Posturing in Bargaining to Influence Outsiders

Andres Perlroth∗

Job Market Paper

January 1, 2020Please click here for the latest version.

Abstract

This paper studies the effect of transparency in negotiations in the presence of outsiders.

I consider a game in which two players (negotiators) engage in bargaining in the presence

of a third player (outsider) who also cares about the outcome of the negotiation. As the

bargaining progresses, the outsider takes a single irreversible action that brings him a payoff

that depends on the allocation that the negotiators will agree to implement. The set of feasible

allocations depends on the state of the world which is known to the negotiators but not to the

outsider. The negotiators care about the bargaining outcome and have misaligned incentives

regarding the outsider’s decisions. When proposals are public (transparent negotiation), there

is an equilibrium where negotiators engage in posturing behavior as a tactic to distort the

outsider’s action. This behavior reduces efficiency because the outsider makes a suboptimal

decision, and sometimes, there is delay in the negotiation. Moreover, I show that these

inefficiencies arise for a general class of equilibria. On the other hand, when proposals are

private (non-transparent negotiation), the negotiators agree immediately, and the terms of

the agreement are highly informative for the outsider. In this situation, I show that every

equilibrium is approximately efficient.

∗Stanford GSB. Email: andperl@stanford.edu. I am grateful to Andy Skrzypacz, Mike Ostrovsky and RobertWilson, for guidance and continuous help. I thank Kyle Bagwell, Anirudha Balasubramanian, Alex Bloedel,Jeremy Bulow, Gabe Carroll, Daniel Chen, Matt Jackson, Fuhito Kojima, Yucheng Liang, Bar Light, Suraj Mal-ladi, Deborah Melnick, Paul Milgrom, Ilya Morozov, Ashutosh Thakur and participants at the Theory LunchSeminar at Stanford for helpful comments. All errors are mine.

1 Introduction

Many important decisions related to social policy and trade agreements are made through nego-

tiations. One dimension along which these negotiations vary is how information is revealed to

parties that are not directly involved in the negotiation. For instance, historically, while most trade

agreement negotiations were conducted behind closed doors, nowadays there is large variation in

the transparency protocols countries adopt (Marceddu, 2018). On the one hand, the recent negoti-

ations between US and Japan over their trade agreement were conducted behind closed doors.1 By

contrast, the European Union (EU) and Australia (UK) conducted their trade negotiations with a

high level of transparency (Drake-Brockman and Messerlin, 2018).

It has become a common trend among scholars and activists to recommend that such negoti-

ations be conducted transparently.2 They argue that transparency gives legitimacy to the nego-

tiations and facilitates accountability of the officials, mitigating agency problems between them

and their constituents (Stasavage, 2004). My paper challenges this presumption by showing that

transparency can be undesirable. When their actions are transparent, negotiators may posture to

influence outsiders, i.e., the parties not directly involved in the transparent negotiation.

In practice, trade agreements often feature an outsider who must make a payoff-relevant decision

on the basis of the negotiation. For example, consider the recent US-Japan trade agreement. Over

the course of the negotiation, car manufacturers in both countries wanted to know whether car-

tariffs would even be an issue covered in this agreement.3 Japanese manufacturers feared the

imposition of high tariffs and would plan to dial back car production in such a scenario. On

the other hand, American manufacturers would benefit from high tariffs to promote domestic

business.4Similarly, in the EU-Australia trade negotiations, the discussion over agricultural tariffs

is an important concern for Australian and European farmers.5

1 Source: https://www.politico.com/story/2019/08/25/us-japan-trade-deal-g7-1474461.2For instance, during the Trans-Pacific Partnership Agreement (TPP) talks, over 30 legal academics wrote a

letter to the United States Trade Representative, Ron Kirk, about their profound concern and disappointment atthe lack of public participation, transparency and open government processes in the negotiation of the intellectualproperty chapter of the Trans-Pacific Partnership Agreement (TPP).

Source: http://infojustice.org/archives/21137.)3“Washington may unveil tariffs on vehicles and automotive parts in the next few months that could have

broad repercussions for Japan’s massive automotive sector.”Source: https://worldview.stratfor.com/article/japans-auto-sector-poised-weather-us-tariff-storm.4“Japan, under Prime Minister Shinzo Abe, wants to head off Trumps threatened penalties on auto exports,

which could tip it into recession, as well as any currency clause directed at the Japanese yen.” (April 2019, For-tune).

5“The EU is notoriously sensitive about maintaining its already significant market protections for farmers.It has already warned Australia that it’s not keen on opening market access to beef, sheepmeat, rice, sugar or

1

To illustrate how the presence of outsiders affects the negotiation, I consider a simplified model.

Specifically, I consider a bargaining game with three players: two negotiators, A and B, and one

outsider O. The negotiators bargain over which allocation to implement and over the division of

the surplus generated by its implementation. The bargaining procedure is a one-sided repeated

proposal game that ends after a (possibly infinite) deadline is reached. The set of feasible allocations

depends on the state of the world. In the constrained state, the only implementable allocation is the

constrained policy in which the scope of the agreement must include the issues that are relevant

for the outsider (e.g. car tariffs). In the unconstrained state, it also feasible to implement the

unconstrained policy in which the agreement terms do not include such issues. The negotiators

know the state but the outsider does not. The outsider decides when and what irreversible action

to take. His payoff depends on the time and kind of action taken, and on the allocation that

negotiators agree to implement. At every period, the outsider trades-off taking the irreversible

action or waiting in the hope of obtaining more information afterwards. The negotiators have

misaligned incentives regarding the outsider’s action. The proposing negotiator A prefers a high

action: the action the outsider O takes when he believes that the unconstrained allocation will

be implemented. The other negotiator B prefers a low action: the action the outsider O takes

when he believes that the constrained allocation will be implemented. Throughout the paper I

assume that the unconstrained-state negotiation gains to A and B are sufficiently larger than their

payoffs arising from O’s action (so that they would not sacrifice the negotiations just to influence

the outsider, at least in the unconstrained state).

The main contribution of the paper is to compare welfare between transparent and non-

transparent negotiations. I define a transparent negotiation as one where the outsider observes all

the proposals that are made, and I define a non-transparent negotiation as one where the outsider

only observes the agreement terms of the negotiation after it has been completed. In negotiations

where A could make proposals frequently, it is typically more efficient to make the negotiation

non-transparent. More precisely, I show that (under standard off-path belief refinements) in every

equilibrium of the non-transparent negotiation game, there is an immediate agreement, and the

outsider takes the informed decision after waiting one period. By contrast, in the transparent

negotiation game, an inefficient equilibrium exists: negotiators engage in posturing behavior as a

tactic to distort the outsider’s action. As a consequence, the outsider makes a suboptimal action.

Furthermore, posturing may lead to a delay in the bargaining, which persists even as the frequency

dairy.”Source: https://www.farmonline.com.au/story/6380868/

eu-deal-is-dead-if-ag-loses-out-birmingham/.

2

of offers increases without bound. In contrast to most of the bargaining literature where inefficient

delay arises due to asymmetric information amongst bargainers, in this paper, delay arises because

bargainers want to influence third party outsiders.

To get an intuition for my results, suppose that the negotiation has an immediate deadline

(T = 0). The order of moves is that A makes a proposal, O decides whether to act immediately or

wait, B either accepts A’s proposal or rejects (and the gains from the negotiation are destroyed),

finally if O has not acted yet, he can take the action incurring a cost for the delay. In the scenario

where the negotiation fails, O believes that the unconstrained allocation will be later implemented

when such policy is feasible; otherwise, the constrained allocation will be implemented.6

When the negotiation is non-transparent, I assert that an equilibrium exists where the ne-

gotiators reach an immediate agreement. Indeed, suppose that the outsider O believes that in

the event of a rejection the constrained policy will be later implemented.7 In the unconstrained

state, the assumption that the gains from negotiation are large implies that A has a proposal such

that (i) B prefers to accept rather than reject and influence O to take B’s preferred action, and

(ii) if the proposal is accepted, the unconstrained policy is implemented. This shows that in the

unconstrained-state, A and B reach an agreement which is informative for O. In the constrained

state, A does not want to make an infeasible proposal since O would take his non-preferred action.

Hence, A makes a constrained-allocation proposal, B accepts, O takes the low action. Therefore,

O prefers to wait until the negotiation reaches an agreement rather than taking the action while

A and B are still negotiating. I conclude that under non-transparency, bargaining is efficient, and

O takes an optimal action with a delay that vanishes as the one-round delay costs are small.

Now suppose that there is an information leak and O somehow is able to observe the proposals

when A and B are privately negotiating (and hence behaving as specified above). In this situation,

O immediately learns the underlying state by observing A’s proposal; therefore, he takes the

optimal action without waiting to observe the final agreement. In this hypothetical situation,

bargaining is efficient and the outsider takes an optimal action without delay. However, if A knows

that O will observe the proposals, he can influence O to make A’s preferable action by making the

unconstrained-allocation proposal even when such allocation is not feasible. Accordingly, when A’s

payoff related to O’s action is larger than the negotiation gains by implementing the constrained

6The outsider believes that in a future negotiation, A and B will avoid to include in the scope of the negoti-ation hard issues from the previous negotiation (e.g. car tariffs), so long as it is feasible to reach an agreementwithout them.

7I show in the paper that this is the only belief consistent with the forward induction refinement, an extensionof the intuitive criterion for multi-sender games.

3

policy, A prefers to influence O at the cost of not reaching an agreement in the constrained-state.

From the previous paragraph, we conclude that when the negotiation is transparent and the

outsider’s action is sufficiently relevant for A, the equilibrium proposals cannot be fully revealing. I

construct a posturing equilibrium where A postures when the state is constrained by proposing, with

positive probability, the unconstrained-allocation proposal even though it is not feasible. Figure 1

describes the posturing probability as a function of the prior probability the state is unconstrained

(p). Notice that when p is greater than a threshold µA, there is strong posturing: A proposes the

unconstrained-allocation proposal with probability one. When p ≥ µA, there is weak posturing: A

mixes between proposing the unconstrained-allocation proposal and proposing a feasible proposal

that reveals the state. In the equilibrium, O does not benefit by waiting a period: B would

reject the unconstrained-allocation proposal as a way to influence O to believe that the only the

constrained policy is feasible. Consequently, O prefers to take a suboptimal action after observing

the proposal. After observing O to take the irreversible action, B accepts every feasible proposal.

Therefore, under transparency, there are two sources of inefficiencies: breakdown of negotiations

because A makes a non-feasible proposal (see Figure 1), and a suboptimal action from the outsider.

Furthermore, I show that these inefficiencies do not vanish as the frequency of the outsider’s actions

increases.

Prior belief pµA

Prob. A makes

unfeasible proposal

Prob. reach

agreement

10

1

Figure 1: Inefficiencies in the posturing equilibrium.

In the paper, I analyze the robustness of the previous intuition by looking at a larger class

of equilibria and considering a general deadline T . When the negotiation is transparent, I show

that under some technical conditions a posturing equilibrium exists. Depending on the deadline

4

T and the prior p, posturing can be strong or weak. When there is an extended deadline T or

the prior p is large, posturing is strong: negotiators A and B do not reveal any information to

the outsider O until he takes the action. In this situation, the outsider takes the uninformed

action immediately and, in the next period, negotiators reach an efficient agreement. Hence, with

frequent offers, bargaining is approximately efficient but the outsider takes a suboptimal action.

When T and p are small, there is weak posturing: Players A and O engage in a war of attrition. In

the constrained-state, A mixes between making an unfeasible unconstrained-allocation proposal or

revealing the underlying state by making a feasible constrained-allocation proposal. The outsider

O, on the other hand, mixes between taking the action or waiting to learn by observing from the

future proposals. Thus, with positive probability the outsider takes a suboptimal action. In this

case, the bargaining-inefficiencies do not vanish as the frequency of offers increases. Therefore,

when the frequency of offers increases, the weak posturing equilibrium shares three sources of

inefficiencies: suboptimal decision from the outsider, delay of the bargaining and breakdown of the

negotiation. I complement the analysis by showing that, under some technical conditions, every

equilibrium is inefficient as the frequency of offers increases.

When the negotiation is non-transparent, I show that a separating equilibrium always exists.

In the equilibrium, negotiators immediately agree, and hence, the outsider takes the full informed

action by waiting for one period. Thus, bargaining is efficient; O takes an optimal decision with

a one period delay. In particular, when the frequency of action increases, the outsider’s decision

is approximately optimal. Furthermore, I show that, under certain conditions, the equilibrium

outcome is unique.

My results share a concern over transparency that applies to a broader class of negotiations

beyond trade agreement negotiations. For instance, such concern may arise in a negotiation between

a labor union and managers, where the scope of the negotiation may include the implementation of

new technology for the company. While managers want suppliers to believe that the new technology

will be implemented, the labor union prefers the other scenario since the new technology would

signify a reduction of the labor force. One example of such conflict is the recent negotiation between

General Motors and the Union of Automobile Workers, which was centered around the company

transitioning to electric vehicle production. Managers were taking a hard position over reopening

shuttered car factories, and hence, influencing Workhorse Group Inc. (the outsider) to invest in

electric vehicle production for future supply to GM.8

8For a description of the timeline of the negotiation see https://www.reuters.com/article/

usa-autos-labor-events/timeline-uaw-members-ratify-new-labor-deal-with-gm-to-end-strike-idUSL2N27A0PF.

5

The paper is organized as follows. Section 2 discusses related literature. Section 3 describes the

model. Section 4 analysis the transparent negotiation game. Section 5 analyzes the game when

the negotiation is non-transparent. Section 6 concludes. All omitted proofs are in the Appendix.

2 Literature Review

The effects of transparency in bargaining have been studied in situations where representatives

(bargainers) and constituents (outsiders) have conflicting goals. Perry and Samuelson (1994) con-

sider a two-period bargaining game where one of the two bargainers represents a constituency who

may intervene and terminate negotiations. Because bargainers may prevent such intervention by

reaching immediate agreement, the level of transparency only affects allocation but not efficiency.

Stasavage (2004) consider a static bargaining game in which bargainers care about the outcome and

also about their constituents beliefs over their conflicting interests. He shows that transparency is

preferable when bargainers’ conflicting interests are strong. His result contrasts with mine because,

in my model, influencing the outsider affects the gains from the negotiation. In a similar model,

Fingleton and Raith (2005) consider a static bargaining model in which bargainers are motivated

by a (pure) career concern motive. Their result shares a similar concern regarding transparency,

although for a different reason. In my model, bargainers care about the outcome of the negotiations

and what the outsiders expect from the negotiations. In Fingleton and Raith (2005), bargainers

only care about the third parties’ perception regarding their bargaining skills.9

Beyond the political economy applications, Chaves (2019) analysis the role of transparency in

a bargaining game between a seller and a privately informed buyer that may be interrupted by

the entry of new buyers. He shows that under transparency, bargaining ends faster because the

seller’s lack of commitment makes him lower prices quickly in order to increase competition among

buyers. This contrasts with my model where bargaining ends faster under non-transparency. The

difference arises because in my model the Coasean force does not appear, and because bargainers

have stronger incentives to influence outsiders.10

My work also connects with the recent literature on price transparency and market performance

(Horner and Vieille, 2009; Kaya and Liu, 2015; Fuchs, Ory, and Skrzypacz, 2016). Those models

9A critical assumption in their model is that the outsiders must have private information about the bargaining.Otherwise, their perception about the bargaining skills does not depend on the transparency protocol.

10In Chaves’ model, when the seller has complete information, the bargainers behave without considering theentry of future buyers (outsiders). In my model, where bargainers have complete information, they still prefer todistort the bargaining outcome to influence the outsider decision.

6

consist of a long-run informed player facing a sequence of short-run uninformed players. In these

papers, the authors conclude that transparency is undesirable from a welfare point of view. The

driving force among these works is that the long-run player may want to delay trade to increase

his continuation payoff upon rejection inefficiently. My model shares the concern for transparency

but differs in that it studies a dynamic game where all players are long-lived. With this, I uncover

novel dynamics of posturing that arise when the negotiation is transparent.

Beyond the transparency agenda, my paper contributes to the bargaining literature in two

areas. First, to understand inefficient delay in bargaining without asymmetric information among

bargainers. With complete information, delay may occur when one of the parties can strike during

the course of the negotiation (Fernandez and Glazer, 1991), when bargainers make simultaneous

offers (Perry and Reny, 1993; Sakovics, 1993), and when bargainers are optimistic about their

bargaining power (Yildiz, 2004). My model adds to this literature by showing that delay may occur

because bargainers may want to influence a third party not involved in the negotiation. Second, the

paper contributes to the literature that rationalizes posturing behavior in negotiations. Following

the seminal work of Kreps and Wilson (1982) in reputation models, Abreu and Gul (2000); Kambe

(1999) show that posturing occurs when bargainers may be stubborn and demand a particular

allocation in the bargaining. Chatterjee and Samuelson (1987) shows that this behavior can happen

when both parties have incomplete information about their opponents’ valuation.

