Mediation in reputational bargaining Jack Fanning∗ July 27, 2016

Abstract In the unique equilibrium of a reputational bargaining model, rational agents imitate behavioral types who are committed to their demands, resulting in inefficient delay. This paper investigates the potential for mediation to improve outcomes. Agents can privately indicate to a mediator that they are willing to compromise (accept less than their behavioral demand). The mediator can then suggest an agreement based on this information. I first consider a mediator who filters information by publicly suggesting an agreement, only if both parties compromise. I show that if the mediator makes her suggestion immediately after joint compromise, then agents never compromise. Adding noise helps. If the mediator sometimes makes no suggestion after joint compromise then she can bring about Pareto improvements when behavioral types are unlikely. If the mediator sometimes delays her suggestion after joint compromise, then she can implement almost any compromise division (between agents’ behavioral demands) when behavioral types are unlikely. Finally, if the mediator can make any suggestion at any time based on her acquired information, and the utility possibility frontier is strictly concave, then she can bring about Pareto improvements regardless of how likely behavioral types are. Keywords: Bargaining, reputation, behavioral types, mediation, delay

1

Introduction

In its broadest definition, mediation refers to any instance in which a third party helps others reach a voluntary agreement. It is distinct from arbitration, which can impose agreement. Mediation is widely used to help parties to resolve disputes ranging from international conflicts and industrial relations to divorce proceedings. For instance, Dixon (1996) finds that mediation occurred in 13% of dispute phases of international conflicts between 1947-1982.1 In a survey of general counsel for Fortune 1000 companies, Stipanowich and Lamare (2013) find that less than 15% of companies “rarely” or “never” use mediation in every category of dispute.2 Use ∗

Brown University. Email: jack [email protected]. Address: Department of Economics, Robinson Hall, 64 Waterman Street, Brown University, Providence, RI 02912. Website: https : \\sites.google.com\a\brown.edu\ j f anning. 1 These dispute phases are distinguished by the level of conflict between parties (e.g. threats of hostilities, open hostilities). 2 By contrast arbitration was “rarely” or “never” used in at least 49% for each category.

1

of mediation increased in every category compared to a 1997 survey. Attesting to mediations benefit’s, Dixon found that mediated disputes were 47% less likely to escalate and 24% more likely to peacefully resolve compared to disputes with no conflict management. More convincing evidence comes from Emery et al. (1991), who find that a treatment group, randomly selected to receive mediation services, settled 89% of contested custody cases out of court, compared to 28% of a control group. Mediation also halved the time spent reaching agreement and increased parties’ satisfaction with the outcome. Why does mediation help? Veteran mediator and Former Secretary of State for Labor, John Dunlop, describes the difficulties of “end-play” negotiations and the benefit of mediation as follows: “The critical problem is that each side would prefer the other to move to avoid a further concession itself, and that any move may create the impression of being willing to move all the way to the position of the other side... In these circumstances a third party may greatly facilitate agreement. The separate conditional acceptance to the mediator by one side of the proposal does not prejudice the position of that side if there is no agreement. It is not unusual for a mediator to secure the separate acceptance of each side of a “package” of the mediator’s design and then to bring the parties together to announce that, even if they do not know it, they have an agreement.”3 The claim is that mediators help by filtering information. Agents may resist proposing a compromise themselves for fear of being identified as a weak type, who is willing to concede entirely to her opponent’s demand. By releasing the information that an agent is willing to accept a compromise only when both agents are willing, the mediator can eliminate this fear. This paper seeks to use economic theory to help understand why and when such mediation techniques can be effective. I do this in the context of the reputational bargaining model of Abreu and Gul (2000) (henceforth AG). In AG’s model, two agents must divide a dollar. They can make frequent offers over the course of an infinite horizon. With positive probability an agent is a behavioral type, who is committed to demanding a fixed share of the dollar and accepting nothing less, otherwise the agent is rational. In equilibrium, an agent identified as rational must concede immediately to a possibly behavioral opponent’s demand. Given this, rational agents must imitate behavioral types and bargaining resembles a war of attrition, with inefficient delay. This reputational model captures many of the difficulties of unmediated negotiations. In particular, agreement is often delayed and negotiators’ fear that small concessions will necessitate larger ones is well justified. An important advantage of considering reputation as opposed to other forms of incomplete information in bargaining is the uniqueness of the equilibrium, absent mediation. This provides a clean benchmark against which to asses mediation’s benefits. Typically, two sided incomplete information dynamic bargaining models exhibit multiple equilibria, with hard to characterize equilibrium payoff sets. Of course, however, reputational 3

Dunlop (1984), p16-24).

2

bargaining may be an imperfect description of many disputes in which mediation is used.4 I model mediation as follows: Each agent can privately indicate to a mediator that she is willing to compromise (accept less than her behavioral demand). The mediator can suggest an agreement based on this information. The mediator has no preferences, but rather follows an exogenous protocol. I initially stick closely to Dunlop’s proposed mediation protocol. A mediator promotes a specific compromise division and suggests an agreement on those terms only if both parties have indicated a willingness to accept them. The first main results are negative. I show that if the mediator suggests agreement immediately after both parties compromise, then mediation always fails. Neither party is willing to compromise and play follows the unique equilibrium of AG. The result is proved by showing that at least one agent who compromised and received the bad news that her opponent did not (and so is more likely to be behavioral) would subsequently immediately concede to her opponent. The prospect of that concession destroys her opponent’s incentive to compromise in the first place. This finding suggests that mediators must not only worry about protecting an agents’ reputations, but also about the effects of bad news (that an opponent has not compromised) on agents who do compromise. Given this, I consider two simple extensions of Dunlop’s mediation protocol, which in addition to filtering information, add noise, and so mitigate the impact of bad news. In the first extension, the mediator sometimes fails to announce a deal even when both parties compromise. This can be interpreted as a mediator mistake or tremble. It allows for a positive result. Such mediation can bring about Pareto improvements over AG’s equilibrium when behavioral types unlikely. In the second extension, the mediator sometimes delays announcing an agreement until some randomly determined time. This allows the mediator to implement any compromise agreement (between behavioral types’ demands) with probability close to one, when behavioral types are unlikely.5 Particular agreements may be desirable for reasons of fairness, or their benefit to other (fourth) parties. Finally, with a more general mediation protocol in which the mediator can make any suggestion based on information she has acquired, I show that if the utility possibility frontier is strictly concave (because at least one agent is risk averse), the mediator can bring about Pareto improvements over AG’s equilibrium regardless of the likelihood of behavioral types. Although the potential for mediators to filter information and in so doing expand payoff sets is well known in economics (for instance, the set of correlated equilibria is typically larger 4 Pre-trial negotiations (highlighted above as an important setting for mediation) typically don’t have an infinite horizon, rather, agents must agree before a deadline (the trial). However, Fanning (2016) shows that the infinite horizon and deadline models do not differ substantively if there is even slight uncertainty about the deadline’s timing (the last time at which agents can strike a deal). 5 This partially unwinds AG’s results that even small reputational forces tightly determine bargaining outcomes.

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than the set of Nash equilibria), the role for mediators in dynamic bargaining settings (where real world mediators practice) has received relatively little attention. The mechanism design bargaining problem has received attention (e.g. Myerson and Satterthwaite (1983)), but such a designer can typically act as an arbitrator, with the power to impose agreements. This abstracts from dynamic issues of compliance with the designers orders. The paper closest in spirit to my own is probably Jarque et al. (2003), which illustrates an equilibrium with a mediator in a continuous time bargaining game with incomplete information about reservation values. However, the benefits offered by mediation in that paper are unclear, because of the difficulty in analyzing the payoff set without a mediator. I discuss their result in greater depth in section 9. It is important to note, that I do not claim to identify an optimal structure for mediation. I do not build a fully general mechanism design problem on top of the reputational bargaining game and attempt to maximize the weighted sum of rational agents payoffs. The optimal mediation problem offers a promising avenue for future research, but is beyond the scope of this paper. My more modest aim is to better understand the difficulties for a simple model of mediation, motivated by practitioners, and how those might be overcome. It is also important to note that there are many reputed benefits of mediation, which I do not consider. These include the acknowledgement of each side’s grievances by a neutral party, the creation of a less confrontational atmosphere for negotiation, and a mediator’s ability to establish commonly accepted facts. The paper is arranged as follows. Section 2 outlines the model; in Section 3 there is no mediator; in sections 4 and 5 the mediator immediately announces a compromise agreement after its joint (provisional) acceptance; in Section 6 the mediator sometimes fails to announce such agreements; in Section 7 the mediator sometimes delays her announcement; in Section ?? the mediator can make arbitrary suggestions; Section 9 discusses the results in relation to the existing literature.

2

The model

The model presented below encompass all the mediation protocols I consider in a consistent way. The setup adapts the the discrete continuous time bargaining protocol advanced by Abreu and Pearce (2007), although for much of the analysis time can be treated as completely continuous. I discuss in Section 9 how results can be generalized to different bargaining protocols. Two bargainers, i = 1, 2, must agree on how to divide a dollar and face an infinite horizon. Bargainers are either rational or behavioral types. If rational bargainer i obtains a share αi ∈ [0, 1] of the dollar at time t then her utility is e−ri t ui (αi ) where her discount factor is ri and the twice continuously differentiable utility function ui satisfies ui (0) = 0, u0i (αi ) > 0 and u00i (αi ) ≤ 4

