i, the conditional distribution of π 1s at t < s is the mixture of
point mass at π1t with probability e−λ(s−t) and Gi with probability 1 − e−λ(s−t) . Define
R
R
π̄ 1 = πdG1 (π) and π̄ 2 = (1 − π) dG2 (π), so that π̄ i is the expected value of π i
according to agent i after each arrival. With this notation, the expected bargaining
power has a simple form:
¡
£ ¤
¢
Eti π is = e−λ(s−t) π it + 1 − e−λ(s−t) π̄ i .
(4)
The agent thinks that the bargaining power in the near future is likely to be similar
12
to the present one. The similarity depends on the arrival rate λ, in particular, the
higher the λ, the bargaining power is more likely to be different than the present
bargaining power.
By (2) and (4), the optimism level at date t for the bargaining power at date s is
¡
¢
yt,s = ȳ 1 − e−λ(s−t)
where
ȳ =
Z
¡ ¢
π dG π 1 −
1
1
Z
(5)
¡ ¢
π 1 dG2 π 1 = π̄ 1 + π̄ 2 − 1
is the level of optimism about the bargaining power in the long run. The level of
optimism for the bargaining powers in the near future is low, as the agents know that
their bargaining powers will be similar to the current ones. As they consider later
dates, the level of optimism grows, for they expect the bargaining power to be shifted
by the underlying reality, about which they hold optimistic beliefs. As s → ∞, yt,s
approaches ȳ.
Solution without a Deadline Now assume that d = ∞, i.e. there is no deadline.
One can easily check–as we do in the Appendix–that the solution to (3) is
Vti =
b
a−b
π it +
π̄ i
a + ȳ (a − b)
a + ȳ (a − b)
(∀i ∈ N, t ∈ T ∗ )
(6)
¡
¢
¡
¢
where a = 1/ 1 − e−r/n and b = 1/ 1 − e−(r+λ)/n , and the agents immediately
reach an agreement, agent i receiving V0i . As n → ∞,
Vti →
r
λ
π it +
π̄ i .
r + λ + ȳλ
r + λ + ȳλ
The share Vti of agent i is a convex combination of the current bargaining power, π it ,
and the expected value of long-run bargaining power, π̄ i . In the limit, the weights are
determined by the ratio of the arrival rate λ of new events that reset the bargaining
13
power to the discount rate r. As we increase λ, the bargaining power becomes less
durable, decreasing the importance of the current bargaining power relative to the
bargaining power in the long run. Accordingly, in the above equations, the weight of
π it decreases, and the weight of π̄ i increases. Finally, Vti is decreasing with the level
ȳ of (the other agent’s) optimism about (her) bargaining power in the long run.
Next, we examine the effect of a stochastic deadline or an increase in durability
of bargaining power starting at a date t∗ . We assume that
R (t) ≡
y 1 − e−(λ+r)t
> 1 for all t ∈ (0, t∗ ].
1 + y 1 − e−rt
In general, the assumption holds if the level of optimism is high enough and t∗ is not
too far. Note that in order the assumption to hold for small values of t, the parties
must hold optimistic beliefs about their own long-run bargaining power:
y
1+y
µ
¶
λ
1+
> 1, i.e., ȳ > r/λ.
r
Deadline Effect Let us now introduce a stochastic deadline that arrives with con∗)
stant hazard rate r̂ − r starting at some t∗ i.e. F (t) = 1 − e−(r̂−r)(t−t
for t ≥ t∗ .
Starting at t∗ , the effective discount factor is δ t,s = e−r̂(s−t) , and the rest of the model
is as before. Hence, at t∗ and thereafter, the agents agree on a division with shares
Vti =
b̂
â − b̂
³
´ π it +
³
´ π̄ i ,
â + ȳ â − b̂
â + ȳ â − b̂
(7)
¡
¢
¡
¢
where â = 1/ 1 − e−r̂/n and b̂ = 1/ 1 − e−(r̂+λ)/n . Now consider any t < t∗ . The
∗ −t)
value of not agreeing until t∗ for agent i is e−r(t
∗ −t)
disagree at t if Et1 [Vt1∗ ] + Et1 [Vt1∗ ] > er(t
Eti [Vti∗ ]. Then, the agents will
. By (5) and (7), the latter condition can
be written as
1 − e−(r̂+λ)/n
< R (t∗ − t) .
1 − e−r̂/n
14
As n → ∞, the condition becomes
1+
λ
< R (t∗ − t) .
r̂
(8)
The expression on the left-hand side is monotonically decreasing in r̂, the arrival rate
of deadline. When
r̂ > λ/ (R (t∗ − t) − 1) ,
there will be disagreement at t. In particular, there is a cutoff value
rc = sup λ/ (R (t) − 1) ,
0<t≤t∗
such that when r̂ > rc the agents will never agree until t∗ and reach an agreement at t∗ .
This is the well-known empirical regularity: the deadline effect. That is, the parties
keep negotiating until the very deadline and reach an "eleventh-hour agreement" "on
the steps of the courthouse". Within a canonical model, our example shows that such
deadline effect will be observed in equilibrium if there is enough optimism (so that
R (t) > 1 for all t ∈ (0, t∗ ]), there is enough certainty about the deadline (so that
r̂ > rc ) and the bargaining power is not completely transient (i.e. λ 6= ∞). On the
other hand, when r̂ < rc , we will not observe the deadline effect: the parties will
reach an agreement before t∗ . This suggests that one could avoid the deadline effect
by making the deadline sufficiently stochastic.
Notice that if we increase λ, keeping everything else constant (and r̂ < ∞),
the deadline effect will eventually disappear. The random proposer models with
independence, as it is usually assumed in applications, take λ = ∞. In that case, (8)
will necessarily fail and there will not be any deadline effect in the continuous-time
limit, as long as the deadline is stochastic (i.e. r̂ < ∞). Indeed, there will be an
immediate agreement, as we will show later more generally. On the other hand, when
the deadline is deterministic (i.e. r̂ = ∞), there will still be a deadline effect, showing
15
a discontinuity between the deterministic and stochastic deadlines. This discontinuity
comes from the extreme discontinuity of bargaining power implicitly assumed in those
models in the continuous time limit. With durability, the discontinuity disappears.