3 Model

The model has three players, two negotiators A and B, and an outsider O. Players A and B bargain

over which policy to implement and how to divide the gains generated from its implementation.

Player O has to make an irreversible decision z ∈ [0, 1].

The set of implementable policies depends on the state of the world Θ ∈ {Θ0,Θ1}. When the

state is Θ0, the only allocation that is feasible to implement is the constrained policy θ0. When the

state is Θ1, it is feasible to implement the unconstrained policy θ1 and also the constrained policy θ0.

Let π(θ) the gains from implementing allocation θ. I assume that π(θ0) ≤ π(θ1). Further, I consider

that Θ is known to Players A and B. Player O assesses a prior probability P[Θ = Θ1] = p.11

Time is discrete t ∈ {0,Λ, 2Λ, . . . , T, T + Λ, . . . , }, where T ≤ ∞ is the deadline for the negoti-

ation. 12 For a period t ≤ T the stage game is described in Figure 2. This is formally represented

11I ignore informational asymmetries among the negotiators to keep a tractable model that cleanly describesthe effect outsiders can have on negotiations.

12For notational simplicity I consider values for Λ and T such that TΛ ∈ N.

7

in three stages:

(t0) Player A proposes (θt, xAt ) ∈ {θ0, θ1} × [0, π(θt)].

(t1) Player O decides whether to take the irreversible action zt ∈ [0, 1] or to wait (w).

(t2) Player B decides whether to accept (a) xAt or to reject (r). If Θ = Θ0 and θt = θ1, Player B

has to reject the proposal.

A proposes ( "#$, &$') ∈ #*, #+ ×[0, /( "#$)] O waits (1) B rejects 2

O decides 3$ ∈ 0,1 ,and leaves

B accepts (5),negotiation ends

Figure 2: Stage game at period t ≤ T .

For t > T , Player O is the only one moving (with the same action space). In this case, the status

quo allocation is θt = ψ(Θ), where

ψ(Θ) =

θ0 if Θ = Θ0

θ1 if Θ = Θ1

.

Thus, if there is a breakdown of the negotiation, the outsider believes that in a future negotiation

the allocation θ1 will be implemented so long as it is feasible.13

I consider that proposals are observable for the outsider making the negotiation transparent.

I latter analyze the case when proposals are non-observable, that is, when the negotiation is non-

transparent.

Let τa = inf0≤t≤T

{Player B plays a at time t}, the time when the negotiation ends, and τz =

inft≥0{Player O plays z ∈ [0, 1] a time t}, the time where Player O takes the irreversible decision.14

The payoffs for each player are

uA = 1{τa<∞}e−rτaxAτa + 1{τz<∞}e

−rτzVA(zτz) where VA : [0, 1]→ R+

uB = 1{τa<∞}e−rτa(θt − xAτa) + 1{τz<∞}e

−rτzVB(zτz) where VB : [0, 1]→ R+,

uO = + 1{τz<∞}e−rτzZ(zτz , θt) where Z : [0, 1]× {θ0, θ1} → R++,

13The results remain unaltered if the outsiders believes that the allocation θ0 will be implemented in case ofbreakdown of negotiations.

14I use the convention inf ∅ =∞.

8

where r > 0 is the common discount rate.

I refer to the functions VA and VB as the externalities functions that Player O induces on each

respective negotiator.

Notice that the payoffs related to Player O’s decision are independent of the outcome of the

negotiation. Hence, if Player O were not strategic and played a fixed strategy, Players A and B

would behave in the negotiation as if no outsider was participating in the game. This non-linkage

in the payoff structure implies that any effect that the outsider could have on the negotiation is

due to informational considerations. Besides, these payoffs are nonnegative. Thus, players prefer

to receive their payoffs earlier in time. Consequently, if there is delay in the game, it comes from

the strategic considerations that the players are inducing on each other.

I assume that the outsider’s optimal action, z∗(µ) = arg maxz∈[0,1] Eµ[Z(z, θ)], is strictly mono-

tone in the belief µ = P[θ = θ1]. Given this monotonicity, without loss of generality, I consider

that z∗(µ) = µ.

Let Z(µ) := Eµ[Z(µ, θ)] the expected payoff for Player O when he takes the optimal action

given the belief µ that θ1 is implemented. I assume that Z is strictly convex, implying that Player

O is information-seeking. I restrict my attention to the case where Z(p) < e−rΛEµ[Z(1{θ=θ1}, θ)].

This assumption means that there is a value for learning: the outsider prefers to wait a period

and learn which allocation is implemented θ rather than taking an immediate action using only his

prior information. For technical reasons, I assume that Z is a continuously differentiable function.

In this paper, I focus on the case where negotiators have misaligned incentives regarding the

outsider’s decision. More precisely, I assume that VA is an increasing function, and VB is a de-

creasing function. Thus, Player A is better off when the outsider believes that the final allocation

is θ1. By contrast, Player B is better off when the outsider believes that the final allocation is θ0.

I define the size of the externalities by

∆VA = VA(1)− VA(0) and ∆VB = VB(0)− VB(1).

I assume that π(θ1) > max{∆VA,∆VB}. That is, the bargaining component of the game, in the

high state, is large enough in comparison with the externalities. This allows me to reduce the set

of equilibria and obtain sharper results. I further assume that VA, VB are differentiable functions.

9

Information structure, strategies, beliefs and solution concept

All actions are observable. Thus, a history htA ∈ H tA for Player A consists of the realization of Θ

and the actions that every player took before t0. A history htB ∈ H tB for Player B consists of the

realization of Θ and the actions that every player took before t2. A history htO ∈ H tO for Player O

consists of the actions that every Player took before t1.

A behavioral strategy for Player A is a sequence of functions σA = (σΘ,tA )t=0,...,T,Θ∈{Θ0,Θ1}, such

that σΘ,tA (htA) ∈ P

({(θ, x) ∈ {θ0, θ1} × [0, π(θt)]| x ≤ π(θ)

})for every htA ∈ H t

A that contains Θ.15

A behavioral strategy for Player B is a sequence of functions σB = (σΘ,tB )t=0,...,T,Θ∈{Θ0,Θ1}, such that

σΘ,tB (htB) ∈ P({a, r}) for every history htB ∈ H t

B that contains Θ. A behavioral strategy for Player

O is a sequence of functions σO = (σtO)t≥0, such that σtO(htO) ∈ P([0, 1] ∪ {w}).Given the information structure, a belief system for Player O is a probability distribution

µ = (µt)t≥0 over {Θ0,Θ1} such that P[Θ = Θ1|htO] is µt(htO) ∈ [0, 1].

For Player i ∈ {A,B,O}, I denote by uti(σ|µ, hti) the expected continuation payoff, at time

t, given hti and strategies σ = (σA, σB, σO) and the belief µ of Player O.16 An assessment (σ, µ)

is sequentially rational if for every Player i, any history hti, and any strategy σ′i, ui(σ|µ, hti) ≥ui(σ

′i, σ−i|µ, hti). An assessment (σ, µ) is a perfect Bayesian equilibrium (PBE) if it is sequentially

rational and beliefs follow Bayes’ rule wherever is possible (on and off the path of play).

Preliminaries

The assumption that Z > 0 means that Player O strictly prefers to take a particular action

earlier (to avoid discounting costs). Because after the negotiation deadline the information that

the outsider has about the state Θ does not change, he prefers to take the action immediately.

Therefore, in equilibrium, τı ≤ τa + Λ.17

The following result shows the role that private information plays in the model.

Lemma 1 Suppose that Player O knows the state Θ. Then, there is an equilibrium where Players

A and B strategies are the same as the equilibrium strategies in a game without an outsider. There

is immediate agreement in the negotiation (τa = 0). Player A extracts all the gains from the

negotiation (θτa = ψ(Θ), xAτa = π(θτa)). Furthermore, if T <∞ or Z(1) ≥ Z(0), the equilibrium is

unique.

15I denote by P(X), the space of probability measures given the Borel σ-algebra on X.16Because Players A and B know the state Θ, their belief system places probability one on Θ at every history

of the game.17In particular, τı ≤ T + Λ. For this reason, I describe the strategies and beliefs only until period T + Λ.

10

Lemma 1 follows from the non-linkage in payoffs between the outsider’s action and the pay-

offs from the negotiation. When Player O knows the state Θ, there is an equilibrium where he

immediately plays z = 1{Θ=Θ1}. The payoffs related to the Player O’s decision, VA, VB, Z, accrue

at that moment. Hence, Player A and B approach the negotiation without taking into account

the presence of the outsider. Therefore, in my model, where Player O does not know Θ, any dif-

ference in the outcome of the negotiation (e.g. allocation and efficiency) is due to informational

considerations.

Given the signaling nature of the game, the set of PBE is potentially large. To address this

issue, I develop equilibrium refinements that impose natural constraints on off-path beliefs. A

reader not interested in the technical details of these refinements may skip the next subsection.

3.1 Equilibrium Refinements

I impose two kinds of refinements. The first refinement is a version of forward induction. It

restricts the belief at an information set htO to assess positive probability only to the states of

the world that are relevant for that information set. That is, the states of the world that achieve

the information set for some sequentially rational strategies given an arbitrary belief at htO. This

refinement generalizes the intuitive criterion to a multi-period game with multiple senders. The

second refinement imposes a monotonicity condition on the beliefs after observing proposals.

Forward induction

To formally define the refinement, I first define a perturbed assessment at htO given an initial belief

system µ.

Definition 1 ((µ, µ, htO)-assessment) Given µ ∈ [0, 1], a belief system µ, and an information

set htO, a (µ, µ, htO)-assessment (σ(µ,htO), µ(µ,htO)) is an assessment where the strategies σ(µ,htO) are

sequentially rational to the belief system

µ(µ,htO)t (htO) =

µ if htO = htO

µt(htO) if htO 6= htO.

The (µ, µ, htO)-assessment contains sequentially rational strategies to beliefs that are a modifi-

cation to the original belief system µ at htO (where the belief is µ instead of µt(htO)).

11

Given an information set ht+ΛO = htO∪{y, (θ, x)}, where (y, (θ, x)) ∈ {a, r}×

({θ0, θ1}×[0, π(θ)]

),

and an assessment (σ, µ), I define the set

D(Θ, (y, (θ, x))|µ, htO) = {µ ∈ [0, 1]| there exists a (µ, µ, htO)-assessment such that

{y, (θ, x)} happens with positive probability given htO and that the state is Θ}

This set contains all the plausible beliefs that Player O could have at htO ∪ {y, (θ, x)} so that the

event (y, (θ, x)) happens with positive probability for strategies that are sequentially rational to

these plausible belief, given htO and that the state of the world is Θ.

Definition 2 (Relevant information set) Given a beliefs system µ, a state Θ is relevant to

ht+ΛO = htO ∪ {y, (θ, x)} if the set D(Θ, (y, (θ, x))|htO, µ) is not empty.

In words, a state Θ is relevant for htO if observing htO when the state is Θ can be explained by

(sequential) rational behavior of the players.

Definition 3 (Forward induction) A PBE satisfies forward induction if for every information

set htO, every element of the support of the belief µt(htO) is relevant for htO.

Govindan and Wilson (2009)[GW] present another version of forward induction, which differs

in the definition of the relevant information set. In GW, a state Θ is relevant for htO if a weak

sequential equilibrium exists such htO happens with positive probability when the state is Θ. 18 My

refinement only requires the existence of some belief at htO that can induce a sequential rational

behavior so that htO happens with positive probability when the state is Θ. This difference parallels

the difference between the solution concepts PBE and sequential equilibrium. The first one does

not impose a rational foundation to off-path beliefs. A sequential equilibrium requires that the

off-path beliefs have to be consistent (i.e., being a limit of a sequence of beliefs that are following

Bayes’ rule).

To see the usefulness of the refinement, consider a one-period version of the game. By construc-

tion, the proposal (θ1, π(θ1) −∆VB − ε) is accepted by Player B independently of the belief that

Player O could have after observing a rejection. In turn, Player A prefers to make such proposal

over proposal that will be later rejected. Thus, the state Θ1 is not relevant for history in which

there is rejection. Hence, if Player O observes that Player B rejects the proposal, the forward

induction refinement imposes that belief to be zero.

18Weak sequential equilibrium differs from a sequential equilibrium by requiring sequential rationality over thepath of the game instead of every information set (See GW for a detailed definition of the concepts).

12

Monotone Belief

Definition 4 (Monotone Belief) A PBE of the game satisfies the monotone belief refinement

if for every history htO, we have that

1. µt+Λ(htO ∪ {r, (θ, x)}) is non-decreasing in x ∈ [0, π(θ)].

2. µt+Λ(htO ∪ {r, (θ1, x)}) ≥ µt+Λ(htO ∪ {r, (θ0, x)}) for every x ∈ [0, π(θ0)].

Intuitively, if Player O observes a higher proposal, he should not think that gains from the negoti-

ations are smaller than when he observes a lower proposal. The refinement imposes a condition on

beliefs on-path and off-path. However, my results do not change if the monotonicity is restricted

only to off-path beliefs.

4 Posturing Equilibrium

In this section, I construct an equilibrium of the game that exhibits posturing. In the equilibrium,

Player A postures by proposing (θ1, π(θ1)) when the state is Θ0 even though the proposal is not

feasible. Similarly, Player B postures by rejecting (θ1, π(θ1)) when the state is Θ1 even though he

will later accept (θ1, π(θ1)) (after Player O takes the action). I then provide a simple characteriza-

tion of the equilibrium shape by looking at a sequence of games with period length Λ converging

to zero.

The equilibrium strategies have the following structure. Consider the subgame where at period

t, Player O has not taken the action yet. Then, when the state is Θ1, Player A proposes (θ1, π(θ1))

with probability one. When the state is Θ0, Player A mixes between proposing (θ1, π(θ1)), with

probability α∗t , and proposing (θ0, π(θ0)), with probability 1 − α∗t . When Player O observes the

proposal (θ1, π(θ1)), he mixes between taking the irreversible decision equals to the current belief

µ∗t , with probability β∗t , or waiting, with probability 1−β∗t . When Player O observes (θ0, π(θ0)), he

learns that the state is Θ0 and takes immediately the optimal decision z∗ = 0. If Player O waits at

t, Player B decides to reject if the proposal is (θ1, π(θ1)) and to accept if the proposal is (θ0, π(θ0)).

If Payer O decides to take the action at period t, the following subgame has a unique solution:

Player B accepts every feasible proposal, and Player A proposes (θ1, π(θ1)) when the state is Θ1

and (θ0, π(θ0)) when the state is Θ0.

A key aspect of these strategies is the mixing probabilities, α∗t and β∗t , which play a crucial role

in a potential delay in the bargaining. When there is an extended deadline T or when the prior p

13

is high, these mixing probabilities are degenerate, i.e., α∗0 = β∗0 = 1. Player A proposes (θ1, π(θ1))

with probability one, independently of the state of the world. Therefore, Player O does not learn by

delaying his decision. Thus, he immediately plays the uninformed action p. I call this equilibrium

a strong posturing equilibrium. On the other hand, when there is a short-term deadline, and the

prior is low, Player A and Player O mixed strategies are not degenerate, i.e., α∗t , β∗t ∈ (0, 1).

Both players engage in a war of attrition. With positive probability Player A compromises by

proposing (θ0, π(θ0)) when the state Θ0, and with positive probability Player O waits to obtain

more information from the subsequent rounds of the negotiation. I call this equilibrium a weak

posturing equilibrium.

To understand the connection between the posturing behavior, the deadline, and the prior,

consider the subgame when the negotiation reaches the deadline T and the outsider has not taken

the action yet.19 In this subgame, when the state is Θ0, Player A faces the following trade-off. By

posturing and proposing (θ1, π(θ1)), he makes an unfeasible proposal, which due to the deadline

limit, implies the destruction of the gains from the negotiations. By proposing (θ0, π(θ0)), he reveals

the non-preferable state Θ0 and gets a payoff of π(θ0) +VA(0). Let µA be the smallest belief µ such

that V (µ) ≥ π(θ0) + V (0).20 Then, when µT−Λ(hT−ΛO ) ≥ µA, Player A strictly prefers to propose

(θ1, π(θ1)) over (θ0, π(θ0)) when the state Θ0. Player O does not learn from the proposal and,

therefore, plays µT−Λ(hT−ΛO ) at T . In this situation, posturing is strong. When µT−Λ(hT−Λ

O ) < µA,

the value for posturing does not overweight the destruction of surplus. Hence, Player A mixes

so that these two forces are equalized. Thus, µT (hT−ΛO ∪ {(θ1, π(θ1))}) = µA. Therefore, when

µT−Λ(hT−ΛO ) < µA, posturing is weak. Given the equilibrium of the subgame, I deduce that when

there is a short-term deadline and the prior is small, Player O can obtain information about the

state Θ by waiting to period T . Thus, Player A must reveal information about the state of the

world. This fact explains why the weak posturing equilibrium depends on the deadline and prior.

