The Opacity of Prices in Multilateral Bargaining Under
Incomplete Information
Fei Li∗
January 29, 2014
Abstract
We study a multilateral bargaining game between one buyer and multiple sellers. The
buyer wants to undertake a project that requires him to purchase the necessary inputs from
multiple sellers. The buyer privately knows the revenue of the project and negotiates with
sellers one-by-one. We study the role of price transparency in which sellers sequentially screen
(bargain with) the buyer via a contingent contract. A high price agreed by the buyer and a
previous seller indicates a small residual share of the revenue and a high value of the buyer.
In the private offer case, each seller charges the same price; in the public offer case, there is a
first-mover advantage through charging a high price. The probability of impasses is greater
in the public offer case under a mild condition.
Keywords: Dynamic Game, Multilateral Bargaining, Opacity (transparency) of Prices,
Public Negotiations, Private Negotiations, Sequential Screening.
JEL Classification Codes: C70, D83, L23
∗
Department of Economics, University of North Carolina at Chapel Hill. Email: [email protected]. I am
grateful to Gary Biglaiser, Qingmin Liu, Mallesh Pai, Maher Said, Can Tian, Haibo Xu and Jidong Zhou.
1
Introduction
In many real-world situations, a buyer has to sequentially bargain with a large number of sellers
to obtain the necessary inputs to undertake a project. For example, a firm wants to produce a
final good that requires many intermediate goods or services provided by different upstream firms.
Other examples include an employer negotiating with each of many workers over wages, Disney
purchasing a large tract of land from several owners for a new Disney World, a stock-market
arbitrager buying up large blockholders to secure a toehold stake in a company, a drug developer
purchasing the rights to use multiple patents, a politician attempting to receive the last several
vote necessary to form a ruling coalition, and Coase’s famous railroad-farmer’s case. In such
situations, it is natural to assume that the profitability of the project is privately known by the
buyer only. By observing the contract signed by the buyer and other sellers, the buyer’s private
value may be partially revealed. As a result, the privacy of the offers plays a central role in
determining the likelihood that the project is undertaken and the transaction prices. This paper
tries to understand the role of price transparency in this scenario.
Since the project is undertaken only if an agreement is reached between the buyer and each
seller, goods owned by different sellers are perfectly complementary. However, there are two ways
in which sellers can affect each other. On the one hand, given the private value of the buyer,
signing a high price contract implies that the remaining surplus for other sellers is small, so other
sellers are inclined to charge the buyer low prices in order to secure the transactions. On the other
hand, by signing a high price contract, the buyer reveals his high private value, so other sellers are
apt to charge the buyer high prices. We show that when the distribution of the buyer’s value has
a decreasing inverse hazard rate, the first effect dominates, and sellers’ pricing behavior exhibits
strategic substitutes. As a result, the first mover has the incentive to price more aggressively since
he believes that the following sellers would respond by charging lower prices which partially offsets
the damage from the demand reduction caused by the aggressive price charged by the first seller.
Furthermore, the total price is higher under the public offer, and therefore, the probability of
agreement is smaller. According to Durden (1993), it was publicly known that James B. Duke was
trying to set up a new university in his neighborhood, so land prices in the community soared.
As a result, Duke had to slightly change the location and hire real-estate agents to acquire the
property in private. Our model provides a simple explanation of Duke’s experience.
In the main model, we focus on the contingent contract which is well studied in the one-tomany bargaining literature. That is to say, we assume that the buyer pays each seller only if he
reaches an agreement with all sellers and he is going to undertake the project. In the literature,
there is another well-studied contract: the binding cash flow contract. Under such a contract, the
2
buyer pays a seller once they reach a bilateral agreement. The key difference is that, under the
binding cash flow contract, a previous payment becomes a sunk cost to the buyer, so there is a
severe hold-up problem. We leave the discussion of the equilibrium under the binding cash flow
contract to section 4.
