Delay in Bargaining with Externalities

Working Paper 2006:29

Department of Economics

Delay in Bargaining with Externalities

Jonas Björnerstedt and Andreas Westermark

Department of Economics Working paper 2006:29

Uppsala University November 2006

P.O. Box 513 ISSN 1653-6975

SE-751 20 Uppsala Sweden

Fax: +46 18 471 14 78

D ELAY IN B ARGAINING WITH E XTERNALITIES

J ONAS B JÖRNERSTEDT AND A NDREAS W ESTERMARK

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Delay in Bargaining with Externalities ^∗

Jonas Björnerstedt ^† Andreas Westermark ^‡ November 24, 2006

Abstract

This paper studies infinite-horizon bargaining between a seller and multiple buyers when externalities are present. We extend the analysis in Jehiel & Moldovanu (1995a), by allowing for both pure and mixed equilibria. This extension is warranted, since under some circum- stances, the complexity of the equilibria with bounded recall they analyze tend to infinity as players become very patient. We show that stationary subgame perfect equilibria always exist. Moreover, a characterization of the stationary subgame perfect equilibria in generic games is presented. Equilibria with delay exist only for strong positive externalities. Since each buyer receives a positive payoff when the seller agrees with some other buyer, posi- tive externalities induces a war of attrition between buyers. Furthermore, the results when analyzing mixed stationary equilibria are different than when focusing on pure strategies with bounded recall as Jehiel & Moldovanu (1995a). Specifically, they find delay only when externalities are negative.

Keywords : Bargaining, externalities, delay JEL Classification : C72, C78, D62.

∗

Our work has benefitted greatly from discussions with Sven-Olof Fridolfsson, Lars Persson and Johan Stennek and from participants at seminars/workshops at ESEM 2006 in Vienna and the International Conference on Game Theory 2006 at Stony Brook. Björnerstedt gratefully acknowledges financial support from the Jan Wallander and Tom Hedelius Foundation and Westermark from the Swedish Council for Working Life and Social Research.

†

Swedish Competition Authority, SE-103 85 Stockholm, Sweden. E-mail: jonas.bjornerstedt@kkv.se.

‡

Corresponding author. Department of Economics, Uppsala University, Box 513, SE-751 20 Uppsala, Sweden.

Fax (+46) 18 471 15 94, phone (+46) 18 471 15 94. E-mail: andreas.westermark@nek.uu.se.

1 Introduction

Many agreements in society are determined through bargaining, e.g., firm worker bargaining over wages. It is also common that, when a good is traded, externalities are imposed on others. As an example, if an exclusive patent right is sold to one of several manufacturing firms, externalities are imposed on the others, if these firms compete with each other and if the patent right affect the cost structure of the buying firm. Another large strand of literature concerns the issue of delay in bargaining. One focus has been on asymmetric information; see e.g., Admati & Perry (1987) Cramton (1992) and others. The possibility of delay with perfect information has also been analyzed by e.g., Fernandez & Glazer (1990), Cai (2000). In a couple of seminal papers, Jehiel

& Moldovanu (1995a and 1995b) has analyzed delay in bargaining with externalities with perfect information. Both games with a finite and infinite horizon are analyzed, and the results depend on whether a final deadline exists. With an infinite horizon, only pure strategy equilibria are analyzed. The focus is on “simple” pure strategy equilibria. Since stationary equilibria in pure strategies not necessarily exist, they focus on strategies with bounded recall, i.e., strategies that are more complicated than stationary but where players have limited memory capacity. However, as pointed out by Jehiel & Moldovanu the complexity tends to infinity in some circumstances as discount factors converge to one. Some recent papers show existence of stationary subgame perfect equilibria, when allowing for both pure and mixed equilibria; see Westermark (2003) and Gomes (2005). To extend the analysis of bargaining with externalities to mixed stationary strategies is clearly motivated, since these strategies are simple, independently of the degree of patience of the players.

In this paper, we analyze infinite-horizon bargaining between a seller and many buyers with perfect information when externalities are present. We analyze delay and provide a charac- terization of generic stationary subgame perfect equilibria in both mixed an pure stationary strategies.

In the model, the seller owns an indivisible object that can be sold to one out of many potential buyers. If one buyer buys the good, externalities are imposed on the other buyers.

The externalities imposed on a buyer that does not acquire the good might depend on the identity

of the buyer that obtains the good. Bargaining takes place as follows. In each period, one of the

buyers is selected randomly to bargain with the seller. Then the proposer is randomly selected

from the chosen buyer and the seller. The selected proposer offers a price and the respondent

accepts or rejects. In case of acceptance, the game ends and otherwise the negotiation proceeds

to the next stage where again a buyer is drawn at random and so on. We assume that there is

no deadline; thus, there is an infinite horizon.

The bargaining model is a generalization of the model in Jehiel & Moldovanu (1995a) with an infinite horizon, allowing for different degrees of bargaining power. First, we show that stationary equilibria in mixed strategies always exist. Second, we find that delay occurs only with positive externalities. Thus, allowing for mixed equilibria gives different result, more in line with the deadline model in Jehiel & Moldovanu (1995a), where they also get delay with positive externalities. The reason is that the price that buyers are willing to pay increases the closer to the deadline one gets. In the last period, all buyers are willing to pay their valuation (Jehiel

& Moldovanu assumes that the seller has all the bargaining power). In the period just before the deadline, buyers are not willing to pay so much, since if some other buyer buys the good in the last period, they end up with a positive payoff, due to the positive externalities. Thus, prices increase the closer one gets to the deadline, inducing the seller to wait. The argument here is fairly similar but is a little different. Although there is no deadline, buyers want to wait to agree since if some other buyer ends up with the good, the buyer still receives a positive payoff. Thus, there is a war of attrition between the buyers, with each potential buyer hoping for some other buyer to purchase the good. Such a war of attrition is not possible to sustain in equilibrium when externalities are negative. Jehiel & Moldovanu also gets delay when there is negative externalities, both when there is a deadline and with an infinite horizon. With an infinite horizon as in Jehiel & Moldovanu (1995a), cyclical delay results. These equilibria are nonstationary and are hence ruled out in this paper. Moreover, as pointed out by Jehiel &

Moldovanu (1995a), complexity tends to infinity as the discount factors converge to one. Our equilibrium strategies are, independently of discount factors, very simple.

In section 2 the model is described and existence is proven. Section 3 defines genericity, section 4 characterizes the equilibria, section 5 describes more conditions for when only “simple”

equilibria exist and finally section 6 concludes. All proofs are relegated to the appendix.

2 The Model

One seller bargains with a set N of buyers with |N| > 1 on the sale of an indivisible good. The surplus of selling to buyer i is π _i > 0, with all other buyers j receiving their externality e _j,i , setting e _i,i = 0. Let Ω ⊂ R ^|N| + × R ^|N|(|N|−1) denote the set of possible pies and externalities.

We assume that in each round all buyer seller pairs meet with equal probability. ¹ The seller

1

Arbitrary matching probabilities complicates notation without qualitatively affecting results. The existence

proof is based on Westermark (2003) that assumes arbitrary matching probabilities. Thus, the existence proof

in this paper could trivially be extended to this case. Moreover, the intuition for the delay result in proposition

5 does not depend on the fact that matching probabilities are identical. The result has to be modified so that

equilibrium probabilities for making acceptable offers are adjusted to taken into account the fact that matching

has discount factor δ S and the buyers discount factor δ B , in line with the assumptions in Jehiel

& Moldovanu (1995a and 1995b). Generalizing the model of Jehiel & Moldovanu (1995a and 1995b), the seller makes a bid with probability η, and with 1 − η the selected buyer does. It is also a generalization in the respect that we allow for equilibria in both pure and mixed strategies.