0.6 Behavioral types have no preferences, but mechanically implement an exogenously defined strategy. A third player is a mediator, i = 3, who is always a behavioral type, with a fixed strategy. Time is discrete-continuous to allow multiple events to occur at the same time in a sequential order. Each time t ∈ [0, ∞) is divided into five different discrete times t1 , t2 , t3 , t4 , t5 . Time follows a natural ordering so that tk < tk+1 , and tk < sl whenever t < s. The set of discrete continuous times is DC = [0, ∞) × {1, 2, ..., 5} ∪ ∞. There is no discounting of payoffs within each time t. The bargaining protocol is as follows: at time 01 each bargainer i simultaneously announces a demand αi (01 ) ∈ [0, 1]; at t1 > 01 each bargainer can concede to her opponent’s existing demand (accept the share (1 − α j (t1 ))), ending the game; at any t2 each bargainer can privately send a message to the mediator indicating that she is rational (and so willing to compromise); at t3 the mediator can publicly suggest a compromise share of the dollar, (m1 (t3 ), m2 (t3 )), on which agents should agree; at t4 each bargainer can simultaneously change her demand to αi (t4 ); at t5 each bargainer can concede to her opponent’s (possibly new) existing demand. If both bargainers concede at the same time then each proposal is selected with probability 12 . At every tk > 01 each bargainer is associated with an existing demand. If bargainer i changes her demand at t4 then she cannot change her demand again until time (t + ∆)4 for some ∆ > 0. That is, if αi (t3 ) , αi (t4 ) then i’s existing demand is αi (sk ) = αi (t4 ) for all sk < (t + ∆)4 . A bargainer can only indicate a willingness to compromise once. The mediator can only propose a compromise once in the game. These restrictions mean that agents’ bargaining environments are relatively stable, allowing strategies to be more easily defined. Agent i is a behavioral type with probability zi ∈ (0, 1), and is otherwise or rational. A behavioral type for agent i initially demands a share αi (01 ) = αi ∈ (0, 1) and never changes this.7 She concedes to her opponent’s demand at tk if and only if (1 − α j (tk )) ≥ αi . She never sends a message to the mediator indicating rationality. The behavioral demands of the two agents are incompatible, α1 + α2 > 1. The intuitive description of the game above does not define an explicit extensive form. I do that below using stopping times. The setup is somewhat technical, and the details are not important for a general interest reader. In later sections, I am able to reduce the strategy space considerably given particular behavioral strategies for the mediator. At each realized (non-terminal) private history hi for agent i, she chooses an action plan ai (hi ). An action plan ai (hi ) = (ti (hi ), xi (hi ), i) for agent i consists of three parts: a future time to take action, t(hi ); an action to take, xi (hi ); and a marker for agent i.8 A private history for agent i 6

Results generalize to any bargaining situation with a concave positive utility possibility frontier. In AG, agents can imitate multiple behavioral types (and announce demands sequentially). The results on mediation can be extended to that setting. 8 Actions planned for t1 , t2 , t5 are unambiguous, but actions at 01 , t3 , t4 must additionally specify a dollar divi7

5

is a finite sequence of action plans which she has observed. A belief system for bargainer i, µi , specifies a belief µi (hi ) ∈ [0, 1] that her opponent is a behavioral type, for each private history. Realized private histories for agent i are ordered h1i , h2i , ..., hiK , where K ∈ N ∪ ∞ and where we may have hki = hk+1 (to be explained shortly). The first private history is the null set, i h1i = ∅. Private history hk+1 is determined by the joint history hk = (hk1 , hk2 , hk3 ) as follows. Let i the time of the first action planned given hk be described by t(hk ) = min{t1 (hki ), t2 (hki ), t3 (hki )} ; let the set of action plans which i observes at t(hk ) be described by Ji (hk ) = { j : t(hk ) = t j (hkj ) and i can observe j0 s actions at t(hk )}; let ti (hki ) be the last time observed in history hki , k ∅ (this will be defined recursively). If Ji (hk ) = ∅ then hk+1 = hki and ti (hk+1 i i ) = ti (hi ). If k 9 k Ji (hk ) = {1, 2} then hk+1 = hki × (ai (hki ), a1 (hk1 ), a2 (hk2 )) and ti (hk+1 i ) = t1 (h1 ). If Ji (h ) = { j} then i k hk+1 = hki × (ai (hki ), a j (hkj )) and ti (hk+1 i i ) = t j (h j ). At time 01 bargainers must declare their initial demands so that ti (∅) = 01 for i ∈ {1, 2}. For an action plan to specify a “future” time to take action implies that ti (hki ) > ti (hki ). Agent i can plan to never take a future action by setting ti (hki ) = ∞. The outcome of the game is determined by a sequence of joint histories (h1 , h2 , ..., hK ) such that if k < K then t(hk ) < {t1 , t5 } for any t ∈ R+ and either t(hK ) ∈ {t1 , t5 } so that someone concedes at t(hK ), where existing demands are well defined; or K = ∞ or t(hK ) = ∞, so that nobody ever concedes. A pure strategy for agent i must specify an action plan for each possible private history. A mixed strategy randomizes between pure strategies. An equilibrium is a profile of strategies and belief systems such that rational bargainer i’s strategy is sequentially rational at each history given her beliefs, and her belief system satisfies µ( hi ) = 0 if hi is inconsistent with an opponent’s behavioral type and is determined by Bayes’ rule otherwise. Although it has been useful for the game’s description to describe the mediator as an agent, henceforth, I use the term agent to refer only to one of the two bargainers.

3

Baseline model

In the Baseline model, the mediator makes no announcements (we can ignore time t2 and t3 ). In this case, the results of AG imply that if agent i is revealed to be rational at time tk in equilibrium (i.e. just after i makes a non-behavioral demand), while agent j may be behavioral, then i’s (present discounted) continuation payoff is less than ui (1−α j ). This mirrors the logic of the Coase conjecture (Coase (1972)), one-sided incomplete information implies an immediate agreement favourable to the informed party. Given AG’s result, it is without loss of generality to assume that rational agents always imitate behavioral types and then simply choose when to concede. We can, therefore, move to fully sion. 9 If i ∈ {1, 2} then ai (hki ) is redundant in this definition, but does no harm.

6

continuous time and simply describe agent j’s strategy with a distribution function F j , with domain [0, ∞) ∪ ∞. Let F j (t) be the total probability that agent j has conceded before time t (not the probability that rational agent j has conceded before time t). Given that behavioral types never concede, the probability that a rational agent j has conceded before time t is then z F j (t) . Agent j’s implied reputation (for being behavioral) at t is then z¯ j (t) := 1−Fjj (t) . H j (t) = 1−z j Given agent j’s strategy, agent i’s expected payoff to conceding at t is: Ui (t, F j ) =

3.1

Z

e−ri s ui (αi )dF j (s) + (1 − F j (t))e−ri t ui (1 − α j ) s
Analysis

The unique equilibrium of the Baseline model is characterized by three properties: (i) at most one agent concedes with with positive probability at time zero; (ii) both agents reach a probability one reputation at the same time, T ∗ < ∞; and (iii) agents are indifferent to conceding at any time on (0, T ∗ ]. This third indifference condition implies that agent j must concede on the interval (0, T ∗ ] at rate: f j (t) ri ui (1 − α j ) = λ j := (1) 1 − F j (t) ui (αi ) − ui (1 − α j ) This implies that 1 − F j (t) = (1 − F j (0))e−λ j t . Next define rational agent j’s exhaustion time, T j , as the time by which she must have conceded even if she did not concede at time zero. This must satisfy e−λ j T j = z j . Condition (i) and (ii) then imply T ∗ = min{T 1 , T 2 }, and finally: λ jT ∗

1 − F j (0) = z j e

(

−λ j λi

= min 1, z j zi

) (2)

Proposition 1 (AG). The Baseline model has a unique equilibrium, characterized by equations 1 and 2. The fact that T ∗ > 0 implies that rational agents sometimes reach agreement only after an inefficient delay. This offers scope for mediation to improve outcomes. Equilibrium payoffs are: UiB = ui (αi )F j (0) + ui (1 − α j )(1 − F j (0)) (3)

4

Immediate One-shot (I1) mediation

In this section (as well as sections 5, 6 and 7), I consider a mediator who seeks agreement on a specific compromise, and suggests agreement on those terms only if both agents provisionally 7

accept it. In this section (and section 5) the mediator announces such a deal immediately when both rational agents have compromised. In this section (unlike in Section 5), agents can only compromise only at time zero, I call this the Immediate One-shot (I1) mediation protocol.10 At time 02 agents can compromise, if both do so then at time 03 the mediator suggests a division (m1 , m2 ). Following such a joint revelation of rationality, any dollar division or even perpetual delay is consistent with sequential rationality (e.g. agent i changes her demand to αi (04 ) ∈ [0, 1] and doesn’t concedes unless j offers her more than that). Nonetheless, it will ultimately be without loss of generality to assume that agents follow the mediator’s suggestion given the eventual negative result.11 Given this, we can again simplify to a continuous time framework. Let ci ∈ [0, 1] be the probability that agent i compromises at 02 (not the probability that a rational agent i compromises). I refer throughout the paper to an agent who compromises as a compromising agent and an agent who does not compromise as a non-compromising agent and use the superscripts c and n indicate this choice. If both agents compromise then agent i obtains the payoff ui (mi ), otherwise she must choose when to concede. Her concession choice is described by two distribution functions Fic and Fin . Let Fic (t) be the probability that agent i has conceded to her opponent before time t conditional on her compromising and no mediator suggestion. Similarly, let Fin (t) be the probability that agent i has conceded before time t in the war of attrition, conditional on her not compromising. Finally, let Fi (t) = ci Fic (t) + (1 − ci )Fin (t) be the probability that agent i has conceded by time t conditional on no mediator suggestion. I have not included subscripts or superscripts on Fi to indicate that it may be distinct from the function used to describe the Baseline model’s equilibrium. This is done in order to reduce notation but should not cause confusion. Agent i’s strategy is summarized by σi = (ci , Fic , Fin ).12 If agent j adopts a strategy σ j then rational agent i’s utility from compromising and then conceding at time t if the mediated makes no suggestion, is: Uic (t, σ j )

=c j ui (mi ) + (1 − c j )

Z

e−ri s ui (αi )dF nj (s) + (1 − F nj (t))e−ri t ui (1 − α j ) s
10

This mediation protocol and that of Section 5 are not-nested. Expected continuation payoffs must be weakly below those associated with some mediator proposal. I show that even the prospect of those higher payoffs cannot incentivize joint compromise. 12 There are potentially relevant, non-degenerate higher order beliefs in this game. If agent i compromised (but j zj didn’t) then at time t, i believes j is behavioral with probability zcj (t) := (1−c j )(1−F n (t)) . If i did not compromise, then 11

at time t she believes j is behavioral with probability znj (t) :=

zj 1−F j (t) .

j

If j did not compromise, then she believes

that i beliefs about her likelihood of being behavioral are zcj (t) with probability

8

ci (1−Fic (t)) 1−Fi (t)

and znj (t) otherwise.