This paper is closely related to the literature on one-to-many negotiations. Cai (2000) and Horn
and Wolinsky (1988) study a fixed order of negotiations with complete information and discuss
the possibility of delay. Noe and Wang (2004), and Krasteva and Yildirim (2012) studied the
endogenous order of negotiations. They focus on the confidentiality of the sequence of bargaining,
which creates strategic uncertainty among sellers. However, we study a sequential screening model
and assume the order of negotiation is common knowledge, but we focus on the transparency of the
price rather than the sequence of the bargaining. The paper is also related to a growing literature
on the transparency of offers in bilateral bargaining, for example Horner and Vieille (2009), Kaya
and Liu (2013), and Kim (2011). Pavan and Calzolari (2009) study a sequential contracting model
where multiple principals sequentially sign irreversible contract with one agent, while we focus on
the contingent contract where sellers receive payment once the final agreement is reached between
the buyer and all sellers. In addition, we compare the price and the probability of impasses
under public negotiation and private negotiation. Lastly, our paper is related to consumer privacy
literature. Curtis (2004) and Calzolari and Pavan (2006) investigate the information sharing
among principals (sellers) about the agent’s (buyer’s) private value.
The rest of the paper is organized as follows. The model is presented in section 2. We study
the equilibrium in both public bargaining and private bargaining in section 3. In section 4, we
discuss some extensions of the main model. All the proofs are provided in the Appendix.
2
Model
There is a single buyer who decides to undertake a profitable project. The project can generate
a revenue v, which is the buyer’s private information. To undertake the project, the buyer has
to purchase two differential intermediate goods from two sellers, respectively. We index the seller
who owns good i as seller-i. Without loss of generality, we assume that the buyer visits sellers
according to their index, i.e., he visits seller 1 first, then seller 2. Each seller obtains 0 utility from
owing the good. Sellers do not know the buyer’s potential revenue v, and they share a common
prior belief on it: v ∈ [0, v̄] with a CDF F (v) and a PDF f (v). We assume that f (v) > 0 for each
(v)
v ∈ [0, v̄], and both f (v) and F (v) are infinitely many times differentiable. Denote J(v) = 1−F
f (v)
as the inverse hazard rate. Moreover, we make the following assumption.
3
Assumption 1. The inverse hazard rate J(v) is strictly decreasing.1
Since we assume that players bargain to sign a contingent contract, the player’s payoff is given
as follows. The buyer’s utility is v −p1 −p2 if the project is undertaken where pi is the price he pays
to seller i = 1, 2; otherwise, his utility is zero. Seller i’s payoff is pi if the project is undertaken,
and zero otherwise.
We make the following two assumptions on the bargaining procedure to rule out unreasonable
equilibria. First, we assume that the buyer sequentially visits sellers by paying a cost c ≥ 0 for
each visit. Since we consider a contingent contract, a buyer can alway “accept” an offer that he
cannot afford from seller 1 and reject the last seller’s offer to obtain a zero payoff. By assuming a
positive visiting cost, we can rule out such an unreasonable best response of players. Second, we
assume that once the buyer and seller i meet, with probability 1 − ∈ [0, 1], the seller is chosen
as the proposer of a take-it-or-leave-it offer and the buyer is the responder; with probability , the
buyer is chosen as the proposer.2 Given the offer made by the proposer, the responder decides
whether to accept it. If he declines the offer, the game ends; otherwise, the two parties sign the
contract, and the buyer moves on. Notice that, in any equilibrium, the buyer’s dominant strategy
is to name a zero price offer. Notice that for each > 0, a buyer with any positive revenue v > 0
strictly prefers to participate in the negotiations. We are interested in the case where both and
c are small. Moreover, we focus on the limit case in which both and c are equal to zero to
approximate the situation where they are both small; hence, in effect, only sellers make offers.
We consider two information regimes: private offers and public offers. In the first regime, seller
2 does not observe the price on which the buyer and seller 1 agreed. The bargaining sequence
is common knowledge, so when seller 2 is bargaining with the buyer, he knows that the buyer
has already taken seller 1’s offer. In the second regime, the whole bargaining history is public
information. Seller i’s (pure) strategy is σi : H → R+ where H = R+ ∪ ∅ is the set of public
history. The buyer’s (pure) strategy is σb1 , σb2 s.t. σb1 : R+ → {0, 1} and σb2 : R2+ → {0, 1}
where σb1 (p1 ) denotes the probability that the buyer accepts seller 1’s offer p1 and σb2 (p1 , p2 ) is the
probability that the buyer accepts seller 2’s offer p2 provided he has already accepted seller 1’s
offer p1 . The solution concept is a (pure strategy) perfect Bayesian equilibrium (PBE) in which
(1) the buyer maximizes his payoff, (2) each seller maximizes his expected payoff given his belief
on the buyer’s revenue, and (3) sellers’ beliefs update according to Bayes’ rule.