Let v S,i and w S,i denote the value to the seller in bidding and receiving a bid from buyer i, and v _i,S and w _i,S denote the value to buyer i of bidding and receiving a bid. Let p _S,i be the probability that the seller gives an acceptable bid to i when bidding and p i,S the probability that i gives an acceptable bid. Defining p _i = (p _S,i + p _i,S ) /2, the value equations are given by

v S,i = (1 − p ^S,i ) w S,i + p S,i (π i − w ^i,S ) , (1)

w _S,i = δ _S

⎛

⎝ η

|N|

X

j∈N

v _S,j + 1 − η

|N|

X

j∈N

w _S,j

⎞

⎠ ,

v _i,S = p _i,S (π _i − w S,i ) + (1 − p i,S ) w _i,S , w i,S = δ B

1

|N| ((1 − η) v ^i,S + ηw i,S ) + δ B

X

j∈N\{i}

p j

|N| e i,j + δ B

X

j∈N\{i}

1 − p ^j

|N| w i,S .

When negotiating with i, in giving an acceptable offer (with probability p S,i ) it is sufficient to offer w _i,S to i. Since w _S,i is the continuation value conditional on disagreement, the value v _S,i in (1) follows. By similar reasoning v i,S is determined. When rejecting a proposal by i, S gets v _S,j with probability η and w _S,j with 1 − η, giving w S,i in (1). When i rejects a proposal, i is selected to bargain with S with probability _|N| ¹ giving (1 − η) v i,S + ηw _i,S . If some other player j is selected, i will receive e _i,j if S and j agree in the next period. With probability 1 − p j they do not, giving w i,S .

Note that, since the left hand side of the value equation for w _S,i is the same for all i, we have w _S,i = w _S,j for all i and j. Hence we can write

w S,i = δ S η 1 − δ ^S (1 − η)

1

|N|

X

j∈N

v S,j (2)

For it to be profitable to make an acceptable offer, it is necessary that

π _a − w a,S ≥ w S,a . (3)

probabilities are asymmetric; specifically, the equilibrium probabilities for making acceptable offers have to be

adjusted so that the actual agreement probabilities in Proposition 5 are unchanged. See also Jehiel & Moldovanu

(1995b) for a motivation for assuming symmetric matching probabilities.

Similarly, for the seller and m to bid with 0 < p m < 1, they have to be indifferent between bidding and not:

π m − w ^m,S = w S,m . (4)

It is also necessary that in negotiations with buyers where p r = 0 that it is profitable to make unacceptable offers, i.e., that

π r − w ^r,S ≤ w ^S,r . (5)

We first show that a stationary subgame perfect equilibrium exists.

Proposition 1 There exists a stationary subgame perfect equilibrium.

All proofs are relegated to the appendix. The argument used closely follows Westermark (2003), see also Gomes (2005).

3 Genericity

It turns out that many equilibria exist only for special parameter configurations. This is formal- ized in detail below. More specifically, the set of π _i and e _i,j that support these equilibria have strictly lower dimensionality than the full parameter space as δ B → 1 and δ ^S → 1.

Example 2 To illustrate this point, consider the case with two buyers, and conjecture an equi- librium with immediate agreement with both buyers along the lines of Horn & Wolinsky (1988).

In the proposed equilibrium p 1 = p 2 = 1. Using this in (1) gives

w _a,S = δ _B (1 − η) (π ^a − w ^S,a ) + e a,j

2 − δ B η ,

for j 6= a and

w _S,a = δ _S η

2 (1 − δ S (1 − η)) − ηδ B

1

2 ((2 − δ B ) (π ₁ + π ₂ ) − δ B (e _1,2 + e _2,1 )) . Using these in the condition for acceptance (3) we get

π a − δ _B

2 − δ ^B e a,j ≥ δ _S η

2 (1 − δ ^S (1 − η)) − ηδ ^B 1

2 ((2 − δ ^B ) (π 1 + π 2 ) − δ ^B (e 1,2 + e 2,1 )) , for j 6= a. We have, in the limit, setting a = 1, j = 2 and a = 2, j = 1 in the expression above

π 1 + e 2,1 ≥ π ² + e 1,2 ,

π 2 + e 1,2 ≥ π ¹ + e 2,1 .

This implies that π 1 + e 2,1 = π 2 + e 1,2 , implying an additional restriction on the parameters for this equilibrium to exist. Hence, the equilibrium is non-generic. Thus, agreement with both sellers with probability one cannot be a generic equilibrium. In order for equilibria with agreement with both buyers with probability 1 to exist for δ _B , δ _S close to one, the seller essentially has to be indifferent between who he agrees with. If not, the seller could simply wait for the best buyer.

Note that, when the discount factors are close to one, the region where the equilibrium exist is arbitrarily small.

Now, let us define generic equilibria. When doing so, it is convenient to distinguish between different equilibrium types. Here, equilibria are classified according to which buyers that, in equilibrium, are associated with agreement probabilities being one, positive but less than one and zero, respectively. Let σ denote a stationary strategy profile. Given σ, let A ⊂ N be the set of buyers that agree with probability one, and let M and R denote the set of buyers that agree with mixed and zero probabilities respectively. Let Φ (σ) = (|A| , |M| , |R|) denote the equilibrium type of σ. Let Σ (ω, δ _B , δ _S ) denote the correspondence from the set of parameters ω ∈ Ω and δ B and δ _S to the (possibly empty) set of stationary equilibria for these parameters.

Define

Ω (u, δ _B , δ _S ) = {ω ∈ Ω : ∃σ ∈ Σ (ω, δ B , δ _S ) such that Φ (σ) = u}

as the set of parameter values generating the equilibrium type u, given δ _B and δ _S . Let λ denote a Lebesque measure of subsets of Ω.

We define genericity in terms of whether equilibria in the limit exist in a subset of Ω with positive measure. Before defining genericity, first note that we below state conditions in terms of equilibrium and deviation payoffs for the existence of different types of equilibria. Note that the payoff distribution might depend on the relationship between δ S and δ B . Specifically, the limit equilibrium payoffs, i.e., the payoffs as we let δ _S and δ _B converge to one, in some cases depend on the limit of the ratio _1−δ ^1−δ

B

. To ensure that limit payoffs exist, we restrict attention to sequences {δ ^t B } and {δ S ^t } converging to one that also have a well defined finite limit of the ratio ^1−δ _1−δ

B

. Thus, the limit of this ratio is nonnegative and is contained in the set [ρ, ¯ ρ] where ρ < ∞ and ρ ≥ 0. Let λ ¯ r denote a Lebesque measure of subsets of [ρ, ¯ ρ]. Although we view the restriction on sequences as not very strong - basically we restrict the rate of convergence of discount factors so that convergence cannot be infinitely faster for the seller than for the buyers - the restrictions can be ignored, if we set η = 1 as in Jehiel & Moldovanu (1995a and 1995b). ²

2

This can be seen from the proof of Propositions 4 and 6 (especially expressions (27) and (31)), since the ratio

1−δS

1−δB

vanishes from equilibrium payoffs, enabling us to use arbitrary sequences.

We define genericity as follows.

Definition 3 Say that the equilibrium type u is generic if there is a set S ⊂ [ρ, ¯ ρ] with λ _r (S) > 0 such that, for all sequences {δ B ^t } and {δ B ^t } with the property that lim t→∞ 1−δ

1−δ

= ρ ∈ S, we have lim _t→∞ λ ¡

Ω ¡

u, δ ^t _B , δ _S ^t ¢¢

> 0.