To pin-down the equilibrium assessment, suppose that Player O is mixing along the equilibrium

path. Because Player O is sequentially rational, the optimal action at period t is to play the current

belief µt(htO), which I denote by µt. In order that Player O is indifferent between playing play µt,

19In the appendix I show that for this subgame, proposals are uniquely determined in every PBE satisfyingforward induction and monotonicity of beliefs.

20The monotonicity and continuity assumptions on VA ensures the existence and uniqueness of µA. Notice thatwhen π(θ0) < ∆VA, µA > 0.

14

or wait a period to learn from Player A’s proposals, the beliefs have to satisfy:Z(µt) = e−rΛ

( µtµt+Λ

Z(µt+Λ)︸ ︷︷ ︸A proposes (θ1,π(θ1)) at t+Λ

+ (1− µtµt+Λ

) Z(0)︸ ︷︷ ︸A proposes (θ0,π(θ0)) at t+Λ

)if t < T

µT = µA

(1)

Equation (1) could have multiple solutions; however, there is only one that satisfies continuity

as Λ→ 0.

Lemma 2 Consider t ≤ T < ∞. Suppose that (µ1s)Ts=t and (µ2

s)Ts=t solve Equation (1) from

s = t, . . . , T and satisfy the following condition

Z ′(µs)µs+Λ ≥ e−rΛ(Z(µs+Λ)− Z(0)) for s < T. (2)

Then, µ1s = µ2

s for every s = t, . . . , T . The solution (µs)Ts=t is increasing over time.

To induce the above sequence of beliefs, Player A proposes (θ1, π(θ1)), when the state is Θ0, with

probability αt = µt(1−µt+Λ)

µt+Λ(1−µt) . In particular, for t = 0, Player A mixes with probability α0 = p(1−µ0)µ0(1−p) .

From this, I observe that α0 < 1 if and only if p < µ0. Thus, if p < µ0 there is weak posturing.

If p > µ0, there is strong posturing along the equilibrium path. When there is strong posturing

and Player O deviates and wait, using Bayes’ rule, I get that the off-path belief equals to the prior

p. Therefore, the strong posturing strategy continues in the off-path game until the games reaches

period t, where µt−Λ < p < µt. For t ≥ t and in the subsequent periods, the off-path behavior

corresponds to the weak posturing behavior where beliefs follow Equation (1).

Given the beliefs (µt), I now formally define the strategies at every history of the game. For

this extent, I require the following notation.

• Let tE := inft≥0{a solution (µs)

Ts=t exists for Equation (1) satisfying Condition (2)}. Period tE

corresponds to the first time where a solution to Equation (1) exists. Notice that tE ≤ T and

tE =∞ when T =∞.

• Let t∗ = inft≥tE{µt ≤ p, where (µt)

Tt=tE

solve Equation (1) satisfying Condition (2)}. Period

t∗ corresponds to the first time where the sequence of beliefs is greater than p (because of

monotonicity µs > p for s > t∗).

15

• Consider the sequence of beliefs

µ∗t =

p if t < t∗

µt if t ≥ t∗.

This sequence generates the belief system of the equilibrium assessment.

• Let α∗t =µ∗t (1−µ∗t+Λ)

µ∗t+Λ(1−µ∗t ), the mixing probability for Player A that induces the sequence (µ∗t )

Tt=0.

• Let β∗t the mixing probability that keeps indifferent Player A between proposing θ1, π(θ1) or

(θ0, π(θ0)) when the state is Θ0. Thus,

π(θ0) + VA(0) = β∗t (e−rΛπ(θ0) + VA(µt))︸ ︷︷ ︸O plays µt after observing (θ1,π(θ1))

+ (1− β∗t ) e−rΛ(π(θ0) + VA(0))︸ ︷︷ ︸O waits and A plays (θ0,π(θ0)) at t+Λ

.

Solving the above equation, I obtain that

β∗t = (1− e−rΛ)π(θ0) + VA(0)

VA(µ∗t )− e−rΛVA(0)if t < T. (3)

When t = T , Player O does not learn by waiting. Hence, β∗T = 1.

• Let

uB(σ∗|Θ1, htB) =

T∑s=t+Λ

β∗s

s−1∏j=t+Λ

(1− β∗j )e−rΛ(s−t)VB(µ∗s),

Player B’s continuation payoff given that the state is Θ1, and that at history htB, Player A

proposes (θ1, π(θ1)) and Player O decides to wait.

• Let x∗t defined as

π(θ1)− x∗t + e−rΛVB(1) = uB(σ∗|Θ1, htB).

Thus, x∗t is the proposal that makes indifferent to Player B at period t between accepting,

an agreeing in policy θ1 or rejecting and continuing along the equilibrium path.

Equilibrium Assessment

The strategies for each player are as follows.

16

Strategy for Player A: consider htA a history such that there is not an agreement before t, and

Player O has not taken the action yet. If state is Θ1, P[σ∗t,Θ1

A (htA) = (θ1, π(θ1))] = 1. If the state is

Θ0, Player A mixes so that P[σ∗t,Θ0

A (htA) = (θ1, π(θ1))] = α∗t and P[σ∗t,Θ0

A (htA) = (θ0, π(θ0))] = 1−α∗t .Notice that α∗t = 1 if and only if t < t∗. Thus, if t < t∗ posturing is strong, and if t ≥ t∗ posturing is

weak. For a history htA where Player O has already taken the action, P[σ∗t,ΘA = (ψ(Θ), π(ψ(Θ))] = 1.

Strategy for Player B: consider a history htB such that there is not an agreement before t. If

Player O takes the action after t2, P [σ∗t,Θ1

B (htB) = a] = 1{xAt ≤x∗t } and P [σ∗t,Θ0

B (htB) = a] = 1{θt=θ0}.

If Player O takes the action before t2, P [σ∗t,ΘB (htB) = a] = 1{θt is feasible}.

Strategy for Player O: consider a history htO such that there is not an agreement before t,

θt = θ1 and xAt ≥ x∗t . If t < t∗ or t = T , P[σ∗tO(htO) = µ∗t ] = 1. If t∗ ≤ t and t < T , Player O

mixes so that P[σ∗tO(htO) = µ∗t ] = β∗t and P[σ∗tO(htO) = w] = 1 − β∗t . If Player O observes θt = θ0

or xAt < x∗t , then P[σ∗tO(htO) = 0] = 1. For a history htO where the negotiation ends before t,

P[σ∗tO(htO) = 1{θτa=θ1}] = 1.

The belief system µ∗ is as follows. For a history htO such there is not an agreement before

t,θt = θ1 and xAt ≥ x∗t , µ∗tO(htO) = µ∗t . For a history htO such there is not an agreement before

t and θt = θ0 or xAt < x∗t , µ∗tO(htO) = 0. For a history htO such there is an agreement before t,

µ∗tO(htO) = 1{θτa=θ1}.

Proposition 1 A δ ∈ (0, 1) exists such that if e−rT > δ or T = ∞, the assessment (σ∗, µ∗) is a

PBE of the game satisfying forward induction and monotonicity of beliefs. If t∗ = 0 and p < µ0

there is weak posturing along the equilibrium path. If not, there is strong posturing and there is an

immediate agreement in the negotiation.

Corollary 1 Suppose that e−rT > δ. Then, P[limΛ→0 e−rτΛ

0 Z(zτΛ0

) < pZ(1) + (1 − p)Z(0)] > 0.

Furthermore, if p < limΛ→0 µΛ0 , then P[limΛ→0 τ

Λa > 0] > 0.

Corollary 1 shows that when the negotiation is transparent an equilibrium exists where the

negotiation is inefficient. The outsider takes, with positive probability, a suboptimal decision.

Furthermore, if in the equilibrium there is weak posturing, there is delay in the bargaining which

is persistent as Λ → 0. Regarding the robustness of Corollary 1, Section 4.2 shows that, under

some conditions on the parameters, that these inefficiencies hold over a larger set of equilibria.

As a side remark, notice that in my model, there is no asymmetric information among the

negotiators and still, there is delay in the bargaining. Just the presence of a relevant outsider, may

affect the negotiators’ behavior with the consequence of destroying surplus of the bargaining.

17

To present a clearer expositions of these results, I analyze the limit of the sequence of the

posturing equilibria as Λ→ 0. The formal proofs can be found in the appendix.

4.1 Continuous time approach: Λ→ 0

I start the analysis by computing the beliefs along the equilibrium path. Consider the sequence

(µΛt ) that solves Equation (1) for Λ > 0. Because Z is continuously differentiable, the limit

µt = limΛ→0 µΛt , is differentiable.21 Thus, Equation (1) turns torZ(µt) = µt

µt(Z ′(µt)µt + Z(0)− Z(µt)) if t < T

µT = µA.(4)

The expression in the left-hand-side correspond the cost of waiting a period. The expression in

the right-hand-side measures the two benefits of waiting a period. The first term Z ′(µt)µt ≈(Z(µt + µtΛ) − Z(µt))/Λ is the marginal benefit of making a more accurate decision conditional

that Player O observes (θ1, π(θ1)). The second term is the benefit by playing 0 instead of µt when

the state is Θ0.

Because Z is strictly convex, Z ′(µt)µt + Z(0) − Z(µt) > 0 for µt > 0.22 Thus, the solution

trivially satisfy Condition (2). The convexity inequality and the smoothness of Z guarantee a

unique solution to Equation (4). Furthermore, because the solution is continuous we have that

tE := limΛ→0 tΛE is strictly less than T . From the Differential Equation (4) I obtain that if r or T

are sufficiently small, then tE = 0 and µ0 converges to µA. Therefore, when e−rT is small enough

the equilibrium exhibits strong posturing when p ≥ µA and weak posturing when p < µA.

Figures 3 and 4 depict the beliefs µt and µ∗t as a function of time for different discount rates.

When the discount rate is large, the slope of the beliefs are steeper. Hence, 0 < tE < t∗ =

limΛ→0 t∗,Λ and µ∗ differs from µ (see Figure 3). Because t∗ > 0, the limit equilibrium presents

strong posturing. Figure 4 analyzes the model when r is small. In this case, tE = t∗ = 0 and

µ∗ = µ. Since p < µ0, the limit equilibrium has weak posturing.

The limit-strategies of the players in the strong posturing region, when t < t∗, are the same

as in the discrete model. Player A proposes (θ1, π(θ1)) with probability one, Player O plays p

immediately, and Player B accepts every proposal after observing Player O taking the action. For

the weak posturing region, the limit-strategies take a simple form that I describe in the following

21The formal proof of the statement is in the appendix22A function f : [0, 1]→ R differentiable is strictly convex if for every x 6= y, f ′(x)(y − x) + f(x) < f(y).

18

Time

µA

p

1

t∗tE T0

µt

µ∗

t

Figure 3: Beliefs for the strong posturingequilibrium for r = 1.

Time

µA

p

1

0 T

}

← atom

µ0

µt = µ∗

t

Figure 4: Beliefs for the weak posturingequilibrium for r = 0.3.

Example for Z(µ) = 2− µ(1− µ) and T = 1.

section.

Weak posturing in continuous time: war of attrition

In the weak posturing equilibrium, Players A and O engage in continuous time war of attrition.

When t∗ = 0 and there is weak posturing along the equilibrium path, Player A places an atom at

t = 0 by playing (θ0, π(θ0)) with probability γ∗0− = 1 − limΛ→0 α∗,Λ0 = 1 − p(1−µ0)

µ0(1−p) (see Figure 4).

When t > 0, the war of attrition strategies of Players A and O converges to a continuous time

distribution. Let FΛA (t) the probability that Player A decides to concede by proposing (θ1, π(θ0))

before t when the state is Θ0. Then,

limΛ→0

1

Λ· F

ΛA (t+ Λ)− FΛ

A (t)

1− FΛA (t)

= limΛ→0

1− α∗,ΛtΛ

=µt

µt(1− µt)=

rZ(µt)

(1− µt) (Z ′(µt)µt + Z(0)− Z(µt)).

Thus, in the continuous time game, Player A proposes (θ0, π(θ0)) according to a continuous distri-

bution with hazard rate

γ∗t =rZ(µt)

(1− µt) (Z ′(µt)µt + Z(0)− Z(µt)).

Similarly, let FΛO (t) the probability that Player O decides to take the action before time t < T .

19

Then,

limΛ→0

1

Λ· F

ΛO (t+ Λ)− FΛ

O (t)

1− FΛO (t)

= limΛ→0

β∗,ΛtΛ

= rπ(θ0) + VA(0)

VA(µt)− V (0).

Thus, in the continuous time game, Player O takes the action at t < T , and playing µt, according

to a continuous distribution with hazard rate

η∗t = rπ(θ0) + VA(0)

VA(µt)− V (0). (5)

If the game reaches T , Player O places an atom at T and takes the action µA.

A common feature that appears in both equilibrium, strong and weak, is that the outsider

takes a suboptimal decision. Furthermore, the outsider’s payoff is bounded away from the payoff

he obtains with full information as Λ approaches zero. Indeed, by a simple computation, I deduce

that

limΛ→0

uO(σ∗,Λ|µ∗,Λ, h0O) = (p+ (1− p)(1− γ+

0−))Z(µ∗0) + (1− p)γ+0−Z(0) < pZ(1) + (1− p)Z(0),

where γ+0− = max{γ0− , 0}. In Section 4.2 I address this inefficiency result by looking at a larger set

of equilibria.

Tackling Proposition 1

Using the continuous-time approach, I address Proposition 1 by analyzing sequential rationality at

the critical histories. Thus, I restrict the attention to histories hti where there is not an agreement

of the negotiation yet, and Player O has not taken the action yet.

The conditions for Players B and O are simple to check. If Player B observes θt = θ0 and rejects,

then in the subgame, Player A never proposes less than (θ0, π(θ0)) afterwards. Thus, Player B’s

payoff by rejecting is less or equal than e−rΛVB(0). By accepting the proposal, he gets a payoff of

at least e−rΛVB(0). Hence, it is optimal for B to accept the proposal. If θt = θ1 and the state is Θ1

(if the state is Θ0, Player B is forced to reject the proposal), the division x∗t is set so that Player

B’s decision is sequentially rational.

Regarding Player O’s decision, notice that after observing xAt ≤ x∗t , his belief is that µt(htO) = 0.

Thus, playing 0 is sequentially rational. Suppose that he observes that θt = θ1, that xAt > x∗t and

that t < t∗. If Player O decides to deviate by waiting, Player B rejects, and in the next period,

Player A proposes (θ1, π(θ1)) with probability one and Player O plays µ∗t+Λ = µ∗t . Thus, under the

20

deviation, there is one period delay, and Player O takes the same action µ∗t . I conclude that Player

O prefers to take the action immediately. The other scenario is that θt = θ1, that xAt > x∗t and that

t ≥ t∗. In this situation, the belief (µ∗s)Ts=t satisfy Equation (4). Therefore, Player O is indifferent

between taking the action µ∗t or waiting. Thus, Player O is sequentially rational.

Regarding Player A’s decision, notice that if t < t∗, following the equilibrium strategies leads

to a payoff of π(ψ(Θ)) + VA(µ∗t ). He obtains the same payoff if he proposes θt = θ1 and xAt > x∗t .

However, if he proposes θt = θ0 or xAt ≤ x∗t his payoff is less or equal than π(ψ(Θ)) +VA(0). Hence,

Player A does not want to deviate. If the state is Θ0 and t ≥ t∗, Player A’s payoff by proposing

either θt = θ1 or xAt ≥ x∗t is π(θ0) + VA(0). If he proposes θt = θ0 or xAt < x∗t , his payoff less or

equal than min{xAt , π(θ0)}+VA(0). Therefore, Player A does not want deviate. Finally, it remains

the case when the state is Θ1 and t ≥ t∗. Suppose that Player A deviates by proposing θt = θ1

and xAt > x∗t , then the outcome is the same as the one by proposing (θ1, π(θ1)), with the caveat

that if Player O takes the action at t, he obtains xAt + VA(µ∗t ) instead of π(θ1) + VA(µ∗t ). Hence,

this deviation is not profitable. Suppose that Player A proposes either θt = θ0 or xAt ≤ x∗t . Then,

Player O immediately takes the action z∗ = 0. Thus, the optimal deviation is to propose θt = θ1

and x∗t . Therefore, a sufficient condition for the equilibrium to hold is that

limΛ→0

uA(σ∗,Λ|Θ1, htA) > x∗t + VA(0) .

where x∗t = limΛ→0 x∗,Λt .