1
Bergstrom and Bagnoli (2005) show that the condition holds if and only if 1 − F (v) is strictly log-concave,
which is satisfied by many well-known distributions.
2
This bargaining protocol is commonly used in the literature; see Marx and Shaffer (2007, 2010) and Krasteva
and Yildirim (2012), for example.
4
3
Analysis
In this section, we derive the equilibrium under private and public offers. Similar to the
standard Coasian bargaining model, the skimming property holds in our model: in any PBE, if
the buyer with value v accepts the offer p1 proposed by the first seller, then the buyer with value
v 0 > v must accept p1 as well. The idea is that the v 0 -buyer can always mimic the v-buyer by
obtaining a higher payoff. As a result, in any PBE, on or off the path of play, seller 2 believes
that the buyer’s value v follows a truncated distribution on [k, v̄] where k ∈ [0, v̄].
3.1
Private Bargaining
We consider the private offer case first and solve the model backwards. In this regime, seller
2 does not observe the agreement between the buyer and seller 1, but he has a belief on the price
on which the buyer and seller agree. Suppose he believes the price is p̃1 . In any equilibrium, the
agreed price cannot be greater than v̄. Hence, p̃1 ≤ v̄, and the cutoff value of the buyer is k = p̃1 .3
Seller 2’s problem is to choose p2 to maximize his expected payoff. When p̃1 = v̄, seller 2’s
expected payoff is zero for every p2 ≥ 0. When p̃1 < v̄, seller 2’s problem is
max+
p2 ∈R
1 − F (p̃1 + p2 )
p2
1 − F (p̃1 )
where the solution p∗2 satisfies the F.O.C., 1 − F (p̃1 + p∗2 ) − p∗2 f (p̃1 + p∗2 ) = 0. Since f (v) > 0 for
every v ∈ [v, v̄], we have
1 − F (p̃1 + p∗2 )
= J(p̃1 + p∗2 ).
(1)
p∗2 =
∗
f (p̃1 + p2 )
Since J(v) is strictly decreasing, v − J(v) is strictly increasing. Because f > 0 for every v, we
1
must have (a) v − J(v)|v=0 = 0 − J(0) = − f (0)
< 0, and (b) v − J(v)|v=v̄ = v̄ − J(v̄) = v̄ > 0.
By the intermediate value theorem and the monotonicity of v − J(v) function, for each p̃1 < v̄,
there is a unique solution p∗2 < v̄ satisfying equation (1). The basic idea is illustrated in figure (1).
Given the believed p̃1 , seller 2 believes that the buyer’s value is v ∈ [p̃1 , v̄]. By charging a price p2 ,
seller 2 expects that the final agreement will be reached only if the buyer’s value is v ≥ p̃1 + p2 .
Seller 2’s optimal price p∗2 is uniquely pinned down by equation (1). Differentiating equation (1)
3
For every c > 0, the buyer with value v < p̃1 would reject the offer for sure. When > 0 and c is arbitrarily
small, the buyer with value v ∈ (p̃1 + O(c), p̃2 + O(c)) would accept price p̃1 since he expects to make a zero price
offer to seller 2 with positive probability, where p̃2 is the price he believes was offered by seller 2. We examine
the limit of such a best response of the buyer by taking c and to zero sequentially. Notice here the sequence of
taking the limits of c and does matter. We are interested in the case in which (1) the buyer’s bargaining power is small, and (2) the visiting cost is even smaller relative to .
5
v − J(v)
p̃1
p2 − J(p̃1 + p2 )
p̃1
p̃1 + p∗2
0
v̄
v
Figure 1: Seller 2’s best response: p∗2 . For any p̃1 , seller 2 believes the present buyer’s value is
v ∈ [p̃1 , v̄], and the buyer accepts a price offer p∗2 only if his private value is not less than k = p̃1 +p∗2 .