Note that genericity for equilibrium types is not defined in the strong sense that it exists for almost all parameter values. It is sufficient that it has positive measure. Non-generic equilibria exists only on sets of measure zero, though. Definition 3 can then be modified as follows:

Definition 3’. Say that the equilibrium type u is generic if we have lim _t→∞ λ ¡ Ω ¡

u, δ _B ^t , δ _S ^t ¢¢

> 0.

4 Equilibrium Characterization

In this section, we characterize the generic equilibria. In general, there are a large number of equilibrium types, since we can divide buyers into three sets A, M and R where agreement occurs with probability one for a ∈ A, with positive probability less than one for m ∈ M and with zero probability for r ∈ R. Any partition of the set of players in three such sets is an equilibrium candidate. As we will show in Proposition 9, the following four cases are the generic equilibrium types.

Single out Φ = (1, 0, |N| − 1)

Hold-up Φ = (0, |M| , |N| − |M|) for |M| > 1 Outside option Φ = (1, 1, |N| − 2)

Type IV Φ = (1, |M| , |N| − |M| − 1) for |M| > 1

Before showing genericity, we establish some conditions that these different equilibrium types satisfy. Let E _R,M be the matrix of externalities e _r,m to r ∈ R when m ∈ M agree, and E M,M the matrix of externalities to m ∈ M. We similarly define E A,M and E _M,A . Let π _M the vector with π _m as the m’th element for m ∈ M with π R similarly defined and let Π _M and Π _R be diagonal matrices with π M and π R respectively on the main diagonals. Let J x,y be a |x| × |y| matrix with all elements 1, I _x the |x| dimensional identity matrix, and j x a |x| vector of ones.

We first focus on pure strategy equilibria. Since, as we show below, equilibria with |A| > 1 are non-generic, the proposition below describes conditions for pure strategy equilibria with

|A| = 1.

Proposition 4 Consider sequences {δ B ^t } and {δ S ^t } with ρ = lim t→∞ 1−δ

1−δ

. There exists a T such that for all δ _B ^t , δ ^t _S with t > T there exists an equilibrium with p a = 1 for some a ∈ N, and p _r = 0 for all r 6= a if

π r − e ^r,a < ηπ a

η + ρ (1 − η) (6)

for all r 6= a.

Since p _a = 1 and p _r = 0 for all other buyers r we can think of the equilibrium payoff as being a situation where the seller only bargains with a and the surplus consists of π a , giving the seller _{η+ρ(1−η)} ^ηπ

π _a . Note that, if ρ = 1, the seller gets a share of the surplus corresponding to the probability of being selected as proposer, i.e., η. If the seller were to deviate and agree with r instead, the net payoff is π _r − e r,a . The condition (6) then says that such deviations are unprofitable.

Note that, if externalities are sufficiently positive, an equilibrium of the type in Proposition 4 exists, since then there must be some a for which (6) holds.

We now turn to mixed equilibria. We first look at mixed equilibria where |A| = 0 and

|M| > 1. The following proposition illustrates conditions for such equilibria to exist.

Proposition 5 There exists a ¯ δ < 1 such that for all δ B > ¯ δ there exists an equilibrium with p _m > 0 for m ∈ M ⊆ N with |M| > 1 and p r = 0 for r ∈ R = N\M if E M,M −Π M ·(J M,M − I M ) is invertible,

(E _M,M − Π M · (J M,M − I M )) ⁻¹ · π M À 0 (7) and

π R ¿ (E ^R,M − Π ^R · J ^R,M ) · (E ^M,M − Π ^M · (J ^M,M − I ^M )) ⁻¹ · π ^M . (8) The equilibrium does not exist if e i,j ≤ 0 for all i, j.

Equilibrium probabilities are, on matrix form, equal to expression (7) times |N| ^1−δ _δ

and expression (8) is the condition for rejecting bids derived above with equilibrium probabilities inserted, on matrix form. In the case with two buyers, the equilibrium can be easily illustrated.

First, since |M| > 1 then R is empty. Second, equilibrium probabilities are

p 1 = 2 1 − δ ^B δ _B

π 2

e _2,1 − π 2

, p ₂ = 2 1 − δ B

δ B

π ₁ e 1,2 − π ¹ .

Probabilities are positive only when e _2,1 > π ₂ and e _1,2 > π ₁ . As pointed out by Jehiel &

Moldovanu (1995a), a model with this property is a situation where a single individual has to

pay for a public goods; see Bliss & Nalebuff (1984) for an analysis of such a model with imperfect

information. This interpretation naturally leads to positive externalities, since if one agents pays

for a public goods, all other agents benefit. In such a setup, it is reasonable that externalities can

be larger than own surpluses. ³ Since externalities are larger than the surpluses, if buyers could choose between getting the entire surplus of agreement or getting the externality, they actually prefer the externality. This generates a hold-up problem, which forces probabilities being close to zero. Also, using (1), equilibrium payoffs can be shown to be zero for the seller and π _m for buyer m. Since externalities are positive for this equilibrium to exist and probabilities converge to zero in the limit, the equilibrium is inefficient and delay is substantial.

The following proposition establishes conditions on parameters satisfied by equilibria where A and M are singletons.

Proposition 6 Consider sequences {δ B ^t } and {δ ^t S } such that ρ = lim t→∞ 1−δ

1−δ

. There exists a T such that for all δ _B ^t , δ ^t _S with t > T there exists an equilibrium with p a = 1, p m > 0 for some a, m ∈ N and p r = 0 for all r 6= a, m if

∞ > (π _m − e m,a ) − _{η+ρ(1−η)} ^η π _a

(π a − e ^a,m ) − (π ^m − e ^m,a ) > 0, (9)

π m > e m,a (10)

and

π _r − e r,a < π _m − e m,a , (11)

for all r 6= a, m.

We have p m → 0 as δ ^B , δ S → 1.

The first condition (9) in the proposition is just the condition that the probability p _m is positive. As δ B , δ S → 1 it can be shown that this probability converges to zero. The equilibria in Proposition 6 can best be understood in relation to bargaining with outside options. From the proof of the proposition, the payoff of S is π m − e ^m,a . Condition 10 then states that the payoff of S is positive and condition 11 that S does not want to deviate and agree with r. The share of the surplus of the agreement that A gets in negotiating with 1, depends on whether the Rubinstein-Ståhl split _{η+ρ(1−η)} ^η π _a is greater or less than the "outside option" agreeing with 2 first: π m − e ^m,a . The condition (6) thus shows that the outside option is not binding if externalities are positive and not too small, leading to existence of pure equilibria.

To understand why p a = 1 and 0 < p m < 1 in the equilibrium, consider the cases where p _m = 1 or p _m = 0 and assume |N| = 2. From the discussion of example 2 above, it is easy to

3

To derive the delay result in the two buyer case, externalities have to be larger than surpluses. Another

motivation for large externalities is given by the discussion (although the example involves negative externalities)

on the Ukrainian nuclear arsenal in Jehiel, Moldovanu & Stachetti (1996).

see that if p m = 1, if π a − e ^a,m > π m − e ^m,a , S will never want to agree with m, as we have π _a − e a,m > π _m − e m,a . Thus, S gains by reducing the probability p _m . In the case where p _m = 0, the payoff to S is _{η+ρ(1−η)} ^η π a . As (6) is violated from (9), S gains by agreeing with m to obtain π _m − e m,a . Thus to ensure that neither of these deviations are profitable, by continuity we have 0 < p m < 1.

Note that, although there is agreement with two buyers in the outside option equilibrium, the equilibrium still satisfied the well-defined buyer property as defined by Jehiel & Moldovanu (1995a) in the limit, since p _m → 0.