Using the continuous time limit characterization of the strategies, I obtain

limΛ→0

uA(σ∗,Λ|Θ, htA) =

∫ T

t

e−(r(s−t)+∫ st ηwdw)ηs(π(θ1) + VA(µs))ds+ e−(r(T−t)+

∫ Tt ηwdw)(π(θ1) + VA(µA)),

limΛ→0

uB(σ∗,Λ|Θ1, htB) =

∫ T

t

e−(r(s−t)+∫ st ηwdw)ηsVB(µs)ds+ e−(r(T−t)+

∫ Tt ηwdw)VB(µA),

x∗t = H − limΛ→0

uB(σ∗,Λ|Θ1, htB) + VB(1),

where htB = hTA ∪ {(θ1, π(θ1)), w}.Thus, the equilibrium condition is equivalent to

∫ T

t

e−(r(s−t)+∫ st ηwdw)ηs(π(θ1) + VA(µs) + VB(µs))ds+ e−(r(T−t)+

∫ Tt ηwdw)(π(θ1) + VA(µA) + VB(µA)) > π(θ1) + VA(0) + VB(1).

The next lemma bounds the left-hand-side of the above inequality.

21

Lemma 3 For every t ≤ T ,the following inequality holds∫ T

t

e−(r(s−t)+∫ st ηwdw)ηs(π(θ1) + VA(µs) + VB(µs))ds+ e−(r(T−t)+

∫ Tt ηwdw)(π(θ1) + VA(µA) + VB(µA))

> (π(θ1) + VA(µA) + VB(µA))

(ηT

r + ηT+ e−(r+ηT )T r

r + ηT

).

Using Lemma 3, I deduce that a sufficient condition so that Player A does not want to deviate

is that

(π(θ1)+VA(µA)+VB(µA))

(π(θ0) + VA(0)

2L+ VA(0)+ e

−rT(

2π(θ0)+VA(0)

π(θ0)

)π(θ0)

2π(θ0) + VA(0)

)≥ π(θ1)+VA(0)+VB(1).

In particular, when e−rT → 1, the right-hand-side converges to π(θ1) +VA(µA) +VB(µA), which

is strictly greater than π(θ1) + VA(0) + VB(1). Therefore, a δ ∈ (0, 1) exists such that if e−rT > δ,

Player A is sequentially rational.

As a final remark, I can dispense the assumption that e−rT > δ if the following condition holds

(π(θ1) + VA(µA) + VB(µA))π(θ0) + VA(0)

2π(θ0) + VA(0)≥ π(θ1) + VA(0) + VB(1). (6)

Condition (6) is satisfied when ∆VA and ∆VB are high enough.23 For instance, consider VA(0) =

0, VB(1) = 0 and VB(µA) + VA(µA) > π(θ1)2

.

4.2 Inefficiency of transparent negotiation

In the previous section, I show that in the posturing equilibrium, there is inefficiency due to a

suboptimal decision from the outsider. I now generalize this result for a broader set of equilibria.

To be precise, under the assumption that e−rTVB(µA) > VB(1), I show that if p ≥ µA, then the

probability that the outsider takes an optimal action without delay is bounded away from one as

Λ approaches zero. Furthermore, strong posturing strategies are the unique proposal strategies

consistent with a PBE satisfying forward induction and monotonicty of beliefs.

Theorem 1 Suppose that e−rTVB(µA) > VB(1), p ≥ µA. A Λ > 0 exists such that if Λ < Λ and

(σ, µ) is a PBE satisfying forward induction and monotonicty of beliefs, then the equilibrium has

strong posturing behavior along the equilibrium path.

23Importantly, Condition (6) also ensures that x∗t < Hπ(θ1).

22

Theorem 1 states that when the length of the period is small, the equilibrium payoff that Player

O obtains is bounded away from the full information payoff. Thus, as Λ approaches zero, the

inefficiency related to Player O’s decision does not vanish. This inefficiency also affects the overall

welfare (including Players A and B) when the outcome of the full information game maximizes

total welfare.24 In such a case, Theorem 1 implies that the welfare-loss in equilibrium is persistent

as the length period Λ converges to zero.

To address the proof, consider the following lemma.

Lemma 4 Suppose that e−rTVB(µA) > VB(1), and consider (σ, µ) a monotone equilibrium satis-

fying forward induction.

Let htO any history such that: there is not an agreement of the negotiation before t < T ; Player

O has not taken the action before t; θt = θ1 and xAt = π(θ1). Then if µt(htO) ≥ µA, Λ > 0 exists

such that if Λ < Λ, the equilibrium is a strong posturing equilibrium. Along the equilibrium path,

Player A proposes (θ1, π(θ1)) independently of the state of the world, and Player O takes the action

immediately after observing (θ1, π(θ1))

To see this intuition behind this lemma, consider t = T . Because µT (hTO) ≥ µA , I show that

for this subgame there must be strong posturing along the equilibrium path.25 That is, Player A

proposes (θ1, π(θ1)) for both states and Player O takes the optimal decision at T . Moving a step

back in time, consider the history at T − Λ where Player A proposes (θ1, π(θ1)) and then Player

O waits. If Player B accepts, he obtains e−rΛVB(1), if he rejects, he obtains e−rΛVB(µT (hT−ΛO ∪

{r, (θ1, π(θ1))})). By Bayes’ rule, µT (hT−ΛO ∪ {r, (θ1, π(θ1))}) < 1. Hence, Player B rejects. Using

Bayes’ rule again, I conclude that µT (hT−ΛO ∪{r, (θ1, π(θ1))}) = µT−Λ(hT−2Λ

O ∪{r, (θ1, π(θ1))}) ≥ µA.

Therefore, after observing (θ1, π(θ1)) at T − Λ, Player O does not obtain information by delaying

his decision. Hence, he immediately takes the action at T − Λ. Using this logic until the initial

period, I conclude that if µT (hT−ΛO ∪ {r, (θ1, π(θ1))}) > µA then in any PBE satisfying forward

induction and monotonicity of beliefs, Player A proposes (θ1, π(θ1)) for both states and Player O

takes the action immediately.

For a general t < T , I proceed by backward induction. Suppose the proposition holds for t+ Λ.

Using the monotonicity of beliefs, the belief after observing a proposal (θ1, π(θ1)) is greater than

the belief after observing (θt, xAt ). Thus, the belief after observing (θ1, π(θ1)) is greater than µt(h

tO).

Because e−rTVB(µA) > VB(1), Player B prefers to reject (θ1, π(θ1)) and disagree until period T ,

24For example, when the externalities payoffs VA and VB are convex functions.25Lemma 7 in the appendix shows that every PBE satisfying forward induction and monotonicity of beliefs is a

posturing equilibrium. In particular, if µT (hTO) ≥ µA posturing is strong.

23

where gets at least VB(µA) rather accepting (θ1, π(θ1)) and obtaining VB(1). Hence, Player B

rejects when he observes (θ1, π(θ1)), independently of the state of the world. The Bayes restriction

over beliefs implies that the belief does not change if, in the next period, Player A proposes

(θ1, π(θ1)). Mathematically, µA = µt(htO) = µt+Λ(htO ∪ {r, (θ1, π(θ1))}). Because µt+Λ(ht+Λ

O ) ≥ µA

for hOt+Λ = htO ∪ {r, (θ1, π(θ1))}, the inductive hypothesis concludes the proof.

A consequence of Lemma 4 is that if the prior p ≥ µA, the equilibrium is unique (by Bayes’ rule

and the monotonicity of beliefs, µ0({(θ1, π(θ1))}) ≥ p. Therefore, I conclude from Lemma 4 that

when the length period Λ is small the outsider takes the uninformed decision with probability one.

5 Efficiency of non-transparent negotiation

The previous section shows that when the negotiation is transparent, that is, when proposals are

observable for the outsider, there is an inefficiency of the game which is persistent as the period

length goes to zero. This section shows that this result heavily relies on the information structure

chosen for the negotiation. In what follows, I consider a setting where the negotiation is non-

transparent: the outsider only observes whether there is an agreement or not, and the terms of

the agreement. Under a non-transparency policy, there is a separating equilibrium of the game,

which is approximately efficient as Λ converges to zero. I provide conditions that guarantee that

every equilibrium satisfies this property.

The non-transparent negotiation game is basically the same model as the transparent negoti-

ation game described in Section 3. The only difference is the information structure for Player O.

Here, a history htO for Player O consists of all the actions made by Player B prior to time t and

the outcome (θτa , xτa) in case τa < t. Regarding the action space for each player, their strategies,

the beliefs, and the solution concept, they remain invariant once is taking into account the new

information structure for Player O. Likewise, I naturally redefine the forward induction refinement

for this particular context.26 I keep the same assumptions made in Section 3.

Before going to the main results of the section, I develop a new refinement that ensures that

every equilibria of the game is approximately efficient.

26Given a belief system µ, the set D(θ, k|µ, htO) consists of all possible beliefs µ at k ∈ ({a, (θ, x)}, {r}) suchthat there are sequentially rational strategies to the perturbed belief system (µ, µ, htO) where k happens with posi-tive probability.

24

Refinement: Belief stable equilibrium

Definition 5 (Belief stable equilibrium) An assessment (σ, µ) is a belief stable equilibrium, if

(σ, µ) is a PBE such that for every history htO a sequence (µn)n≥0 exists, satisfying

1. That µn ∈ [0, 1] \ {µt(htO)} and µn → µt(htO).

2. For every player i ∈ {A,B,O}, we have that uni (hti) → ui(hti). Where uni (hti) is the expected

continuation payoff for player i at hti given the (µn, µ, htO)-assessment.

A belief stable equilibrium is an equilibrium where the payoffs that players obtain can be

approximated by the payoffs they would receive under some sequentially rational strategies to

some perturbed beliefs. Mathematically, it imposes that the correspondence from beliefs to all the

payoffs that are sequentially rational to those beliefs has to be upper-hemicontinuous.27

Separating Equilibrium

The first result provides the existence of a separating equilibrium. In a separating equilibrium,

Player A makes a different proposal for each state of the world. Player B immediately accepts the

respective proposal and, hence, he reveals the underlying state. Thus, Player O waits to observe

the agreement terms at the initial period, and takes the efficient action after time zero.28

Proposition 2 There exists a separating equilibrium satisfying forward induction and belief sta-

bility with the following properties:

I. When π(θ1)(1− e−rΛ) > e−rΛ(∆VB −∆VA), Player A proposes (θ1, H − e−rΛ∆VB) when the

state is Θ1, and proposes (θ0, π(θ0)) when the state is Θ0.

II. When π(θ1)(1−e−rΛ) ≤ e−rΛ(∆VB−∆VA), Player A proposes (θ1, π(θ1)−e−rΛ[e−r(T−Λ)VB(0)−VB(1)]+) when the state is Θ1 ,and proposes (θ0, π(θ0)) when the state is Θ0.29

The full characterization of the equilibrium assessment, and the proof of Proposition 2 are

delegated to the appendix. In what follows, I provide an intuition for the result.

27Kohlberg and Mertens (1986) develop a similar refinement, called, stable equilibrium. A stable equilibriumrequires that correspondence from games to equilibrium strategies must be upper-hemicontinuous.

28The assumption that there is a value for learning implies that Player O prefers to wait a period rather thantaking the action only using his prior information p.

29I use the notation [x]+ = max{x, 0}.

25

First, consider a history hTO where the negotiation reaches T , and Player O has not taken the

action yet. Suppose that the belief µT (hTO) is such that Player O has value for learning. Let

xB = π(θ1)−, H − e−rΛ∆VB. Then for any belief Player O could have after rejection we have that

Player B prefers to accept (θ1, xB − ε). Indeed,

uB(a|htB,Θ1) = π(θ1)− (xB− ε) + e−ΛVB(1) > π(θB)−xB + e−ΛVB(1) > e−ΛVB(0) ≥ uB(r|htB,Θ1),

Taking ε→ 0, I conclude that Player A can secure at least π(θ1)−xB+e−rΛVA(1) when the state is

Θ1. Thus, for any potential belief that Player O could have at hTO∪{r}, the negotiation never ends

in disagreement when the state is Θ1. In other words, Θ1 is not relevant to hTO ∪ {r}. Therefore,

the forward induction refinement imposes that, in case of rejection at T , Player O believes that the

state is Θ0. Consequently, Player A proposes (θ1, xB) when the state is Θ1, and proposes (θ0, π(θ0))

when the state is Θ0. Player B accepts every proposal. Thus, Player O prefers to wait a period

and then learn the state through the agreement terms. This reasoning shows that for the subgame

hTO, separating strategies are sequentially rational. As a final remark, notice this proof shows a

stronger result. In any equilibrium where Player O waits at T with probability one, the negotiation

ends with an agreement. Moreover, the agreement terms fully reveal the state of the world.

When t < T , a similar logic applies, however, it depends on his belief after observing a rejection.

For Scenario I, forward induction imposes that after observing a rejection at t, Player O believes

that the state is Θ0. Hence, this rejection-threat forces Player A to make an initial proposal of

(θ1, xB) when the state is Θ1. For Scenario II, however, this refinement does not longer apply. In

particular, I construct an equilibrium where the off-path belief after a rejection equals to the initial

prior p. Thus, in case of observing a rejection, Player O waits assuming that, in the next period,

there is an agreement in the negotiation which reveals the underlying state. As a consequence,

for Scenario II, the rejection-threat consists of reaching period T without an agreement. For

that reason, when the state is Θ1, Player A proposal is such that Player B is indifferent between

accepting or letting the negotiation to reach T .

Observe that, in this equilibrium, there is no delay in the negotiation, and the outsider takes

the optimal decision with one period delay. Thus, as the length period shrinks, the equilibrium

converges to the full information outcome. This result contrasts with the posturing equilibrium of

Section 4. There, the outsiders takes a suboptimal decision, moreover, if there is weak posturing

there is delay in the negotiation that is persistent when Λ → 0. Accordingly, when the full

information benchmark achieves the first best, the separating equilibrium converges to the efficient

outcome as Λ→ 0.

26

Nonetheless, the continuous-time limit of this separating equilibrium differs from the full in-

formation benchmark in one aspect: the pie distribution. When ∆VA ≥ ∆VB (Scenario I), or

∆VB > ∆VA and T is small (Scenario II), Player B obtains part of the gains from the negotiation

when the state is Θ1. This is because the gains from the negotiation in the high state are signif-

icant enough compare to the externalities. Thus, when the state is Θ1, negotiators always prefer

to agree rather than breaking the negotiation. Hence, a breakdown of negotiations should only

happen when the state is Θ0. This further generates a credible threat that Player B can pull when

the game is close to the deadline T . Thus, to avoid the surplus-destruction, Player A compensates

Player B by allocating to him a fraction of the gains from the negotiation. Moreover, the smaller

is the size of the externalities for Player B (∆VB), the lower is what Player A obtains from the

negotiation. The reason is that, in this case, the cost of delaying the negotiation is greater the

incentive cost that Player A needs to pay for an earlier agreement. Thus, observing a rejection

means that the state is Θ0. By the previous logic, Player B’s bargaining power increases since just

a delay of the negotiation triggers Player to believe that the state Θ0.

The following conditions guarantee that every equilibrium of the non-transparent negotiation

game is a separating equilibrium.

Assumption 1 Suppose that VA and VB satisfy the following properties:

1a. If VA(µ)− VA(0) ≥ π(θ0) then VB(µ)− VB(1) ≤ π(θ0).

1b. V ′A(0) < π(θ0)(Z(1)−Z(0)−Z′(0)

Z(0)

)and V ′B(1) < π(θ0)

(Z(0)−Z(1)−Z′(1)

Z(1)

).

Assumption 1a. means that the gains from the negotiations in the low state are significant in

comparison to the externalities VA and VB. That is if the belief µ is such that VA(µ)−VA(0) ≥ π(θ0)

the belief has to be sufficiently close to one so that VB(µ)− VB(1) ≤ π(θ0).