Durability effect Now assume that there is no deadline, but at t∗ bargaining power
becomes more durable, i.e., the arrival rate of the process that resets the bargaining
power decreases to some λ̂ < λ. For example, at t∗ we switch to a more stable world
as it would happen after a new election. As we discussed in Section 2, such a change
is equivalent to having a deadline starting at t∗ . Indeed, as in the case of a deadline,
one can easily show that in the continuous time limit (i.e. when n is large) the agents
will disagree at any t < t∗ if
1+
λ̂
< R (t∗ − t) .
r
They will agree at t∗ and thereafter. In the continuous time limit, there is a cutoff
value
λc = sup r (R (t) − 1) ,
0<t≤t∗
such that when λ̂ < λc , the agents will wait until time t∗ to reach an agreement. For
example, if it is expected that new elections (or joining to say the European Union)
will bring more stability to a country, then there will be a period before the elections
where the decisions are postponed to the election day. Such inactivity period will be
longer when elections bring more stability. On the other hand, when λ̂ > λc , there
will be agreement before t∗ .
4
Effect of Optimism, Durability and Deadlines in
Two Extreme Cases
In order to motivate our main result, in this section, we will consider two extreme
cases. We will first show that if the bargaining power becomes extremely durable
16
at a given date t∗ or there is a deterministic deadline at t∗ , then sufficient optimism
about the bargaining power at t∗ will entice agents to wait until t∗ and agree at t∗ .
We will then show that under the usual independence assumptions about bargaining
power, which are also extreme and unrealistic assumptions, slight uncertainty about
the deadline will lead to immediate agreement in the continuous time limit.
We write
β (t, t∗ ) =
δ t,t∗ (1 + yt,t∗ ) − 1
,
δ t,t∗
(9)
for the normalized excessive optimism, and assume that it is positive in the sequel.
Similar to the condition on R (t) that we have defined for the Poisson model in Section
3, this condition on β (t, t∗ ) captures the fact that there should be some optimism on
the bargaining power for there to be delay in agreement.
Under this assumption, our first result shows that if the deadline is deterministic
and arrives at t∗ (or the bargaining power becomes constant after t∗ ), then there will
be a deadline effect:
Proposition 2. Let t∗ = k/n be such that
(C) π it (ω) = π it∗ (ω) for all (t, ω) with t ∈ [t∗ , d (ω)] (e.g. the deadline deterministically arrives at t∗ ) and
(O) β (t, t∗ ) > 0 for each t < t∗ .
Then, (i) the agents disagree at each t < t∗ and (ii) agree at t∗ with Vti∗ = π it∗ .
This proposition generalizes a two-period example (Example 0) of Yildiz (2003),
without making any assumption about the process of bargaining power, such as independence, the parties learning about their bargaining powers, and so on. Condition C
states that bargaining power does not change after t∗ , and hence each agent gets his
bargaining power if they wait until t∗ , as stated in (ii). But, as we have seen above,
condition O states that agents are willing to wait to get their bargaining power at t∗ ,
leading to (i).
17
In particular, when the deadline is deterministic, condition C is vacuously satisfied.
Then, the proposition states that optimistic agents will wait for the deadline to settle,
as in the deadline effect. This is intuitive: just before the deadline, the cost of delay is
high, and hence the parties’ relative bargaining power affects the outcome greatly. If
the agents are highly optimistic about their bargaining power at the last minute and
the deadline is not too far away, then there will be no feasible agreement before t∗ that
satisfies both parties’ (highly optimistic) expectations. Notice that the required level
of optimism for delay goes to zero as we approach the deadline. This is important
because in general, as in the Poisson model, the agents plausibly have less uncertainty
and lower level of optimism for the near future, compared to the long run.
Next, we will show that, when the beliefs about the future are independent of
the past and the deadline is stochastic, there will be an immediate agreement in the
continuous time limit. In other words, the deadline effect is discontinuous as a function
of the deadline arrival process. We need the following independence assumption to
state our result.
Assumption IND. For any i, t < s and π i ∈ [0, 1], the conditional probability
Fsi (πi ) ≡ P rti (π is ≤ π i ) is independent of ω.
That is, the beliefs about the future bargaining powers does not depend on the
past observations, including the past bargaining powers. Assumption IND implies
that the optimism at t < s for s is a function of s alone, i.e.
yt,s =
Et [π 1s ]
+
Et [π 2s ]
−1 =
Z
π
π 1 ∈[0,1]
1
dFs1 (π 1 )
≡ Y (s),
+
Z
π 2 ∈[0,1]
π 2 dFs2 (π2 ) − 1
where the last line defines new notation. This further implies that whether there will
be agreement at date t is deterministic.
The following proposition provides conditions under which the deadline effect
disappears when the deadline becomes stochastic.
18
Proposition 3. Assume that the deadline is stochastic with continuous distribution
(i.e. F is continuous). Under Assumption IND, assume also that either (i) Y (t) ≥ 0
for all t, or (ii) Y (t) is a continuous function. Then, for any
such that if n > n, the agents agree before
> 0, there exists n
(in states in which the deadline does not
arrive before date ).
That is, under Assumption IND, if the deadline is stochastic, then there will be an
immediate agreement in continuous-time limit, so long as agents are optimistic or the
level of optimism is a continuous function of time. Under the independence assumption, this result shows the intuitive rationale for deadline effect given by Proposition 2
is fragile. Any uncertainty about the deadline makes the deadline effect disappear and
leads to an immediate agreement. More generally it shows that, under Assumption
IND, the equilibrium outcome may be discontinuous with respect to the distribution
of the deadline (as the uncertainty of deadline vanishes). Nevertheless, although the
independence assumption is sometimes useful and assumed frequently, it is an extreme
assumption. It assumes that the bargaining power is completely transient in the sense
that the bargaining power now has no effect on the bargaining power a moment later.
It thus assumes that the bargaining power is extremely discontinuous. As we have
seen in the Poisson model, under the continuity assumption of that model, there is no
discontinuity and the result of Proposition 2 can be generalized in an intuitive way.