As p̃1 increases, (1) the lower bound of the buyer’s type p̃1 increases (information externality), (2)
for each v ∈ [p̃1 , v̄], the residual revenue v − p̃1 decreases (payoff externality), and (3) the cutoff
type k strictly increases.
with respect to p1 yields the finding that the strategic link between two sellers’ pricing decision
can be expressed as follows:
∂p∗2
J 0 (p̃1 + p∗2 (p̃1 ))
=
<0
∂ p̃1
1 − J 0 (p̃1 + p∗2 (p̃1 ))
(2)
where the strict inequality is implied by the fact that J(v) is strictly decreasing. Notice that an
increment of p̃1 plays a dual role:
1. information externality: it pushes up seller 2’s belief on the revenue of the buyer he is facing,
so seller is apt to charge higher price, and
2. payoff externality: for every v, the remaining share v− p̃1 is lower, so seller 2 has the incentive
to lower the price to secure a transaction.
When the inverse hazard rate is decreasing, the second effect dominates, so that sellers’ pricing
behavior is strategic substitute.
Now consider seller 1’s problem:
max+ [1 − F (p1 + p∗2 [p̃1 ])]p1
p1 ∈R
6
where p∗2 [p̃1 ] is the price that seller 1 believes is charged by seller 2. Notice that seller 1 cannot
manipulate seller 2’s pricing behavior under a private offer, so he has to treat p∗2 as given. As long
as p̃1 < v̄, p∗2 [p̃1 ] < v̄ as well, so there is a unique solution of p∗1 < v̄ such that
p∗1 = J(p∗1 + p∗2 [p̃1 ]).
(3)
When p̃1 = v̄, any p∗2 (p̃1 ) ≥ 0 is seller 2’s best response.
In equilibrium, seller 2’s belief is correct, so p̃1 = p∗1 . Therefore, we have the following proposition.
Proposition 1. In the private bargaining, there is a PBE in which seller i offers p∗i for i = 1, 2,
p∗1 = p∗2 = J(p∗1 + p∗2 ).
(4)
where p∗1 + p∗2 < v̄, and the agreement is reached only if the buyer’s value v ≥ v ∗ = p∗1 + p∗2 .
Since the transaction price is not observable, seller 1 cannot manipulate seller 2’s belief in
private bargaining. The equilibrium is symmetric: two sellers charge the same price, so there is
no advantage or disadvantage to be the first mover, and the probability of agreement is positive.
Notice that there is another equilibrium in which p∗1 = p∗2 = v̄. In this equilibrium, each seller’s
payoff is zero, and the agreement is reached with probability zero. However, this equilibrium is not
robust to the perturbation of > 0. When > 0, seller 1 can deviate to any price 0 < p1 < v̄ and
expect a positive payoff. The reason is that the buyer is chosen as the proposer in his negotiation
with seller 2 so that the project is undertaken and seller 1 gets p1 . Since our goal is to understand
the situation where c, are positive but very small, we do not consider such an equilibrium.
3.2
Public Bargaining
Now we consider the public offer case. Seller 2’s problem is similar except that he observes the
price agreed by the buyer and seller 1 in this regime, i.e. p̃1 = p1 . His best response again satisfies
p∗2 = J(p1 + p∗2 ).
for p1 < v̄. When p1 ≥ v̄, any p2 ≥ 0 belongs to the set of best response and his expected payoff
is zero regardless of the choice of p2 .
Back to seller 1’s problem,
max+ [1 − F (p1 + p∗2 (p1 ))]p1
p1 ∈R
It is clear that for p1 ≥ v̄, the expected payoff is zero. By choosing a price p1 < v̄, seller 1
expects p1 + p2 < v̄ as well, so the probability of agreement and his expected payoff is strictly
7
positive. As a result, we can focus on interior solutions. In this case, seller 1 understands that his
pricing decision has a direct effect on seller 2’s pricing decision and the probability that the final
agreement is reached. The F.O.C. is given by
∂p∗2
1 − F (p1 + p∗2 )
− p1 [1 +
]=0
f (p1 + p∗2 )
∂p1
(5)
Since seller 1’s offer is observable, seller 2’s optimal price p2 directly depends on seller 1’s choice
∂p∗
p1 . As a result, in the F.O.C, ∂p21 6= 0, which captures seller 1’ strategic impact on seller 2’s pricing
decision. Consequently, seller 1 has the incentive to manipulate seller 2’s behavior through his
pricing decision.
