# Bargaining with endogenous deadlines

**ABSTRACT** We develop a two-person negotiation model with complete information that makes endogenous both the deadline and the level of surplus destruction after the deadline. We show that the undominated Nash equilibrium outcome is always unique but might be inefficient. Moreover, as the bargaining period becomes short or as the players become very patient, the unique undominated Nash equilibrium outcome is always inefficient.

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**ABSTRACT:**Alternating offers bargaining has been extensively used to model two-sided negotiations. The celebrated model of Rubinstein [Econometrica 50(1):97–109, 1982] has provided a formal justification for equitable payoff division. A typical assumption of these models under risk is that the termination event means a complete and irrevocable breakdown in negotiations. In this paper, the meaning of termination is reinterpreted as the imposition to finish negotiations immediately. Specifically, bargaining terminates when the last offer becomes definitive. While Rubinstein’s model predicts an immediate agreement with stationary strategies, we show that the same payoff allocation is attainable under non-stationary strategies. Moreover, the payoffs in delayed equilibria are potentially better for the proposer than those in which agreement is immediately reached.International Journal of Games Theory 02/2008; 37(4):457-474. · 0.58 Impact Factor - SourceAvailable from: Juan Vidal-Puga[Show abstract] [Hide abstract]

**ABSTRACT:**A typical assumption of the standard alternating-offers model under risk is that the breakdown event means a complete and irrevocable halt in negotiations. We reinterpret the meaning of breakdown as the imposition to finish negotiations immediately. Specifically, after breakdown the last offer becomes definitive. A full characterization of the set of subgame perfect equilibrium payoffs is provided. We show that Rubinstein's allocation (1/(1+?),?/(1+?)) is obtained under non- stationary strategies. Moreover, the payoffs in delayed equilibria are potentially better for the proposer than those in which agreement is immediately reached.03/2005;

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Journal of Economic Behavior & Organization

Vol. 54 (2004) 321–335

Bargaining with endogenous deadlines

Ana Mauleona, Vincent Vannetelboschb,∗,1

aLABORES (URA 362, CNRS), Université catholique de Lille, France

bFNRS, CORE, and IRES, Université catholique de Louvain, Voie du Roman Pays 34,

B-1348 Louvain-la-Neuve, Belgium

Received 25 October 2001; received in revised form 22 July 2002; accepted 15 January 2003

Available online 5 January 2004

Abstract

We develop a two-person negotiation model with complete information that makes endogenous

both the deadline and the level of surplus destruction after the deadline. We show that the un-

dominated Nash equilibrium outcome is always unique but might be inefficient. Moreover, as the

bargaining period becomes short or as the players become very patient, the unique undominated

Nash equilibrium outcome is always inefficient.

© 2003 Elsevier B.V. All rights reserved.

JEL classification: C78; J50; J52

Keywords: Bargaining; Alternating-offers; Deadlines; Complete information

1. Introduction

Negotiations often take place under the pressure of a deadline that may be exogenously

imposed, or one of the parties to the negotiation may have chosen the deadline and made a

credible commitment to it. Recent work has focused on bargaining models with exogenous

deadlines after which there is no surplus to be divided (see Fershtman and Seidmann, 1993;

Ma and Manove, 1993). A key feature of our paper is a first attempt to endogenize the

deadline in negotiation models. We consider a more general definition of a deadline as a

point in time after which the surplus to be shared is permanently reduced.

Moreover, we allow the players to choose the level of surplus destruction after the dead-

line. Thus, in case an agreement is not reached before the deadline, the value of the un-

∗Corresponding author. Tel.: +32-10-474142; fax: +32-10-474301.

E-mail address: vannetelbosch@core.ucl.ac.be (V. Vannetelbosch).

1Vincent Vannetelbosch is Chercheur Qualifi´ e at the Fonds National de la Recherche Scientifique, Brussels.

0167-2681/$ – see front matter © 2003 Elsevier B.V. All rights reserved.

doi:10.1016/j.jebo.2003.01.007

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A. Mauleon, V. Vannetelbosch/J. of Economic Behavior & Org. 54 (2004) 321–335

derlying relationship will be permanently reduced, and the level of surplus destruction will

depend on the actions of the players. Then, the surplus available to the players once an

agreement is reached may actually be lower than it is at the beginning.

Inoneexampleofbargaininginthefaceofsuchendogenousdeadlines,wagenegotiations

between unions and employers, both parties may have the opportunity to set a deadline

after which starts either a strike or a lockout. The deadline may be subject to labor laws

: it is common that a strike or a lockout requires few days’ notice in order to be legal.

However, a conflict may reduce permanently the profitability of the relationship itself by

affecting, for example, the future demand for the firm’s products. Indeed, customers may

decide to buy from now on from some competitor. In order to avoid the loss of customers,

the firm may decide to move (partially or entirely) the production to another plant or to

use replacement workers. However the firm is not alone in having actions at its disposal

that directly influence the profitability of the future relationship. For example, the union

may jeopardize production equipment by the lack of scheduled maintenance or skilled

operators.2

In this paper, we develop a two-stage negotiation model with complete information be-

tween a firm and a union. In the first stage, the deadline in force during the wage bar-

gaining is chosen. That is, the firm and the union choose, respectively, a lockout date

(and the intensity of the lockout) and a strike date (and the intensity of the strike). In

addition, we allow both parties to choose no deadline. In the second stage, both parties

bargain over the division of a surplus that is time dependent. Indeed, before the deadline

we have a peaceful bargaining, where in each period until a new agreement is reached,

both parties continue to produce and the value added is shared following the old wage

contract. After the deadline we have an open-conflict bargaining (a strike or a lockout

has occurred), where in each period until a new agreement is reached both parties get

nothing and the value added in later periods (once an agreement is reached) will be af-

fected by the intensity of the conflict occurred. The wage bargaining proceeds following

Rubinstein’s (1982) alternating-offer bargaining procedure with the firm making the first

offer.