Example 7 Nonexistence of a hold-up, single out or outside option equilibrium. We assume that

π _N =

⎛

⎜ ⎜

⎜ ⎝ 5 4 3

⎞

⎟ ⎟

⎟ ⎠ and E _N,N =

⎛

⎜ ⎜

⎜ ⎝

0 1 4

5 2 0 1 1 1 0

⎞

⎟ ⎟

⎟ ⎠

Assume that η < 0.2. It is easily verified that (6) is violated for all a ∈ N and all r 6= a. Also, we have π _i > e _i,j for all i ∈ N and j 6= i implying that (7) is violated. Thus, there are no single out and hold-up equilibria. To check whether there are outside option equilibria, since (11) holds and there is agreement with a with probability 1, buyer m must be the buyer solving

m = arg max

i π _i − e i,a . The matrix of payoff differences π i − e ^i,j is

⎛

⎜ ⎜

⎜ ⎝

0 4 1

3 2 0 3 2 2 0

⎞

⎟ ⎟

⎟ ⎠ . (12)

Thus, if a = 1 then m = 3, if a = 2 then m = 1 and if a = 3 then m = 2. In addition, (9) must hold. Since η < 0.2 the numerator of the ratio in (9) is positive. However, for all possible choices of a, the denominator is negative, implying that there is no outside option equilibrium.

As can be easily checked, the following candidate is an equilibrium (when δ _B , δ _S → 1); p ¹ = 1, p ₂ = ¹ ₄ and p ₃ = ¹ ₂ . To check this, rearrange (1) as in expression (42) in the appendix with a = 1 and take limits. The other possible candidates gives (when a = 2) p 3 = 2 from the first equation and (when a = 3), from the second equation p ₁ = 2, a contradiction.

In equilibria of type IV, A is a singleton and M has more than one member. We have the

following result.

Proposition 8 Consider sequences {δ ^t B } and {δ ^t S } and suppose, for t > T there exists an equilibrium with |A| = 1 and |M| > 1. Then we have, in the limit,

w _S,i = p _M · (π A · J A,M − E A,M ) p _M · J M,A

, (13)

((Π _M − w S,i I _M ) · (J M,M − I M ) − E M,M ) · p M = E _M,A − (Π M − w S,i I _M ) · J M,A (14) and

π _r − w r,S = π _r − e r,a + p M · E ^r,M

p _M · J M,A ≤ p M · (π ^A · J ^A,M − E ^A,M ) p _M · J M,A

. (15)

Note that we do not explicitly state conditions that guarantees existence of an equilibrium in the proposition, as for the other equilibrium types in Propositions 4-6. The reason is that the conditions (13) and (14) are a nontrivial nonlinear system of m + 1 equations and unknowns.

Therefore, we do not explicitly solve for equilibrium probabilities and values as functions of parameters, and hence we cannot explicitly state conditions on parameters ensuring existence.

In addition, (15) corresponds to (5) that ensures that it is not profitable to deviate and make acceptable offers to workers in R.

Another feature of this equilibrium is that, generically, all probabilities p _m cannot converge to zero in the limit. Condition (14) for m ∈ M can be rewritten as (with w ^S,m = w S,i for all i) ⁴

π _m − e m,a + P

j∈M\m p _j (π _m − e m,j ) 1 + P

j∈M\{m} p _j = w _S,m . (16)

Letting p _m → 0 for all m ∈ M then gives π m − e m,a = w _S,m in the limit. Hence, we get, for m, n ∈ M that

π _m − e m,a = π _n − e n,a ,

establishing non-genericity. Generically, there is thus agreement with positive probability with at least two buyers in the limit, violating the well-defined buyer property. Thus, in contrast to the stationary equilibria in Jehiel & Moldovanu (1995a), there is no well defined buyer that the seller agrees with.

The next proposition not only shows that the equilibrium types in the propositions above exist generically, but also that no other equilibrium types do so.

4

See expression (42) in the appendix.

Proposition 9 The generic equilibrium types are the following;

1. Single out: u _S = (1, 0, |N| − 1) for all i ∈ N.

2. Hold-up: u _H = (0, i, |N| − i) for all 1 < i ≤ |N|.

3. Outside option: u _O = (1, 1, |N| − 2) for all i ∈ N and j 6= i.

4. Type IV: u _IV = (1, i, |N| − i − 1) for some 1 < i ≤ |N|.

Note that any equilibrium type in Proposition 9, that entails agreement with probability one prescribes agreement with probability one with exactly one buyer. This is proven by using that, in the limit, the conditions for acceptance (3) holds with equality for all buyers in A, rendering additional restrictions on the parameter space if |A| > 1; see example 2. Thus, using that the conditions for acceptance (3) holds with equality in the limit dramatically reduces the equilibrium type candidates. The reason for why equilibria with |M| > 1 cannot be ruled out as with the equilibria with |A| > 1 is the following. Recall that the reason for the non-genericity of the equilibria with |A| > 1 is that the conditions for acceptance holds with equality as players become infinitely patient. These equilibria also restrict equilibrium probabilities to be exactly one, leading to non-genericity. For mixed equilibria with |M| > 1 the condition for making acceptable offers, (4), also hold with equality. However, probabilities are not constrained to be one, implying enough degrees of freedom for adjustment when parameters are changed.

The reason for delay in Jehiel & Moldovanu (1995a) with positive externalities and a finite

horizon is that then the price that buyers are willing to pay increases the closer to the deadline

one gets. In the last period, all buyers are willing to pay their valuation (Jehiel & Moldovanu

assumes η = 1). In the period just before the deadline, buyers are not willing to pay as much,

since if some other buyer gets the good in the last period, the buyer ends up with a positive

payoff, due to positive externalities. Prices increases the closer to the deadline one gets, inducing

the seller to wait, if patient enough. The argument here is slightly different. Although there

is no deadline, buyers want to wait to agree since if some other buyer ends up with the good,

the buyer still receives a positive payoff, since externalities are positive. Thus, there is a war

of attrition between the buyers. Jehiel & Moldovanu also gets delay when there is negative

externalities. The reason is that, as illustrated by Example 3.1 in Jehiel & Moldovanu (1995b)

with three buyers. In the example, buyer 3 suffers no externalities at all while the first two

buyers suffer large externalities if the third buyer obtains the object. The seller would then like

to threaten the first two buyers with selling to buyer 3 to obtain a higher price from the first

two buyers. However, this threat is not credible until the last period. Buyers 1 and 2 are then

willing to pay a fairly high amount in the period before the last. Buyer 1 and 2 then faces a war of attrition, each trying to wait for the other buyer to buy the object. As our result shows there is no delay with negative externalities and thus the existence of a deadline is crucial for such a war of attrition to occur. With an infinite horizon as in Jehiel & Moldovanu (1995a), there is cyclical delay. The intuition for this is very similar to the deadline effect with a finite horizon. These equilibria are nonstationary and are hence ruled out in this paper. Moreover, as pointed out by Jehiel & Moldovanu (1995a), complexity tends to infinity as the discount factors converge to one. Here, the equilibrium strategies are, independently of discount factors, very simple.

In a model where all externalities are zero, and pies π _i are generic, the equilibrium allocation entails no delay and is Walrasian in the limit, in the sense that the buyer with the largest valuation buys the good. ⁵ To see this, note that there cannot be m, n such that (16) holds, since we generically have π _n 6= π m . Also, the conditions for existence in proposition 5 is violated since e _i,j = 0. Furthermore, if a single-out equilibrium exists, then there must be agreement with the buyer i ^∗ with the largest surplus. Otherwise, condition (6) is violated, since _{η+ρ(1−η)} ^η ≤ 1.