Assumption 1b. implies that for small Λ, each negotiator prefers to reveal the non-preferable

state and to obtain at least π(θ0) of the pie, rather than waiting a period to also obtain π(θ0),

and to induce a belief with a mild noise over the non-preferable state, so that Player O does not

have a value of learning. Thus, under Assumption 1b., the cost of delaying π(θ0) a period, which

is approximately rπ(θ0), is greater than the benefit generated by slightly changing the outsider’s

belief, which I show that is bounded by V ′i (ξθ)rZ(1−ξθ)−Z(ξθ)−Z′(ξθ)

Z′(ξθ), where ξθ = 1{θ=Θ1}.

Theorem 2 Suppose that Assumption 1 holds and that T < ∞. A Λ > 0 exists such that if

Λ < Λ, then every belief stable equilibrium satisfying forward induction is a separating equilibrium.

Furthermore, if π(θ0) < ∆VA, the on-path behavior is uniquely determined.

27

The following corollary is an immediate consequence of Theorem 2.

Corollary 2 Suppose that T <∞ and that Assumption 1 holds. Then, the outcome of every belief

stable equilibrium satisfying forward induction is approximately efficient as Λ→ 0.

Theorem 2 shows that along the equilibrium path, negotiators do not signal information about

the state through rejections. To see this, assume that if the negotiation reaches the deadline, the

unique equilibrium is separating – I address this point in the next paragraph. Consider now a

period previous to the deadline. If there is rejection along the equilibrium path, in the next period

T , Player O waits since he knows that negotiators are going to agree at T . Thus, a rejection at

T − Λ does not persuade Player O’s decision. Hence, it is optimal for negotiators to avoid an

unnecessarily delay by making an informative agreement at T − Λ. Therefore, the negotiators

play separating strategies at T − Λ as well. This logic generates an unraveling process, leading to

separating strategies at the initial period. Critical for this argument is that in case of a rejection,

Player O has a value of learning. Assumption 1b. ensures that, when the off-path beliefs is such

that Player O does not have a value of learning, there is no value for the negotiators to disagree

at that period.

The reason for the uniqueness of equilibrium for the subgame reaching period T is as follows.

Without loss of generality, suppose that Player O’s belief is such that he has value of learning. If

Player O waits with probability one, the same logic as for Proposition 2 implies that the equilibrium

has separating strategies. Hence, suppose for the sake of a contradiction that Player O plays at

T with positive probability. Because Player O is information-seeking, the posterior belief after

observing rejection at T must be noisy. Hence, Player B rejects with positive probability for

both states Θ0 and Θ1. Thus, Player A’s proposal when the state is Θ1 is rejected by Player

B. This implies that in equilibrium, Player A prefers to propose (θ1, π(θ1)) over the acceptable

proposal (θ1, π(θ1) − e−rΛ(VB(µT+Λ(hTO ∪ {r})) − VB(1)). This decision is optimal for Player A

if: (i) the probability that Player O waits at T is low; (ii) if the incentive cost for an agreement,

e−rΛ(VB(µT+Λ(hTO ∪{r}))− VB(1)), is high. However, conditions (i) and (ii) are impossible to hold

if Player B rejects when the state is Θ0. Using forward induction, I show that Player A has to

make an unfeasible proposal (θ1, x) when the state is Θ0. This behavior is optimal if: (iii) the

probability that Player O waits is high and if e−rΛVA(µT+Λ(hTO ∪{r})) > π(θ0) + e−rΛVA(0). Using

Assumption 1a. the last inequality turns implies that (iv) e−rΛ(VB(hTO ∪ {r}) − VB(1)) ≤ π(θ0).

Thus, Player A has a low cost to pay for an agreement when the state is Θ1. In the appendix, I

formally show that conditions violates (i) and (ii) are impossible to hold with conditions (iii) and

(iv). This shows that the subgame at T has a unique separating equilibrium.

28

6 Concluding remarks

There are several extensions of my framework that have not been addressed. First, how the

results would change if the asymmetric information among the negotiators is important. When

the negotiation is non-transparent, it could be that it is too costly for the outsider to wait until

negotiators reach an agreement for taking the action. Hence, transparency could be preferable

since it reveals some information to the outsiders earlier on time.

Another issue to be concerned about is the robustness of my results to the bargaining protocol.

In the model, proposals and offers are not a synonym. Similarly, to propose a proportion of

the surplus is different than proposing a particular allocation. In those cases, the incentives for

(strong) posturing are even higher when the negotiation is transparent. An equilibrium exists

where Player A obtains all the surplus of the negotiation, and the outsider takes the uninformed

action immediately. In turn, when the negotiation is non-transparent, an (approximately) efficient

equilibrium can be sustained by threating Player A that if there is no immediate agreement, Player

O will play z = 0.

Third, I assume throughout the paper that negotiators can only communicate their private

information to the outsider through the negotiation. If other communication channels are costless

(cheap talk), the same logic of our results will imply that the outsider will not obtain more infor-

mation. However, if negotiators somehow can “burn” money to send some signal, it may be the

case that a separating equilibrium could be sustained in equilibrium.

Beyond these extensions, it is important to notice that the conclusions of the paper do not

contradict why many negotiations are transparent. My results only claim what would be more

efficient, but does not provide a theory of how bargainers decide what transparency policy to

implement in the negotiation. Furthermore, my theory does not consider other aspects of the

negotiations that may be relevant as well. For instance, when negotiators represent multiple

constituents, transparency could be important since it provides legitimacy to the outcome of the

negotiation. Furthermore, it may also help to avoid moral hazard conflicts between the negotiators

and their constituents.

References

Abreu, D. and F. Gul (2000): “Bargaining and reputation,” Econometrica, 68, 85–117.

Chatterjee, K. and L. Samuelson (1987): “Bargaining with two-sided incomplete informa-

29

tion: An infinite horizon model with alternating offers,” The Review of Economic Studies, 54,

175–192.

Chaves, I. (2019): “Privacy in Bargaining: The Case of Endogenous Entry,” working paper,

Northwestern.

Drake-Brockman, J. and P. A. Messerlin (2018): Potential benefits of an Australia-EU free

trade agreement: Key issues and options, Adelaide: University of Adelaide Press.

Fernandez, R. and J. Glazer (1991): “Striking for a Bargain Between Two Completely

Informed Agents,” The American Economic Review, 240–252.

Fingleton, J. and M. Raith (2005): “Career concerns of bargainers,” Journal of Law, Eco-

nomics, and organization, 21, 179–204.

Fuchs, W., A. Ory, and A. Skrzypacz (2016): “Transparency and distressed sales under

asymmetric information,” Theoretical Economics, 11, 1103–1144.

Fudenberg, D. and J. Tirole (1991): Game Theory, Cambridge, MA: MIT Press.

Govindan, S. and R. Wilson (2009): “On forward induction,” Econometrica, 77, 1–28.

Horner, J. and N. Vieille (2009): “Public vs. private offers in the market for lemons,” Econo-

metrica, 77, 29–69.

Kambe, S. (1999): “Bargaining with imperfect commitment,” Games and Economic Behavior,

28, 217–237.

Kaya, A. and Q. Liu (2015): “Transparency and price formation,” Theoretical Economics, 10,

341–383.

Kohlberg, E. and J.-F. Mertens (1986): “On the strategic stability of equilibria,” Economet-

rica: Journal of the Econometric Society, 1003–1037.

Kreps, D. M., P. Milgrom, J. Roberts, and R. Wilson (1982): “Rational cooperation in

the finitely repeated prisoners’ dilemma,” Journal of Economic theory, 27, 245–252.

Kreps, D. M. and R. Wilson (1982): “Reputation and imperfect information,” Journal of

economic theory, 27, 253–279.

30

Marceddu, M. L. (2018): “Implementing Transparency and Public Participation in FTA Nego-

tiations: Are the Times a-Changin?” Journal of International Economic Law, 21, 681–702.

Milgrom, P. and J. Roberts (1982): “Predation, reputation, and entry deterrence,” Journal

of economic theory, 27, 280–312.

Perry, M. and P. J. Reny (1993): “A non-cooperative bargaining model with strategically

timed offers,” Journal of Economic Theory, 59, 50–77.

Perry, M. and L. Samuelson (1994): “Open-versus closed-door negotiations,” The RAND

Journal of Economics, 348–359.

Sakovics, J. (1993): “Delay in bargaining games with complete information,” Journal of Eco-

nomic Theory, 59, 78–95.

Stasavage, D. (2004): “Open-door or closed-door? Transparency in domestic and international

bargaining,” International organization, 58, 667–703.

Yildiz, M. (2004): “Waiting to persuade,” The Quarterly Journal of Economics, 119, 223–248.

A Appendix

Proof of Lemma 1.

The proof that an equilibrium exists such that: A proposes (θ, π(θ)), O playes z = 1{Θ=Θ1}

and B accepts, is straightforward from the standard one-sided proposal bargaining game and is

therefore omitted (See Section 4.4. in Fudenberg and Tirole (1991)). In what follows, I show the

uniqueness of the equilibrium.

Notice that when the state is Θ = Θ0, O immediately proposes z = 0. Because the external-

ities payments VA(0) and VB(0) accrue immediately, A and B face a standard one-sided proposal

bargaining game. Hence, when Θ = Θ0, negotiators behave is if there is not outsider present.

When Θ = Θ1 and T < ∞. The proof is by backward induction. For the subgame consider

the last period T , where the negotiators have not agreed prior to T . If the outsider has taken

the action. Clearly, A proposes (θ1, π(θ1)) and B accepts. If O has not taken the action, A can

guarantee at least π(θ1)− ε+VA(1). Indeed, if he proposes (θ1, π(θ1)− ε), it is optimal for O plays

z = 1 (he knows that for t > T , θt = θ1). Therefore, B prefers to accept and obtain ε + VB(1)

31

rather than rejecting and obtaining VB(1). Because this result hold for every ε, I conclude that A

proposes (θt, π(θt)), O plays z = 1, and B accepts. Using the same reasoning for T − Λ, . . . , 0, I

conclude that A always proposes (θt, π(θt)), O does not wait and plays z = 1, and B accepts.

For the case when Θ = Θ1, T = ∞, and Z(1) ≥ Z(0), I claim that can A guarantee at least

π(θ1) − ε + VA(1). Suppose A proposes (θ1, π(θ1) − ε). Then if O plays z = 1, VA(1) and VB(1)

are sunk, and therefore, B prefers to obtain ε of the bargaining pie rather than rejecting, and

given subgame perfection, obtain 0 from the pie. Because, Z is convex an Z(1) ≥ Z(0), O strictly

prefers to play z = 1 rather wait (or playing z < 1) and obtaining a payoff which is bounded by

e−rΛ maxµ ∈ [0, 1]Z(µ) = e−rΛZ(1). Therefore, A can get at least a payoff of π(θ1)− ε+ VA(1) for

every ε > 0. I conclude that A proposes (θt, π(θt)), O plays z = 1, and B accepts.

A.1 Proofs of Section 4

Proof of Lemma 2. At period T , µ1T = µ2

T = µA. Let

f(x|µs+Λ) = Z(x)− e−rΛ(

x

µs+Λ

(Z(µs+Λ)− Z(0)) + Z(0)

).

Notice that f ′′(x|µs+Λ) = Z ′′(x). Thus, f is strictly convex. The function f satisfies f(µs+Λ|µs+Λ) =

Z(µs+Λ)(1− e−rΛ) > 0 and

f ′(µs+Λ|µs+Λ) = Z ′(µs+Λ)µs+Λ − e−rΛ (Z(µs+Λ)− Z(0)) > (1− e−rΛ) (Z(µs+Λ)− Z(0)) > 0,

where the first inequality comes from the strict convexity of Z. Therefore, f(x|µs+Λ) > 0 for

x ≥ µs+Λ. This implies that any solution to Equation (1) is monotone.

Regarding uniqueness of the solution, since f(0|µs+Λ) = Z(0)(1 − e−rΛ) > 0, f(x|µs+Λ) can

cross zero at most twice. In particular, there is unique solution such that f ′(x|µs+Λ) ≥ 0. That is,

given µs+Λ, there is a unique solution µt solving Equation (1) and Condition (2). Because µT = µA,

by a simple backward induction argument I deduce the uniqueness result.

Proof of Proposition 1.

Simple inspection shows that beliefs are monotone. Also, by construction of the beliefs, they

follow Bayes’rule wherever is possible. To show sequentially rationality, notice that for every history

such that either there is already an agreement in the negotiation, or Player O has already taken

the action, the strategies are (trivially) the unique subgame PBE. Thus, in what follows I consider

a period t such that the there is not an agreement before period and Player O has not taken the

32

action yet. I use the one-shot deviation principle to prove sequential rationality.

For these histories notice that the equilibrium has the following memoryless property: the

decision at period t does not depend on the history before period t. I now proceed to show

sequential rationality for each player.

Regarding Player A, consider the case where t < t∗. If he proposes xtA < x∗t , he gets a payoff

of min{π(θ), xAt } + VA(0). If he proposes xAt ≥ x∗t he gets a payoff of min{π(θ), xAt } + VA(µ∗t ).

Thus, the optimal decision is to propose (θ1, π(θ1). For the case t ≥ t∗, the only relevant deviations

are θt = θ0 or xAt < x∗t (for θt = θ1 and xAt ≥ x∗t , the behavior of every player is as (θ1, π(θ1))).

When the state is Θ0, Player A is indifferent between proposing (θ1, π(θ1)) or proposing (θ0, π(θ0)),

and getting π(θ0) + VA(0). By proposing θt = θ0 or xAt ∈ [0, x∗t ) his payoff is less or equal than

max{e−rΛπ(θ0),min{π(θ0), xAt }}+ VA(0). Therefore, Player A prefers to propose (θ0, π(θ0)) which

is the same payoff as proposing (θ1, π(θ1)). Finally, consider the case when the state is Θ1. If Player

proposes θt = θ0 or xAt < x∗t , he obtains a payoff bounded by xAt +e−rΛVA(0). The equilibrium payoff

uA(σ|Θ1, htO) can be bounded by e−rT (π(θ1) + V (p)). This is because µ∗t ≥ p and τz ≤ T (a.s.).

Let δ such that δ(π(θ1) + V (p)) > xAt + VA(0). Then if e−rT > δ, then uA(σ|Θ1, htO) > xAt + VA(0).

Therefore, Player A is sequentially rational.

Regarding Player B if he observes (θ0, xAt ) and rejects, given (σ∗, µ∗), his payoff by rejecting is

less or equal than e−rΛVB(0). By accepting the proposal, he gets a payoff of at least e−rΛVB(0).

Thus, is optimal for B to accept the proposal. If the state is Θ1, by construction of x∗t Player

B decision is sequentially rational. Finally, to ensure that x∗t is well-defined, I take δ so that

uB(htB|Θ1) > e−rΛVB(1).

Regarding Player O, if he observes θt = θ0 or xAt < x∗t , µt(htO) = 0. Thus, playing 0 is

sequentially rational. Suppose that θt = θ1 and xAt ≥ x∗t , t < t∗,and that Player O deviates and

decides to wait (clearly, a deviation to make a different action is not profitable). In such situation,

Player B always rejects. If t + Λ < t∗, Player A plays (θ1, π(θ1)) and Player O takes the same

action at t + Λ than the optimal action at t. Because he does not learn at all by waiting, this

deviation is not profitable. If t + Λ ≥ t∗, in that case Player A mixes when the state is Θ0. If

Player O would have a profitable deviation then

Z(µ∗t ) < e−rΛ(

µ∗tµ∗t+Λ

Z(µ∗t+Λ) +

(1− µ∗t

µ∗t+Λ

)Z(0)

). (7)

If t∗ = tE, then f(x|µ∗t+Λ), defined as in the proof of Lemma 2, is strictly greater than zero (in th

lemma I show that f is convex, with f(0|µ∗t ) > 0 and f(µ∗t+Λ|µ∗t+Λ) > 0). Therefore, f(µ∗t |µ∗t+Λ) > 0,

33

this contradicts Inequality (7). If tE < t∗, then µt < µ∗t . Because (µt) satisfies Condition (2), and

µ∗t+Λ = µt+Λ, from the proof of Lemma 2 f ′(µt|µ∗t+Λ) > 0. Because f is convex, f ′(x|µ∗t+Λ) > 0 for

every x > µt. Because µ∗t > µt, I conclude that f(µ∗t |µ∗t+Λ) > 0, this contradicts Inequality (7).

Therefore, Player O does not have a profitable deviation when xAt > x∗t and t ≤ t∗. If xAt ≥ x∗t

and t ≥ t∗, then the belief (µ∗s)Ts=t satisfy Equation (1). Therefore, Player O is indifferent between

taking the action µ∗t or waiting. Thus, Player O is sequentially rational.

Proof of Lemma 3.

Because beliefs are increasing over time, I obtain that ηw = r π(θ0)+VA(0)VA(µs)−V (0)

> ηT for every w ≤ T .