We show that the undominated Nash equilibrium outcome of our negotiation model

is always unique but might be inefficient. The condition to get inefficiency is satisfied

whenever the old wage is relatively small, each player has at his disposal both actions that

reduce substantially the value added in the future and actions that have only a minor impact

on the future value added, and the players are patient. Which is the intuition behind the

result? Both players would like the other player to be the last mover at the deadline and

preferablyfacingthethreatofaconflictofastrongintensity.Astrongconflictsimplymeans

thatafterthedeadline,thevalueaddedwillbereducedsubstantially.Then,inordertoavoid

having to accept a very low wage offer facing the threat of a severe lockout, where the firm

would grab most of the surplus, it becomes optimal for the union to strike immediately and

2Cutcher-Gershenfeldetal.(1998)haveexamined,fortheUS,pressuretacticsusedbyunionsandemployersto

influence the process in collective bargaining and its outcomes. In the past, the threats of a strike and the imminent

contract expiration deadline have been central features motivating the parties to reach agreements. But in recent

years, the observations suggest that management threats regarding replacement workers and plant closings or

movings are now also a key part of the collective bargaining landscape.

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323

to destroy part of the future value added, but not too much. So, at equilibrium we observe

both players competing to be the one who will make an offer just before the deadline, but

alsotryingtoavoidhavingtomoveatthedeadlineduetothethreatofaverystrongconflict.

This can lead to one party launching the conflict immediately and to the conclusion of

a Pareto-dominated agreement. Finally, as the bargaining period becomes short or as the

players become very patient, the unique undominated Nash equilibrium outcome is always

inefficient.

Our two-stage negotiation model is related to papers that derive bargaining inefficiency

under complete information,3e.g. Van Damme et al. (1990), Fernandez and Glazer (1991),

Haller and Holden (1990). One important difference is that inefficiency and delay can arise

intheseothermodelsbecausethereexistmultipleefficientequilibria,whiletheundominated

equilibrium is always unique but sometimes inefficient in our model.4

Anotherstrandofrelatedliteratureincludespapersonbargainingmodelswithexogenous

deadlines. In fact, any finite horizon bargaining model can be interpreted as such. Fersht-

manandSeidmannstudyacompleteinformationbargainingmodelwitharandomproposer

and an exogenous deadline beyond which there is no surplus to divide. They assume that a

playercannotacceptalowershareofthesurplusthanshehaspreviouslyrejectedduringthe

bargaining session. This endogenous commitment assumption together with the deadline

imply that, for patient players, there is a unique equilibrium where agreements are delayed

untilthedeadline.Thisresultdependsontheinteractionbetweentheexistenceofadeadline

and endogenous commitment. Absent the endogenous commitment, the other assumption

cannotexplaindelay.MaandManoveconstructabargainingmodelwithcompleteinforma-

tion,whoseuniqueequilibriumissuchthatearlyinthegameoffersarepostponedandlatein

the game agreements are reached or the deadline is missed with positive probability. To ob-

tainsuchequilibrium,twoassumptionsareintroducedtothefinite-horizonalternating-offer

bargaining model. The first one is strategic delay. An alternating-offer model incorporates

strategic delay if a player is permitted to postpone the implementation of her move without

losing her turn. The second assumption is imperfect player control over the timing of offers

during the bargaining session. Offers and counter-offers are exchanged with exogenous

random delay.

In these papers just mentioned, the deadline is exogenously determined. Here, we show

that once the deadline is endogenous, no other assumptions such as endogenous com-

mitment or strategic delay are needed to get a unique and inefficient equilibrium. More-

over, our result may also justify the existence of Pareto-inferior phenomena other than

strikes or lockouts, such as tariff wars, debt moratoria, break-up of cease-fires or wars in

general.

The next section presents the basic negotiation model and some preliminary results.

In Section 3, we characterize the equilibrium of the deadline stage game and show that

3Another source for agreements reached with delay is incomplete information (see e.g. Watson, 1998)

4Van Damme et al. (1990) have considered an extension of Rubinstein’s bargaining game wherein there is a

smallest money unit (i.e. the number of feasible agreements is finite). In Vannetelbosch (1999a), it is shown that

this discrete bargaining game can have multiple undominated subgame perfect equilibrium outcomes. The same

holds for Fernandez and Glazer (1991) and Haller and Holden (1990) models.

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Fig. 1. The timing of the negotiation model.

the undominated Nash equilibrium outcome is always unique but might be inefficient. In

Section 4, we discuss some of the assumptions of the model. Section 5 concludes.

2. The negotiation model

Wedevelopatwo-stagenegotiationmodel;itstimingisdepictedinFig.1.Inthefirststage,

before the wage bargaining starts at time 0, the firm and the union choose simultaneously

a lockout date df ∈ {0,1,2,3,... ,∞} and its intensity γf ∈ {γ

du∈ {0,1,2,3,... ,∞} and its intensity γu∈ {γ

to choose no deadline: df= ∞ simply means that the firm decides to not choose a lockout

date. Afterwards, the players are committed to the deadline and its intensity they have

chosen. A deadline rule that seems reasonable and fits with wage negotiations follows. Let

(d,γ)bethedeadlineanditsintensityinforceduringthewagebargainingstage.Itisgivenby

¯

, ¯ γ} and a strike date

¯

, ¯ γ}, respectively. We allow both parties

(d,γ) =

(df,γf)

if df< du,

if du< df,

if du= df.

(du,γu)

(du,min{γf,γu})

(1)

That is, the deadline in force and its intensity (d,γ) are determined by the minimum of the

deadline choices of the players, and in case of ties, ties are broken in favor of the more

intense conflict. This simple deadline rule implies that the wage bargaining will take place

facing either the threat of a lockout or the threat of a strike. Finally, (∞,·) denotes the case

where both parties decide to choose neither a strike date nor a lockout date.5

In the second stage of the negotiation model, the deadline d and its intensity γ set are

common knowledge, and both parties begin to negotiate. There is an infinite number of

periods, and in each period of normal production the firm has a value added of one unit of a

good that the firm and the union can divide between them. The union’s share is W ∈ [0,1];

the firm’s share is 1 − W. Two bargaining phases are distinguished. Before the deadline

we will have a peaceful bargaining, where in each period until a new agreement is reached

5We can also interpret the commitment assumption to start a conflict at the deadline in terms of negotiators’

reputation.Imaginethatnoagreementhasbeenreachedandthattheuniondecidesnottogoonstrikeatthechosen

deadline. Since the strike date is public knowledge (due to labor laws), the union would lose most of its reputation

for the on-going negotiation as well as for future ones. In other words, the results we obtain are robust to the case

where the commitment is revocable but the cost of revoking is large enough.

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325

both parties continue to produce and the value added is shared following the old contract.