Alternatively, rational i’s utility if she does not compromise and concedes at time t is: Uin (t, σ j )

4.1

=

Z

e−ri s ui (αi )dF j (s) + (1 − F j (t))e−ri t ui (1 − α j ) s
Analysis

It is clear that the Baseline model’s equilibrium can still be an equilibrium here, indeed this is the case in all mediation protocols considered. If agent j does not compromise with positive probability then agent i has no incentive to do so either. It is also clear that there can be no equilibrium in which rational agent i always compromises and j does so with positive probability. If there was then agent j would learn for sure that i was behavioral if the mediator made no announcement, and so would subsequently immediately. Knowing this, a rational agent i would optimally reject the compromise unless mi ≥ αi . But if m j ≤ 1 − αi then U cj (t, σi ) < U nj (t, σi ) for all t. Any equilibrium with mediation, therefore, must involve rational agents mixing between compromising and not compromising. The next proposition says that there is no such equilibrium. Proposition 2. The distribution of outcomes in any equilibrium of the I1 model are identical to those in the unique equilibrium of the Baseline model.13 The explanation for this result is similar to why it is impossible for rational agents to always compromise. I pprove that if the mediator does not suggest an agreement (at 03 ), then at least one compromising agent, say j, must immediately concede with probability one (F cj (0) = 1). I loosely sketch the proof of that claim below. Such concession destroys the incentive for her opponent to compromise in the first place. Suppose neither compromising agent immediately concedes with probability one. Standard arguments show that concession behavior after time zero must be continuous. I then show that if a compromising agent i is willing to concede at any time on some interval (s, t), then so must a non-compromising agent i. Indifference conditions for these agents respectively require that f n (t) a non-compromising agent j must concede at rate 1−Fj n (t) = λ j and total concession for agent j j

must be at rate f jc (t)

f j (t) 1−F j (t)

= λi . In turn that implies a compromising agent j must concede at rate

= λ j . But such a bounded concession rate would imply that a (rational) compromising agent j never concedes with probability one in finite time. Such behavior cannot be optimal for a rational agent. 1−F cj (t)

13

I refrain from saying that the equilibria are identical only because strategies are described by different objects.

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5

Immediate Infinite (I∞) mediation

One concern about the negative result in the I1 model is that mediation is discontinuous, happening once and for all at time zero. It might be thought that allowing agents to compromise continuously over time might allow for greater success. After all, in practice, mediators often hold multiple conferences with disputing parties before helping them reach a settlement. This section considers such a model of mediation, I call it the Immediate Infinite (I∞) mediation model. Agents can now compromise at any t2 with t ∈ [0, ∞). If agent i compromises at time s2 and agent j accepts the compromise at t2 ≥ s2 , then at t3 the mediator suggests the agreement (m1 , m2 ) with mi . I focus on what I call I∞ equilibria. An I∞ equilibrium is an equilibrium of the I∞ model in which rational agents follow the mediator’s suggestion (changing their demands to (m1 , m2 ) at t4 ). Focussing on such equilibria may entail some loss of generality because this imposes constant continuation payoffs following an announcement by the mediator (we could imagine that such continuation payoffs change over time and depend on which agent compromised first). My focus on such equilibria is for reasons of tractability. In an I∞ equilibrium, it is without loss of generality to assume mi ∈ (1 − α j , αi ) because if mi ≥ αi then rational i will compromise with probability one at 02 (as this can only increase her payoff) causing the game to quickly unravel into a standard war of attrition. In an I∞ equilibrium, agent i’s strategy reduces to choosing a time to compromise and a time to concede (to her opponent’s behavioral demand). It is without loss of generality to assume that an agent never concedes at t1 but only at t5 , and compromises before she concedes (because doing so strictly increases her payoff whenever it affects the game’s outcome). We can again, therefore, analyze the game in continuous time. Agent i’s strategy is described by two distribution functions, σi = (Fic , Fid ). Let Fic (t) be the probability that agent i has compromised before time t, and Fid (t) be the total probability that agent i has conceded before time t (c=compromised, d=defeated) where Fic (t) ≥ Fid (t). If agent j adopts strategy σ j then rational agent i’s utility from planning to compromise at time s and concede at time t ≥ s is: Ui (s, t, σ j ) =

Z −ri v

e

ui (αi )dF dj (v)

+

v
+ (1 −

5.1

Z e−ri v ui (mi )dF cj (v) v∈(s,t]

F cj (t))e−ri t ui (1

− α j ) + (F cj (s) − sup F dj (v))e−ri s ui (mi ) v
Analysis

Again, clearly the Baseline model’s equilibrium, still exists here when setting Fic (t) = Fid (t). Despite agents always compromising before conceding, there is zero probability that the compromise is agreed. Are there any other I∞ equilibria, however? The next proposition says

10

no. Proposition 3. The distribution of outcomes in any I∞ equilibrium are identical to those in the unique equilibrium of the Baseline model. The proof of this result is structurally similar to the proof of Proposition 2, if somewhat more involved. It shows that if ever Fic (t0 ) > Fid (t0 ), then this must be true on an non-degenerate interval [t0 , t00 ) where Fic (t00 ) = Fid (t00 ). Concession and compromising behavior must be increasing and continuous on (t0 , t00 ]. The conditions for a compromising agent and non-compromising agent i to be indifferent regarding their compromise and concession times, imply linear ODE that govern F cj and F dj . Those imply that F cj (t) − F dj (t) > 0 on (t0 , t00 ]. In turn, that implies that rational agents won’t concede in finite time, a contradiction.

6

Partial One-Shot (P1) mediation

In both the I1 and I∞ an agent who compromises immediately receives an unambiguous signal about whether her opponent done likewise. If the mediator does not suggest an agreement that signal is bad news: an opponent is more likely to be behavioral. This causes the agent to subsequently immediately concede, destroying the incentive for her opponent compromise in the first place. This section presents a simple way to limit such bad news by having the mediator sometimes fail to make an announcement even when both parties compromise. I call this the Partial One-Shot (P1) mediation model. An equivalent interpretation of this model is that the mediator always suggests when she knows that both agents have compromised, but sometimes agents’ messages, indicating a willingness to compromise, go astray or are misinterpreted. As in the I1 model agents can only compromise at 02 . If both agents compromise then at 03 the mediator suggests the agreement (m1 , m2 ) with probability b ∈ (0, 1), and otherwise remains silent. I focus attention on what I call P1 equilibria. A P1 equilibrium is an equilibrium of the P1 game in which rational agents always compromise and implement the suggestion of the mediator. As in the I1 model, the assumption that agents implement the mediator’s suggestion is with minimal loss of generality. If the mediator makes no suggestion, then agents must decide when to concede. We can again describe strategies in continuous time. As before, let ci ∈ [0, 1] describe the probability that agent i compromises (so that ci = 1 − zi in a P1 equilibrium). If both agents compromise, then agent i obtain a payoff ui (mi ). Let Hic be a distribution function such that Hic (t) is the probability that a rational agent i has conceded before time t conditional on compromising and the mediator making no suggestion. Similarly, let Hin (t) be the probability that a rational agent i has conceded before time t, conditional on not compromising. Agent i’s strategy is summarized by σi = (ci , Hic , Hin ). 11

In a P1 equilibrium, agent i’s utility if she compromises and concedes at time t is: Uic (t, σ j )

!

Z

=(1 − z j ) bui (mi ) + (1 − b) e s
ui (αi )dH cj (s)

Agent i’s utility if she does not compromise and then concedes at time t is: Uin (t, σ j )

6.1

Z

  e−ri s ui (αi )dH cj (s) + (1 − z j )(1 − H cj (t)) + z j e−ri t ui (1 − α j ) s
=(1 − z j )

Analysis

In a P1 equilibrium, if the mediator does not make a suggestion (at 03 ) behavior in the continuation game must resemble the unique equilibrium of the Baseline model but with initial zi instead of zi . As noted previously, this equilibrium is characterized reputations of z¯i = 1−(1−z i )b by three conditions: (i) at most one agent concedes with with positive probability at time zero; (ii) both agents reach a probability one reputation at the same time, T ∗ < ∞; and (iii) agents are indifferent to conceding at any time on (0, T ∗ ]. Let F j (t) = (1 − z¯ j )H cj (t) be the probability that (a compromising) agent i believes that j will concede before t conditional on no mediator announcement. Condition (iii) implies that agent f (t) i must expect j to concede at rate 1−Fj j (t) = λ j on (0, T ∗ ]. Agent j’s exhaustion time is now T j = − λ1j ln(¯z j ). To ensure conditions (i) and (ii) are satisfied, we must have T ∗ = min{T 1 , T 2 } n z¯ j − λλ j
o c and 1 − H j (0) = max 1, 1−¯z j z¯i i − 1 . Rational concession generally satisfies: 1 − H cj (t) = (1 − H cj (0))e−λ j t −

z¯ j (1−e−λ j t ) . 1−¯z j

Given such behavior, a rational agent i who did not compromise (off the equilibrium path) will subsequently find it in her interest to wait until T ∗ and then concede. This is because the probability that she expects j to concede before t is larger if she compromised than if she did U n (t,σ ) not (1 − z j )H cj (t) ≥ (1 − z¯ j )H cj (t), implying i dt j > 0 on [0, T ∗ ) Hence we must have: Ui∗n

:=

max Uin (t, σ j ) t

= (1 − z j )

Z

∗

e−ri s ui (αi )dH cj (s) + z j e−ri T ui (1 − α j ) s
When agent i’s compromises, she is subsequently indifferent to conceding at any t ∈ (0, T ∗ ] so

12

that: Ui∗c

:=

max Uic (t, σ j ) t

= (1 − z j ) bui (mi ) + (1 − b)

!