Finally, suppose (6) is violated for the buyer with the highest valuation. Then, letting j ^∗ denote the buyer with the second highest valuation we see that (9) in proposition 6 is satisfied for a = i ^∗ and m = j ^∗ and (11) is satisfied for m = j ^∗ and r 6= i ^∗ , establishing existence of an outside option equilibrium. Moreover, it is the only outside option equilibrium that can exist. First, if r = i ^∗ in proposition 6, condition (11) is violated. Thus, the only other possible outside option equilibrium is when m = i ^∗ and a = j ^∗ . Since π _i

> π _j

, (9) is violated. Thus, any SSPE prescribes agreement with the buyer with the highest valuation and no agreement with other buyers, in the limit.

5 Simple equilibria

The equilibria in proposition 8 are much more complicated than the equilibria in the previous propositions 4 - 6. It is therefore interesting to derive conditions that eliminates such equilibria.

Note from expressions (13) and (16) that probabilities serve as a weighting function of the payoff differences π _i − e i,j ; see also (12). Then, it is easy to see that if there is some player k such that π _k − e k,l > π i − e ^i,j for all l, i, j ∈ N then there is no equilibrium as in proposition 8.

Another assumption that leads to elimination of the equilibria in 8 is when externalities are type independent; if the object is sold to some buyer j, the externality imposed on i is the same,

5

This observation is also mentioned by Jehiel & Moldovanu (1995b)

irrespectively of the identity of j, i.e., e i,j = e i for all j. If this is the case, the equilibrium structure is easy to analyze. There are generically no mixed equilibria of type 4 in Proposition 9. From (16) we have

π _m − w S,i = e _m . Since |M| > 1 there are m, n ∈ M such that

π _m − e m = π _n − e n ,

establishing non-genericity.

6 Conclusion

In the paper, a model where a seller bargains with multiple buyers when externalities are present

is analyzed. Restricting attention to generic equilibria and allowing for mixed strategies makes

the analysis of bargaining with externalities simple. We show existence of a stationary subgame

perfect equilibrium and proceed to characterize the equilibria. We show that delay occurs only

when externalities are positive, in contrast with Jehiel & Moldovanu (1995a). The reason for

delay is that buyers prefer the externalities to agreeing with the seller, thus generating a war of

attrition.

A Proofs

Proof of Proposition 1:

We now show existence of equilibrium. To do this, we demonstrate that the following strategy profile is an equilibrium. First, respondents always accept proposals equal to their continuation payoff. Second, the firm and workers optimally choose whether to make acceptable offers or not, allowing for mixed strategies. Thus, the firm and worker, respectively, choose p ^S _i ∈ [0, η] and p ^B _i ∈ [0, 1 − η]. A probability p ^S i ∈ (0, η) or p ^B i ∈ (0, 1 − η) is interpreted to imply that the firm or worker makes an acceptable offer with some positive probability less than one.

Here, let p ^S = ¡ p ^S _i ¢

i∈N , p ^B = ¡ p ^B _i ¢

i∈N and p = ¡

p ^S , p ^B ¢

. Let P ^S = {p ^S ∈ R ^|N| + | p ^S i ≤ η}

and P _i ^B = {p ^B i ∈ [0, 1 − η]} denote the set of possible probabilities for the seller and buyer i, respectively.

Now we define a mapping that enables us to find an equilibrium. Let Q = [0, k ₂ ] × [k 1 , k ₂ ] ^|N|

where

k 1 = − |N| max{e i,k , 0} − X

i∈N

π i

k ₂ = X

i∈N

π _i + |N| max{e i,k , 0}.

and X = × i∈N ([0, η] × [0, 1 − η]). Furthermore, let D = Q × X. Note that D is compact and convex. For some q ∈ Q and x ∈ X we define

µ S (q, x) = max

p

∈P

X

i∈N

p ^S _i [δ S π i − δ ^S q i ] + Ã

1 − X

i∈N

p ^S _i

!

δ S q S (17)

and

µ _i (q, x) = max

p

∈P

p ^B _i [δ _B π _i − δ B q _S ] + x ^S _i δ _B q _i +

⎛

⎝1 − X

j6=i

£ x ^S _j + x ^B _j ¤

⎞

⎠ δ B q _i (18)

− £

x ^S _i + p ^B _i ¤

δ B q i + X

j6=i

£ x ^S _j + x ^B _j ¤ δ B e i,j .

Note that these are continuation payoffs before the proposer has been selected. Also, let

α ^S (q, x) = arg max

p

≤P

X

i∈N

p ^S _i [δ _S π _i − δ S q _i ] + Ã

1 − X

i∈N

p ^S _i

!

δ _S q _S , (19)

let α ^S _i (q, x) be the i th element of α ^S (q, x) and let

α ^B _i (q, x) = arg max

p

∈P

p ^B _i [δ _B π i − δ B q _S ] + x ^S _i δ _B q i +

⎛

⎝1 − X

j6=i

£ x ^S _j + x ^B _j ¤

⎞

⎠ δ _B q i (20)

− £

x ^S _i + p ^B _i ¤

δ B q i + X

j6=i

£ x ^S _j + x ^B _j ¤ δ B e i,j .

Finally, we let

Φ (q, x) =

³

(µ i (q, x)) _i∈N , ¡

α ^S _i (q, x) , α ^B _i (q, x) ¢

i∈N

´ .

Thus, Φ is a mapping consisting of continuation payoffs and optimal choices of the players, given q and x.

We claim that a fixed point of Φ gives equilibrium continuation payoffs and strategies. To see that a fixed point of Φ gives equilibrium continuation payoffs and strategies, first note that, if w S,i = µ S (q, x) = q S , w i,S = µ i (q, x) = q i , p S,i = α ^S _i (q, x) = x ^S _i and p i,S = α ^B _i (q, x) = x ^B _i for all i ∈ N, then (17) and (18) are the same as the value equations (for respondent payoffs).

Second, (17), (18), (19) and (20), respectively, imply that the firm and workers optimally choose whether to make an acceptable offer or not. Moreover, by construction, respondents choose optimally.

By the maximum theorem under convexity, α ^S (q, x) and α ^B _i (q, x) are upper-hemicontinuous, convex-valued and compact-valued correspondences on D and µ _i (q, x) is a continuous function on D for all i ∈ N. Thus, Φ is an upper-hemicontinuous, convex-valued and compact-valued correspondence. Then the Kakutani fixed-point theorem implies that a fixed point exists. ¥

Now, let us turn to equilibrium characterization in section 4.

In negotiations with r ∈ R, m ∈ M and a ∈ A we have, from the value equations (1) and (4),

v _S,r = w _S,r (21)

v _r,S = w _r,S ,

v _S,m = w _S,m (22)

v m,S = π m − w ^S,m

and

v S,a = π a − w ^a,S (23)

v a,S = π a − w ^S,a .