Initially the wage level in the old contract is W0> 0. After the deadline we will have an

open-conflict bargaining (a strike or a lockout has occurred), where in each period until a

new agreement is reached both parties get zero and the value added in later periods (once a

new agreement is reached) will be affected by the intensity of the conflict, γ (where γ will

be equal to γ

¯¯

For simplicity, both parties are assumed to have linear utility functions, so their payoffs

can be represented by the discounted sum of future shares. For the union this is

or ¯ γ with γ< ¯ γ < 1).

U =

∞

?

t=0

δtut,

(2)

where ut= W0if t < d and an agreement has yet to be reached, ut= 0 if t ≥ d and no

agreement has been reached, and ut= W for t ≥ s if an agreement is reached at period s

on W. The common discount factor is δ ∈ (0,1). For the firm we have correspondingly

∞

?

t=0

where vt= 1 − W0if t < d and an agreement has yet to be reached, vt= 0 if t ≥ d and

no agreement has been reached, vt= 1 − W for t ≥ s if an agreement is reached at period

s < d, and vt= γ − W for t ≥ s if an agreement is reached at period s ≥ d.

The bargaining proceeds following Rubinstein’s alternating-offer bargaining procedure.

The players are assumed to make offers alternately, one offer per period, and without loss

of generality the firm is assumed to make an offer in the beginning of period 0. The union

can then accept or reject this offer. If the union accepts, the bargaining ends. If the firm’s

offer is rejected the union makes a new offer in the next period, which the firm accepts

or rejects. If the firm accepts, the bargaining ends. If the union’s offer is rejected the firm

makes a new offer in the next period, and so on until an agreement is reached. Both parties

are assumed to have perfect information in the bargaining stage.

We denote B(d,γ) the bargaining stage, in other words the alternating-offer bargaining

game where it is common knowledge that the deadline is d and its intensity is γ. As in

Rubinstein, one can show that the alternating-offer bargaining game B(d,γ) possesses a

unique subgame perfect equilibrium (SPE) and that an agreement is reached without delay

at period 0. When the deadline in force is an odd period the SPE wage W∗(d odd,γ) and

payoffs are

δdγ

1 + δ,

U∗(d odd,γ) =1 − δd

1 − δ−1 − δd

When the deadline in force is an even period the SPE wage W∗(d even,γ) and payoffs

are

V =

δtvt,

(3)

W∗(d odd,γ) = (1 − δd)W0+

(4)

1 − δW0+

1

δdγ

1 − δ2,

(5)

V∗(d odd,γ) =

1 − δW0−

δdγ

1 − δ2.

(6)

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A. Mauleon, V. Vannetelbosch/J. of Economic Behavior & Org. 54 (2004) 321–335

W∗(d even,γ) = (1 − δd)W0+δd(1 + δ − γ)

1 + δ

,

(7)

U∗(d even,γ) =1 − δd

1 − δW0+δd(1 + δ − γ)

1

1 − δ−1 − δd

1 − δ2

,

(8)

V∗(d even,γ) =

1 − δW0−δd(1 + δ − γ)

1 − δ2

.

(9)

When the deadline in force is set at period 0, we enter immediately an open-conflict that

affects forever the value added to be shared, and the SPE wage W∗(0,γ) and payoffs are

W∗(0,γ) =

δγ

1 + δ,U∗(0,γ) =

δγ

1 − δ2,V∗(0,γ) =

γ

1 − δ2.

(10)

When no deadline is set, the SPE wage W∗(∞,·) and payoffs are (see Haller and Holden)

W∗(∞,·) = W0,U∗(∞,·) =

W0

1 − δ,V∗(θ,·) =1 − W0

1 − δ.

(11)

Comparing the expressions here and above, we obtain the next lemma, giving us some

ideas about the preferences of the players over the intensity of the conflict at equilibrium

given a deadline d.

Lemma 1. For any deadline d odd, W∗(d,γ

V∗(d,γ

¯

U∗(d, ¯ γ), and V∗(d,γ

¯

Lemma 1 tells us that, the union prefers the negotiation facing the threat of a conflict

of a weak (strong) intensity rather than facing the threat of a conflict of a strong (weak)

intensity whenever the deadline is odd (even). A conflict of a strong intensity is simply the

case where γ = γ

¯

the firm prefers the negotiation facing the threat of a conflict of a strong (weak) intensity

rather than facing the threat of a conflict of a weak (strong) intensity whenever the deadline

is odd (even).

¯

) < W∗(d, ¯ γ), U∗(d,γ

¯

) < U∗(d, ¯ γ), and

) > W∗(d, ¯ γ), U∗(d,γ ) > V∗(d, ¯ γ). For any deadline d ?= 0 even, W∗(d,γ

) < V∗(d, ¯ γ).

¯¯

) >

, and a conflict of a weak intensity is the case γ = ¯ γ. On the contrary,

3. Endogenous deadlines

We turn back to stage one of the negotiation model, where the firm and the union choose

simultaneously a lockout date df ∈ {0,1,2,3,... ,∞} and its intensity γf ∈ {γ

a strike date du∈ {0,1,2,3,... ,∞} and its intensity γu∈ {γ

strategy for the firm is denoted by (df,γf) and a strategy for the union is denoted (du,γu) in

the deadline stage game. Remember that the deadline in force during the wage negotiation

is the earliest date among the lockout date and the strike date. In case of a tie, we simply

assume that the conflict with the strongest intensity will start at the chosen deadline. This

¯

, ¯ γ} and

¯

, ¯ γ}, respectively.6Thus, a

6The results we obtain are not modified if we allow more than two levels of intensity of the conflict.

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327

Table 1

The strategic choice of a deadline

0,γ

¯

0, ¯ γ

γ

¯

¯ γ − W∗(0, ¯ γ)

¯ γ − W∗(0, ¯ γ)

¯ γ − W∗(0, ¯ γ)

¯ γ − W∗(0, ¯ γ)

1, ¯ γ

γ

¯

¯ γ − W∗(0, ¯ γ)

1 − W∗(1,γ

1 − W∗(1, ¯ γ)

1 − W∗(1, ¯ γ)

2,γ

¯

∞,·

γ

¯

¯ γ − W∗(0, ¯ γ)

1 − W∗(1,γ

1 − W∗(2, ¯ γ)

1 − W0

0,γ

0, ¯ γ

1,γ

2, ¯ γ

∞,·

¯

γ

¯

γ

¯

γ

¯

γ

¯

γ

¯

− W∗(0,γ

− W∗(0,γ

− W∗(0,γ

− W∗(0,γ

− W∗(0,γ

¯

¯

¯

¯

¯

)

)

)

)

)

− W∗(0,γ

¯

)

− W∗(0,γ

¯

)γ

¯

¯ γ − W∗(0, ¯ γ)

1 − W∗(1,γ

1 − W∗(2,γ

1 − W∗(2,γ

− W∗(0,γ

¯

)

− W∗(0,γ

¯

)

¯¯

)

¯

¯

¯

)

)

)

¯

)

assumption to break ties is quite reasonable in case of wage negotiations. Nevertheless,

the results we obtain are qualitatively robust to a more general assumption where ties are

broken with high probability in favor of the strongest conflict.