Z e s
−ri s

ui (αi )dH cj (s)

∗

+ z j e−ri T ui (1 − α j )

A necessary and sufficient condition for a P1 equilibrium to exist therefore is: U ∗c − Ui∗n Qi := i = ui (mi ) − (1 − z j )b

Z s
e−ri s ui (αi )dH cj (s) ≥ 0

(4)

That is, the share proposed by the mediator must be better than the stream of payoffs from a known rational agent’s concession on [0, T ∗ ). The main result of this section, below, shows that when agents’ reputations are sufficiently small, a P1 equilibrium always exists which Pareto dominates the equilibrium of the Baseline model. Proposition 4. For any given ri , ui , αi for i = 1, 2, b ∈ (0, 1) and fixed K ≥ 1, there exists z > 0 such that if zi ≤ z and K ≥ zz12 ≥ K1 , then there is a P1 equilibrium, which rational agents strictly prefer to the Baseline equilibrium. Some intuition for the result comes from examining equation 4. If agent i does not compromise agent she sacrifices an immediate payoff of ui (mi ) in return for the stream of payoffs R e−ri s ui (αi )dH cj (s). That stream of payoffs comes relatively slowly when initial reputations s
by compromising an agent effectively gains an immediate payoff of ui (mi ) by loses a delayed payoff of ui (αi ). When zi ≈ 1, however, there is little delay, whether or not agents compromise. For large initial reputations, therefore, equation 4 requires that mi ≈ αi for both agents, which is impossible. Proposition 5. For any given ri , ui , αi for i = 1, 2 there exists z < 1 such that if z1 ≥ z, then there is no P1 equilibrium. There are further limits on what the mediator can achieve in a P1 equilibrium, even when behavioral are unlikely. The proof of Proposition 6 constructs a particular compromise division (m1 , m2 ) which ensures that both agents are willing to compromise. The next Proposition, shows that if agent j’s Baseline equilibrium concession rate exceeds that of agent i, λ j ≥ λi , then the mediator must offer agent i at least as high a payoff in a mediated outcome as she could obtain in the Baseline model’s equilibrium. Proposition 6. If λ j ≥ λi , then in any P1 equilibrium, the mediator must offer agent i at least her payoff from the Baseline equilibrium, ui (mi ) > UiB . The result is established by showing that when λ j ≥ λi concession by rational agent j is faster in a P1 equilibrium than in the Baseline equilibrium (in terms of first order stochastic dominance of concession times). This in turn means that i’s payoff when she does not compromise, Ui∗n , must exceed her Baseline equilibrium payoff, UiB . Unless ui (mi ) > UiB , therefore, agent i will not compromise. Although, behavioral agents have no preferences, this faster speed of rational agent concession when λ j ≥ λi , suggests that behavioral agent i would “prefer” a P1 equilibrium to the Baseline equilibrium. When λi = λ j all agents of all types would then prefer mediation.

7

Delayed One-shot (D1) mediation

Proposition 6 illustrated that mediators in the P1 model face significant constraints on which compromise divisions can be implemented. This section shows how that such constrains can be relaxed if the mediator sometimes delays suggesting an agreement. I call this the Delayed One-shot mediation (D1) model. As in the I1 and P1 models, agents can only compromise at time 02 . If both agents compromise, then at 03 the mediator suggests an agreement (m1 , m2 ) with probability b ∈ [0, 1), and conditional on not doing so makes this same suggestion at some randomly determined time t3 > 03 (so long as neither agent has revealed rationality prior to t3 ).14 The distribution of mediator suggestion after 03 is described by the continuous distribution function G so that so that G(t) 14

If either agent changes her demand prior to t3 then the mediator remains silent for the rest of the game.

14

is the probability that the mediator has suggested agreement before t3 conditional both agents compromising and the mediator not suggesting agreement at 02 . Let g(t) be the associated density of suggestions at t. I focus attention on what I call D1 equilibria. A D1 equilibrium is an equilibrium of the D1 game in which rational agents always compromise, implement any mediator suggestion, and don’t concede if there remains some possibility that the mediator may still suggest an agreement. That is, let T ∗ = min{t : G(t) = 1}, then a rational agent will not concede to a behavioral opponent before T ∗ . Focussing on D1 equilibria involves some loss of generality, in particular because this imposes a constant continuation payoff after a mediator’s suggestion. Given the focus on such equilibria, however, it is without loss of generality to assume that mi ∈ (1 − α j , αi ).15 G Rational agent i’s strategy in a D1 equilibrium can be described by the same three objects, σi = (ci , Hic , Hin ), which describe her strategy in a P1 equilibrium. In a D1 equilibrium agent i’s expected utility if she compromises and then concedes at time t ∈ [0, ∞) (whether at t1 or t5 ): Uic (t, σ j )

Z

! ui (mi )dG(s)

=(1 − z j ) bui (mi ) + (1 − b) e s≤t   + (1 − z j )(1 − b)(1 − G(t)) + z j e−ri t ui (1 − α j ) −ri s

I assume that in a D1 equilibrium, that agent j concedes at exactly T 1∗ (when there is zero probability of a mediator announcing a deal),16 which implies that the payoff to agent i if she does not compromise and concedes at some time t5 is:   −r t ∗   ui (1 − α j )e i if t < T n Ui (t, σ j ) =    e−ri T ∗ (1 − z j )ui (αi ) + e−ri t z j ui (1 − α j ) if t ≥ T ∗

7.1

Analysis

The main result of this section (Proposition 7 below) shows that a mediator can implement any compromise division with probability arbitrarily close to one, if behavioral types are sufficiently unlikely. Proposition 7. For any given ri , ui , αi , mi ∈ (1 − α j , αi ) for i = 1, 2, b ∈ [0, 1), there exists z > 0 such that whenever zi ≤ z, there exists some distribution of delayed mediated agreements G such that there is a D1 equilibrium. Because if mi ≤ 1 − α j then agent i will receive a payoff that is strictly less than ui (1 − α j ) if she doesn’t concede until T ∗ > 0. 16 This assumption ensures that agent i’s (off equilibrium path) continuation strategy is well defined when she does not compromise. 15

15

The logic behind this result is that by promising an agent a steady stream of compromise agreements (if she faces a rational opponent), the mediator can persuade her not to concede for a long period of time (T ∗ can be made large when behavioral types are unlikely). This ensures that more of the bargaining surplus is shared between agents who “do the right thing”, to the exclusion of those who do not compromise. Faced with such exclusion, a non compromising agent may subsequently find that the best she can do is to concede immediately after time zero. A mediator therefore, only needs to promise the agent slightly more than she could get by conceding immediately. To develop these ideas further, notice that a necessary condition for a D1 equilibrium to exist is that a compromising agent finds it optimal to subsequently concede at T ∗ . To satisfy this U c (t,σ ) necessary condition it is sufficient that i dt j ≥ 0 on (0, T ∗ ), which implies zj g(t) ≥ γ j (1 − G(t)) + (1 − z j )(1 − b) where γ j :=

! (5)

ri ui (1 − α j ) . ui (mi ) − ui (1 − α j ) z

j To understanding equation 5 better, recall that z j = 1−(1−z is compromising agent i’s belief j )b that j is behavioral immediately after observing no mediator suggestion at time zero. Let F j (t) = G(t)(1 − z j ), be the probability that a compromising agent i believes that the mediator will make an announcement before t conditional on no announcement at 02 . Equation 5 simply f (t) says that the expected agreement rate from i’s perspective must be 1−Fj j (t) ≥ γ j .

n

A second necessary condition for a D1 equilibrium to exist is for Uic (T ∗ , σ j ) ≥ U i := maxt∈[05 ,∞) Uin (t, σ j ). n ∗ Notice that U i ≤ max{ui (1−α j ), e−ri T ui (αi )}. Agent i’s payoff if she compromises in a D1 equilibrium is certainly greater than ui (1 − α j ). For a D1 equilibrium to exist therefore, it is sufficient to find some G that satisfies equation 5 ∗ and implies T ∗ such that e−ri T ui (αi ) ≤ ui (1 − α j ). For sufficiently small z j , this is always posg(t) = max{γ1 , γ2 } sible. To see this, notice that if z j = 0 then a constant announcement rate 1−G(t) ∗ ∗ would satisfy equation 5 and imply T = ∞. For zi ≈ 0 therefore, T is arbitrarily large. Proposition 7 does not guarantee that rational agents prefer D1 equilibria to the Baseline equilibria. By extending the model to allow the mediator to make different suggestions at different times (in particular at time zero and afterwards) it is possible to demonstrate the existence of a D1 equilibrium, which is a Pareto improvement. Taking things in a different direction, it is possible to demonstrate the existence of a D1 equilibrium which is Pareto inferior to a D1 equilibrium. For instance, suppose that α1 = 0.9, α2 = 0.6, zi = z < 0.2, ri = 1 and ui (x) = x for i = 1, 2. For such parameters, there is a D1 z equilibrium with b = 0,g(t) = (1 − G(t)) + 1−z and m1 = 1 − m2 = 0.8 that gives each player i

16

an equilibrium payoff of 1 − α j . In the Baseline equilibrium, by contrast, U2B > 1 − α1 because agent 1 concedes with positive probability at time zero. The final result of this section establishes that P1 equilibria cannot exist when the probability of behavioral types is large. The intuition and proof of this result are comparable to Proposition 5. Proposition 8. For any given ri , ui , αi for i = 1, 2, there exists z < 1 such that if z1 ≥ z, then there is no D1 equilibrium.

8

General One-shot (G1) mediation

The two positive results above for the P1 and D1 models (Propositions 4 and 7) require that behavioral types are unlikely. Those mediation protocols stuck to the basic logic of mediation outlined by Dunlop: a mediator seeks support for a compromise proposal and (sometimes) announces an agreement only if both sides provisionally accept it (in private). In this final section I consider a more general strategy set for the mediator, who can suggest any agreement at any time regardless of whether or not both sides have compromised.17 I call this the General One-shot (G1) mediation model. I show that whenever the utility possibility frontier is strictly concave, then mediation allows for Pareto improvements and reduced delay, regardless of the likelihood that agents are behavioral types. As in the I1, P1 and D1 models, agents can only compromise at time 02 . If both agents compromise, then at some randomly determined t3 the mediator suggests some agreement (so long as neither agent has revealed rationality prior to t3 ). The distribution of announcement times conditional on joint compromise is described by GR , so that GR (t) is the probability that the mediator suggests an agreement before t3 (R=rational). Conditional on joint compromise and suggesting an agreement at t3 the mediator suggest that player i should get a randomly determined dollar share xit . The conditional distribution of xit is described by the distribution function S it , so that S it (x) is the probability that the mediator suggests that i gets less that x. If agent i compromises and agent j does not, then the mediator suggests an agreement (1 − α j , α j ) at some (randomly determined) time t3 . The distribution of announcement times conditional on agent i alone compromising is described by Gi , so that Gi (t) is the probability that the mediator suggests an agreement before t3 . If neither agent compromises, then the mediator stays silent. I focus on G1 equilibria. A G1 equilibrium is an equilibrium of the G1 game in which rational agents always compromise, and subsequently don’t concede until the mediator suggests an agreement. Focussing on such equilibria involves minimal loss of generality, given the generality of the mediator’s strategy. Rational agent i’s strategy in a D1 equilibrium can be described 17

There are still more strategic options for a mediator than those considered here, such as sometimes providing partial information about the likelihood that an opponent is behavioral.