Using (21), (22) in (4) and (23) gives

w _r,S =

δ _B ³P

j∈A e _r,j + P

j∈M p _j e _r,j ´

|N| (1 − δ B ) + δ _B |A| + δ B P

j∈M p _j (24)

w _m,S =

δ _B (1 − η) (π m − w S,m ) + δ _B ³P

j∈A e _m,j + P

j∈M p _j e _m,j ´

|N| (1 − δ B ) + δ _B − δ B η + δ _B |A| + δ B P

j∈M\{m} p _j w _a,S =

δ _B (1 − η) (π a − w S,a ) + δ _B ³P

j∈A e _a,j + P

j∈M p _j e _a,j ´

|N| (1 − δ B ) − δ B η + δ _B |A| + δ B P

j∈M p _j Using (24), (21), (22) and (23) in (2) gives, letting

L (δ _B , δ _S ) = |A| ηδ S (δ _B (|A| − 1) + |N| (1 − δ B )) + |N| (1 − δ S ) ((|A| − η) δ B + |N| (1 − δ B ))

and using some straightforward but tedious algebra,

w _S,i = δ _S η ¡P

a∈A

¡ π _a ¡

|N| (1 − δ B ) + δ _B ¡

|A| − 1 + P

m∈M p _m ¢¢¢¢

L (δ B , δ S ) + δ B (|N| (1 − δ ^S ) + δ S η |A|) P

m∈M p m

(25)

−

δ S η ³P

a∈A

³ δ B ³P

j∈A\{a} e a,j + P

m∈M p m e a,m

´´´

L (δ _B , δ _S ) + δ _B (|N| (1 − δ S ) + δ _S η |A|) P

m∈M p _m

Proof of Proposition 4: With |A| = 1 and |M| = 0 the condition for acceptance (3) for a is satisfied for all δ B and δ S since, using (24) and (25)

v S,a − w ^S,a = (π a − w ^S,a ) |N| (1 − δ ^B )

|N| (1 − δ B ) + δ _B (1 − η) (26)

= π _a |N| (1 − δ S ) |N| (1 − δ B )

ηδ S (|N| (1 − δ ^B )) + |N| (1 − δ ^S ) ((1 − η) δ ^B + |N| (1 − δ ^B )) > 0.

There will not be acceptance in negotiations with r when (5) is fulfilled. Inserting solutions for w _r,S and w _S,i gives

π _r − δ _B e r,a

|N| (1 − δ B ) + δ _B ≤ δ _S η ηδ S + ^1−δ _1−δ

B

((1 − η) δ ^B + |N| (1 − δ ^B )) π _a . (27)

From condition (6) in the statement of the proposition, there exists a T such that (27) holds

for all t > T. As w S,a > 0, firms also make a non-negative profit. Thus the conditions for the equilibrium to exist are satisfied for t > T. ¥

Proof of Proposition 5: Using |A| = 0 and |M| > 1 in (24), (21) and (22), expression (2) is

w S,i = δ _S η 1 − δ ^S (1 − η)

1

|N|

X

j∈N

w S,j ⇐⇒ w ^S,i = 0. (28)

Using (22), (4) and w S,m = w S,r = 0 in the value equation for w m,S gives

|N| 1 − δ ^B

δ _B π m = X

j∈M\{m}

e m,j p j − π ^m X

j∈M\{m}

p j .

Note that the condition |M| > 1 in the proposition follows, since when |M| = 1 we have (1 − δ B ) π _m = 0, contradicting π _m > 0 by assumption.

In matrix form, the above expression is

|N| 1 − δ ^B

δ _B π M = (E M,M − Π ^M · (J ^M − I ^M )) · p ^M (29) Since the matrix on the right hand side is invertible by assumption, we get

p _M = |N| 1 − δ B

δ _B (E _M,M − Π M · (J M − I M )) ⁻¹ · π M . (30) Thus from assumption (7), p _M À 0 for all δ B δ _S < 1. Note that, as δ _B converges to one, p _M converges to zero. Thus there exists a ¯ δ B < 1 such that p m < 1 for all m ∈ M and δ ^B > ¯ δ B .

If externalities are non-positive then E _M,M − Π M · (J M − I M ) is a matrix with zero on the main diagonal and negative off diagonal elements. Then, for any p M ≥ 0, the right hand side of (29) is non-positive and hence no hold-up equilibrium exists.

In order for all r not to make acceptable bids, (5) holds. As w S,r = 0 by (28) and using w r,S

in (24) when |A| = 0 and |M| > 1 we have

|N| 1 − δ B

δ _B π _r ≤ X

k∈M

p _k e _r,k − π r

X

k∈M

p _k .

In matrix form this condition becomes, using the solution for probabilities (30)

π _R ≤ (E R,M − Π R · J R,M ) · (E M,M − Π M · (J M − I M )) ⁻¹ · π M

which holds by condition (8) in the proposition. ¥

Proof of Proposition 6: From indifference (4) for m, we get, using (24) with |A| = 1 and

|M| = 1

π _m − w S,m = δ _B

|N| (1 − δ B ) + δ _B e _m,a and hence using (25) with |A| = 1 and |M| = 1 and solving for p m gives

p _m = (1 − δ B ) ∆ (31)

∆ =

³³

π _m − _|N|(1−δ ^δ

_)+δ

e _m,a ´ ³

ηδ _S |N| + |N| _1−δ ^1−δ

((1 − η) δ B + |N| (1 − δ B )) ´

− δ S ηπ _a |N| ´

³

δ _S η (π _a δ _B − δ B e _a,m ) − δ B (|N| (1 − δ S ) + δ _S η) ³

π _m − _|N|(1−δ ^δ

_)+δ

e _m,a ´´

Note that

δ

lim ,δ

→1 ∆ = |N| η + ρ (1 − η) η

(π m − e ^m,a ) − _{η+ρ(1−η)} ^η π a

(π _a − e a,m ) − (π m − e m,a ) > 0 (32) Since p _m = (1 − δ B ) ∆ and since lim _t→∞ (1 − δ B ) ∆ = 0, by condition (9) and the continuity of

∆, there exists a T ₁ such that, for all t > T ₁ we have 0 < p _m < 1.

Then, using p _m = (1 − δ) ∆ in (25) and (24) with |A| = 1 and |M| = 1

π _r − w r,S − w S,r = π _r − δ _B (e r,a +p m e r,m )

|N| (1 − δ B ) + δ _B + δ _B p _m

− δ S η (π a (|N| + δ ^B ∆) − δ ^B ∆e a,m )

ηδ _S |N| + |N| ^1−δ _1−δ

((1 − η) δ B + |N| (1 − δ B )) + δ _B (|N| (1 − δ S ) + δ _S η) ∆ In the limit, this expression is, using (32),

π r − w ^r,S − w ^S,r = π r − e ^r,a − (π ^m − e ^m,a )

By (11), this expression is strictly positive. Then, there exists a T 2 such that (5) holds for all t > T ₂ .

Using (23) and (25) with |A| = 1 and |M| = 1 we have

v _S,a − w S,a = (|N| (1 − δ ^B ) + δ B p m ) π a − δ ^B p m e a,m

|N| (1 − δ B ) − δ B η + δ _B + δ _B p _m

× |N| (1 − δ S ) ((1 − η) δ B + |N| (1 − δ B )) + δ _B (|N| (1 − δ S )) p _m

ηδ S |N| (1 − δ ^B ) + |N| (1 − δ ^S ) ((1 − η) δ ^B + |N| (1 − δ ^B )) + δ B (|N| (1 − δ ^S ) + δ S η) p m

Note that the second ratio and the denominator of the first ratio are positive for δ B , δ S < 1.The numerator of the first ratio can be rewritten as

(1 − δ B ) ((|N| + δ B ∆) π _a − δ B ∆e _a,m )

If

(|N| + δ B ∆) π _a − δ B ∆e _a,m > 0

then (3) is satisfied. As δ _B , δ _S → 1, the left hand side of this expression is, using (32),

|N| (η + ρ (1 − η)) η

à (π m − e ^m,a ) − _{(η+ρ(1−η))} ^η π a

(π _a − e a,m ) − (π m − e m,a ) + 1

!

(π _m − e m,a )

From (10), there exists a T 3 such that (3) holds for all t > T 3 .