Since we allow the firm to choose (df = 0,γf) and the union to choose (du= 0,γu),

possible inefficiency is not excluded a-priori. In Table 1, we represent (very partially) the

matrix of the deadline stage game, where the firm is the row player and the union is the

column player.

In the matrix, we only give the per-period SPE payoff for the firm; the per-period SPE

payoff for the union is simply the SPE wage. For solving the deadline stage game, a natural

conceptistheNashequilibrium(NE).However,thisconceptfailstoexcludestrategyprofiles

that seem implausible such as {(0,γ

¯¯

In most of the equilibrium selection literature, it is widely accepted that Nash equilibria

involving weakly dominated strategies are not reasonable. Indeed, every perfect, proper,

essential, strongly stable or regular equilibrium is undominated (see Van Damme, 1991).

Moreover, every element of a stable set (Kohlberg and Mertens, 1986) is undominated too.

WeproposetousetheundominatedNashequilibriumconceptasadevicetoselectplausible

outcomes among the Nash ones. A strategy profile ((df,γf),(du,γu)) is an undominated

NE if and only if it is a NE and neither (df,γf) nor (du,γu) is weakly dominated with

respect to {0,1,2,... ,∞} × {γ

¯

if the player has another strategy at least as good no matter what the other player does and

better for at least some strategy of the other player.7

From Lemma 1 and the weak dominance concept, it is obvious that (dfodd, ¯ γ) is weakly

dominated by (dfodd,γ

¯

(dfodd,γ

¯

the strategy (dfodd,γ

¯

du= df, the strategy (dfodd,γ

¯

doing equally well if γu= γ

¯

nated by (dfeven, ¯ γ), (duodd,γ

¯

is weakly dominated by (du ?= 0even,γ

¯

(du= 0, ¯ γ).

),(0,γ)}.

, ¯ γ}. We say that a player’s strategy is weakly dominated

). Indeed, (i) against any (du,γu) such that du> df, the strategy

)isdoingstrictlybetterthan(dfodd, ¯ γ),(ii)againstany(du,γu)suchthatdu< df,

) is doing as well as (dfodd, ¯ γ), (iii) against any (du,γu) such that

) is doing strictly better than (dfodd, ¯ γ) if γu= ¯ γ and is

. Similarly, one can show that (dfeven,γ

) is weakly dominated by (duodd, ¯ γ), (du ?= 0even, ¯ γ)

), and (du = 0,γ

¯

) is weakly domi-

¯

) is weakly dominated by

7A formal definition of the undominated NE concept for normal-form games can be found in Van Damme

(1991) or Salonen (1996). In finite normal form games, there always exists an undominated NE. Salonen has

shown a more general result: an undominated NE exists if (a) strategy sets are convex polytopes in Rnwhere n is

the number of players and (b) utility functions are affine with respect to each player’s own strategy.

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A. Mauleon, V. Vannetelbosch/J. of Economic Behavior & Org. 54 (2004) 321–335

Lemma 2. The strategies (dfodd, ¯ γ), (dfeven,γ

(du= 0,γ

¯

Throughout the paper we focus on the case where the players are sufficiently patient (δ

large), which can be reinterpreted as though the interval between offers and counteroffers

is short.8More precisely, we focus on the case where δ > ¯ γ > γ

¯

), (duodd,γ

¯

), (du ?= 0even, ¯ γ) and

) are all weakly dominated.

¯

.

Assumption 1. δ > ¯ γ.

Lemma 3. The strategies (dfeven, ¯ γ), (duodd, ¯ γ) are all weakly dominated.

Indeed, if δ > ¯ γ then (df?= 0even, ¯ γ) is weakly dominated by (df− 1,γ

is weakly dominated by (df= 1,γ

¯

Throughout the paper the proofs related to weak dominance results are similar to the one of

Lemma2and,henceforth,theseproofsareomitted.Whatwillbetheequilibriumoutcomes?9

FromLemmas2and3,weknowthattheunionmaychooseeither(du= 0,γu= ¯ γ)or(du?=

0even,γu= γ

¯

egyprofiles((0,γ

¯¯

(0,γ

¯

egy for the firm. For instance, (df= 0,γf= ¯ γ) is weakly dominated by (df= 1,γf= γ

Choosing (df= 0,γf= ¯ γ) instead of (df= 1,γf= γ

payoffagainst(du= 0,γu)butastrictlyworstpayoffagainstany(du> 0,γu).Thefactthat

aplayerdoesnotwanttotakeunnecessaryriskiscapturedbythenotionofundominatedNE.

¯

), (df= 0, ¯ γ)

), and (duodd, ¯ γ) is weakly dominated by (du+ 1,γ

¯

).

),andthefirmmaychoose(dfodd,γf= γ

),(0,γ ))and((0, ¯ γ),(0, ¯ γ))cannotbeundominatedNEprofilesbecause

)isaweaklydominatedstrategyforbothplayersand(0, ¯ γ)isaweaklydominatedstrat-

¯

).So,the”unreasonable”NEstrat-

¯

).

¯

) will give the firm an equally well

Lemma 4. If W0 < (1 + δ − γ

(du= ∞,·) are weakly dominated.

¯

)/(1 + δ) then the strategies (du ≥ 4 and even, γ

¯

) and

Lemma 5. If W0> (γ

are weakly dominated.

¯

)/(1 + δ) then the strategies (df≥ 3 and odd, γ

¯

) and (df= ∞,·)

Could it be that the no deadline situation is an undominated NE outcome? No. Indeed,

from Lemma 2–5 we already know that, the firm will choose (1,γ

the union will choose between (0, ¯ γ) and (2,γ

characterize the undominated NE outcome fully we distinguish three cases.