Consider two distributions X and Y over [t, T ], where P[X > s] = e−(r(s−t)+∫ st ηwdw) and P[Y >

s] = e−(r+ηT )(s−t) for s ∈ [t, T ). Because ηw > ηT , I obtain that P[Y ≥ s] > P[X ≥ s] for every

s ≥ T . Hence, Y first order stochastic dominates X. Consider the function

g(s) =ηs

r + ηs(π(θ1) + VA(µs) + VB(µs)).

I assert that g is non-increasing. Indeed, since g(s) = π(θ0)+VA(0)π(θ0)+VA(µs)

(π(θ1) + VA(µs) + VB(µs)), I can

rewrite g as composition g(s) = h(µs), where h(µ) = π(θ0)+VA(0)π(θ0)+VA(µ)

(π(θ1) + VA(µ) + VB(µ)). Notice

that h is decreasing since

h′(µ) = (π(θ0) + VA(0))V ′B(µ)(π(θ0) + VA(µ))− V ′A(µ)(π(θ1) + VB(µ))

(π(θ0) + VA(µ))2< 0.

Using that µs is increasing over time, I conclude that g is a non-increasing function.

Because Y first order stochastic dominates X and g is non-increasing, I conclude that EX [g] ≥EY [g]. The above expectations are equivalent to∫ T

t

e−(r(s−t)+∫ st ηwdw) ηs(π(θ1) + VA(µs) + VB(µs))︸ ︷︷ ︸

(r+ηs)g(µs)

ds+ e−(r(T−t)+∫ Tt ηwdw) ηT

r + ηT(π(θ1) + VA(µA) + VB(µA))

≥∫ T

t

e−(r+ηT )(s−t)(r + ηT )g(µs)ds+ e−(r+ηT )(T−t) ηTr + ηT

(π(θ1) + VA(µA) + VB(µA)).

34

Therefore,∫ T

t

e−(r(s−t)+∫ st ηwdw)ηs(π(θ1) + VA(µs) + VB(µs)) + e−(r(T−t)+

∫ Tt ηwdw)(π(θ1) + VA(µA) + VB(µA))

=∫ T

t

e−(r(s−t)+∫ st ηwdw)ηs(π(θ1) + VA(µs) + VB(µs)) +

ηTr + ηT

e−(r(T−t)+∫ Tt ηwdw)(π(θ1) + VA(µA) + VB(µA)) +

r

r + ηTe−(r(T−t)+

∫ Tt ηwdw)(π(θ1) + VA(µA) + VB(µA))

≥∫ T

t

e−(r+ηT )(s−t)(r + ηT )g(µs)ds+ηT

r + ηTe−(r+ηT )(T−t)(π(θ1) + VA(µA) + VB(µA)) +

r

r + ηTe−(r(T−t)+

∫ Tt ηwdw)(π(θ1) + VA(µA) + VB(µA))

≥∫ T

t

e−(r+ηT )(s−t)(r + ηT )g(µT )ds︸ ︷︷ ︸g(µT )(1−e−(r+ηT )(T−t))

+ηT

r + ηTe−(r+ηT )(T−t)(π(θ1) + VA(µA) + VB(µA)) +

r

r + ηTe−(r(T−t)+

∫ Tt ηwdw)(π(θ1) + VA(µA) + VB(µA))

= (π(θ1) + VA(µA) + VB(µA))

(ηT

r + ηT+ e−(r+ηT )T r

r + ηT

).

The first inequality comes from EX [g] ≥ EY [g]. The second inequality comes from the fact that

g is decreasing for s ≤ T .

A.1.1 The continuous time limit

The following lemma shows that continuous time limit is well-defined.

Lemma 5 Let (µΛt )Tt=tΛE

the solution to Equation (1). Then, tE = limΛ→0 tΛE, t∗E exists. Also,

µt = limΛ→0 µΛtΛ

, where tΛ is any period in the game indexed by Λ that satisfies tΛ → t, exists.

Furthermore, µt is differentiable for t ∈ (tE, T ).

Proof. Consider the function f(x|µs+Λ,Λ) defined as in the proof of Lemma 2. Notice that

d

dΛf(x|µs+Λ,Λ) = re−rΛ

(x

µs+Λ

(Z(µs+Λ)− Z(0)) + Z(0)

)> 0,

d

dµs+Λ

f(x|µs+Λ,Λ) =−e−rΛ

µ2s+Λ

(Z ′(µs+Λ)µs+Λ − (Z(µs+Λ)− Z(0))) > 0.

where the second inequality holds due to the strict convexity of Z. A simple induction argument

shows that f(x|µΛ′

s+Λ′ ,Λ′) < f(x|µΛ

s+Λ,Λ) for every x and Λ′ < Λ. Therefore, µΛ′s > µΛ and tΛ

′E < tΛE

for Λ′ < Λ and s ≥ tΛE. Therefore, µs = limΛ→0 µΛs and tE = limΛ→0 t

ΛE are well-defined. I assert

that for µs > 0, µs is differentiable.

By definition µΛt , I obtain that

Z(µΛt+Λ)− Z(µΛ

t ) = (1− e−rΛ)Z(µΛt+Λ) + e−rΛ

(1− µΛ

t

µΛt+Λ

)(Z(µΛ

t+Λ)− Z(0)). (8)

35

From the convexity of Z, I get that Z(µΛt+Λ) − Z(µΛ

t ) ≥ Z ′(µΛt )(µΛ

t+Λ − µΛt ). Plugging this into

Equation (8), I derive that

(µΛt+Λ − µΛ

t )(Z ′(µΛ

t )µΛt+Λ − e−rΛ(Z(µΛ

t+Λ)− Z(0)))≤ (1− e−rΛ)Z(µΛ

t+Λ).

From Condition 2, I obtain that

|µΛt+Λ − µΛ

t | = µΛt+Λ − µΛ

t ≤ (1− e−rΛ)µΛt+ ΛZ(µΛ

t+Λ)

Z ′(µΛt )µΛ

t+Λ − e−rΛ(Z(µΛt+Λ)− Z(0))

. (9)

Using (again) the convexity of Z, I get that Z(µΛt+Λ)−Z(µΛ

t ) ≤ Z ′(µΛt+Λ)(µΛ

t+Λ−µΛt ). Plugging

this inequality into Equation (8), I derive that

(µΛt+Λ − µΛ

t )(Z ′(µΛ

t+Λ)µΛt+Λ − e−rΛ(Z(µΛ

t+Λ)− Z(0)))≥ (1− e−rΛ)Z(µΛ

t+Λ).

Because Z is convex Z ′(µΛt+Λ) > Z ′(µΛ

t ). Hence, from Condition 2, I obtain that

Z ′(µΛt+Λ)µΛ

t+Λ − e−rΛ(Z(µΛt+Λ)− Z(0)) > 0.

Thus,

|µΛt+Λ − µΛ

t | = µΛt+Λ − µΛ

t ≥ (1− e−rΛ)µΛt+ ΛZ(µΛ

t+Λ)

Z ′(µΛt+Λ)µΛ

t+Λ − e−rΛ(Z(µΛt+Λ)− Z(0))

. (10)

Let > Tt > tE such that µt > 0, then Z ′(µt)µt + Z(0) − Z(µt) > 0 and Z ′(µΛt+Λ)µΛ

t+Λ −e−rΛ(Z(µΛ

t+Λ) − Z(0)) > ξ for Λ small enough and ξ > 0. Using Inequality 9 and Inequality 10, I

obtain that µt is differentiable with

µt =µtZ(µt)

Z ′(µt)µt + Z(0)− Z(µt).

Furthermore because Z is strictly convex, Z ′(µt)µt + Z(0) − Z(µt) is bounded from zero for

every t > tE. Thus, using the Picard-Lindelof Theorem, I conclude that (µt) is the unique solution

of the o.d.e (4) for t ∈ [tE + ε, T ]. Because the result holds for every ε > 0, I conclude the proof

by taking ε→ 0.

36

A.1.2 Uniqueness of the subgame at period T

Consider the subgame at period T , where the negotiation has not reached an agreement yet and

Player O has not taken the action yet. I show that for this subgame, there is a unique monotone

equilibrium that satisfy forward induction.

The existence is shown in the proof of Proof of Proposition 1. Therefore, it remains to show

uniqueness.

Lemma 6 In any monotone equilibrium (σ, µ) we have that P[σTO(hT−ΛO ∪ {(θ1, π(θ1))}) = w] = 0.

Proof of Lemma 6. I assert that from Player O’s perspective, waiting does not provide any extra

information. If µT (hT−ΛO ∪ {(θ1, π(θ1))}) is degenerate, Player O takes the action immediately. If

µT (hT−ΛO ∪ {(θ1, π(θ1))}) ∈ (0, 1) from by Bayes’ rule I obtain that µT+Λ(hT−Λ

O ∪ {(θ1, π(θ1)), r}) <1.30 From the sequential rationality of Player B, Player B rejects for both states Θ0 and Θ1. Hence,

µT+Λ(hT−ΛO ∪ {(θ1, π(θ1)), r}) = µT (hT−Λ

O ∪ {(θ1, π(θ1))}). Thus,

uO(w|hT−ΛO ∪ {(θ1, π(θ1))}) = e−rΛZ(µT+Λ(hT−Λ

O ∪ {(θ1, π(θ1)), r}))

< Z(µT (hT−ΛO ∪ {(θ1, π(θ1))}))

= uO(µT (hT−ΛO ∪ {(θ1, π(θ1))})|hT−Λ

O ∪ {(θ1, π(θ1))}).

Therefore, Player O never waits after observing proposal (θ1, π(θ1)).

Lemma 7 Consider a monotone equilibrium (σ, µ) satisfying forward induction. Then, P[σT,Θ1

A (hTA) =

(θ1, π(θ1))] = 1.

Proof.

Consider a monotone equilibrium (σ, µ) and suppose, for the sake of a contradiction, that

(θ, x) ∈ supp(σT,Θ1

A (hTA)) exists such that θ = θ0 or x < π(θ1). Consider the case µT (hT−ΛO ∪

{x}) = 1. Because beliefs are monotone, for ε > 0 such that π(θ1) − ε > π(θ0), the beliefs are

µT (hT−ΛO ∪{π(θ1)−ε}) = 1. If A proposes π(θ1)−ε when the state is Θ1, we have that O plays z = 1

at period T , and B accepts the proposals. Thus, uA((θ1, π(θ1)− ε)|Θ1, hTA) = π(θ1)− ε+ VA(1) >

x+ VA(1) ≥ uA((θ, x)|Θ1, hTA). This is a contradiction.

Consider the case µT (hT−ΛO ∪ {(θ, x)}) < 1, from Bayes’ rule I conclude that with positive

probability Player A plays (θ, x) when the state is Θ0. This further implies that is sequentially

rational for Player A to play (θ, x) when the state is Θ0.

30With positive probability the state is Θ0, and therefore, Player B rejects.

37

If θ = θ1, notice that uA((θ, x)|Θ0, hTA) ≥ VA(µT (hT−Λ

O ∪{(θ, x)})). Furthermore, from Lemma 6,

uA((θ1, π(θ1))|Θ0, hTA) = VA(µT (hT−Λ

O ∪ {(θ1, π(θ1))})) ≥ VA(µT (hT−ΛO ∪ {(θ, x)})). Where the

inequality comes from the monotonicity of beliefs. The optimality condition requires that µT (hT−ΛO ∪

{(θ1, π(θ1))}) = µT (hT−ΛO ∪ {θ, x}) and that uA((θ, x)|Θ0, h

TA) = VA(µT (hT−Λ

O ∪ {(θ, x)})). Hence,

Player O plays at T with probability one after observing (θ, x). Thus, uA((θ, x)|Θ1, hTA) = x +

VA(µT (hT−ΛO ∪ {(θ, x)})). Consider the proposal (θ1, π(θ1) − ε), I claim that O does not wait. If

he waits, B always rejects.31 Therefore, uA((θ1, π(θ1) − ε)|Θ1, hTA) = π(θ1) − ε + VA(µT (hT−Λ

O ∪{(θ, x)})) > uA((θ, x)|Θ1, h

TA). This is a contradiction.

If θ = θ0, B always accepts when the state is Θ1. He obtains at least VB(0) while he obtains

VB(µT (hT−ΛO ∪ {(θ, x)} ∪ {r}) < VB(0). Hence, uA((θ, x)|Θ1, h

TAleπ(θ0) + VA(0). Suppose that

uA((θ, x)|Θ0, hTA) ≥ VA(1). Then, by Forward Induction, (or simply the intuitive criterion) I

obtain that µT (hT−ΛO ∪ {(θ1, x)}) = 1 for every x ∈ [0, π(θ1)]. Henceforth, Player A prefers to

propose (θ1, π(θ1)− ε), getting π(θ1)− ε+ VA(1) rather than proposing (θ, x), and getting at most

π(θ0) + VA(0). If uA((θ, x)|Θ0, hTA) < VA(1), from the sequential rationality of Player A, when

the state is Θ0, I obtain that µT (hT−ΛO ∪ {(θ1, π(θ1))}) < 1. Hence, for every ε > 0, µT (hT−Λ

O ∪{(θ1, π(θ1)− ε)}) ≤ µT (hT−Λ

O ∪{(θ1, π(θ1))}) < 1. Thus, for ε small enough Player O does not wait

when he observes the proposal (θ1, π(θ1)− ε). Therefore,

π(θ1)−ε+VA(µT (hT−ΛO ∪{(θ1, π(θ1)−ε)})) = uA(θ1, π(θ1)|Θ1, h

TA) ≤ uA((θ, x)|Θ1, h

TA) ≤ π(θ0)+VA(0) .

Taking ε→ 0, this implies that π(θ1) < π(θ0). This is a contradiction.

Proposition 3 Consider a monotone equilibrium (σ, µ) satisfying forward induction. For the sub-

game at period T , where the negotiation has not reached an agreement yet and Player O has

not taken the action yet, the equilibrium is a posturing equilibrium. Thus, when the state is Θ1,

Player A plays (θ1, π(θ1)); when the state is Θ0, A mixes so that µT (hT−ΛO ∪ {(θ1, π(θ1))}) =

max{µT−λ(hT−ΛO ), µVA}. Moreover, Player O plays with probability one at T .

Proof. From Lemma 7, I obtain have that P[σT,Θ1

A (hTA) = (θ1, π(θ1))] = 1. When the state

is Θ0, clearly A does not propose (θ1, x) with x < π(θ1). On the other hand for every ε > 0,

uA((θ0, π(θ0)− ε)|Θ0, hT−Λ) = π(θ0)− ε+VA(0). This is because B prefers to accept such proposal,

31The logic is the same as in Lemma 7. If B rejects, he obtains

VB(µT (hT−ΛO ∪ {(θ1, π(θ1)− ε), r})) ≥ VB(µT (hT−Λ

O ∪ {(θ1, π(θ1)− ε)})) ≥ ε+ VB(1),

where the inequalities hold for ε small enough.

38

independently on whether O waits or takes the action before. Thus, if π(θ0) ≥ ∆VA. There is a

separating equilibrium, where A proposes (θ, π(θ)) for θ ∈ {θ0, θ1}. B accepts each proposal. And

O plays accordingly.

When π(θ0) < ∆VA, we have that in equilibrium uA((θ0, π(θ0))|Θ0, hT−Λ) ≤ uA((θ1, π(θ1))|Θ0, h

T−Λ).

That is, π(θ0) +VA(0) ≤ VA(µT (hT−ΛO ∪{(θ1, π(θ1))}). Therefore, if µT (hT−Λ

O ∪{(θ1, π(θ1))}) > µA,

A proposes (θ1, π(θ1)) with probability one when the state is Θ0. If µT (hT−ΛO ∪{(θ1, π(θ1))}) = µA,

A mixes between both proposals. In both cases O takes the action immediately, and B accepts

whenever is feasible.

A.2 Proof of Section 4.2

Proof of Lemma 4. The proof is by backward induction for t = T, . . . , 0.

I prove the base case in Lemma 8. For the inductive step, consider a history htO such that

(θt, xAt ) = (θ1, π(θ1)) and µt(h

tO) ≥ µA. Under that history, if B rejects with probability one, then

µt(htO ∪{r, (θ1, π(θ1))}) ≥ µA. Henceforth, by the inductive hypothesis, the equilibrium is a strong

posturing equilibrium. If Player B accepts so that µt(htO ∪ {r, (θ1, π(θ1))}) < µA, I claim that

Player B has to be indifferent. If he strictly prefers to accept, then µt(htO ∪{r, (θ1, π(θ1))}) = 0. In

such case, Player B prefers to reject when the state is Θ1. Hence, Player B’s continuation payoff

by rejecting is exactly e−rΛVB(1). However, this cannot be true. By rejecting the proposal and

waiting to T he obtains at least VB(µA). By the assumption e−rΛVB(µA) > VB(1), this is profitable.