0
2
(v)] 4
Proposition 2. Suppose that J 00 (v) ≥ − [1−J
. In the public bargaining, there is a unique
J(v)
?
PBE in which seller i offers a price pi for i = 1, 2 s.t.
p?1 = J(p?1 + p?2 )[1 − J 0 (p?1 + p?2 )]
p?2 = J(p?1 + p?2 ).
where p?1 > p?2 , p?1 + p?2 < v̄ and agreement is reached only if the buyer’s value v ≥ v ? = p?1 + p?2 .
Since sellers’ pricing behavior is strategic substitutes, seller 1 has stronger incentive to charge
a high price, since he believes that seller 2 would respond by charging a lower price which would
compensate the reduction in demand caused by seller 1’s high price. In equilibrium, p?1 > p?2 , and
seller 1 has the first mover advantage, which is similar to the standard Stackelberg model in which
sellers choose quantity instead of price sequentially.
3.3
Comparison
We compare the probability of an agreement being reached under private and public offers. To
do so, we only need to compare the cutoff type who is indifferent about reaching agreement with
sellers under private and public offers. First, suppose two sellers can form a coalition to bargain
with the buyer. Apparently, they would charge the optimal monopoly price pc = v c s.t.
v c = J(v c )
and share the total payoff once agreement is reached. Since the buyer would accept the offer as
long as his revenue v ≥ v c , the probability of impasses is F (v c ). In the following proposition, (1)
4
This condition ensures the second-order condition of seller 1’ optimization problem holds. A stronger condition
is that J(.) is convex. It is satisfied by many distributions, such as a uniform distribution and Weilbull distributions
with a CDF F (x) = 1 − exp(−xλ ) where x ≥ 0 and the parameter λ > 1
8
we show that, in both the public and private negotiation, the probability of impasses is higher
than the probability of impasses in the case where they can coordinate and bargain with the seller
together, and (2) we compare the probability of agreement under coordination, in both the private
and the public bargaining.
Proposition 3. Under both private and public bargaining, sellers price more aggressively than in
the case of coordination, i.e., v ? , v ∗ > v c , and the probability of agreement in private bargaining is
greater than that in public bargaining, i.e., v ? > v ∗ .
As we mentioned, seller 1 takes the first mover advantage and charges a high price, while seller
2 charges a low price to partially make up for the reduction in demand caused by seller 1. However,
the compensation is only partial, so the total price in the public offer case is higher than in the
private offer case.
Example 1 (Uniform Distribution). Suppose that v ∼ U [0, 1]. The equilibrium outcome under
private offers and public offers is given as follows. Under private offers, p∗1 = p∗2 = 1/3 and the
probability that an agreement is reached is 1/3. Under public offers, p?1 = 1/2, p?2 = 1/4 and the
probability that an agreement is reached is 1/4.
4
Discussions
A key assumption in the main model is that, the inverse hazard rate is strictly decreasing. This
assumption is commonly used in mechanism design and non-linear pricing literature. However, it
is theoretically interesting to know how this assumption drives the result. If we assume increasing
inverse hazard rate and increasing virtual value v − J(v), sellers pricing behavior will not be
2
> 0, so in the public bargaining case,
strategic substitutes anymore. Equation (2) implies that dp
dp̃1
?
?
we will have p1 < p2 , and the probability of impasse is higher in the private bargaining case.