¯

) if W0> γ

)/(1 + δ). In order to

¯

/(1+δ) and

¯

) if W0< (1 + δ − γ

¯

Case 1. (1 + δ − γ

equilibrium, the union could choose either the strategies (0, ¯ γ) or (2,γ

choose either the strategies (df≥ 1odd,γ

unique best response for the firm is (1,γ

¯

)/(1 + δ) > (γ

¯

)/(1 + δ) ≥ W0. It follows from Lemma 2–5 that, at

¯

) and the firm could

), then the

¯

). If the union chooses (0, ¯ γ), then best responses

) or (∞,·). If the union chooses (2,γ

¯

¯

8The discount factor can also be expressed by the formula δ = exp(−r?), where r > 0 is discount rate and ?

is the length of a single bargaining period.

9It can be shown that, when δ > ¯ γ > γ

¯

iff W0 > δ(¯ γ − γ

¯

outcomes are reasonable ones if ever.

, NE outcomes are (d = 0,γ = γ

¯

), (d = 0,γ = ¯ γ), and (d = 1,γ = γ

¯

)

)/(1 − δ2). The concept of undominated NE permits us to say under which conditions NE

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329

for the firm are (df≥ 1odd,γ

then the unique best response for the union is (2,γ

unique best response for the union is (0, ¯ γ) if W0 < [(δ)/((1 + δ)(1 − δ))](¯ γ − γ

otherwise(2,γ

¯

reverts to W0< [δ/((1 + δ)(1 − δ))](¯ γ − γ

NE is {(1,γ

¯

Case 2. (1 + δ − γ

¯

¯

equilibrium, the union could choose either the strategies (0, ¯ γ) or (2,γ

choose the strategy (1,γ

¯

(0, ¯ γ) if W∗(0, ¯ γ) = [δ/(1 + δ)]¯ γ > (1 − δ)W0+ (δ/(1 + δ))γ

W0< [δ/((1 + δ)(1 − δ))](¯ γ − γ

¯

Case 3. W0 ≥ (1 + δ − γ

¯¯

equilibrium, the union could choose either the strategies (0, ¯ γ) or (du ≥ 2even,γ

(∞,·)andthefirmisgoingtochoosethestrategy(1,γ

(1,γ

¯

W∗(1,γ

¯

(du≥ 2even,γ

¯

Havingcharacterizedtheequilibriumstrategieswecanshowthatthedeadlinestagegame

has a unique undominated NE.10Indeed, if W0 > [δ/((1 + δ)(1 − δ))](¯ γ − γ

unique undominated NE outcome is d = 1 and γ = γ

union will start the wage bargaining under the threat of a severe lockout at period 1. The

firm will make a wage offer W∗(d = 1,γ

¯

the union will accept this offer immediately.

¯

) and (∞,·). If the firm chooses (df≥ 3odd,γ

¯

).Indeed,W∗(0, ¯ γ) = [δ/(1+δ)]¯ γ > (1−δ)W0+[δ/(1+δ)]γ

¯

),(0, ¯ γ)} if W0< [δ/((1+δ)(1−δ))](¯ γ −γ

¯

) or (∞,·),

¯

¯

= W∗(1,γ

). If the firm chooses (1,γ), then the

) and

¯¯

)

). As a consequence, the unique undominated

) and is {(1,γ

¯¯

),(2,γ

¯

)} otherwise.

)/(1 + δ) > W0> (γ )/(1 + δ). It follows from Lemma 2–5 that, at

¯

) and the firm will

), the union will choose

= W∗(1,γ

¯

¯

). Hence, given that the firm chooses (1,γ

¯

¯

). That is, if

). Otherwise, the union will choose (2,γ).

)/(1 + δ) > γ/(1 + δ). It follows from Lemma 2–5 that, at

¯

) or

¯

).Hence,giventhatthefirmchooses

),theunionwillchoose(0, ¯ γ)ifW∗(0, ¯ γ) = [δ/(1+δ)]¯ γ > (1−δ)W0+[δ/(1+δ)]γ

). That is, if W0< [δ/((1 + δ)(1 − δ))](¯ γ − γ

) or (∞,·).

¯

=

¯

). Otherwise, the union will choose

¯

) then the

¯

. So, at equilibrium, the firm and the

) = (1 − δ)W0+ (δγ

¯

)/(1 + δ) at period 0, and

Proposition1. IfW0> [δ/((1+δ)(1−δ))](¯ γ−γ

and efficient undominated Nash equilibrium outcome. The deadline (a lockout threat of a

strongintensity)issetatperiod1andanagreementonW∗(d = 1,γ

at period 0.

¯

)thenthenegotiationmodelhasaunique

¯

)isreachedimmediately

However,ifW0< [δ/((1+δ)(1−δ))](¯ γ −γ

profile where the firm chooses (df = 1,γ

W0 < [δ/((1 + δ)(1 − δ))](¯ γ − γ

γ = ¯ γ, and it is an inefficient outcome since a strike of a weak intensity occurs immediately

at the start of the wage bargaining.

That is, if the equilibrium wage in case of an immediate strike of a weak intensity is

greater than the equilibrium wage in case of the union moving at the deadline and facing

the threat of a lockout of a strong intensity, then the union will choose at equilibrium to

implement immediately a strike of a weak intensity. Since the old wage contract is small

enough, the difference between ¯ γ and γ

¯

¯

)thereisauniqueundominatedNEstrategy

) and the union chooses (du = 0, ¯ γ). So, if

) the unique undominated NE outcome is d = 0 and

¯

¯

is large enough and the players are patient enough,

10Notice that all the results we get throughout the paper can be obtained using rationalizability concepts as

defined in Herings and Vannetelbosch (1999, 2000) and Vannetelbosch (1999b).

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A. Mauleon, V. Vannetelbosch/J. of Economic Behavior & Org. 54 (2004) 321–335

it becomes optimal for the union to go immediately on strike and to destroy part of the

available surplus. Indeed, such a conflict of a weak intensity allows the union to avoid

having to accept a very low wage in the face of a severe lockout that would allow the firm to

grab most of the surplus. So, at equilibrium, the union goes into conflict immediately, and a

Pareto-dominatedagreementfollows.Infact,thisequilibriumoutcomeisPareto-dominated

bytheequilibriumoutcomeofB(du= 1, ¯ γ).Atequilibrium,thefirmwillmakeawageoffer

W∗(d = 0, ¯ γ) = (δ¯ γ)/(1+δ) at period 0, and the union will accept this offer immediately.