In the next section, we establish this result more generally.
5
Optimism, Durability, and Deadlines
In this section, for our general framework, we develop our notions of durability of
bargaining power and show that if agents are sufficiently optimistic about their bargaining power at some t∗ and the bargaining power becomes sufficiently durable at
t∗ , then they delay settlement until t∗ . In particular, if bargaining power is somewhat
durable (i.e. not completely transient), then there will be deadline effect whenever
19
the uncertainty about the deadline is sufficiently small. We start by formally defining
our notions of durability of bargaining power.
Definition 4 (Strong Durability). We say that the bargaining power is strongly
durable on [t1 , t2 ] with inverse rate of durability c > 0 if there exist c0 , c00 ≥ 0 such
that c0 + c00 ≤ c and for all t1 ≤ t ≤ s ≤ t2 , for all i, and for all ω ∈ Ω,
¯
¯
¢
¡
P rti w0 | ¯π 1s0 (w0 ) − π 1t (w0 )¯ ≤ c0 (s0 − t), ∀ s0 ∈ [t, s] | w ≥ 1 − c00 (s − t).
That is, in each agent’s view, the bargaining power is Lipschitz-continuous in time
with the exception of some rare events which happen with a rate slower than c00 . This
is stronger than the following notion:
Definition 5 (Weak Durability). We say that the bargaining power is weakly
durable on [t1 , t2 ] with inverse rate of durability c > 0 if for all t1 ≤ t ≤ s ≤ t2 , for
all i, and for all ω ∈ Ω,
|Eti [π1s ] (ω) − πit (ω) | ≤ c(s − t).
That is, in expectation, the bargaining power is Lipschitz-continuous in time. In
weak durability, the inverse rate of durability c captures the rate by which agents’
expectations about the bargaining power in the near future can deviate from the
current bargaining power. Strong durability implies weak durability:
|Eti [π1s ] − π it | ≤ (1 − c00 (s − t))c0 (s − t) + c00 (s − t) ≤ (c0 + c00 )(s − t) ≤ c(t − s). (10)
Example 1 (Durability in the Poisson Model). In the Poisson model, the bargaining
power is strongly durable with inverse rate of durability λ. This can be seen by taking
c0 = 0 and c00 = λ, since for any t ≤ s,
P rti (π is0 = π it , ∀s0 ∈ [t, s]) ≥ 1 − e−λ(t−s) ≥ λ(s − t).
20
This is intuitive since for higher λ, the bargaining power jumps more frequently and
hence is less durable.
We let ta (ω) be the first date with an agreement regime, i.e. the settlement date,
at state ω. we use the convention that ta (ω) = ∞ when they fail to agree before the
deadline. We also let
ΩA = {ω | ta (ω) ≤ d (ω)}
be the set of states at which the agents reach an agreement before the deadline.
Proposition 2 has established that if the agents are optimistic about t∗ (condition
O) and the bargaining power becomes constant at t∗ until the deadline arrives (condition C), then the agents will wait until t∗ to settle. We will now generalize this result
by relaxing the extreme condition C and by strengthening condition O slightly.
Condition O* The dates t̂ and t∗ with t̂ ≤ t∗ are such that
¢
¡
β̄ t̂, t∗ ≡ inf β (t, t∗ ) > 0,
t<t̂
where β (t, t∗ ) = (δ t,t∗ (1 + yt,t∗ ) − 1) /δ t,t∗ , as defined in (9).
When t̂ < t∗ , condition O* is implied by condition O and a continuity assumption
on y. When t̂ = t∗ , it becomes a strong assumption, as it implies that yt,t∗ is discontinuous in t at t = t∗ . We will take t̂ less than t∗ but close to t∗ in our interpretations.
We next weaken condition C in two ways, using weak and strong durability.
Condition D Date t∗ and
> 0 are such that the bargaining power is weakly
durable on [t∗ , te ] with inverse rate of durability c for some c and te with δ t∗ ,te ≤ /4
and c(te − t∗ ) ≤ /4.
Condition SD Date t∗ and
> 0 are such that the bargaining power is strongly
durable on [t∗ , te ] with inverse rate of durability c for some c and te with δ t∗ ,te ≤ /4
21
and c(te − t∗ ) ≤ /4.
The durability condition D requires that bargaining power is sufficiently durable
for a sufficiently long while. As decreases, the bargaining power needs to be durable
for a longer period of time and with higher rate of durability (i.e. with lower c). The
strong durability condition SD requires in addition that bargaining power must be
strongly durable. The next proposition, which is our main result, establishes that
under the durability and optimism condition agents will wait until sometime near t∗
to settle.
Proposition 6. For any t̂, t∗ ∈ T ∗ and
¡
¢
∈ (0, β̄ t̂, t∗ ] that satisfy the optimism
condition O* and the durability condition D, the agents disagree at each t < t̂. If
in addition t∗ and
satisfy the strong durability condition SD, then at t∗ , each agent
assigns high probability that they will reach an agreement before the deadline arrives:
P rti∗ (ΩA ) ≥
π t∗ −
πt∗ +
(∀i ∈ {1, 2}) .
That is, if the bargaining power is sufficiently durable and the agents remain
sufficiently optimistic about t∗ until t̂, then they will settle after t̂ before the deadline
arrives with high probability.
Our proof, which is presented in the appendix, relies on the following lemma:
Lemma 7. Assume that the bargaining power is weakly durable with inverse rate of
durability c on some [ts , te ] with ts , te ∈ T ∗ . Then, for each t ∈ [ts , te ] ∩ T ∗ and i,
¢
¡
¢
¡ i
πt − c(te − t) (1 − δ t,te ) ≤ Vti ≤ π it + c(te − t) (1 − δ t,te ) + δ t,te .
That is, when the bargaining power is durable, the payoffs at t are within a
bounded interval around the bargaining power. If δ t,te is small and c(te − t∗ ) is small,
i.e. there is enough durability for a long enough time, then the payoff of an agent at
t will be close to her bargaining power at t. We obtain the lower bound by applying
22
backward induction using the smallest possible values for V i and largest possible
values for V j under the durability assumption. Upper bound is obtained similarly.