In the main model, we assume that players sign a contingent contract: the seller gets money
only if the project is finally undertaken. In the multilateral bargaining model with complete
information, there is another well-studied contract: the binding cash flow contract. The key
difference is that, under the binding cash flow contract, once a payment is made, it becomes a
sunk cost. In our model, if players sign a binding cash flow contract, there is no equilibrium with
trade under private offers. Consider the uniform distribution case as an example. Suppose there
is an equilibrium with trade and the equilibrium prices are (p∗1 , p∗2 ). The buyer accepts seller 1’s
offer p∗1 only if he expects he will accept seller 2’s offer p∗2 and will get a non negative surplus by
undertaking the project, i.e. v ≥ p∗1 + p∗1 . Hence, when seller 2 is bargaining with the buyer, he
should believe that the buyer’s revenue v ≥ p∗1 + p∗2 . However, since p∗1 is a sunk cost, seller 2
9
has the incentive to charge a price p02 ≥ p∗1 + p∗2 . When p∗1 > 0, p02 > p∗2 , there exists a profitable
deviation, which implies that the hypothetical strategy profile is not an equilibrium. When p∗1 = 0,
seller 2 may not have the incentive to deviate from the equilibrium. However, under private offers,
seller 1 has the incentive to deviate by charging a non-negative price to obtain positive payoff,
which is a contradiction again. In the public offer case, since seller 1’s deviation is observable, and
one can impose a belief threat off the path of play, there exists pure strategy equilibria with trade.
For example, there is a PBE in which (1) seller 1 charges p∗1 = 0, (2) seller 2 charges p∗2 = J(p∗2 )
if p1 = 0 and charges v̄ otherwise, and (3) the buyer accepts the offer only if v > p∗2 and p1 = 0.
The strategy profile is a PBE because once seller 1 deviates by charging p1 > 0, both the buyer
and seller 2 understand they are off the path of play. The demand is zero since no buyer expects
non negative surplus by accepting p1 > 0 and p̃2 = v̄. Seller 2 is supposed to see no demand after
seller 1’s deviation. If he sees the buyer, he believes the buyer deviates as well. His belief off the
path of play is that the buyer’s type is v̄, so he would charge a price v̄. As a result, seller 1 has
no incentive to deviate from the zero price in the first place.
In the main model, we assume there are two sellers. We can extend our intuition into a model
in which there are N > 2 sellers. The main difference is that, in the public offer case, seller i’s
pricing behavior has not only a direct effect but also an indirect influence on seller i+1, i+2, ....N ’s
pricing behavior. Consider a numerical example where the seller’s revenue v ∼ U [0, 1] and N = 3.
Given seller 1’s price p1 , the continuation game played by the buyer and seller 2 and seller 3 is
identical to our main model. Simple algebra implies that p3 = 1−p21 −p2 , and p2 = 23 [1 − p1 − p3 ].
Notice that both the optimal p2 and p3 depend on p1 , so seller 1 can directly affect seller 2 and
3’s pricing decision. In addition, the optimal p2 and p3 depend on each other, so seller 1 can also
affect p2 (and p3 ) indirectly through influencing p3 (and p2 ). In the public offer case where N = 3,
one can show that the equilibrium price is p1 = 1/2, p2 = 1/4, p3 = 1/8, and the probability
of impasse is 7/8; while in the private offer case, pi = 1/4 for i = 1, 2, 3, and the probability
of impasse is 3/4. Thus, the probability of transaction is higher under private bargaining. One
can show that this conclusion remains for finite N in the uniform distribution model, and show
that the probability of transition is higher under private bargaining for each N . As N → ∞, the
probability of transaction goes to zero in both case, but the probability of transaction in the public
bargaining goes to zero faster than that in the private bargaining. For general distributions, the
presence of higher order effect requires restrictions on higher order derivatives of J(v) to ensure
the existence and uniqueness of the equilibrium.
10
Appendix
The Proof of Proposition 1. First, J 0 < 0, so v − 2J(v) is strictly increasing in v and there is
a unique v ∗ s.t.
v ∗ = 2J(v ∗ )
(6)
Let p∗1 = p∗2 = v ∗ /2 = J(v ∗ ). It is clear that, p∗i satisfies seller i’s F.O.C. The second order
condition (S.O.C) of each seller can be simplified as (J 0 − 1)f + f 0 (J − p) where the second term
is zero at v ∗ and J 0 (v) − 1 < 0 by assumption. Hence, no seller has the incentive to deviate from
v ∗ /2. The Proof of Proposition 2. The result directly comes from the combination of equation (5)
and (2). Let p1 + p2 = v, and define v ? s.t.