Proposition2. IfW0< [δ/((1+δ)(1−δ))](¯ γ−γ

and inefficient undominated Nash equilibrium outcome. A conflict of a weak intensity starts

immediatelyattime0followedbyanimmediateagreementonW∗(d = 0, ¯ γ).Theper-period

efficiency loss is equal to 1 − ¯ γ.

¯

)thenthenegotiationmodelhasaunique

The firm at equilibrium chooses (df= 1,γ

because otherwise the firm faces serious repercussions: the union would choose another

deadline, leaving the firm with the threat of a severe strike that would allow the union to

grab most of the surplus.

The condition W0< [δ/((1 + δ)(1 − δ))](¯ γ − γ

is relatively small, each player has at his disposal both actions that reduce substantially the

value added in the future and actions that have only a minor impact on the future value

added11(in other words, the difference between ¯ γ and γ

are patient (δ is large enough).

Before discussing some of the assumptions, we will also consider the limit case of fully

patient players as δ goes to one.

¯

) either because it is a dominant strategy or

¯

) is satisfied whenever the old wage W0

¯

is large enough), and the players

Corollary 1. As δ goes to one or as the interval between offers and counteroffers vanishes,

thenegotiationmodelhasauniqueandinefficientundominatedNashequilibrium.Aconflict

of a weak intensity starts at time zero followed by an immediate agreement on W∗(d =

0, ¯ γ) = (1/2)¯ γ.

Notice that we have taken the limit δ → 1, assuming that the players can still reduce

permanently the future value added. This assumption may be questionable once we rein-

terpret the limit as the interval between offers and counteroffers vanishes. An alternative

assumption is to suppose that the level of surplus destruction would decline as ? decreases.

Nevertheless, we still obtain the inefficiency result if we assume that the intensities of a

conflictareboundedbelowone.Thatis,ifγ

¯

lim?→0¯ γ(?) < 1.

(?) < ¯ γ(?) < δforall?andlim?→0γ

¯

(?) <

11The actions as well as their impact on the future value added may depend on factors such as the competition

on both the product market and the labor market (see Cutcher-Gershenfeld et al., 1998). Indeed, if there is a strong

competition in the product market, a labor conflict may induce a big loss in market power and future revenues to

be generated. However, if there is a strong competition and mobility in the labor market, the impact of a conflict

could be reduced since the firm may more easily find temporary replacement workers.

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331

4. Discussion

Before concluding we comment upon some of the assumptions of the model. Section 4.1

contains a discussion about the assumption that the players are committed to engage in

conflictatthedeadline.InSection4.2,wediscusstheassumptionthatthesurplusdestruction

duetoaconflictdoesnotdependonthedurationoftheconflict.Finally,Section4.3isdevoted

to lockouts.

4.1. The commitment assumption

If at least one of the parties has chosen a deadline, the wage bargaining will take place

facingeitherthethreatofalockoutorthethreatofastrike.Itisassumedthatifnoagreement

isreachedpriortothedeadline,thenaconflictwilloccurwithprobabilityoneatthedeadline.

In other words, one of the parties is committed to start a conflict at the deadline. One

interpretation for such commitment assumption is in terms of negotiators’ reputation. For

instance,theunioncouldlosemostofitsreputationfortheon-goingnegotiationandfutures

ones if it decides not to go on strike at the chosen deadline. An alternative interpretation

for the commitment assumption has to do with certain rules and rituals followed by unions.

Labordisputesareoftenprecededbystrikeballots.GoerkeandHoller(1999)haveanalyzed

theconsequencesofstrikeballotsinanoncooperativemodelofnegotiationsbetweenaunion

and a firm over wage increases. The firm possesses private information about its revenues.

The union can only obtain a wage increase if it makes a rejection of a wage demand costly

tothefirmduetoastrike.However,itmightbethatstrikethreatsareemptybecausetheyare

not credible. Goerke and Holler have shown that strike ballots can remedy this feature since

ballots can provide a commitment to adhere to a strike decision. Thus, ballots can make

strike threats credible. For this effect to occur, the timing of voting is crucial: ballot should

take place before the first demand can be rejected. They have also derived the minimum

level of commitment that a strike ballot must have if a worker is to support a strike in the

vote assuming that a strike ballot imposes positive costs on a strike breaker.

As the analysis of Goerke and Holler has made clear, if we allow the union to choose a

strike date and make a strike ballot at time zero, then if costs imposed on strike breakers are

sufficiently high, the ballot will make credible the threat to strike at the chosen deadline.

Thereby we conclude that making strike ballots and deadline notices compulsory can lead

to inefficiencies, even in an environment of complete information.

4.2. The severity of labor conflicts

We have allowed the players to choose the deadline after which the surplus to be shared

is permanently reduced as well as the level of surplus destruction after the deadline. The

union can choose between a “light” strike and a “severe” strike. An interpretation for light

strikes are partial strikes or work to rule. One might argue that long lasting damage from

labor conflicts (such as a permanent loss of customers or suppliers) is typically caused

by prolonged conflicts rather than high intensity of a short conflict whereas duration does

not matter in our model. It is assumed that the surplus destruction after the deadline, in

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A. Mauleon, V. Vannetelbosch/J. of Economic Behavior & Org. 54 (2004) 321–335

case no agreement is reached before the deadline, does not depend on the duration of

the conflict. An alternative would have been both to fix the per-period level of surplus

destruction due to strikes and lockouts, and to allow the parties to choose the length or

duration of the conflict. Then, both players would like the other player to be the last mover

at the deadline and preferably facing the threat of a conflict of a long duration. It might still

be optimal for the union to go on strike immediately for a short time, thus not having to

accept a very low wage offer or to face the threat of a long lasting lockout where the firm

would grab most of the surplus. Thereby, inefficiency losses, but smaller ones, might still

occur.

However, in many labor conflicts the main cost of the negotiation appears to be its

lasting effects on the profitability of the firm rather than the foregone revenue during the

strike or the lockout. For instance, the union could disclose some information about the

firm’s products that could negatively affect the demand beyond the date of an agreement.