Under condition D, for t = t∗ , the lower bound becomes (π it∗ − /4) (1 − /4). Un¡
¢
der condition O* and ≤ β̄ t̂, t∗ , this implies that at any t < t̂, δ t,t∗ (Et1 [Vt1∗ ] + Et2 [Vt2∗ ]) >
1, and hence it is impossible to satisfy both agents’ expectations from waiting until
t∗ . Hence the first statement in the proposition.
There is a simple intuition for the second statement. If the continuation value of
an agent i at t∗ is high, then he must assign a high probability P rti∗ (ΩA ) on reaching
an agreement before the deadline (in the near future) because he gets 0 if they fail
to agree before the deadline (or a small payoff if they wait). Unfortunately, this
argument does not lead to a very high lower bound for P rti∗ (ΩA ). It leads to a lower
bound for P rt1∗ (ΩA ) + P rt2∗ (ΩA ) that is nearly 1. In the Appendix, using more subtle
arguments based on strong durability, we show that each of these probabilities are
nearly 1.
The durability conditions D and SD of Proposition 6 could be satisfied in two
distinct plausible scenarios. First, settlement will be delayed before an event, such as
elections, after which c will be very small for very long time. Second, settlement will
also be delayed if the bargaining power is durable with some rate c and the deadline
arrives sufficiently fast in a neighborhood of t∗ . For in this case, te will be very close
to t∗ , and even if c is high, c(te − t∗ ) might be low and might satisfy D or SD. We
will next spell out these two cases.
Durability Effect
As a first application of Proposition 6, we consider an event
after which the bargaining power becomes durable for a long time. Such an event could
be elections that determine who will run a country until the next elections. Another
example of such an event is whether a country joins a union, such as European Union.
The relative bargaining powers in the country may heavily depend on whether they
join. For example, military may have a strong position if they fail to join, while the
23
civil government may have a strong position if they join. The following proposition
provides sufficient conditions for the length and inverse rate of durability to generate
settlement delay.
Corollary 8. For any t̂, t∗ ∈ T ∗ that satisfy optimism condition O*, assume that
¡
¢
for some ∈ (0, β̄ t̂, t∗ ] the bargaining power is weakly durable on [t∗ , t∗ + 1/n +
log (4/ ) /r] with inverse rate of durability c such that
c≤r
/4
.
log(4/ ) + r/n
Then, the agents disagree at each t < t̂. If the bargaining power is strongly durable
with above parameters, then at t∗ , each agent assigns high probability that they will
reach an agreement before the deadline arrives:
P rti∗ (ΩA ) ≥
π it∗ −
π it∗ +
(∀i ∈ {1, 2}) .
∗
Proof. Let te = max {t ∈ T ∗ |t∗ < t ≤ t∗ + 1/n + log (4/ ) /r}. Then, δ t∗ ,te ≤ e−r(te −t ) ≤
/4. Moreover, since te − t∗ ≤ log(4/ )/r + 1/n,
c(te − t∗ ) ≤ r
/4
[log(4/ )/r + 1/n] = /4.
log(4/ ) + r/n
The result then follows from Proposition 6.
Corollary 8 provides intuitive comparative statics for sufficient conditions which
generate durability effect. The higher the optimism before event date (in which case
we can choose a higher ) and the higher the discount rate (keeping the optimism
constant), the stronger the durability effect becomes, i.e. the less durable the bargaining power could be over a shorter interval and still generate the durability effect.
A higher level of optimism about bargaining power translates itself into a higher level
of optimism about payoffs, which makes the durability effect stronger. The rationale
for the comparative statics with respect to the discount rate is somewhat deeper. A
24
higher discount rate increases the scope of trade at t∗ , which brings the (worst case)
payoffs at t∗ closer to the bargaining powers at t∗ (cf. Lemma 7). Hence, keeping the
optimism for t∗ constant, a higher discount rate makes durability effect stronger.
Deadline Effect
Using Proposition 6, we will now establish generally that the
deadline effect arises in equilibrium if the bargaining power is durable and the uncertainty about the deadline is sufficiently small.
∗ −t)
Proposition 9. Consider any t̂, t∗ ∈ T ∗ with t̂ < t∗ and inf t<t̂ e−r(t
(1 + yt,t∗ ) > 1.
Assume that the bargaining power is weakly durable on [t∗ , t∗ + γ] with inverse rate of
durability c for some γ, c > 0. Then, there exist ∈ (0, 1), τ ∈ (t∗ , t∗ + γ) and n̄ < ∞
such that the agents disagree at each t < t̂ whenever
F (τ ) − F (t∗ ) > 1 −
(11)
and n > n̄. Moreover, if the bargaining power is strongly durable and Pri0 (π it∗ > 0) =
1, then for any μ > 0, we can pick
so that agent i assigns at least probability 1 − μ
that they will reach an agreement before the deadline arrives (and before τ ).
The condition (11) quantifies the notion that the deadline arrives around t∗ and
the uncertainty about the deadline is small. When the distribution of the deadline
converges to the point mass at t ∈ (t∗ , τ ), this condition is satisfied for any . Under
this condition, the probability that the deadline will arrive before t∗ is quite small,
less than . Since the agents remain optimistic about their bargaining power at t∗
∗ −t)
(i.e. inf t<t̂ e−r(t
(1 + yt,t∗ ) > 1), this implies that the optimism condition O* of
Proposition 6 is satisfied. Moreover, under (11), δt∗ ,τ ≤ (1 − F (τ )) / (1 − F (t∗ )) <
/ (1 − ) is also small. Since we can choose τ arbitrarily close to t∗ so that c (τ − t∗ )
is as small as needed, the durability condition D of Proposition 6 is also satisfied.
Proposition 6 then implies that the agents will not settle before t̂. Likewise, the
second statement follows from the second part of Proposition 6. Under the above
25
conditions Proposition 6 states that, at t∗ , an agent will assign high probability of
reaching an agreement before the deadline arrives. Since the probability of arrival of
a deadline before t∗ is small, the Bayes rule then implies that the agent will assign a
high probability of reaching an agreement.