To ensure that all conditions hold, we choose T = max{T 1 , T ₂ , T ₃ }. ¥ Proof of proposition 8. From (24) we have, using (5), (4) and (3)

π _r − w S,i ≤

δ _B ³P

j∈A e _r,j + P

j∈M p _j e _r,j ´

|N| (1 − δ B ) + δ _B + δ _B P

j∈M p _j (33)

π _m − w S,i =

δ _B ³P

j∈A e _m,j + P

j∈M p _j e _m,j ´

|N| (1 − δ B ) + δ _B |A| + δ B P

j∈M\{m} p _j π _a − w S,i ≥

δ _B ³P

j∈A e _a,j + P

j∈M p _j e _a,j ´

|N| (1 − δ B ) + δ _B P

j∈M p _j (34)

Taking limits and rearranging gives the result ¥

Proof of Proposition 9: To prove the proposition, we need to show both that the equilib- rium types stated in the proposition are generic, and that any other equilibrium type only exists non-generically. We begin by showing that the equilibrium types stated in the proposition are generic. Then we continue to show that any other equilibrium type only exists non-generically.

Lemma 10 The single out equilibria in proposition (4) are generic.

Proof: Consider the single out equilibrium type u _S where |A| = 1, |M| = 0 and |R| = N −1.

Suppose k > _{η+¯ ρ(1−η)} ^η and that

π r − e ^r,a < kπ a . (35)

for all r 6= a for some a. When k < _{η+¯ ρ(1−η)} ^η see Lemma 12 for other types of equilibria. Then there is some ρ ∈ (ρ, ¯ ρ) such that (6) holds. Since ρ < ¯ ρ then (6) holds for all sequences {δ B ^t } and {δ ^t S } where lim _1−δ ^1−δ

B

= ρ ⁰ ≤ ρ. In the remainder of the proof, we restrict attention to sequences where lim ^1−δ

1−δ

= ρ ⁰ ≤ ρ. Clearly, λ r

¡ [ρ, ρ) ¢

> 0. Given ρ, condition (6) in Proposition 4 holds for some parameter value ¯ ω ∈ Ω. Then there exists a closed ball B (¯ ω) with radius ε around the parameter vector ¯ ω such that the condition holds for all ω ∈ B (¯ ω).

Also, from the proof of Proposition 4, the condition for acceptance (26) for a holds for all

δ B and δ S . Rewriting (27) we get

π r − δ B e r,a

|N| (1 − δ B ) + δ _B ≤ δ S η ηδ _S + ^1−δ _1−δ

B

((1 − η) δ B + |N| (1 − δ B )) π a . (36) Rearranging and letting

L 1 (δ B , δ S ) = µ

(δ S − δ ^B ) ρ ⁰ + δ B

µ

ρ ⁰ − 1 − δ ^S 1 − δ B

¶¶

(1 − η) δ ^B +

µ ¡

δ _S η + δ _S ρ ⁰ (1 − η) ¢

− δ B 1 − δ ^S 1 − δ B

¶

(|N| (1 − δ B ))

we get

|N| 1 − δ B

δ _B π _r + L ₁ (δ _B , δ _S ) 1 δ _B

η

η + ρ ⁰ (1 − η) π _a ≤ η

η + ρ ⁰ (1 − η) π _a − (π r − e r,a ) (37) Note that (37) converges to (6) with ρ = ρ ⁰ .

For a given ω, let T (ω) denote the largest t such that (37) holds for all t > T (ω) and all r 6= a. Let ˆ T denote the largest T (ω) for all ω ∈ B (¯ ω). By continuity of (26) and (6) and since lim ^1−δ _1−δ

B

= ρ ⁰ ≤ ρ, ε can be chosen such that ˆ T < ∞. Then B (¯ ω) ⊆ Ω ¡

u S , δ ^t _B , δ _S ^t ¢

for all t > ˆ T . Then lim _t→∞ λ ¡

Ω ¡

u _S , δ _B ^t , δ ^t _S ¢¢

≥ λ (B (¯ ω)) > 0 establishing that u _S is generic. ¥

Lemma 11 The hold-up equilibria in proposition (5) are generic.

Proof: Note first, from proposition 5 that the limit of ^1−δ _1−δ

B

does not affect the conditions for existence. To show that the hold-up equilibrium type is generic, consider the case where e _j,i = e _j for all i 6= j and assume that e i > π _i for all i ∈ N. In addition let all m have identical pies and externalities π m = α and e m = β for all m ∈ M and similarly π ^r = θ and e r = τ for all r ∈ R. Let ¯ ω denote this parameter vector. We also assume

θ

τ − θ < |M|

|M| − 1 α

β − α . (38)

Note that, for any parameter vector ω, there exists a δ _B (ω) < 1 such that probabilities are smaller than one for all δ B > δ B (ω) .

Then the invertibility condition in proposition 5 is satisfied and, using E _M,M = β (J _M,M − I M )

(E _M,M − Π M · (J M,M − I M )) ⁻¹ · π M = 1

|M| − 1 α β − α j _M

This is positive as β > α by assumption and hence condition (7) is satisfied. Furthermore, using

E R,M = τ J R,M

π _R = θj _R ¿ (τ − θ) |M|

|M| − 1 α β − α j _R . Then, since (38) holds, condition (8) is satisfied.

Since the invertibility condition is satisfied for the parameter vector ¯ ω and the determinant is a continuous function of ω ∈ Ω, there exists a ball B (¯ ω) with radius ε around the parameter vector ¯ ω such that the matrix E M,M − Π ^M · (J ^M,M − I ^M ) is invertible and the conditions (7) and (8) still hold. Let ˆ δ _B denote the largest δ _B (ω) such that probabilities are smaller than one for all δ B > δ B (ω) for all ω ∈ B (¯ ω). Then a hold-up equilibrium exists for all ω ∈ B (¯ ω) for all δ ˆ _B < δ _B < 1, establishing that u _H is generic. ¥

Lemma 12 The outside option equilibria in proposition (6) are generic.

Proof: Suppose, without loss of generality, that the denominator of (9) is positive, that k < _{η+ρ(1−η)} ^η and that

(π _m − e m,a ) − kπ a

(π _a − e a,m ) − (π m − e m,a ) > 0

and that conditions (10)-(11) hold. When k ≥ _{η+ρ(1−η)} ^η see Lemma 10 for other types of equilibria. Then there is some ρ ∈ (ρ, ¯ ρ) such that (9). Since ρ > ρ then (9) holds for all sequences {δ ^t B } and {δ S ^t } where lim _1−δ ^1−δ

B

= ρ ⁰ ≥ ρ. In the remainder of the proof, we restrict attention to sequences where lim _1−δ ^1−δ

B

= ρ ⁰ ≥ ρ. Clearly, λ ^r ([ρ, ¯ ρ]) > 0. Consider the outside option equilibrium type u _O where |A| = 1, |M| = 1 and |R| = N − 2. Given that condition (9) holds for some parameter value ¯ ω ∈ Ω, there exists a closed ball B ^m (¯ ω) with radius ε ^m around the parameter vector ¯ ω such that the condition holds for all ω ∈ B ^m (¯ ω).

Since condition (9) holds, from the proof of Proposition 6 we have p m = (1 − δ ^B ) ∆ > 0.

Moreover, (1 − δ B ) ∆ converges to zero. Since ∆ is continuous in δ _B , δ _S there is some T (ω) such that p m is smaller than one for all t > T (ω). Let T ^m denote the largest T (ω) such that p m is smaller than one for all t > T (ω) for all ω ∈ B ^m (¯ ω).