v ? = J(v ? )[2 − J 0 (v ? )]
(7)
0 2
)
implies that v = J(2 − J 0 ) has a unique solution. Let p?1 =
The condition J 00 ≥ − (1−J
J
J(v ? )[1 − J 0 (v ? )] and p?2 = J(v ? ). Simple algebra implies that the second-order conditions are
satisfied for both seller i at p?i for i = 1, 2. Hence, we get the unique equilibrium. The Proof of Proposition 3. We first show that v ∗ < v ? . Rewrite equations (6) and (7) as
follows
v ∗ − J(v ∗ ) = J(v ∗ )
v ? − J(v ? ) = J(v ? )[1 − J 0 (v ? )]
Taking the difference of them implies that
v ∗ − J(v ∗ ) − [v ? − J(v ? )] = J(v ∗ ) − {J(v ? )[1 − J 0 (v ? )]}
(8)
Since J 0 (·) < 0, it is clear that v ∗ 6= v ? ; otherwise, the left-hand side (LHS) of equation (8) is
zero but the right-hand side (RHS) is not. Now suppose that v ∗ > v ? . The LHS of equation (8) is
strictly positive since v −J(v) is strictly increasing. The RHS becomes J(v ∗ )−J(v ? )+J(v ? )J 0 (v ? ).
Since J 0 (·) < 0 and v ∗ > v ? , we must have J(v ∗ ) − J(v ? ) < 0 and (v ? )J 0 (v ? ) < 0. Hence, the RHS
of equation (8) is strictly negative, which is a contradiction! Hence, in the strategic substitutes
case, v ∗ < v ? .
Since v c = J(v c ). Suppose v c ≥ v ∗ , then we must have v c − v ∗ = J(v c ) − 2J(v ∗ ) >≥ 0 and
J(v c ) > J(v ∗ ) since J(·) > 0 for v < v̄. However, since J(·) is strictly decreasing, we have a
contradiction! Hence, v c is less than v ∗ and v ? . 11
References
[1] Bergstrom, Ted, and Mark Bagnoli. ”Log-concave probability and its applications.” Economic
Theory 26 (2005): 445-469.
[2] Cai, Hongbin. ”Delay in multilateral bargaining under complete information.” Journal of Economic Theory 93.2 (2000): 260-276.
[3] Calzolari, Giacomo, and Alessandro Pavan. ”On the optimality of privacy in sequential contracting.” Journal of Economic Theory 130.1 (2006): 168-204.
[4] Durden, R., 1993. The Launching of Duke University, 1924-1949. Duke University Press,
Durham, NC.
[5] Horn, Henrick, and Asher Wolinsky. ”Bilateral monopolies and incentives for merger.” RAND
Journal of Economics (1988): 408-419.
[6] Horner, Johannes, and Nicolas Vieille. ”Public vs. private offers in the market for lemons.”
Econometrica 77.1 (2009): 29-69.
[7] Kaya, Ayca, and Qingmin Liu. ”Transparency and price formation.” (2013).
[8] Kim, Kyungmin. ”Information about sellers past behavior in the market for lemons.” University
of Iowa, unpublished manuscript (2011).
[9] Krasteva, Silvana, and Huseyin Yildirim. ”On the role of confidentiality and deadlines in
bilateral negotiations.” Games and Economic Behavior 75.2 (2012): 714-730.
[10] Marx, Leslie M., and Greg Shaffer. ”Rent shifting and the order of negotiations.” International
Journal of Industrial Organization 25.5 (2007): 1109-1125.
[11] Marx, Leslie M., and Greg Shaffer. ”Break-up fees and bargaining power in sequential contracting.” International Journal of Industrial Organization 28.5 (2010): 451-463.
[12] Noe, Thomas H., and Jun Wang. ”Fooling all of the people some of the time: A theory of
endogenous sequencing in confidential negotiations.” Review of Economic Studies 71.3 (2004):
855-881.
[13] Pavan, Alessandro, and Giacomo Calzolari. ”Sequential contracting with multiple principals.”
Journal of Economic Theory 144.2 (2009): 503-531.
12
[14] Taylor, Curtis R. ”Consumer privacy and the market for customer information.” RAND
Journal of Economics (2004): 631-650.
13