To sum up from an agent’s point of view, if the uncertainty about the deadline
is sufficiently small and the agents are sufficiently optimistic about their bargaining
power around the deadline, then they will wait sometime very close to the deadline
but they will settle before the deadline arrives with very high probability. Hence the
deadline effect arises in equilibrium. This is, of course, provided that the bargaining
power is durable around the deadline. Under the usual independence assumption
there is no deadline effect with uncertainty about the deadline, and indeed there is an
immediate agreement under canonical conditions (Proposition 3). This is the extreme
case in which the bargaining power is completely transient. Proposition 9 states that
under any amount of durability we are in the continuous case in which deadline effect
arises for small amounts of uncertainty.
How much uncertainty is small enough though? Our next result, under regularity assumptions, quantifies how fast the deadline arrival rate should be to generate
the deadline effect. The deadline effect prevails when the rate of deadline arrival is
sufficiently larger than the inverse rate of durability.
¤
£
∗
Proposition 10. Let t̂, t∗ ∈ T ∗ be such that β ≡ inf t<t̂ 1 + yt,t∗ − er(t−t ) / (1 − F (t∗ )) >
0. For some
[t∗ , t∗ + 1/n +
∈ (0, β], assume that the bargaining power is weakly durable on
log(4/ )
]
r
with the inverse rate of durability c. If the moving aver-
age hazard rate for the arrival of deadline is at least r̂ − r ≥ 0 over the interval
[t∗ , t∗ + 1/n +
log(4/ )
]
r
∗)
(more specifically, (1 − F (t)) / (1 − F (t∗ )) ≤ −e−(r̂−r)(t−t
each t ∈ [t∗ , t∗ + 1/n +
log(4/ )
])
r
for some r̂ with
r̂ ≥
c log(4/ )
,
/4 − c/n
26
for
then the agents disagree at each t < t̂. In addition, if the bargaining power is strongly
durable, then for each i ∈ {1, 2},
P rti∗ (ΩA ) ≥
π t∗ −
.
π t∗ +
Note that, in the continuous time limit, i.e. as n → ∞, the sufficient condition
for deadline effect becomes
r̂
log(4/ )
≥
.
c
/4
The condition depends only on the ratio of arrival rate of deadline to the inverse
rate of durability of the bargaining power. When this ratio is above a cutoff value
(i.e. arrival rate is sufficiently higher than the inverse rate of durability), the deadline
effect prevails. How much faster should the deadline arrive than the inverse rate of
durability is determined by the optimism about t∗ . When the optimism for t∗ is
higher, we can choose
higher and the cutoff becomes lower. In that case, a less
durable bargaining power or a slower deadline arrival rate will suffice to generate a
deadline effect.
6
Bargaining with a Common Prior
In this section, we compute the continuation values under (CPA) and show that
durability of the bargaining power does not affect the agreement decisions of agents
and has only an indirect effect on the continuation values at any time t.
Under CPA, St+1/n is identically 1 before the deadline, so that the agents agree
at each date (if they have not yet agreed), and (3) simplifies to
£ i
¤
,
Vti = π it (1 − δ t ) + δ t Et Vt+1/n
(12)
where Et denotes the common conditional expectation operator. The solution to this
27
difference equation and its limit as n → ∞ are easily computed as in the following
result. The proof is omitted.
Proposition 11. Under (CPA), for each t ∈ T ∗ ,
" ∞
#
X
¡
¢
1
Vti =
Et
e−r(k/n−t) (1 − F (k/n)) 1 − δ k/n π ik/n
1 − F (t)
k=tn
(a.s).
If in addition the sample paths t 7→ π it (ω) are piecewise continuous for almost all ω,
then
lim
n→∞
Vti
≡
V̄ti
1
=
Et
1 − F (t)
∙Z
s≥t
−r(s−t)
e
(f (s) + r (1 − F
¸
(s))) π is ds
(∀t) (a.s.) .
This proposition shows that the overall bargaining power of an agent is the expected value of a weighted sum (or integral) of his future instantaneous bargaining
powers. The agents weigh their bargaining power at different situations differently.
Firstly, as expected, they discount the future bargaining powers in the same way as
they discount their future payoffs, using the effective discount rate δ t . Secondly, they
put higher weight to the bargaining power at the times at which it is more likely that
¡
¢
they will face a deadline. This is reflected by the term 1 − δk/n in the discrete sum
and f (s) in the limiting integral. The rationale for this is somewhat deeper. When
the probability of deadline is high, so is the cost of delay. Hence, there is a wide
range of individually rational divisions, which can be traced as we vary the agents’
relative bargaining power. In that case, the bargaining power has a large impact on
the share agent would get if the negotiation continues until that day. Therefore, at
the beginning of the negotiation, the agents put a high weight on their instantaneous
bargaining power at that time. In particular, if there is a firm deadline, the agents’
bargaining power just before the deadline will have a large impact.
28
By Proposition 11, the share of agent i is
V0i
=
∞
X
t∈T ∗
£ ¤
e−rt (1 − F (t)) (1 − δ t ) E π it
where E denotes the unconditional expectation operator. That is, the share of an
agent is an additive function of the expected value of his instantaneous bargaining
powers in the future, and it is independent of how the bargaining powers at different
dates interact with each other, e.g., whether the instantaneous bargaining power
changes frequently, or whether the agents will learn about their future bargaining
power from past, and so on. The durability of bargaining power does affect the
parties’ shares through its effect on the future bargaining power. For example, in the
Poisson model, under the common-prior assumption (i.e. ȳ = 0), the share of agent
i is
Vti = (b/a) πit + (1 − b/a) π̄ i ,
which converges to
r
λ
π it +
π̄ i ,
r+λ
r+λ
a convex combination of the current bargaining power and the long run bargaining
power where the weights are given by the ratio r/λ. As bargaining power becomes
more durable, the weight of the current bargaining power increases.