Therefore, Player B rejects when he observes (θ1, π(θ1)). Thus, µt+Λ(htO ∪ {r, (θ1, π(θ1))}) ≥ µA.

From the inductive hypothesis there is strong posturing and t + Λ. The same argument as in

Lemma 8 shows that there is strong posturing at t as well.

Lemma 8 Consider T < ∞ and a (σ, µ) monotone equilibrium, satisfying forward induction.

Suppose that there is some history hTO = hT−ΛO ∪ {(θT , xAT ), w, r} and µT (hT−Λ

O ) ≥ µA. Then, if

Λ < Λ for some Λ > 0, the equilibrium is a strong posturing equilibrium.

Proof.

The proof is backward induction.

Using Lemma 7, I conclude that µT (hTO) ≥ µA. Given that µT (hT−ΛO ) ≥ µA, Bayes’ rule imples

that in the subgame at t = T , there is strong posturing in the equilibrium.

Suppose that this result holds for the subgame from period t+ Λ, I want to show that there is

strong posturing at t.

39

Let (θ, x) ∈ supp(σt,Θ1

A ) and suppose that either θ = θ0 or x < π(θ1). If θ = θ0, then B

accepts and Player A obtains no more than π(θ0) + VA(0). I claim that if he deviates and offer

(θ1, π(θ1) − ε) for small ε, he obtains at least π(θ1) − ε + e−rΛVA(0). Suppose that this is not

the case. Therefore, µt(ht−ΛO ∪ {(θ1, π(θ1) − ε)}) ≤ µΛ < 1, where µΛ is the highest belief where

Player O weakly prefers to wait (See Equation 11 for a formal definition). Hence, by rejecting

(θ1, π(θ1)− ε Player B obtains at least e−rΛVB(µΛ) (by the inductive hypothesis) which is strictly

greater than the payoff by accepting, ε+e−rΛVB(1), if ε small enough. By the inductive hypothesis,

Player O does not learn by waiting and takes the action immediately. Hence, Player A obtains at

least π(θ1)− ε+ e−rΛVA(0). Because, π(θ1) > π(θ0), we have if Λ and ε are sufficiently small then

π(θ0) +VA(0) < π(θ1)− ε+ e−rΛVA(0).32 Therefore, A prefers to deviate and propose (θ1, π(θ1)− ε)over (θ, x). This is a contradiction.

For the case θ = θ1 and x < π(θ1), notice that µt(ht−ΛO ∪ {(θ, x)}) < 1: otherwise, by mono-

tonicity, µt(ht−ΛO ∪ {(θ1, π(θ1,−ε)}) = 1, and Player A deviates to (θ1, π(θ1,−ε)) when the state

is Θ1. Because µt(ht−ΛO ∪ {(θ, x)}) < 1, this implies that A proposes (θ, x) when the state is Θ0.

Because µt−Λ(ht−ΛO ) ≥ µA, using Bayes’ rule we can assume that µt(h

t−ΛO ∪ {(θ, x)}) ≥ µA. Hence,

if B rejects, the subgame has strong posturing. I assert that B rejects (θ, x) with probability one.

Suppose not, then uA((θ, x)|Θ0, htA) < e−rΛπ(θ0) + VA(µt(h

t−ΛO ∪ {(θ, x)})). However, if A deviates

and proposes (θ1, π(θ1)), tomorrow there is strong posturing. Thus, B rejects for sure if O waits

(e−rTVB(µa) > VB(1)). Consequently, O takes the action immediately. Therefore,

uA((θ1, π(θ1)|Θ0, htA) = e−rΛπ(θ0) + VA(µt(h

t−ΛO ∪ {(θ1, π(θ1))}))

≥ e−rΛπ(θ0) + VA(µt(ht−ΛO ∪ {(θ, x)})) > uA((θ, x)|Θ0, h

tA).

Hence, A prefers to propose (θ1, π(θ1)) over (θ, x). This is a contradiction. Therefore, B rejects for

sure when he observes (θ, x), and consequently, O takes the action immediately. However, A would

deviate when the state is Θ1 and propose (θ1, π(θ1)− ε) for ε small enough. By the same argument,

there is strong posturing if B rejects such proposal and, hence, O takes the action immediately.

Therefore,

uA((θ1, π(θ1)− ε|Θ1, htA) = π(θ1)− ε+ VA(µt(h

t−ΛO ∪ {(θ1, π(θ1)− ε)}))

≥ x+ VA(µt(ht−ΛO ∪ {(θ, x)})) = uA((θ, x)|Θ1, h

tA).

32Because p > µA, this means that π(θ0) < ∆VA < π(θ1).

40

This is a contradiction.

From the above, I conclude that P[σt,Θ1

A (htA) = (θ1, π(θ1))] = 1. If Player B after observing

(θ1, π(θ1)) and Player O waits decides to accept, he obtains er−ΛVB(0). If he rejects, he obtains

e−rΛVB(µt = t(htO)). From Bayes’ rule, if Player B rejects µt+Λ(htO ∪ {r, (θ1, π(θ1))}) < µt+Λ(htO ∪{a}).33 Hence, by the inductive hypothesis, Player O does not learn by waiting. Hence, he takes the

action immediately. Furthermore, using Bayes’ rule µt(htO) = µt+Λ(htO ∪ {r, (θ1, π(θ1))}), which by

inductive hypothesis, equals µT (hT−ΛO ) ≥ µA. Hence, uA((θ1, π(θ1))|htO,Θ0) ≥ e−rΛπ(θ0)+V (µA) >

π(θ0) + VA(0) ≥ uA((θ, x)|htO,Θ0) if (θ, x) 6= (θ1, π(θ1)). I conclude that there is strong posturing

at t.

A.3 Proofs of Section 5

Proof of Proposition 2. Suppose that prior to time t Player B has rejected and O has not

taken the action yet. The strategy for Player A is

σt,Θ0

A (htA) = (θ0, π(θ0)), σt,Θ1

A (htA) =

(θ1, xB) if π(θ1)(1− e−rΛ) > e−rΛ(∆VA −∆VB)

(θ1, π(θ1)− e−rΛ[e−r(T−t−Λ)VB(0)− VB(1)

]+) if π(θ1)(1− e−rΛ) ≤ e−rΛ(∆VB −∆VA).

Regarding the strategy for Player B. If Player O waits at t, then P[σt,Θ0

B (htB) = a] = 1 accepts if

θt = θ0,P[σt,Θ0

B (htB) = r] = 1 if θt = θ1, P[σt,Θ1

B (htB) = a] = 1 if xAt ≤ σt,Θ1

A (htA), P[σt,θ1B (htB) = r] = 1

if xAt > σt,Θ1

A (htA). When Player O takes the action at t, Player B accepts every feasible proposal.

Regarding the strategy for Player O, P[σtO(htO) = w] = 1.

For a history where there is an agreement prior t and Player O has not taken the action,

P [σtO(htO) = 0] = 1 if θτa = θ0 and P [σtO(htO) = 1] = 1 if θτa = θ1.

Finally for a history where Player O has taken the action before t, the equilibrium follows the

strategies from the bargaining game with one side proposal (see Section 4.4. in Fudenberg and

Tirole (1991)).

For history htO of rejections, the beliefs are:

µt(htO) =

0 if t > T or π(θ1)(1− e−rΛ) > e−rΛ(∆VB −∆VA)

p if t ≤ T or π(θ1)(1− e−rΛ) ≤ e−rΛ(∆VB −∆VA)..

33Because µt(htO) > 0, Bayes’ rule implies that µt(h

tO) > 0.

41

For a history htO where there is an agreement before t, the beliefs are

µt(htO) =

0 if θτa = θ0

1 if θτa = θ1

By simple inspection, the beliefs are consistent. Regarding sequential rationality, by the con-

struction of the strategies, I only need to consider histories htO where Player B has rejected from

0, . . . , t−Λ and O has not taken the action yet. Notice that if Player A proposes less than expected,

he obtains the same externality payoff but less from the negotiation pie. If he offers more than

σt,Θ1

A (ht), B rejects, and he has to wait a period to obtain less from the negotiation pie and less

from the negotiation outcome. Thus, Player A is sequentially rational. For Player B the same

reasoning applies. For Player O, by the value of information assumption, it is optimal for Player

O to wait a period and learn the state of the world. Thus, every player is sequentially rational.

Regarding belief stability, when π(θ1)(1−e−rΛ) ≤ e−rΛ(∆VA−∆VB) notice that same strategies

are sequentially rational if the belief at a history htO is any µ ∈ (µΛ, µΛ) (See Equations (11)and (12)

for a definition). Thus, the equilibrium is belief stable. When π(θ1)(1−e−rΛ) > e−rΛ(∆VA−∆VB),

consider µt(htO) = ε and let γ = e−rΛ(VB(0) − VB(ε)). Then, for ε small enough, consider the

strategies as σt,Θ1

A (htA) = (θ1, xB +γ), P[σt,Θ1

B (htB) = a] = 1 if xAt ≤ xB +γ and P[σt,Θ1

B (htB) = r] = 1

if xAt > xB + γ, and the other set of strategies defined as before. Then, such strategies, are

sequentially rational given the perturbed beliefs. Taking ε → 0, I obtain the equilibrium payoffs

for each player. Therefore, the assessment is belief stable.

Regarding the forward induction argument. Consider an off-path history where Player B has

rejected until t and Player O has not taken the action yet. From the Step 0. and Step 8. of

Lemma 9, I obtain that µt+Λ(htO∪{(θ, x)}) = 0 when θt = θ0 and µT+ΛO (hTO∪{r}) = 0, respectively.

Let

µΛ

= inf{µ ∈ [0, 1] s.t. Z(µ) < e−rΛ (µZ(1) + (1− µ)Z(0))},

the smallest belief where Player O prefers to wait a period. I define

η = maxµ∈[0,µ

Λ]VA(µ) + VB(µ)− (VA(0) + VB(0)).

From Lemma 10 I get that if π(θ1)(1− e−rΛ) > (∆VB−∆VA) + η, by forward induction, µt+Λ(htO ∪

42

{r}) = 0. However, when π(θ1)(1− e−rΛ) ≤ e−rΛ(∆VB−∆VA) +η, a belief µ ≤ µΛ

exists such that

π(θ1)(1− e−rΛ) ≤ e−rΛ(VB(µ)− VB(1)) + e−rΛ(VA(µ)− VA(1)) ⇐⇒ π(θ1)− e−rΛ(VB(µ)− VB(1)) + VA(1) ≤ e−rΛπ(θ1) + e−rΛVA(µ).

Because is sequentially for PlayerO to play µ at htO∪{r}. Then under that strategy, Player A prefers

to propose (θ1, π(θ1)), that B rejects, and to reach agreement at t + Λ. Thus, µ ∈ D(Θ1, r|µ, htO)

for t < T . Also if e−rΛ ≥ π(θ0)∆VA

, I obtain that π(θ0)(1− e−rΛ) > e−rΛ(VA(µ)− VA(0)). Thus, Player

A prefers to propose a non-feasible offer when the state is Θ0. Thus, µ ∈ D(Θ0, r|µ, htO) and the

forward induction argument does not refine the off-path beliefs.

I conclude that the assessment (σ, µ) is a PBE satisfying forward induction and belief stability.

Proof of Theorem 2

The proof is a consequence of a general result. For this purpose, I define

µΛ = sup{µ ∈ [0, 1] s.t. Z(µ) < e−rΛ (µZ(1) + (1− µ)Z(0))} (11)

µΛ

= inf{µ ∈ [0, 1] s.t. Z(µ) < e−rΛ (µZ(1) + (1− µ)Z(0))} (12)

η = maxµ∈[0,µ

Λ]VA(µ) + VB(µ)− (VA(0) + VB(0)). (13)

The beliefs µΛ and µΛ

are the threshold beliefs such if at some point µt(htO) ∈ [0, 1] \ [µ

Λ, µΛ],

Player O takes the action without taking into account what happens in the negotiation. Because

Z is strictly convex, if µ ∈ (µΛ, µΛ) Player O prefers to delay a period and learn the state rather

than taking the action µ. Notice that the information-value assumption implies that p ∈ (µΛ, µΛ).

Theorem 3 (Theorem 2 extended) Suppose that Assumption 1 holds and that T < ∞. A

Λ > 0 exists such that if Λ < Λ every equilibrium satisfying forward induction and belief stability is

a separating equilibrium. Furthermore, if π(θ0) < ∆VA the equilibrium path is uniquely determined.

Lemma 11 shows that limΛ→0 µΛ= 0. Hence, is straightforward to conclude that Theorem 2 is

a direct consequence of Theorem 3. Now I proceed to demonstrate Theorem 3.

Proof of Theorem 3. First, because p ∈ (µΛ, µΛ) using Lemma 9 I obtain that for Λ < Λ every

equilibrium that satisfies forward induction is a separating equilibrium. Moreover, Player A offers

(θ0, π(θ0)) when the state is Θ1, and Player B always accepts any proposal.

43

When π(θ1)(1 − e−rΛ) > e−rΛ(∆VB − ∆VA) + η and π(θ0) < ∆VA, Lemma 10 shows that in

the case of a rejection, Player O believes that the state is Θ1. Hence, after a history of rejections,

Player A proposes (θ1, xB) when the state is Θ1. Player B accepts if the proposal is such that

xAt ≤ xB, and rejects otherwise. In particular, this condition holds when ∆VA ≥ ∆VB. This shows

that the equilibrium is unique.

When π(θ1)(1 − e−rΛ) > e−rΛ(∆VB − ∆VA) − η, I claim that for a history of rejections ht),

we have that µt(htO) ∈ (µ

Λ, µΛ) and O waits with probability one at t. This claim implies that

on the equilibrium path, B accepts every proposal x ≤ π(θ1) − e−rΛ[e−r(T−t−Λ)VB(0)− VB(1)

]+.

Therefore, A proposes π(θ1)− e−rΛ[e−r(T−t−Λ)VB(0)− VB(1)

]+which proves the last statement of

the proposition.

I know proceed to prove the claim. Suppose that µt(htO) > µΛ. Because µΛ → 1 as Λ → 0

and π(θ0) < ∆VA, a Λ exists such if Λ < Λ we have that ı(θ0) > VA(1)− VA(µΛ) and e−rΛπ(θ0) +

VA(µt(htO) > π(θ0) + VA(0). Thus, Player A prefers to deviate by proposing an unfeasible offer.

This violates Lemma 9. If µt(htO) < µΛ, using that π(θ1)(1 − e−rΛ) > e−rΛ(∆VB − ∆VA) − η,

we have that π(θ1) − e−rΛVB(µt(htO) + e−rΛVA(1) < e−rΛπ(θ1) + e−rΛVA(µt(h

tO). Thus, A prefers

to propose (θ1, π(θ1)) at t − 1, and that B rejects. This violates Lemma 9. From the previous

discussion, we have that if µt(htO) ∈ {µ

Λ, µΛ}, Player O has to wait with positive probability at

t. Because the equilibrium is belief stable, the same argument I use in the inductive hypothesis of

Lemma 9 implies that O waits if µt(htO) = µΛ. This shows the assertion, which finishes the proof.

Lemma 9 Suppose Assumption 1 holds. A Λ > 0 exists such that if Λ < Λ, then every equilibrium

satisfying forward induction is a separating equilibrium. Furthermore, if π(θ0) < ∆VA the for a

history hT+ΛO where B always rejects, the belief satisfy µT+Λ(hT+Λ

O ) = 0.

Proof. The proof is by (backward) induction from t = T to t = 0. To simplify notation, let htOthe history where Player B rejects from 0 to t − Λ and Player O has not taken the action yet. I

denote µt = µt(htO) and βt = P[σtO(htO) 6= w].

I demonstrate the lemma by showing the following statement. Consider a history htO where

Player B rejects from 1, . . . , t and Player O has not taken the action yet. If µt ∈ (µΛ, µΛ) then

Player B accepts in every state and reveals the state θ,and Player O waits to t + Λ. Notice that

this statement clearly proves Lemma 9 since p ∈ (µΛ, µΛ). Moreover, Player B obtains e−rΛVB(0)

when the state is Θ0 and a payoff between e−r(T−t−Λ)VB(0) and e−rΛVB(0) when the state is Θ1.

In what follows, I fix a PBE (σ, µ) satisfying forward induction.