Moreover, empirical evidence has suggested that most of the “news” in a strike seems to

be incorporated in the Stock Market valuation very early in the strike. Dinardo and Hallock

(2002) have examined the Stock Market’s responsiveness to strikes by looking at strike

actions that labor historians generally view as the major ones occurring in the USA in

the years 1925–37. They have found that strikes had large, negative effects on industry

stock valuation. In addition, longer strikes, violent strikes, strikes won by the union, strikes

leading to union recognition, industry-wide strikes, and strikes that led to wage increases

were associated with larger negative share price reactions than other strikes. Industry stock

price movements around the start and the end of the strike have also been studied. Their

analysis has shown that much of the “news” generated by the typical strikes seems to have

been registered by the Stock Market very early in the strike. A negative share price reaction

of 3 percent around the start of the strike was observed.12

4.3. Lockouts

Firms have an almost as powerful weapon in their negotiating arsenal through the mech-

anismofalockout.13FollowingFernandezandGlazer,wehavetreatedstrikesandlockouts

symmetrically. We have assumed that the firm can give the union a deadline notice of the

commencement of the lockout, at which time a protected lockout would begin, permitting

the firm to lock all workers out of the workplace. In another type of lockout, a lockout

in response to a strike, the firm may wish to take the upper hand in seeking a defensive

lockout during a strike; this results in the union and workers being able to bring the strike

to an end, but then having to face the lockout. In this case, a specification where strikes

could be followed by defensive lockouts may be more appropriated. We next incorporate

the possibility of defensive lockouts into our model.

In the first stage, before the wage bargaining starts at time 0, the firm and the union still

choose simultaneously a lockout date df∈ {0,1,2,3,... ,∞} and its intensity γf∈ {γ

¯

, ¯ γ}

12Other related studies have suggested a loss of between 1 and 4 percent around the start of the strike; see e.g.

Becker and Olson (1986), Nelson et al. (1994).

13Fisher (2001) has developed an asymmetric information model of collective bargaining where the firm has the

bargaining power and the union the private information. Results show that the firm may use lockouts to induce the

union to reveal its private information.

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333

and a strike date du ∈ {0,1,2,3,... ,∞} and its intensity γu ∈ {γ

Afterwards,theplayersarecommittedtothedeadlineandtheintensitytheyhavechosen.In

the second stage of the negotiation model, the deadlines (df,γf) and (du,γu) are common

knowledge and both parties begin to negotiate with the firm making the first offer. The firm

can now use a lockout in two ways: either to preempt the occurrence of a strike (preemptive

lockout) or to bring to an end an on-going strike (defensive lockout). The possibility of

defensive lockouts is assumed to modify the surplus to be shared each period as follows:

¯

, ¯ γ}, respectively.

1. du> df.Then,bothplayerswillnegotiatefacingthethreatofalockout.Ifanagreementis

reachedbeforedf,thenthesurplustobesharedeachperiodisequalto1.Ifanagreement

is reached at period t ≥ df, then for t?< dfthe surplus is equal to 1, for df≤ t?< t the

surplus is equal to 0, and for t?≥ t the surplus is equal to γf.

2. du < df. Then, both players will negotiate facing the threat of a strike followed by a

defensive lockout. If an agreement is reached before du, then the surplus to be shared

each period is equal to 1. If an agreement is reached at period t ∈ [du,df), then for

t?< duthe surplus is equal to 1, for du≤ t?< t the surplus is equal to 0, and for t?≥ t

the surplus is equal to γu. If an agreement is reached at period t ≥ df, then for t?< du

the surplus is equal to 1, for du≤ t?< t the surplus is equal to 0, and for t?≥ t the

surplus is equal to min{γf,γu}.

3. du= df.Then,ifanagreementisreachedbeforedu,thesurplustobesharedeachperiod

is equal to 1. If an agreement is reached at period t ≥ du, then for t?< duthe surplus is

equal to 1, for du≤ t?< t the surplus is equal to 0, and for t?≥ t the surplus is equal to

min{γf,γu}.

The rest of the model is left unchanged, and we still assume that δ > ¯ γ > γ

firm has the possibility of using defensive lockouts, it can be shown that the negotiation

model has a unique undominated NE outcome that is always efficient. The intuition behind

this result is that striking cannot revoke the threat of a severe “defensive” lockout. It is still

optimal for the firm to choose the threat of a severe lockout at period 1, (df= 1,γf= γ

Facing such a threat, the union has no incentive to go on strike at period 0 because an

immediate light strike would not preempt the occurrence or the threat of a severe lockout at

period 1. The strategy (du= 0,γu= ¯ γ) is weakly dominated by (du= 2,γu= γ

at equilibrium, the firm and the union will start the wage bargaining under the threat of a

severe lockout at period 1. The firm will make a wage offer (1 − δ)W0+ (δγ

period 0, and the union will accept this offer immediately.

However,iftheunionmakesthefirstofferinthewagebargain,thenaninefficientoutcome

is not excluded. The union has a weakly dominant strategy, which is (du = 1,γu = γ

Facingthethreatofastrikeatperiod1,thefirmwillchooseanimmediatepreemptivelockout

if V∗((df= 0,γf= ¯ γ),(du= 1,γu= γ

¯

which reverts to W0> 1 − (δ(¯ γ − γ

¯

then the negotiation model with defensive lockouts and with the union making the first

offer has a unique and inefficient undominated Nash equilibrium outcome. A preemptive

lockout of a weak intensity occurs immediately at the start of the wage bargaining followed

by an immediate agreement on ¯ γ/(1 + δ). The per-period efficiency loss is equal to 1 − ¯ γ.

Intuitively, when the old wage contract is not too low, it becomes optimal for the firm

immediately to start a lockout and to destroy part of the available surplus. Such a conflict of

¯

. Once the

¯

).

¯

). Thus,

¯

)/(1 + δ) at

¯

)),

).

)) > V∗((df= 2,γf= γ

))/(1 − δ2). So, if W0> 1 − (δ(¯ γ − γ

¯

),(du= 1,γu= γ

¯

¯

))/(1 − δ2),

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aweakintensityallowsthefirmtoavoidhavingtoacceptahigherwageofferfacingthethreat

of a severe strike, where the union would grab most of the surplus. Thus, at equilibrium,

the firm goes into conflict immediately and a Pareto-dominated agreement follows.