Appendix
Proof of Equation (6 ). By Proposition 1, it suffices to verify that V is indeed a solution to
(3). We check it using mathematical induction backwards on time. Check that, by (6) for
t + 1/n and (4),
h
i
i
=
Eti Vt+1/n
a − be−λ/n i
be−λ/n
π it +
π̄ .
a + ȳ (a − b)
a + ȳ (a − b)
29
(13)
Since π 1t + π 2t = 1 and π̄ 1 + π̄ 2 = ȳ + 1, this yields
St+1/n =
Et1
¢
¡
£ 1 ¤
£ 2 ¤ a + ȳ a − be−λ/n
2
.
Vt+1 + Et Vt+1 =
a + ȳ (a − b)
Thus,
1 − δ t = 1 − e−r/n St+1/n = (a + ȳ (a − b))−1 > 0,
(14)
i.e., the agents agree at t. In order to verify (3), write
Vti =
b
a−b
πi +
π̄ i
a + ȳ (a − b) t a + ȳ (a − b)
¡
¢
a − be−λ/n e−r/n i
1
be−(r+λ)/n i
i
π +
π +
π̄
=
a + ȳ (a − b) t a + ȳ (a − b) t
a + ȳ (a − b)
h
i
©
ª
i
= π it max 1 − δ t St+1/n , 0 + δ t Eti Vt+1/n
,
where the first equality by (6), the second equality by definitions of a and b, and the last
equality by (14) and (13).
Proof of Proposition 2. Note that δ t∗ = 0 implies δ t∗ St∗ +1/n = 0 < 1, hence the agents agree
at t∗ with Vti∗ = π it∗ , proving the second part of the proposition. To prove the first part,
note that at any t < t∗ ,
∗ −t)
e−r(t
¡
Et1 [Vt1∗ ] + Et2 [Vt2∗ ]
¢
∗ −t)
= e−r(t
∗ −t)
= e−r(t
¢
¡ 1 1
Et [π t∗ ] + Et2 [π 2t∗ ]
(1 + yt,t∗ ) > 1,
where the first equality follows from the second part of the proposition and the second
inequality follows from condition O. Therefore, the agents expect to be better of waiting
until t∗ hence they disagree at t < t∗ , completing the proof.
Proof of Proposition 3. Under Assumption IND, St+1/n is deterministic, and hence
St = δ t St+1/n
30
(15)
if δ t St+1/n > 1 and
St = 1 + Y (t)(1 − δ t St+1/n )
(16)
otherwise. For each n, let tn be the first date on which the agents agree, which is deterministic by Assumption IND. Whenever tn > 0, we have
¢
¡
1 < δ 0,tn E01 [Vt1n ] + E02 [Vt2n ] = δ 0,tn Stn ,
(17)
where the first inequality follows from the definition of tn and the second equality follows
from Assumption IND. Since Stn ≤ 2, (17) implies that tn ≤ [0, log 2/r], thus tn is a bounded
sequence. Hence it suffices to show that 0 is the only limit point of the sequence tn . Let
t∗ be any limit point of this sequence and consider a subsequence that converges to t∗ . We
will show that t∗ = 0.
Since there is agreement at tn , by Assumption IND, (16) holds at t = tn . We consider
the two conditions separately. First assume (i). Then, if δ tn +1/n Stn +2/n ≤ 1, then (16) and
Y (tn + 1/n) ≥ 0 implies that Stn +1/n ≥ 1. If δ tn +1/n Stn +2/n > 1, then this time (15) implies
that Stn +1/n ≥ 1. Using this and Y (t) ∈ [0, 1] in (16), we obtain
Stn ≤ 2 − δ tn .
Together with (17), this implies that
δ 0,tn (2 − δ tn ) > 1.
By taking the limit n → ∞ on this inequality, we obtain
δ 0,t∗ ≥ 1,
which implies that t∗ = 0, as desired. Here, since F is continuous, δ 0,t = δ t (1 − F (t)) is continuous in t, and δ 0,tn → δ 0,t∗ as tn → t∗ , and also δ tn = e−r/n (1 − F (tn + 1/n)) / (1 − F (tn )) →
1.3
3
This is where the proof fails under a deterministic deadline.
31
Next assume (ii) without assuming (i). If Y (tn ) ≤ 0, then Stn ≤ 1 by (16), which, by
(17) implies tn = 0. Thus consider the subsequence of (tn ) which satisfies Y (tn ) > 0. If
Y (tn + 1/n) ≥ 0 for all but finitely many n, then the above proof shows that t∗ = 0. So
assume Y (tn + 1/n) < 0 for infinitely many values of n. This subsequence, which we denote
by tn0 , also converges to t∗ , and satisfies
Y (tn0 + 1/n0 ) < 0 < Y (tn0 ).
Taking limits in the previous inequality, we obtain Y (t∗ ) = 0. By (16), we have Stn ≤
1 + Y (tn ) → 1 + Y (t∗ ) = 1, which, by (17) implies that t∗ = 0, as desired.
Proof of Lemma 7. For t = te , δ t,te = 1, and the inequalities in the lemma are trivially
satisfied. Towards an induction, assume that for each i,
´¡
³
¢
i
π it+1/n − c(te − (t + 1/n)) 1 − δ t+1/n,te ≤ Vt+1/n
³
´¡
¢
≤ π it+1/n + c(te − (t + 1/n)) 1 − δ t+1/n,te + δ t+1/n,te .
By the durability assumption, Eti [π it+1/n ] ∈ [π it − c/n, π it + c/n], thus
LHS i ≡
h
i
¡ i
¢¡
¢
i
π t − c(te − t) 1 − δ t+1/n,te ≤ Eti Vt+1/n
¡
¢¡
¢
≤ π it + c(te − t) 1 − δ t+1/n,te + δ t+1/n,te ≡ RHS i .
(18)
Recall that
h
i
h
io ¡
h
i
n
¢
j
i
i
Vti = π it max 1 − δ t Etj Vt+1/n
, δ t Eti Vt+1/n
+ 1 − π it δ t Eti Vt+1/n
.
Combining the last two statements we obtain the necessary bounds. To find an upper
32
bound, we write
©
ª ¡
¢
Vti ≤ π it max 1 − δ t LHS j , δ t RHS i + 1 − π it δ t RHS i
= π it (1 − δ t ) + δ t RHS i
¡
¢
= π it + c(te − t) (1 − δ t,te ) + δ t,te − c(te − t)(1 − δ t )
¢
¡
≤ π it + c(te − t) (1 − δ t,te ) + δ t,te .