17

by the same three objects, σi = (ci , Hic , Hin ), which describe her strategy in a P1 equilibrium. In a G1 equilibrium agent i’s expected utility if she compromises and then concedes at t5 (if there has been no mediator announcement) is:

Uic (t, σ j )

=(1 − z j )

Z

Z

e

ui (xis )dS s (xis )dGR (s)

+ zj

Z

e−ri s ui (1 − α j )dGi (s) s≤t s≤t   −ri t R + e ui (1 − α j ) (1 − z j )(1 − G (t)) + z j (1 − Gi (s)) −ri s

Agent i’s expected utility if she does not compromise but concedes at t ∈ [0, ∞) is: Uin (t, σ j )

= (1 − z j )

Z

  e−ri s ui (αi )dG j (s) + e−ri t ui (1 − α j ) (1 − z j )(1 − G j (t)) + z j ) s≤t

8.1

Analysis

I first claim that we can implement the Baseline model’s distribution of equilibrium outcomes ∗ zi in a G1 equilibrium. Set T ∗ := min{− λ11 ln(z1 ), − λ12 ln(z2 )}, 1−Gi (t) = 1−z (eλi (T −t) −1), 1−GR (t) = i (1 − G1 (t))(1 − G2 (t)) and finally:     0 if xit < 1 − α j      S it (xit ) =  qi (t) if xit ∈ [1 − α j , αi )       1 otherwise where qi (t) :=

dGi (t)(1 − G j (t)) dGi (t)(1 − G j (t)) + dG j (t)(1 − Gi (t))

(6)

That is, conditional on agreement at time t, agent i gets (1 − α j ) with probability qi (t) and αi otherwise. The expression for Gi (t) is exactly the probability that rational agent i has conceded before time t in the Baseline equilibrium (expressed in a different way). Hence, in the Baseline equilibrium the probability of agreement before t conditional on both agents being rational is GR (t) with the distribution of agreements at t described by S it , just as here. Given this, it is dU n (t,σ ) dU c (t,σ ) simple to verify that i dt j = i dt j = 0 for t ∈ [0, T ∗ ] and Uic (0, σ j ) = Uin (0, σ j ) = UiB so that we do indeed have a G1 equilibrium. Given this, it is fairly easy to see how the mediator can bring about Pareto improvements when the utility possibility frontier is strictly concave. Conditional on suggesting an agreement at R time t between rational types, suppose the mediator always suggested mti = xit dS t (xit ). If R each agent i has a strictly concave utility function, then ui (mti ) > ui (xit )dS t (xit ) for t ∈ (0, T ∗ ]. Making only this change, while keeping Gi , G j and GR as before, rational agents would have a strict incentive to compromise and then not concede on (0, T ∗ ), unless suggested to do so by the 18

dU c (t,σ )

dU n (t,σ )

mediator. That is, Uic (t, σ j ) > Uin (t, σ j ) = UiB and i dt j > i dt j = 0 for t ∈ (0, T ∗ ). Such compromise announcements, therefore, allow for a G1 equilibrium with strictly larger payoffs than the Baseline equilibrium. Given strict incentives for rational agents to compromise and not concede (unless instructed) on (0, T ∗ ), the mediator can increase GR (0) and reduce gR (t) on (0, T ∗ ) (without changing mti or Gi ) while preserving incentives to compromise and not concede. That is, the mediator can use the slack created by mitigating the inefficiency caused by extreme allocations, to reduce delay (the other form of equilibrium inefficiency). The paper’s final result establishes that the mediator can improve the payoffs of both rational agents and reduce delay, even when only one agent is risk averse (by giving the risk neutral agent slightly more of the dollar). Proposition 9. Suppose that u01 (1 − α2 ) > u01 (α1 ) then there is a G1 equilibrium with strictly less delay and strictly higher payoffs for rational agents, than the equilibrium of the Baseline model.

9

Discussion and literature review

There is an inherent difficulty for mediators in trying to persuade bargainers to compromise, even if in private. If an agent refuses to compromise, then an opponent will learn she is more likely to be committed to her bargaining position, increasing that opponent’s incentive to fully concede (the agent’s preferred outcome). The paper has shown that in a reputational bargaining model, this problem is so severe that if a mediator announces an agreement immediately whenever both parties compromise, then mediation is completely ineffective (Propositions 2 and 3). While these results have an intuitive rationale, they are somewhat surprising. Jarque et al. (2003) illustrate an equilibrium with mediation when there is incomplete information about reservation values and a dynamic bargaining protocol similar to my I∞ model. In their continuous time model, agent i gets utility e−rt (xi − si ) when she gets a share xi of the dollar at time t and has reservation value si ∈ [siL , siH ]. A war of attrition equilibrium always exists in their setting, which focusses only on two possible agreements (agent i demands xi > siH ). They consider a mediator who announces a compromise agreement immediately (giving agent i, mi ∈ (1 − x j , xi )) when both parties have privately indicated to the mediator that they are willing to accept it.18 Their main result is that when fundamentals are symmetric, there is an equilibrium with mediation (where which types with mi > si eventually compromise) if and only if 18

The model allows for any finite number compromise alternatives, with agents successively indicating partial acceptance to the mediator, but results with more than one compromise alternative are only provided for a uniform ˇ c and Ponsat´ı (2008) extend the model setup to allow for a continuum of distribution of reservation values. Copiˇ possible agreements, and illustrates the existence of an existence of an equilibrium which is ex-post efficient.

19

the fraction of types willing to concede in the war of attrition is sufficiently small (siL ≈ 1 − x j ). This can provide an (ex-ante) Pareto improvement over the war of attrition equilibria. A first thing to note about Jarque et al. (2003)’s results, is that they suggest something quite different to my results about when mediation is successful. They demonstrate successful mediation only when the fraction of “flexible” agents (with low reservation values) is small. Propositions 4 and 7, by contrast, which demonstrate mediation in the P1 and D1 models, require that the fraction of flexible (rational) agents is large. A more important difference, is that is possible for a mediator to be successful by announcing agreements immediately (when both parties have accepted) in the reservation value model, but not in the reputational model. Why? The reason is that in the reservation value model, the mediator facilitates agreement between types who never agree in a war of attrition (with only two alternatives). Suppose that xi = 0.7 then types s1 = s2 = 0.4 will never concede in the war of attrition. By introducing the alternative mi = 0.5, such types can reach agreement. The extra payoffs created by such agreements add enough grease to the system to overcome the inherent difficulties of mediation. In the reputational model, by contrast, introducing a compromise does not expand the set of types who are ultimately willing to agree. An important draw back of the reservation value model is that it is not clear what the set of equilibria are without a mediator. Ponsati (1997) shows that if the game rules allow only three alternatives agreements and strategies are Markov, then at least one alternative will not be used (i.e. there is a war of attrition). However, both constraints on that result are substantive. In particular, she also shows that it is possible to construct equilibria with non-Markov strategies, in which all three alternatives are used (i.e. agents compromise) and that these provide equilibria can be ex-ante Pareto superior to the war of attrition. Jarque et al. (2003) do not compare such non-Markov compromise equilibria to their equilibrium with mediation. The uniqueness of the equilibrium in the reputational model, highlights the role of a mediator more clearly. It seems intuitive that a mediator should announce an agreement immediately if both parties compromise. The problem, highlighted by my negative results, is that this gives an agent who compromises an unequivocal signal that her opponent has not. Propositions 4 and 7 show that simple extensions, which add different forms of noise to this protocol (sometimes making no announcement or delaying an announcement) can help resolve this problem. The claim that a mediator may need to add noise in addition to filtering information is not new. For instance, Goltsman et al. (2009) investigate the potential for mediation, arbitration and negotiation (finitely many rounds of communication with no discounting) to improve receiver payoffs in a cheap talk game. They show that both mediators and arbitrators should optimally filter information, but mediators should also add noise. Goltsman et al. (2009) further show that arbitration is (generically) more effective than mediation, while mediation is only sometimes more effective than communication. H¨orner et al. (2015) by contrast show that arbitration and

20

mediation are equally effective at deterring conflict in a simple game in which parties choose whether to go to war. In light of those papers, what can we say about how mediation compares to arbitration in the reputational bargaining game? A problem answering that question, is that it is not clear how an arbitrator should interact with behavioral types or what constraints she should face. Typically bargainers have to be willing to accept the arbitration process (before they know the arbitrator’s recommendation). The difficulties involved in getting both parties to sign up to that process might then seem entirely comparable to those of mediation (if either party refuses arbitration, that may signal that she is committed to her demand). One advantage of mediation over arbitration, is that it may not need to be done through a formal process. Instead, the mediator may simply conduct private conversations with the interested parties, which they can choose to ignore. The expectation that an agent will talk (in private) to a mediator, can then make it worthwhile for her opponent to do likewise. Indeed, as highlighted by Proposition 7, this means that agents are not guaranteed larger expected payoffs than they could obtain without mediation. Let us suppose, however, that an arbitrator can fully determine the terms and timing of agreement, except that she cannot impose an agreement on a (claimed) behavioral type i, which gives her less than αi . Further suppose that behavioral types report their type truthfully. In that case, it would seem that a mediator could never be as effective as an arbitrator. An arbitrator seeking to minimize delay would impose immediate agreement if both agents reveal rationality, as this can only improve the incentive to reveal rationality. A mediator could never suggest that because it would mean that a rational agent would know that her opponent was behavioral immediately after time zero (and so concede). By pretending to be behavioral, therefore, each rational agent i could guarantee (1 − z j )ui (αi ) + z j ui (1 − α j ), which is jointly infeasible. Proposition 4, 7 and 9 together show that giving the mediator more tools allows her to do more. While not perhaps surprising as a general conclusion, the results do also highlight some different techniques which can be effective. The G1 model moves slightly away from Dunlop’s description of a mediator’s role: in addition to seeking agreement on a compromise, the mediator suggests when one party should concede. This additional role does seem in line with the practice of some mediators who spend time spend time persuading parties to be more “realistic”, because an opponent will not (is unlikely to) back down. Proposition 9 highlights a source of inefficiency in war of attrition which is typically neglected: extreme proposals. Each agent sometimes concedes to her opponent and is sometimes conceded to, when both may strictly prefer a compromise for sure. By mitigating this source of inefficiency, the mediator can reduce the other source of inefficiency, delay. In the dollar division setup, a strictly concave utility frontier comes from risk aversion, but the result is more general. It seems to match up with the importance mediators place on finding “win-win” agreements (not compared to an agent’s first best demand, but compared to an expected outcome).