Similarly, given that (10) holds at ¯ ω, there exists a closed ball B ^a (¯ ω) with radius ε ^a such that the condition holds for all ω ∈ B ^a (¯ ω). Let T ^a denote the largest T (ω) such that the condition for acceptance (3) holds for all t > T (ω) for all ω ∈ B ^a (¯ ω). A similar argument using (11) establishes the existence of T ^r and B ^r (¯ ω) where T ^r is the largest T (ω) such that, for all r, the condition for rejection (5) holds for all t > T (ω) for all ω ∈ B ^r (¯ ω).

Letting ˆ T = max{T ^a , T ^m , T ^r } and B (¯ ω) = B ^a (¯ ω) ∩ B ^m (¯ ω) ∩ B ^r (¯ ω). By continuity of (31) and the solutions for the values in Proposition 6 and since lim ^1−δ

1−δ

= ρ ⁰ ≥ ρ, ε can be chosen such

that ˆ T < ∞. Then, for t > ˆ T and ω ∈ B (¯ ω) the conditions (3) and (5) hold with 0 < p _m < 1.

Then B (¯ ω) ⊆ Ω ¡

u O , δ ^t _B , δ _S ^t ¢

for all t > ˆ T and hence lim _t→∞ λ ¡ Ω ¡

u O , δ ^t _B , δ _S ^t ¢¢

≥ λ (B (¯ ω)) > 0 establishing that u _O is generic. ¥

Lemma 13 The equilibria in proposition (8) are generic.

Proof: Consider arbitrary sequences {δ B ^t } and {δ ^t S }. Note first, from proposition 8 that the limit of _1−δ ^1−δ

B

does not affect the conditions for existence. Consider the mixed equilibria of type IV, denoted u _IV , where |A| = 1, |M| > 1. Consider example 7. Suppose payoffs for sellers 1, 2 and 3 are as in the example and that, for all sellers r = 4, ..., n we have

π r − e ^r,j < 1 2

fro j ∈ N. Then (15) is satisfied for this parameter specification. Let the specification be denoted ¯ ω. There exists a closed ball B (¯ ω) with radius ε such that the conditions in proposition 8 are satisfied. This follows by continuity of the conditions in proposition 8. For a given ω, let T (ω) denote the largest t such that (15) holds for all t > T (ω) and all r 6= a. Let ˆ T denote the largest T (ω) for all ω ∈ B (¯ ω). By continuity of the conditions in proposition (8), ε can be chosen such that ˆ T < ∞. Then B (¯ ω) ⊆ Ω ¡

u _IV , δ _B ^t , δ ^t _S ¢

for all t > ˆ T and hence lim _t→∞ λ ¡

Ω ¡

u IV , δ ^t _B , δ _S ^t ¢¢

≥ λ (B (¯ ω)) > 0 establishing that u IV is generic for {δ B ^t } and {δ ^t S }.

Since this holds for all sequences {δ B ^t } and {δ S ^t }, u IV is generic. ¥

Lemma 14 Equilibria with |A| > 1 and |M| = 0 are non- generic.

Proof: Consider arbitrary sequences {δ ^t B } and {δ S ^t }. To show that equilibria with |A| > 1 are non-generic, note that w _S,i in (25) is well defined for δ _B = δ _S = 1, since L (1, 1) = |A| η (|A| − 1).

Using |M| = 0 in (24), to eliminate w a,S and w _r,S in (3) and (5)

π _a − w S,a ≥ δ _B

(|N| (1 − δ B ) − δ B + δ _B |A|) X

k∈A

e _a,k (39)

π r − w ^S,r ≤ δ B P

k∈A e r,k

|N| (1 − δ B ) + δ _B |A| .

For ω ∈ Ω and δ B , δ _S ≤ 1 let ψ : D ³ R ⁺ where D ⊂ Ω × [0, 1] ² be the correspondence satisfying (39) and (25) with |M| = 0. Thus, ψ maps payoff parameters ω and δ S , δ _B to seller respondent values. The correspondence ψ is upper-hemicontinuous (uhc), see Border 1985: If for t = 1, 2,. . . we have p ^t ∈ ψ ¡

ω ^t , δ _B ^t , δ ^t _S ¢ and ¡

ω ^t , δ _B ^t , δ _S ^t ¢

→ (ω, δ B , δ _S ) as t → ∞, and

p = lim _t→∞ p ^t then, since (39) and (25) with |M| = 0 define closed sets, we have p ∈ ψ (ω, δ ^B , δ S ),

establishing that ψ is uhc.

For δ B , δ S ≤ 1, let the correspondence ϕ (δ ^B , δ S ) be the set of ω, such that ψ (ω, δ B , δ S ) is non-empty. ϕ (δ _B , δ _S ) is uhc: Let δ _B ^t → δ B , δ _S ^t → δ S and ω ^t → ω such that ω ^t ∈ ϕ ¡

δ _B ^t , δ ^t _S ¢ . Then there exist p ^t such that p ^t ∈ ψ ¡

ω ^t , δ ^t _B , δ _S ^t ¢

and since ψ is uhc p ^t → p ∈ ψ (ω, δ ^B , δ S ) . Thus ω ∈ ϕ (δ B , δ _S ).

Using the solution for w S,i from (25) with |M| = 0 in (39) gives, when |A| > 1

π _a − 1

|A| − 1 X

k∈A

e _a,k ≥ 1

|A|

X

h∈A

Ã

π _h − 1

|A| − 1 X

k∈A

e _h,k

!

(40)

Since (40) holds for all a, it holds for the a that minimizes the left-hand side. As the minimal element is weakly greater than the average over all a, then π a − _|A|−1 ¹ P

j∈A e a,j is the same for all a. Then ϕ (1, 1) is defined as, for all a ∈ A and r ∈ R,

π _a − 1

|A| − 1 X

k∈A

e _a,k = K

π r − w ^S,r ≤ P

k∈A e _r,k

|A| . Thus, λ (ϕ (1, 1)) = 0. Suppose that lim _t→∞ λ ¡

ϕ ¡

δ ^t _B , δ _S ^t ¢¢

> 0. Then there exists a sequence

¡ ω ^t , δ _B ^t , δ _S ^t ¢

→ (ω, 1, 1) such that ω ^t ∈ ϕ ¡ δ ^t _B , δ _S ^t ¢

for all δ _B ^t , δ ^t _S but ω / ∈ ϕ (1, 1). This contradicts the upper-hemicontinuity of ϕ, establishing non-genericity. ¥

Lemma 15 Equilibria with |A| > 1 and |M| ≥ 1 are non- generic.

Proof: Consider arbitrary sequences {δ ^t B } and {δ ^t S }. To show that equilibria with |A| > 1 and |M| ≥ 1 are non-generic, using (24) in the indifference equation (4) and rearranging gives

(π _m − w S,m )

⎛

⎝|N| (1 − δ B ) + δ _B |A| + δ B

X

j∈M\{m}

p _j

⎞

⎠ = δ _B

⎛

⎝ X

j∈A

e _m,j + X

j∈M

p _j e _m,j

⎞

⎠ (41)

When |A| > 1 w S,i is well defined for δ _B = δ _S = 1. Using (24) in (3) and (24) in (5) and (41) gives

π _a − w S,i ≥

δ _B ³P

j∈A e _a,j + P

j∈M p _j e _a,j ´

|N| (1 − δ ^B ) + δ B (|A| − 1) + δ ^B P

j∈M p j

(42)

π m − w ^S,i =

δ B ³P

j∈A e m,j + P

j∈M p j e m,j

´

|N| (1 − δ ^B ) + δ B |A| + δ ^B P

j∈M\{m} p j

π _r − w S,i ≤

δ _B ³P

j∈A e _r,j + P

j∈M p _j e _r,j ´

|N| (1 − δ ^B ) + δ B |A| + δ ^B P

j∈M p j

.