Here, the first inequality is by (18), the next equality by RHS i + LHS j = 1, and the last
equality and inequality by simple algebra. Similarly,
©
ª ¡
¢
Vti ≥ π it max 1 − δ t RHS j , δ t LHS i + 1 − π it δ t LHS i
= π it (1 − δ t ) + δ t LHS i
¡
¢
= π it − c(te − t) (1 − δ t,te ) + c(te − t)(1 − δ t )
¢
¡
≥ π it − c(te − t) (1 − δ t,te ) .
Proof of Proposition 6. To prove the first part of the proposition, note that, using Lemma
7 with [ts , te ] = [t∗ , te ], we have for each i
Vti∗ ≥ (1 − δ t∗ ,te )(π it∗ − c(te − t∗ ))
¢
¢
¡
¡
≥ (1 − β̄ t̂, t∗ /4)(π it∗ − β̄ t̂, t∗ /4).
(19)
Then, we have,
¢
¡
¡
¡
¡
¢
¡ £¡
¢ ¢¤
£¡
¢ ¢¤¢
δ t,t∗ Et1 [Vti∗ ] + Et2 [Vt2∗ ] ≥ δ t,t∗ (1 − β̄ t̂, t∗ /4) Et1 π 1t∗ − β̄ t̂, t∗ /4 + Et2 π 2t∗ − β̄ t̂, t∗ /4
£
¡
¢
¢
¤
¡
= δ t,t∗ (1 − β̄ t̂, t∗ /4)(1 + yt,t∗ − β̄ t̂, t∗ /2)
¡
£
¢¤
> δ t,t∗ 1 + yt,t∗ − β̄ t̂, t∗
≥ 1,
where the first inequality uses (19), the second inequality uses yt,t∗ ≤ 1 and the last in-
33
¢
¡
equality uses the definition of β̄ t̂, t∗ . Therefore the agents expect to be better off waiting
until t∗ and they disagree at t, proving the first part of the proposition.
¢ ¡
¢
¡
¢ ¡
To prove the second part, we will show that Pr ΩA,te ≥ π it∗ − / π it∗ + where
ΩA,te = {ω | ta (ω) ≤ min(te , d (ω))} ⊆ ΩA
is the set of states at which agents reach an agreement before the deadline and before time
te . To this end, let
¯
¯
©
ª
ΩNJ = w ∈ Ω | ¯π 1t − π 1t∗ ¯ ≤ c0 (t − t∗ ) , ∀ t ∈ [t∗ , te ]
be the set of states in which the bargaining power "does not jump" between t∗ and te , and
¡ ¢
let ΩJ = Ω \ ΩNJ . By the strong durability assumption, P rti∗ ΩJ ≤ c00 (te − t∗ ). We will
find a lower bound for P rti∗ (ΩA,te ∩ ΩNJ ). Note that
(π it∗ − c(te − t∗ ))(1 − δ t∗ ,te ) ≤ Eti∗ [Vti∗ ] = Eti∗ [Vti∗ | ΩA ]
= Eti∗ [Vti∗ |ΩA,te ∩ ΩNJ ]P r(ΩA,te ∩ ΩNJ )
+Eti∗ [Vti∗ |ΩA,te ∩ ΩJ ]P r(ΩA,te ∩ ΩJ )
+Eti∗ [Vti∗ |ΩA,te \ΩA,te ]P r(ΩA \ΩA,te )
≤ Eti∗ [Vti∗ |ΩA,te ∩ ΩNJ ]P r(ΩA,te ∩ ΩNJ ) + c00 (te − t∗ ) + δ t∗ ,te , (20)
where the first inequality follows from Lemma 7 and the last inequality follows since
P rti∗ (ΩA,te ∩ ΩJ ) ≤ P rti∗ (ΩJ ) ≤ c00 (te − t∗ ), Vti∗ ∈ [0, 1], and
Et∗ [Vti∗ |ΩA \ΩA,te ] = Et∗ [δ t∗ ,ta (w) Vtia (w) | ΩA \ΩA,te ] ≤ δ t∗ ,te .
Therefore, towards finding a lower bound for P r(ΩA,te ∩ ΩNJ ), we will find an upper bound
for Eti∗ [Vti∗ |ΩA,te ∩ ΩNJ ].
34
Note that, we have
Eti∗ [Vti∗ |ΩA,te ∩ ΩNJ ] = Eti∗ [δ t∗ ,ta (w) Vtia (w) |ΩA,te ∩ ΩNJ ]
¢
¡¡
¢
≤ Eti∗ [δ t∗ ,ta π ita + c(te − ta ) (1 − δ ta ,te ) + δ ta ,te |ΩA,te ∩ ΩNJ ]
¡
¢
= Eti∗ [ π ita + c(te − ta ) (δ t∗ ,ta − δ t∗ ,te ) |ΩA,te ∩ ΩN J ] + δ t∗ ,te
¡
¢
≤ Eti∗ [ π ita + c(te − ta ) (1 − δ t∗ ,te ) |ΩA,te ∩ ΩNJ ] + δ t∗ ,te
¡
¢
≤ Eti∗ [ π it∗ + c0 (ta − t∗ ) + c(te − ta ) (1 − δ t∗ ,te ) |ΩA,te ∩ ΩNJ ] + δ t∗ ,te
¡
¢
≤ Eti∗ [ π it∗ + c(te − t∗ ) (1 − δ t∗ ,te ) |ΩA,te ∩ ΩNJ ] + δ t∗ ,te
¡
¢
= π it∗ + c(te − t∗ ) (1 − δ t∗ ,te ) + δ t∗ ,te ,
where the first inequality uses Lemma 8 to bound Vtia from above, the next equality follows
by simple algebra using the fact that δ t1, t2 δ t2 ,t3 = δ t1 ,t3 , the second inequality follows since
δ t∗ ,ta ≤ 1, the third inequality follows by definition of ΩN J , and the last inequality follows
by simple algebra and the fact that c0 ≤ c. Then, using (20), we have
P rti∗ (ΩA,te ∩ ΩNJ ) ≥
≥
≥
≥
(π it∗ − c(te − t∗ ))(1 − δ t∗ ,te ) − c00 (te − t∗ ) − δ t∗ ,te
£ i
¤
π t∗ + c(te − t∗ ) (1 − δ t∗ ,te ) + δ t∗ ,te
π it∗ − /4 − (π it∗ − c(te − t∗ ))δ t∗ ,te − /4 − /4
¤
£ i
π t∗ + /4 + /4
π it∗ − /4 − /4 − /4 − /4
π it∗ + /4 + /4
π it∗ −
,
π it∗ +
where the inequalities after the first one follow since c(te −t∗ ) ≤ /4, π it∗ ≤ 1 and δ t∗ ,te < /4.