21

Two of the paper’s positive results (Propositions 4 and 7) require behavioral types to be unlikely. In AG’s model, if the fraction of behavioral types is small then (generically) agents agree with high probability at time zero even without a mediator. This might seem to suggest that the benefits of mediation may be small. That conclusion is not clear, however, as the facts are consistent with a large benefit of mediation when behavioral types are unlikely but not vanishingly unlikely. Moreover, Fanning (2015) shows that if there is uncertainty about agents future costs of delay, then even tiny reputational forces can cause substantial delay. My results about the benefits of mediation generalize to that setting and can certainly be substantial.19 The discrete continuous time bargaining model considered in the paper allows for relatively easy analysis. It may be thought that this involves a substantial loss of generality using this protocol because after the mediator reveals that both agents are rational any surplus division is consistent with equilibrium behavior, but with many discrete time bargaining protocols, this is not the case. In an alternating offer model with period length ∆, for instance, it seems agent 1−e−r j ∆ i should revert to her Rubinstein demand, αRi (e.g. αRi = 1−e −(ri +r j )∆ with risk neutral agents). While I cannot claim that there is no loss of generality, it seems that the substantive results will go through for discrete time models with frequent offers. I sketch below how a mediator who knows whether agents are rational can approximately implement any compromise division in an alternating offer model with frequent offers. Suppose the mediator wants to implement a division with m1 ∈ (1 − α2 , αR1 ) when both agents are rational, and wants agent i to concede if she alone is rational. If agent 1 is behavioral, the mediator announces this fact before period 1. If agent 2 is behavioral then the mediator announces this before period 1 with probability (1 − ε) for ε > 0 small. Furthermore, if agent 1 makes a demand m1 in period 1, then the mediator will reveal whether 2 is rational before period 2, and otherwise remain silent. Anticipating this, rational agent 2 will accept 1 − m1 as u2 (1 − m1 ) > e−r2 ∆ u2 (αR2 ). If the mediator makes no announcement before period z jε ≈ 0. Hence, 1 will demand 1, then 1 believes 2 is behavioral with probability z j = z j ε+(1−z j) m1 because this gives her a payoff of (1 − z2 )u1 (m1 ) + z2 e−r1 ∆ u1 (1 − α2 ) ≈ u1 (m1 ), whereas making any other demand will leave a game of one-sided incomplete information, which gives her only marginally more than u1 (1 − α2 ) when offers are frequent (due to the logic of the Coase conjecture). Beyond the papers already mentioned, the literature on mediation in bargaining is small. One avenue of research considers the possibility a third party can provide additional resources to the bargainers. For instance, Manzini and Ponsati (2006) show that bargainers may delay agree19

It might also seem problematic that for my positive results the mediator needs to be committed to not always immediately suggesting an agreement when both sides compromise. However, even if the mediator does have an ex-post incentive to announce an agreement, it may be possible for her to gain a repeated game reputation for not doing so (mediators typically are long-run players). Moreover, mediators are typically paid by the hour and not by performance, hence the incentive for an early agreement is unclear. Finally, it is possible the the noise needed for mediation to work may come from agents’ trembles (highlighted in Section 6) or just the incompetence of some mediators.

22

ment in a complete information alternating offer model until a third party, with has a stake in the outcome, arrives. They do this in order to extract additional resources from the stakeholder. My mediator has no additional resources, however, to the extent that she can improve an agent’s payoff above what she could get from immediately conceding, bargainers may choose to hold out initially for the prospect of future mediation. Basak (2016) explores a different question related to third-party intervention in a model that is very similar to reputational bargaining. His third party can reveal her own exogenous information about the likelihood that an agent is committed to her demand. He finds that if the third party only has access to moderately informative signals, then the intervention may increase expected delay.

10

Appendix

Proof of Proposition 2. Suppose there is an equilibrium σ = (σ1 , σ2 ) with ci c j > 0. Let Aci = {t : Uic (t, σ j ) = max s Uic (s, σ j )} and Ani = {t : Uin (t, σ j ) = max s Uin (s, σ j )}. Since σ is an equilibrium, Ani , ∅ , Aci . Define T ic as the final time by which a compromising agent i concedes to her opponent: T ic = inf{t : Fic (t) = 1}. Similarly, define T in as the final time a rational, non-compromising agent i concedes to her opponent: T in = inf{t : ci (1− Fin (t)) = zi }. Finally, define T ∗ = max{T cj , T nj , T ic , T in } and min{T ic , T cj } = T c . (a) We must have T cj ≤ T in < ∞ To establish T in = T cj suppose instead that T in < T cj then after time T in a compromising agent j knows that she faces a behavioral opponent, and so would prefer to concede immediately rather than wait until T cj . To establish T in < ∞, let πtj be the conditional probability that agent j continues to act consistent with a behavioral type on the interval [s, s + t) for arbitrary s. For agent i not to concede at s it must be that: ui (1 − α j ) ≤(1 − πtj ) + πtj e−ri t πtj ≤

u j (α j ) 1 − e−ri t

where the second line simply rearranges the first. Fix δ ∈ (u j (α j ), 1), and consider K such that δK < zi and u (α ) t0 such that δ = 1−ej −rji t0 . Suppose agent i did not to concede on the interval [0, t0 K) then it must be that the 0 probability j acts consistent with a behavioral type on that interval is less than (πtj )K ≤ δK < zi , but this contradicts the fact that a behavioral type acts like itself. And so rational agent i will always concede by T in ≤ t0 K (b) We must have T in ≤ max{T cj , T nj }. Suppose that T in > max{T cj , T nj } then after time max{T cj , T nj } non-compromising agent i knows that she faces a behavioral opponent, and so would prefer to concede immediately rather than wait until T in . (c) There is no jump in Fic at t ∈ (0, T ∗ ]. Suppose Fic jumped at t ∈ (0, T ∗ ], then F nj is constant on [t − ε, t] for some ε > 0, as non-compromising agent j would prefer instead to concede an instant after t rather than on the interval [t − ε, t]. But in which case, a compromising agent i would prefer to concede at t − ε rather than wait until t. (d) There is no jump in Fin at t ∈ (0, T ∗ ]. Suppose that Fin did jump at t ∈ (0, T ∗ ], then F j is constant on [t − ε, t] for some ε > 0, as a rational agent j would prefer instead to concede an instant after t rather than on the interval [t − ε, t]. But in which case, a non-compromising agent i would prefer to concede at t − ε rather than wait until t.

23

(e) If Fic and Fin are continuous at t then U cj (s, σ j ) and U cj (s, σ j ) are continuous at t. This follows by their definition. (f) If T ∗ ≥ t00 > t0 then Fi (t00 ) > Fi (t0 ). Suppose not, then let ti∗ = sup{t : Fi (t) = Fi (t0 )} ≥ t00 . It is clear 0 that no rational agent j will concede on s ∈ (t0 , ti∗ ) because this is strictly worse than conceding at s+t . The 2 ti∗ +t0 ∗ continuity of F j on (0, T ] then means a rational agent i would strictly prefer to concede at 2 rather than wait to concede at or just after ti∗ , contradicting the definition ti∗ . (g) If T cj ≥ t00 > t0 then Fin (t00 ) > Fin (t0 ). Suppose not, then let ti∗n = sup{t : Fin (t) = Fin (t0 )} ∈ [t00 , ∞). A t0 +t∗n

compromising agent j will not concede on s ∈ (t0 , ti∗n ) because this is strictly worse than conceding at 2 i . Given (f) this must mean that F nj (t00 ) > F nj (t0 ) and Fic (t00 ) > Fic (t0 ). The latter implies that Aci is dense in (t0 , ti∗n ]. From (d) and (e) U cj (t, σ j ) is continuous on this interval and hence constant on this interval. This means Uic (t, σ j ) is differentiable with concede at implies for

f n (t) rate 1−Fj n (t) j t ∈ (t0 , ti∗n ]:

dUic (t,σ j ) dt

= 0, which implies that the non-compromising agent j must

= λ j . Notice, however, that because t < T cj we have c j (1 − F cj (t)) > 0 which in turn

(1 − c j ) f jn (t) f jn (t) f j (t) = < = λj 1 − F j (t) (1 − c j )(1 − F nj (t)) + c j (1 − F cj (t)) 1 − F nj (t) f (t)

A concession rate of exactly 1−Fj j (t) = λ j would make a non-compromising agent i indifferent to concession at any t ∈ (t0 , ti∗n ). The continuity of F j therefore means that conceding at or just after ti∗n delivers a nont0 +t∗n compromising agent i a strictly lower expected payoff than if she conceded at 2 i meaning that ti∗n cannot be the supremum, a contradiction. f (t)

(h) If T cj > 0, then agent j must concede at rate 1−Fj j (t) = λ j rate on (0, T cj ]. If T cj > 0, then (g) implies that Ani is dense in [0, T cj ]. From (c), (d), and (e) it follows that Uin (t, σ j ) is continuous on (0, T cj ] and hence Uin (t, σ j ) is constant on this interval. Hence Uin (t, σ j ) is constant on this interval, and so differentiable with which implies that agent j concedes at rate λ j .

dUin (t,σ j ) dt

= 0,

f (t)

(i) If T cj < T ∗ , then 1−Fj j (t) = λ j on (T cj , T ∗ ]. Notice that a compromising and non-compromising agent i must have identical beliefs about j’s likelihood of being behavioral on [T cj , T ∗ ], and so Ani ∩[T cj , T ∗ ] = Aci ∩[T cj , T ∗ ]. From (f) if follows that Ani is dense in (0, T cj ]. From (c), (d), and (e) it follows that Uin (t, σ j ) is continuous on (0, T cj ] and hence Uin (t, σ j ) is constant on this interval. Hence Uin (t, σ j ) is constant on this interval, and so differentiable with

dUin (t,σ j ) dt

= 0, which implies that agent j concedes at rate λ j .