Labor conflicts, strikes, and lockouts have not been uncommon in professional sports

in the USA (see Leeds and von Allmen, 2002). Major sports have experienced lockouts

over the last decades.14For instance, the National Hockey League (NHL) prepared to enter

the 1994–95 season without having signed a new collective bargaining agreement with the

NHL Players Association. Fearful of another stoppage at playoff time, the league staged a

preemptive lockout, the NHL’s owners choosing to lock out the players at the beginning of

the season because the timing of a lockout can be crucial. In professional sports, players

have incentive to strike near the end of the season when fan interest is high and network

coverageismostintense.Becauseanystoppageofplayatthistimewouldhaveasignificant

impact on both gate revenue and media revenue, the owners are highly vulnerable to a

strike. In addition, the players are best able to withstand a strike at the end of the season

because they have already received most of their salaries. Owners understand this, and they

have the incentive to lock players out early in the season, minimizing the costs to them and

maximizing the costs to the players. Thus, if we invert the bargaining protocol (the union

makesthefirstofferinsteadofthefirm),theNHLlockoutstoryisconsistentwithourmodel

predictions.

5. Conclusion

We have developed a two-person negotiation model with complete information that is a

firstattempttomakeendogenousboththedeadlineandthelevelofsurplusdestructionafter

the deadline. We have shown that the undominated Nash equilibrium outcome is always

unique but might be inefficient. Moreover, as the bargaining period becomes short or as the

playersbecomeverypatient,thisuniqueoutcomeisalwaysinefficient.Ourmodelmayalso

justify the existence of Pareto-inferior phenomena other than labor conflicts, such as tariff

wars, debt moratoria, break-up of cease-fires or wars in general.15

Oneveryinterestingextensionwouldbetoconsideramoregeneralbargainingprocedure

as in Perry and Reny (1993) that allows the players to choose when and whether to make

an offer.

14Major League Baseball owners have locked players out of training camps over a 32-day period in 1989.

National Hockey League has experienced an interruption of the season 94–95, when 468 games were lost over a

103-day lockout. The 1998–99 National Basketball Association season was truncated by a 202-day lockout.

15Garfinkel and Skaperdas (2000) have obtained similar conclusions but in a totally different framework. They

have demonstrated that conflict can ensue even without misperceptions or incomplete information when the

antagonists take a long-run view. Despite the short-run incentives to settle disputes peacefully, there can be long

term, compounding rewards to going to war when doing better relative to one’s opponent today implies doing

better tomorrow. Peaceful settlement involves not only sharing the pie available today, but also foregoing the

possibility, brought about by war, of gaining a permanent advantage over one’s opponent into the future. Garfinkel

and Skaperdas have shown how war emerges as an equilibrium outcome in a model that takes these considerations

into account. War is more likely to occur, the more important is the future.

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335

Acknowledgements

We wish to thank an anonymous referee for helpful comments. The research of Ana

MauleonhasbeenmadepossiblebyafellowshipoftheFondsEuropéenduDéveloppement

Economique Régional (FEDER). Financial support from the Belgian French Community’s

program Action de Recherches Concertée 99/04-235 (IRES, Université catholique de Lou-

vain) is gratefully acknowledged.

References

Becker, B.E., Olson, C.A., 1986. The impact of strikes on share holder equity. Industrial and Labor Relations

Review 39, 425–438.

Cutcher-Gershenfeld, J., Kochan, T., Calhoun Wells, J., 1998. How do labor and management view collective

bargaining. Monthly Labor Review 121, 23–31.

Dinardo, J., Hallock, K.F., 2002. When unions mattered: the impact of strikes on financial markets, 1925–1937.

Industrial and Labor Relations Review 55, 219–233.

Fernandez, R., Glazer, J., 1991. Striking for a bargain between two completely informed agents. American

Economic Review 81, 240–252.

Fershtman, C., Seidmann, D.J., 1993. Deadline effects and inefficient delay in bargaining with endogenous

commitment. Journal of Economic Theory 60, 306–321.

Fisher,T.C.G.,2001.Anasymmetricinformationmodeloflockouts.BulletinofEconomicResearch53,153–159.

Garfinkel, M.R., Skaperdas, S., 2000. Conflict without misperceptions or incomplete information. Journal of

Conflict Resolution 44, 793–807.

Goerke, L., Holler, M.J., 1999. Strike ballots as a commitment device. Journal of Institutional and Theoretical

Economics 155, 487–504.

Haller, H., Holden, S., 1990. A letter to the editor on wage bargaining. Journal of Economic Theory 52, 232–236.

Herings, P.J.J., Vannetelbosch, V.J., 1999. Refinements of rationalizability for normal-form games. International

Journal of Game Theory 28, 53–68.

Herings,P.J.J.,Vannetelbosch,V.J.,2000.TheequivalenceoftheDekel–Fudenbergiterativeprocedureandweakly

perfect rationalizability. Economic Theory 15, 677–687.

Kohlberg, E., Mertens, J.-F., 1986. On the strategic stability of equilibria. Econometrica 54, 129–144.

Leeds, M., von Allmen, P., 2002. The Economics of Sports. Addison-Wesley, Boston.

Ma, C.A., Manove, M., 1993. Bargaining with deadlines and imperfect player control. Econometrica 61, 1313–

1339.

Nelson, M., Amoako-Adu, B., Smith, B., 1994. Impact of labor strikes on equity values: Canadian evidence.

Journal of Economics and Business 46, 153–165.

Perry, M., Reny, P.J., 1993. A non-cooperative bargaining model with strategically timed offers. Journal of

Economic Theory 59, 50–77.

Rubinstein, A., 1982. Perfect equilibrium in a bargaining model. Econometrica 50, 97–109.

Salonen,H.,1996.On the existence of undominated Nash equilibria in normal-form games.Games and Economic

Behavior 14, 208–219.

Van Damme, E., 1991. Stability and Perfection of Nash Equilibria, second ed. Springer-Verlag, Heidelberg.

Van Damme, E., Selten, R., Winter, E., 1990. Alternating bid bargaining with a smallest money unit. Games and

Economic Behavior 2, 188–201.

Vannetelbosch, V.J., 1999a. Alternating-offer bargaining and common knowledge of rationality. Theory and

Decision 47, 111–137.

Vannetelbosch,V.J.,1999b.Rationalizabilityandequilibriuminn-personsequentialbargaining.EconomicTheory

14, 353–371.

Watson, J., 1998. Alternating-offer bargaining with two-sided incomplete information. Review of Economic

Studies 65, 573–594.

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