Since P rti∗ (ΩA ) ≥ P rti∗ (ΩA,te ) ≥ P rti∗ (ΩA,te ∩ ΩN J ), this completes the proof of the second
part of the proposition.
Proof of Proposition 9. Define
∗ −t)
α = inf e−r(t
t<t̂
(1 + yt,t∗ ) − 1 > 0.
35
Pick any
∈ (0, α/ (5 + α)) with
4 / (1 − ) < (1 − ) (1 + α) − 1.
This is possible because the strict inequality is satisfied at
τ
(21)
= 0. Let
= t∗ + min {0.8 /c, γ/2} ∈ (t∗ , t∗ + γ)
n̄ = 1/ (min { /c, γ} − min {0.8 /c, γ/2}) < ∞.
Define also
te = min {t ∈ T ∗ |τ ≤ t ≤ t∗ + min { /c, γ}} ,
which is well defined since we consider n > n̄. Assume also that F (τ ) − F (t∗ ) > 1 − . We
will check that conditions O* and D are satisfied. The first statement then follows from
Proposition 6. To this end, first note that
F (te ) ≥ F (τ ) > 1 − ,
F (t∗ ) <
.
By definition (see Condition O*),
¡
¢
β̄ t̂, t∗ ≥ (1 − F (t∗ )) (1 + α) − 1 > (1 − ) (1 + α) − 1 ≡ β ∗ > 0,
where the last inequality follows from
< α/ (5 + α). Hence, condition O* is satisfied. We
now check that condition D is satisfied by t∗ and
0
≡ min (β ∗ , 4 / (1 − )) (as the value of
in the condition). Firstly,
δ t∗ ,te ≤ (1 − F (te )) / (1 − F (t∗ )) < / (1 − ) < min (β ∗ /4, / (1 − )) = 0 /4
where the last inequality is given by (21). Secondly,
c(te − t∗ ) ≤ c min { /c, γ} ≤ < min (β ∗ /4, / (1 − )) = 0 /4,
36
where the last equality holds because
< α/ (5 + α). This shows that t∗ and
0
satisfies
condition D.
The last statement in the proposition also follows from Proposition 6. We have
¡ ¢
¡
¡ ¢¢
Pri0 ΩA ≥ (1 − F (t∗ )) E0i Prit∗ ΩA
∙ i
0¸
i π t∗ −
≥ (1 − ) E0 i
.
π t∗ + 0
As → 0, we also have
0
→ 0, and the last expression goes to 1 by the bounded convergence
theorem.
Proof of Proposition 10. First note that
h
i
¡
¢
∗
β t̂, t∗ = inf 1 + yt,t∗ − er(t−t ) (1 − F (t)) / (1 − F (t∗ )) > β,
t<t̂
n
hence t̂,t∗ satisfy condition O*. Let te = min t ∈ T ∗ | t ≥ t∗ +
log(4/ )
r̂
∗
∗)
δ t∗ ,te = e−r(t−t ) (1 − F (t)) / (1 − F (t∗ )) ≤ e−r̂(te −t
Also, since te ≤ t∗ + 1/n +
log(4/ )
,
r
o
.We have
≤ /4.
we have that the bargaining power is weakly durable on
[t∗ , te ] with rate c that satisfies
¶
µ
log(4/ )
≤ /4,
c (te − t ) ≤ c 1/n +
r̂
∗
where the last inequality follows by the upper bound on r̂. Hence, t∗ and satisfy condition
D with c and te , and the first part of the result follows by Proposition 6. If, in addition,
the bargaining power is strongly durable, then condition SD also holds and the second part
of the result also follows from Proposition 6.
References
[1] Ali, S. Nageeb M. (2003): “Multilateral Bargaining with Subjective Biases: Waiting
To Settle,” Stanford University, mimeo, 2003.
37
[2] 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] Ma, Ching-to Albert and Michael Manove (1993): “Bargaining with Deadlines and
Imperfect Agent Control,” Econometrica 61, 1313-1339.
[5] Merlo, Antonio and Charles Wilson (1995): “A Stochastic Model of Sequential Bargaining with Complete Information,” Econometrica 63, 371-399.
[6] Roth, Alvin, Keith Murnighan, and Francoise Schoumaker (1988): "The Deadline
Effect in Bargaining: Some Experimental Evidence," American Economic Review 78,
806-823.
[7] Roth, Alvin and Axel Ockenfels (2002): "Last-Minute Bidding and the Rules for Ending
Second-Price Auctions: Evidence from eBay and Amazon Auctions on the Internet,"
American Economic Review, 92, 1093-1103.
[8] Rubinstein, Ariel (1982): “Perfect Equilibrium in a Bargaining Model,” Econometrica
50 , 97-110.
[9] Spier, Kathryn E. (1992): “The Dynamics of Pretrial Negotiation,” The Review of
Economic Studies, 59 (1), 93-108.
[10] Stahl, I. (1972): Bargaining Theory (Stockholm School of Economics).
[11] Yildiz, Muhamet (2003): “Bargaining without a Common Prior – An Agreement
Theorem,” Econometrica 71, 793-811.
[12] Yildiz, Muhamet (2004):“Waiting to Persuade,” Quarterly Journal of Economics 119,
223-248.
38