44

Base case

Fix a history hTO such that no agreement has been reached prior to T . I claim that if µT ∈ (µΛ, µΛ)

then A proposes (θ0, π(θ0)) when the state is Θ0 and proposes (θ1, π(θ1)− e−rΛ(VB(µT )− VB(1)))

when the state is Θ1. Player B’s payoff is e−rΛVB(0) when Θ = Θ0 and e−rΛVB(µT ) when Θ = Θ1.

Player O learns θ and waits with probability one. Moreover, if π(θ0) < ∆VA then µT+Λ = 0.

I split the argument into the following steps.

I claim that if βT < 1, Player B accepts with probability one when the state is Θ0. The

proof is by contradiction. For the following six steps, suppose that Player B rejects with positive

probability when Θ = Θ0.

Step 1. I claim that µT+Λ > 0. If µT+Λ = 0, because B rejects with positive probability when

Θ = Θ0, we have that uA(hTA|Θ0) < π(θ0) +βTVA(µT ) + (1−βT )e−rΛVA(0). Consider the deviation

where A proposes (θ0, π(θ0)−ε). Then, conditional that O waits, B gets ε+e−rΛVB(0) by accepting

the offer. This is strictly greater than B’s payoff by rejecting. Thus, B accepts (θ0, π(θ0) − ε).

Taking ε small enough we have that uA(hT−ΛA ∪ {(θ0, π(θ0) − ε)}|Θ0) = π(θ0) − ε + βTVA(µT ) +

(1− βT )e−rΛVA(0) > uA(hTA|Θ0). I conclude that A prefers to deviate by proposing (θ0, π(θ0)− ε),which is a contradiction. Hence, µT+Λ > 0.

Step 2. I claim that B rejects with positive probability when the state is Θ1, and that βT > 0.

Because µT+Λ > 0 (Step 1) by Bayes’ rule B rejects when the state is Θ1. If βT = 0, then A strictly

prefers to propose (θ1, xB − ε). This proposal is accepted by B. This implies, by Bayes’ rule that

µT+Λ = 0. This contradicts Step 1.

Step 3. When the state is Θ1, I assert that A proposes (θ1, π(θ1)) and B rejects it. From Step

2, a proposal (θ, x) exists such that B rejects it when Θ = Θ1. Suppose that B accepts (θ, x) with

positive probability, then we have that θ = θ1 and x ≥ xB. Hence, B reveals that the state is Θ1

when he accepts (θ, x). Because B is rejecting (θ, x) with positive probability, A’s payoff is strictly

less than x+ βTVA(µT ) + (1− βT )e−rΛVA(1). However, for ε > 0 small enough, A has a profitable

deviation by proposing (θ1, x − ε): B accepts with probability one, and there is no destruction

of the gains from the negotiation. Therefore, B rejects (θ, x) with probability one. From Step 2,

notice that pT > 0. From the optimality of A’s proposal this implies that (θ, x) = (θ1, π(θ1)).

Step 4. I claim that the following inequality holds

π(θ1)− e−rΛ(VB(µT+Λ)− VB(1)) + βTVA(µT ) + (1− βT )e−rΛVA(1) ≤ βT (π(θ1) + VA(µT )) + (1− βT )e−rΛVA(µT+Λ). (14)

Indeed, if A proposes (θ1, π(θ1)− e−rΛ(VB(µT+Λ)− VB(1))− ε), B accepts. Step 3 implies that

45

A prefers to proposes (θ1, π(θ1)) rather than proposing (θ1, π(θ1) − e−rΛ(VB(µT+Λ) − VB(1)) − ε),for every ε > 0. Taking ε→ 0, I derive Inequality (14).

Step 5. I claim that the following inequality holds

π(θ0) + βTVA(µT ) + (1− βT )e−rΛVA(0) ≤ βT (VA(µT )) + (1− βT )e−rΛVA(µT+Λ) ⇐⇒ (1− βT ) ≥ π(θ0)

e−rΛ(VA(µT+Λ)− VA(0)). (15)

By the same reasoning as in Step 3, I get that if the state is Θ0, then Player B accepts any

offer (θ0, x) for x < π(θ0). In particular, if A proposes (θ0, π(θ0)− ε), B accepts. Taking ε → 0, I

obtain the desired inequality.

Step 6. I claim the Inequality (14) and Inequality (15) are mutually exclusive. Indeed, replacing

Inequality (15) into Inequality (14), I get that

(1− βT )(π(θ1) + e−rΛ∆VA) ≤ e−rΛ(VB(µT+Λ)− VB(1)) + π(θ0) .

Because βT > 0 (Step 3), Inequality (15) implies that e−rΛ(VA(µT+Λ) − VA(0)) > π(θ0). Assump-

tion 1 (a.) implies that e−rΛ(VB(µT+Λ)−VB(1)) ≤ π(θ0). Hence, (1−βT )(π(θ1)+e−rΛV ) ≤ 2π(θ0).

However, this contradicts

(1− βT )(π(θ1) + e−rΛ∆VA) > 2(1− βT )(e−rΛ∆VA) ≥ 2π(θ0)(e−rΛ∆VA)

e−rΛ(VA(µT+Λ)− VA(0))> 2π(θ0) .

The first inequality comes from π(θ1) > ∆VA. The second inequalities comes from Inequality (15).

Therefore, if βT < 1, Player B accepts with probability one when Θ = Θ0.

Step 7. Suppose that µT ∈ (µΛµΛ), I claim that βT < 1. If βT = 1, then by optimality of

A we have that σT,ΘA (hTA) = (θ, π(θ)). Consider a deviation, where O waits instead. If µT+Λ > 0,

then B accepts (θ0, π(θ0)). Hence, Player O learns θ by waiting. Therefore, waiting is a profitable

deviation for Player O. If µT+Λ = 0, Player B strictly prefers to reject when Θ = Θ1. This violates

Bayes’ rule for µT+Λ.

From Step 7, I conclude that if µT ∈ (µΛ, µΛ), then Steps 1- 6 holds. Hence, Player B does not

reject when Θ = Θ0. Thus, A offers (θ0, π(θ0)) when Θ = Θ0. By the same logic as Step 3, B does

not reject when Θ = Θ1. Hence, O learns θ and βT = 0. B’s payoff is less or equal than e−rΛVB(0)

for every Θ ∈ {Θ0,Θ1}.Step 8. I assert that if π(θ0) < ∆VA then µT+Λ = 0 and B’s payoff is e−rΛVB(0). Consider a

belief µ for Player O at T + Λ after observing a rejection. From Step 1 to Step 7, this implies that

βT = 0. Thus, when state is Θ1, Player A strictly prefers to propose (θ1, xB − ε), and obtaining

46

xB − ε + e−rΛVA(1), rather than making a proposal that is going to be rejected, and obtaining

(e−rΛVA(µ)). Hence, D(Θ1, r|µ, hT ) = ∅. On the other hand, notice that π(θ0) + e−rΛVA(0) <

e−rΛVA(1). Therefore, µ ∈ D(Θ0, r|µ, hT ) for µ sufficiently close to 1. Using forward induction,

I conclude that µT+Λ = 0. Hence, Player A proposes (θ1, xB) when the state is Θ1. Therefore,

Player B obtains e−rΛVB(0).

This concludes the base case.

Inductive Step

Consider a history htO where Player B rejects from 0 to t−Λ and Player O has not taken the action

yet and that µt ∈ (µΛ, µΛ).

I claim that uB(htB ∪ {r}|Θ) ≤ e−rΛVB(0) for every Θ ∈ {Θ0,Θ1}. Consider a belief µ instead

of µt+Λ at t+ Λ. For µ < µΛ

or µ > µΛ, Player O does not wait and takes the action immediately.

Thus, B gets e−rΛVB(µ) ≤ e−rΛVB(0) at period t. If µ ∈ (µΛ, µΛ) by the induction hypothesis

B gets a payoff of e−2rΛVB(0) at time t. Because the equilibrium is belief stable, I conclude that

uB(htB ∪ {r}|Θ) ≤ e−rΛVB(0).

To conclude the proof, I divide the argument in three cases.

Case 1. That µt+Λ ∈ (µΛ, µΛ). I claim that for Θ = Θ0, A proposes (θ0, π(θ0)) and B accepts

it. Using the inductive hypothesis, if Player B rejects he obtains a payoff that is less or equal than

e−r2ΛVB(0). Therefore, Player B accepts (θ0, π(θ0)) and reveals θ to O. On the other hand, the

continuation payoff for Player A after a rejection is e−rΛ(π(θ0) + e−rΛVA(0)) < π(θ0) + e−rΛVA(0).

Therefore, when the state is Θ0, Player A proposes (θ0, π(θ0)) and B accepts it. Consequently,

when the state is Θ1, I assert that Player A proposes (θ1, x) with x ≥ xB and B accepts it. Suppose

that with positive probability Player B rejects such proposal. Then, by Bayes’ rule, µt+Λ = 1. This

contradicts the assumption that µt+Λ ∈ (µΛ, µΛ). Summarizing, Player B accepts with probability

one when Θ = Θ1. From the optimality of A, Player A proposes (θ1, π(θ1) − uB(htB ∪ {r}|Θ1).

Because, π(θ1)− uB(htB ∪ {r}|Θ1) > xB, Player B gets a payoff strictly less than e−rΛVB(0).

Case 2. That µt+Λ ≤ µΛ. I claim that Player A proposes (θ0, π(θ0)) when the state is Θ0, then

Player B accepts it, getting a payoff of e−rΛVB(0). From the same argument as the one used in

the Step 7 of the base case, I derive that Player O waits with positive probability at t, i.e., βt < 1.

Consider the proposal (θ0, π(θ0)− ε) when the state is Θ0. Because uB(htB ∪ {r}|Θ0) < e−rΛVB(0),

Player B strictly prefers to accept it since it reveals that the state is Θ0. Thus,

uA(htA∪{(θ0, π(θ0)−ε)}|Θ0)−uA(htA∪{r}|Θ0) > (1−βt)(π(θ0)+e−rΛVA(0)−e−rΛ(π(θ0)+e−rΛVA(µΛ)).

47

From Lemma 11, I obtain that

limΛ→0

e−rΛVA(µ

Λ))− V (0)

Λ= V ′A(0)

rZ(0)

Z(1)− Z(0)− Z ′(0).

Using Assumption 1, I conclude that

limΛ→0

e−rΛVA(µ

Λ))− V (0)

Λ= V ′A(0)

rZ(0)

Z(1)− Z(0)− Z ′(0)< rπ(θ0).

Because limΛ→0π(θ0)−e−rΛπ(θ0)

Λ= rπ(θ0), a Λ exists such that if Λ < Λ, then

π(θ0)+e−rΛVA(0)−e−rΛ(π(θ0)+e−rΛVA(µΛ)) > 0 =⇒ uA(htA∪{(θ0, π(θ0)−ε)}|Θ0)−uA(htA∪{r}|Θ0) > 0.

Therefore, Player A strictly prefers to propose (θ0, π(θ0) − ε) over an unfeasible proposal, for ε

small enough. In such case, Player B accepts with probability one when the state is Θ0 and gets

e−rΛVB(0). Hence, there is no rejection when the state is Θ0. Because µt+Λ < µt, I conclude that

µt+Λ is an off-path belief. Therefore, Player B accepts with probability one when the state is Θ1.

Case 3. That µt+Λ ≥ µΛ. If there is rejection with positive probability, because µt+Λ > µt

I conclude that, with positive probability, there is an agreement at t when the state is Θ0 and

there is a rejection when the state Θ1. Consider a deviation where A proposes (θ1, x) where

x = π(θ1)− e−rΛ(VB(µt+Λ)− VB(1)) at period t when Θ = Θ1. Clearly, B accepts. Moreover,

uA(htA ∪ {(θ1, x)}|Θ1)− uA(htA ∪ {r}|Θ1) > π(θ1)(1− e−rΛ)− e−rΛ(VB(µt+Λ)− VB(1)) ≥ π(θ0)(1− e−rΛ)− e−rΛ(VB(µt+Λ)− VB(1)).

Using that µt+Λ ∈ [µΛ, 1), Lemma 11 and Assumption 1, the same procedure as in Case 2 allow

me to conclude that Player A prefers to deviate and propose (θ1, x) at t and that Player B accepts

with probability one. This is a contradiction.

Lemma 10 Suppose that π(θ1)(1−e−rΛ) > e−rΛ(∆VB−∆VA)+η, π(θ0) < ∆VA and that Assump-

tion 1 holds. A Λ exists such if Λ < Λ and (σ, µ) is an equilibrium satisfying forward induction,

then µt+Λ(htO ∪ {r}) = 0 for every history htO where B rejects until t.

Proof. The proof is by (backward) induction from the last period t = T + Λ to t = 0. The base

case T + Λ corresponds to Step 8. in Lemma 9.

For the inductive hypothesis, consider a belief µ at the history ht+ΛO = htO ∪ {r}. For µ > µΛ, I

assert that Player A prefers to propose an unfeasible offer when the state is Θ0. Indeed, take Λ so

48

for Λ < Λ we have e−rΛ ≥ π(θ0)∆VA

. Then,

utA(htA ∪ {(θ0, π(θ0))}|Θ0) = π(θ0) + e−rΛVA(0) < e−rΛπ(θ0) + e−rΛVA(µ) = utA(htA ∪ {r}|Θ0).

Therefore, µ ∈ D(Θ0, r|µ, htO).

I assert that D(Θ1, r|µ, htO) = ∅. Indeed, for µ ∈ (µΛ, µΛ), and using the inductive hypothesis

I obtain that µt+2O (htO ∪ {r} ∪ {r}) = 0. Thus,

utA(htA ∪ {r}|Θ1) = e−rΛ(xB + e−rΛVA(1)) < xB + e−rΛVA(1) = utA(htA ∪ {(θ1, xb)}|Θ1).

I conclude that µ /∈ D(Θ1, r|µ, htO) if µ ∈ (µΛ, µΛ).

Consider µ ∈ [0, 1] \ [µΛ, µΛ], the following inequality holds when Λ < Λ:

π(θ1)− e−rΛ(VB(µ)− VB(1)) + VA(1) > e−rΛπ(θ1) + e−rΛVA(µ) ⇐⇒ π(θ1)(1− e−rΛ) > e−rΛ(VB(µ)− VB(1)) + e−rΛ(VA(µ)− VA(1)).

(16)

Indeed, for µ > µΛ, Lemma 11 and Assumption 1 implies that for Λ < Λ, π(θ1)(1 − e−rΛ) >

e−rΛ(VB(µ)− VB(1)). For µ < µΛ, notice that

π(θ1)(1−e−rΛ) > e−rΛ(∆VB −∆VA)+η =⇒ π(θ1)(1−e−rΛ) > e−rΛ(VB(µ)−VB(1))+e−rΛ(VA(µ)−VA(1)).

Using Inequality (16), I conclude that

utA(htA ∪ {r}|Θ1) = e−rΛ(π(θ1) + e−rΛVA(µ)) < π(θ1)− e−rΛ(VB(µ)− VB(1)) + e−rΛVA(1)

= utA(htA ∪ {(θ1, xB)}|Θ1).

Thus, µ /∈ D(Θ1, r|µ, htO).

Finally, for µ ∈ {µΛ, µΛ}, the payoff the Player A obtains in a rejection is a convex combination

of the previous two cases. Therefore, D(Θ1, r|µ, htO) = ∅.The forward induction criterion implies that µt+Λ

O (htO ∪ {r}) = 0.

Lemma 11 The following holds limΛ→0µ

Λ

∆= rZ(0)

Z(1)−Z(0)−Z′(0)and limΛ→0

1−µΛ

∆= rZ(1)

Z(0)−Z(1)−Z′(1).

Proof. Because Z is continuous, I conclude that

Z(µΛ) = e−rΛ

(µ

ΛZ(1)− (1− µ

Λ)Z(0)

).

49

Using that Z is strictly convex I conclude that limΛ→0 µΛ→ 0. Also,

Z(µΛ)− Z(0) = (e−rΛ − 1)Z(0) + e−rΛ(Z(1)− Z(0))µ

Λ. (17)

From the convexity of Z I obtain the following inequality:

Z ′(0)µΛ≤ Z(µ

Λ)− Z(0) ≤ Z ′(µ

Λ)µ

Λ.

Plugging this inequalities on Equation (17), I obtain that

µΛ

Λ

(Z ′(0)− e−rΛ(Z(1)− Z(0))

)≤ e−rΛ − 1

ΛZ(0) ≤

µΛ

Λ

(Z ′(µ

Λ)− e−rΛ(Z(1)− Z(0))

).

Taking Λ → 0, and using that limΛ→0 µΛ= 0 and Z ′ is continuous, concludes the proof for µ

Λ.

The case for µΛ is analogous, and is therefore omitted.

50