(j) If T c ≥ t00 > t0 , and F cj (t00 ) = F cj (t0 ) then F cj (t00 ) = Fic (t00 ) = 0. To see that Fic (t00 ) = Fic (t0 ) notice that if F cj (t00 ) = F cj (t0 ) then to ensure agent j on average concedes at rate λ j , as required by (h), a non-compromising agent j must concede at rate:   f jn (t) c j (1 − F cj (t))     = λ j 1 + (7) 1 − F nj (t) (1 − c j )(1 − F nj (t))  For t < T c , however, c j (1− F cj (t)) > 0 and so this rate is strictly greater than λ j , which implies a compromising agent i would strictly prefer to concede at t00 rather than on the interval (t0 , t00 ). Next, define ti∗∗ = inf{s : 0 Fic (s) = Fic (t0 )} ≤ t0 . The previous argument implies ti∗∗ =∗∗ j ∈ [0, t ]. Anticipating that a non-compromising ∗∗ 00 agent j will concede at the rate specified in equation (7) on (ti , t ), a compromising agent i won’t concede on [ti∗∗ − ε, t00 ) for some ε > 0, preferring to concede at t00 ) instead, which implies ti∗∗ = 0 and Fic (t0 ) = 0. (k) Suppose T c > 0, let t∗c := inf{t : Fic (t) > 0 for i = 1, 2}, and suppose t∗c ≤ t0 < t00 ≤ T c , then Fic (t00 ) > Fic (t0 ). First notice that T c > t∗c follows from (c), the continuity of Fic on (0, ∞). The claim then follows immediately from (j), (c) again, and the fact that either Fic (T c ) = 1 or F cj (T c ) = 1.

24

f n (t)

f c (t)

(l) If T c > 0 then 1−Fj n (t) = 1−Fj c (t) = λ j on (t∗c , T c ], where t∗c is defined in (k). First notice that by (k) Aci must be j j dense in [t∗c , T c ]. From (d) and (e) Uic (t, σ j ) is continuous on (0, T c ]. Hence Uic (t, σ j ) is constant on (t∗c , T c ], f n (t) dU c (t,σ ) and so differentiable with i dt j = 0, which implies a non-compromising concession rate 1−Fj n (t) = λ j on this j

interval. By (h) we must also have a total concession rate rates hold, then: λj =

f j (t) 1−F j (t)

= λ j on the interval. If both these concession

c j f jc (t) + (1 − c j ) f jn (t) c j f jc (t) + (1 − c j )(1 − F nj (t))λ j f j (t) = = 1 − F j (t) c j (1 − F cj (t)) + (1 − c j )(1 − F nj (t)) c j (1 − F cj (t)) + (1 − c j )(1 − F nj (t))

This rearranges to give

f jc (t) 1−F cj (t)

= λ j.

(m) We must have T c = 0. Suppose not, and so T c = T cj > 0 for some agent j, and F cj (0) < 1. If t ∈ [t∗c , T c ] (where t∗c is as in (k)), then F cj (t) = 1 − (1 − F cj (0))e−λ j t < 1 for all t, but this contradicts T cj < ∞. We are almost done. Notice that if T cj = 0, then (0, T ∗ ] ⊆ Aci = Ani by (i), hence a compromising agent i who concedes at t ∈ Aci must get the payoff:   Uic (t, σ) = c j ui (mi ) + (1 − c j ) F nj (0)ui (αi ) + (1 − F nj (0))ui (1 − α j ) Whereas a non-compromising agent i’s who concedes at t ∈ Ani must get the payoff:   Uin (t, σ) = c j ui (αi ) + (1 − c j ) F nj (0)ui (αi ) + (1 − F nj (0))ui (1 − α j ) Therefore, if c j > 0 we must have mi ≥ αi , or agent i would not find it optimal to compromise. However, if m j ≤ 1 − αi then U cj (t, σi ) < U nj (t, σi ) for all t, so j would not find it optimal to compromise, a contradiction.  Proof of Proposition 3. Suppose there is an equilibrium σ = (σ1 , σ2 ). In this setup, I refer to agent j who has compromised but not yet conceded, as a compromising agent. Let Ai = {(s, t) : Ui (s, t, σ j ) = maxv,w Ui (v, w, σ j )}. Since σ is an equilibrium, Ai , ∅. Finally, define T id = inf{t : Fid (t) = 1 − zi } and T ∗ = max{T 1d , T 2d }. (a) We must have T id = T ∗ < ∞. This follows for the reasons as outlined in the proof of Proposition 2, point (a). We have T id = T dj , because if an agent knows she faces a behavioral opponent she will concede immediately. We have T dj < ∞ because if the agent does not concede at some t, she must expect her opponent to be concede soon and thus must eventually become convinced that her opponent is behavioral. (b) If Fid jumps at t ∈ (0, T ∗ ] then F cj is constant on [t − ε, t] for some ε > 0. This follows because if agent j has not compromised before t − ε, she would strictly increase her payoff by compromising an instant after t compared to slightly before (giving her u j (α j ) rather than u j (m j ) with positive probability). (c) If Fic jumps at t = (0, T ∗ ] then F dj is constant on [t − ε, t) for some ε > 0. This follows because agent j would prefer to concede an instant after t rather than slightly before (to get u j (m j ) rather than u j (1 − αi ) with probability Fic (t) − sup s t0 . If Fic (t00 ) = Fic (t0 ) then either F dj (t0 ) = F dj (t00 ) or F cj (t) = F dj (t) for t ∈ [t0 , t00 ). Suppose not, then F dj (t0 ) < F dj (t00 ) and F dj (t000 ) < F cj (t000 ) for some t000 ∈ [t0 , t00 ). I first claim that this implies F dj (t000 ) = F dj (t00 ). Otherwise, there is some s ≤ t000 and some t ∈ (t000 , t00 ] such that (s, t) ∈ A j . Given that Fic (t00 ) = Fic (t000 ) the alternative strategy of conceding at 12 (t000 + t) and still compromising at s is more profitable (it moves the payoff u j (1 − αi ) forward in time). Define tˇ = sup{t : Fic (t) = Fic (t0 )}. I claim that Fic is continuous at tˇ. Using the above argument again, we have F dj (t000 ) = sup s F dj (t000 ), it must be that if i would get

25

a higher payoff by compromising at 21 (tˇ + t000 ) instead of tˆ without changing her concession time (this brings the payoff ui (mi ) forward in time). Hence, we must have F dj (t000 ) = F dj (tˇ). But in which case compromising an instant after tˇ cannot be optimal for i either, contradicting the definition of supremum tˇ. (e) Let T ∗ ≥ t00 > t0 . If Fid (t00 ) = Fid (t0 ) and Fic (t0 ) > Fid (t0 ) then F cj (t0 ) = F cj (t00 ) Suppose not so that F cj (t0 ) < F cj (t00 ). Then there exists (s, t) ∈ A j such that s ∈ (t0 , t00 ]. However, the alternative plan of compromising at sˆ = 21 (t0 +s) while still conceding at t would give j a higher payoff as then, with probability (Fic (t0 )−Fid (t0 )) > 0, she gets the payoff ui (mi ) at an earlier date, without affecting the distribution of payoffs from concession. (f) There is no jump in Fid at t ∈ (0, T ∗ ]. Suppose not, then by (b) F cj is constant on [t − ε, t] for some ε > 0. Hence, by (d) either Fid (t) = Fid (t − ε) (a direct contradiction) or Fic (s) = Fid (s) for s ∈ [t − ε, t). It must then be that Fic also jumps at t, because we must have sup s 0 (the same ε without loss of generality). Given that Fic and Fid jump at t, we must have (t, t) ∈ Ai . However, the alternative strategy for i of both compromising and then conceding at tˆ = t − 2ε delivers strictly higher expected profits as she gets the payoffs (F cj (t − ε) − F dj (t − ε))ui (mi ) and (1 − F cj (t − ε))ui (1 − α j ) > 0 at an earlier date, without affecting the distribution of other payoffs. (g) If Fid is continuous at s ≤ t then Ui (s, t, σ j ) is continuous at s, and if Fic is continuous at t then Ui (s, t, σ j ) is continuous at t. This follows from how Ui (s, t, σ j ) is defined. For claims (h)-(l) suppose that F1c (t0 ) > F1d (t0 ) for some t0 ∈ [0, ∞) (symmetric arguments apply if F2c (t0 ) > F2d (t0 )). Define t1 = inf{t ≥ t0 : F1c (t) = F1d (t)} and t1 = inf{t : F1c (s) < F1d (s) ∀s ∈ [t, t0 ]}. By (b), the continuity of F1d , must have t1 > t0 ≥ t1 and F1c (t1 ) = F1d (t1 ). We then have F1c (t) > F1d (t) for t ∈ (t1 , t1 ). Let t1 ≥ t000 > t00 > t1 . (h) We must have F2c (t000 ) > F2c (t00 ). Suppose not. Let tˇ2 := sup{t : F2c (t) = F2c (t00 )} ≥ t000 . I first claim that either F1d (t00 ) = F1d (tˇ2 ) or F1c (t) = F1d (t) for t ∈ [t00 , tˇ2 ). Suppose not then F1d (t00 ) < F1d (tˇ2 ) and there is some t ∈ [t00 , tˇ2 ) such that F1d (t) < F1c (t). By (f), the continuity of F1d , we must then have F1d (t00 ) < F1d (tˇ2 − ε) for all ε > 0 sufficiently small. We must also have F1d (t) < F1c (t) for some t ∈ [t00 , tˇ2 − ε) for ε sufficiently small (because F1d (t) < F1c (t) for some t ∈ [t00 , tˇ2 )). Hence, we have F2c (t00 ) = F2c (tˇ2 − ε), F1d (t) < F1c (t) for some t ∈ [t00 , tˇ2 − ε) and F1d (t) < F1c (t) for all t ∈ [t00 , tˇ2 − ε), which contradicts claim (d). By assumption we have F1c (t) > F1d (t) for t ∈ (t1 , t00 ] so we must have F1d (t00 ) = F1d (tˇ2 ). This implies t1 > tˇ2 because F1d (tˇ2 ) = F1d (t00 ) < F1c (t00 ) ≤ F1c (tˇ2 ) and F1c (t1 ) = F1d (t1 ). I next claim that (tˇ2 , t) < A2 for any t ≥ tˇ2 . To see this, notice that 2’s alternative plan of compromising at tˆ = 12 (tˇ2 + t00 ) and conceding at t must deliver strictly larger payoffs by bringing forward the payoff u2 (m2 ), with probability F1c (tˆ) − F1d (tˆ) > 0, without affecting the distribution of payoffs from concession. Given (f), the continuity of F1d , this argument similarly implies that compromising an instant after tˇ2 cannot be optimal plan for agent 2 either, contradicting the definition of the supremum tˇ2 . (i) We must have F1d (t000 ) > F1d (t00 ). Suppose not, then let tˇ1 = sup{t : F1d (t) = F1d (t00 )} ≥ t000 . Given (f) we have F1d (tˇ1 ) = F1d (t00 ). Given F1d (t00 ) < F1c (t00 ) we must have tˇ1 < t1 . Hence, by (e) we must have F2c (tˇ1 ) = F2c (t00 ) which contradicts (h), that F2c is increasing on (t1 , t1 ]. (j) We must have F2d (t000 ) > F2d (t00 ). Suppose not then F2d (t000 ) = F2d (t00 ). Given that F2c is increasing on the interval [t00 , t000 ] by (h), we must have F2c (t) > F2d (t) for t ∈ (t00 , t000 ]. But then switching the notation for 1 and 2, (h) implies F1c (t000 ) > F1c (t00 ) and (i) then implies F2d (t000 ) > F2d (t00 ), a contradiction. (k) We must have F1c (t000 ) > F1c (t00 ). Suppose not, and so F1c (t000 ) = F1c (t00 ). Let tˇ1 = inf{t : F1c (t) = F1c (t00 )}. Clearly, we have tˇ1 ≥ t1 . By (d), we must have either F2d (t000 ) = F2d (tˇ1 ), which contradicts (j), or F2c (t) = F2d (t) for t ∈ [tˇ1 , t000 ). I claim that (tˇ1 , t) < A1 for any t ≥ t000 . The alternative strategy of compromising at t000 while still conceding at t gives agent 1 a strictly higher payoff of u1 (α1 ) instead of m1 from the positive concession of agent 2 on the interval [tˇ1 , t000 ). That is:

26

U1 (t000 , t, σ2 ) − U1 (tˇ1 , t, σ2 ) ≥

Z tˇ1 ≤s≤t000

(u1 (α1 ) − u1 (m1 ))e−r1 s dF2c (s)

000

≥ e−r1 t (u1 (α1 ) − u1 (m1 ))(sup F2c (s) − F2c (tˇ1 )) > 0 s
where the first inequality follows from F2c (t) = F2d (t) on [tˇ1 , t000 ), the second from t000 ≥ s ∈ [tˇ1 , t000 ] and the third from (h). For the same reason, compromising an instant before tˇ1 cannot be optimal either. This either contradicts the definition of tˇ1 or contradicts F1c (t0 ) > F1d (t0 ). (l) Fic is continuous on (t1 , t1 ]. If Fic did jump at t ∈ (t1 , t1 ] then by (c), F dj is constant on (t − ε, t) for some ε > 0, contradicting either (i) or (j). We are almost done. Because F1c , F1d are increasing on (t1 , t1 ) and F1d (t) < F1c (t) in this interval, it follows that there is some s0 ∈ (t1 , t1 ) such that A1 is dense in the set {(s0 , t) : t, ∈ [s0 , t1 ]}. Notice that regardless of whether the agent concedes at s0 or s > s0 , the agent faces the same continuation payoffs conditional on not having conceded before s. From the continuity of F2c on (t1 , t1 ] it follows that U1 (s0 , t, σ2 ) is constant on [s0 , t1 ], and hence differentiable 0 with respect to t with zero partial derivative, U1 (s∂t,t,σ2 ) = 0. Rearranging this zero derivative condition gives: f2c (t) r1 u1 (1 − α2 ) = λc2 := c 1 − F2 (t) m1 − u1 (1 − α2 ) c

For t ∈ [t1 , t1 ], this implies (1 − F2c (s)) = φc2 e−λ2 (s−t1 ) where φcj = (1 − F cj (t1 )). By the same reasoning there must be some s00 ∈ (t1 , t1 ) such that A1 is dense in the set {(s, s00 ) : s, ∈ [t1 , s00 ]}. The continuity of F2d on (t1 , t1 ] then implies that U1 (s, s00 , σ2 ) is constant on (t1 , s00 ], and hence differentiable with 00 ,σ2 ) respect to s with zero partial derivative, ∂U1 (s,s = 0. Rearranging this zero derivative condition gives: ∂s     c f1d (s) = λd2 (1 − F2d (s)) − (1 − F2c (s)) = λd2 (1 − F2d (s)) − φc2 e−λ2 (s−t1 ) where λd2 :=

r1 u1 (m1 ) u1 (α1 )−u1 (m1 ) .

Solving this linear ODE gives:

  d −λd (s−t ) d −λc (s−t ) c −λd (s−t ) c   m2 e 2 1 + φ2 θ2 (e 2 1 − e 2 1 ) if λ2 , λ2 d (1 − F2 (s)) =    (md + λd φc (s − t ))e−λd2 (s−t1 ) if λd = λc 2

where: θ2 :=

λd2 λd2 −λc2

2 2

1

2

2

and φd2 = (1 − F2d (t1 )) ≥ (1 − F2c (t1 )) = φc2 . Define the gap between F2c and F2d as d2 (s) =

F2c (s) − F2d (s), and take transformations of this gap to give: eλ2 (s−t1 ) φd2 − θ2 φc2 d c = + e(λ2 −λ2 )(s−t1 ) θ2 − 1 θ2 − 1 c eλ2 (s−t1 ) θ2 − 1 d c =e(λ2 −λ2 )(s−t1 ) + d d2 (s) d c φ2 − θ2 φ2 φ2 − θ2 φc2 d

d2 (s)

d2 (s)eλ2 (s−t1 ) =φd2 − φc2 + λd2 φc2 (s − t1 ) d

if

λd2 > λc2

if

λd2 < λc2

if

λd2 = λc2 λc2 λd2 −λc2 from φd2

I claim that each of these transformations is positive. Notice that θ2 − 1 = λc

d c 2 φd2 − θ2 φc2 ≥ −φc2 λd −λ c > 0 when λ2 < λ2 , where the first inequality follows 2

2

> 0 when λd2 > λc2 . Similarly ≥ φc2 . Each to the transformed

gaps is strictly increasing in s, implying that d2 (s) > 0 for s ∈ (t1 , t1 ]. Define t2 = inf{t > t1 : F2c (t) = F2d (t)}. We can now repeat the above arguments with the roles of agent 1 and 2 reverse to find that d1 (s) > 0 for s ∈ (t1 , t2 ]. Let t = min{t1 , t2 }. Suppose t < ∞, then we have an immediate contradiction to di (t) > 0. On the other hand t = ∞ contradicts (a) that T ∗ < ∞. We then must have Fic (t) = Fid (t) for t ∈ [0, ∞). Given this, the unique equilibrium

27

must match that of the Baseline model by standard arguments (see AG).  Proof of Proposition 4. First observe that:     Ui∗c = (1 − z j ) bui (mi ) + (1 − b)H cj (0)ui (αi ) + z j + (1 − z j )(1 − b)(1 − H cj (0)) ui (1 − α j ) because agent i is indifferent to conceding an instant after time 0. This in turn implies: ∗

Z s
e−ri s ui (αi )dH cj (s) = ui (1 − α j ) + H cj (0)(ui (αi ) − ui (1 − α j )) +

z j (1 − e−ri T )ui (1 − α j ) (1 − z j )(1 − b)

(8)

And so, Qi reduces to: ! ∗ z j (1 − e−ri T )ui (1 − α j ) c Qi = ui (mi ) − ui (1 − α j ) + + H j (0)(ui (αi ) − ui (1 − α j )) (1 − z j )(1 − b) Suppose that T ∗ = T j ≤ T i , and hence H cj (0) = 0. Substituting in for T ∗ and Hic (0) gives:  zj 1 − Qi =ui (mi ) − ui (1 − α j ) −    zi 1 −

Q j =u j (m j ) − u j (α j ) −

zj 1−(1−z j )b

 λri ! j

ui (1 − α j )

(1 − z j )(1 − b)  λr j  j  zj  u (1 − α ) i  j 1−(1−z j )b (1 − zi )(1 − b)

+ (u j (α j ) − u j (1 − αi ))

  − λλi   j zj  zi  1−(1−z j )b − 1 (1 − zi )(1 − b)

Define mi as the mediation share that causes Qi = 0.     λri !   j zj z j 1 − 1−(1−z j )b ui (1 − α j )      mi := u−1 ui (1 − α j ) + i   (1 − z j )(1 − b)   Notice that mi → 1 − α j as z j → 0. Setting m j = 1 − mi ) we then have:

Q j u j (1 − mi ) − u j (α j ) = − zi zi Notice that

u j (1−mi )−u j (α j ) zi

≥K

   1 −

u j (1−mi )−u j (α j ) zj

zj 1−(1−z j )b

 λr j  j   u (1 − α ) i  j

(1 − z j )(1 − b)

+ (ui (αi ) − ui (1 − α j ))

  

zj 1−(1−z j )b

− λλi

j

  − 1

(1 − zi )(1 − b)

(9)

by assumption. Taking the limit of the right hand side as z j → 0 using

l’Hopital’s rule and the inverse function theorem gives limK Q

u j (1−mi )−u j (α j ) zj

u0 (α j )

= −K u0 (1−αj i )(1−b) > −∞. This ensures i

that lim zij = ∞ as the final expression in equation 9 explodes as z j → 0. Hence, there exists z0 > 0 such that if z j ≤ z0 we must have Q j ≥ 0 and Qi ≥ 0 and so there is an P1 equilibrium. It remains to show that such equilibrium can strictly improve the payoff of both agents. If λ j ≥ λi and z j ≥ zi then clearly T i ≥ T j in any P1 equilibrium and in the Baseline equilibrium. Alternatively, suppose that λ j > λi (and

28

possibly z j < zi ). Let z be such that for z j ≥ z we have 00

2

zj 1 − (1 − z j )b

!λi



zj 1−(1−z j )b

 λλi −1 j

≥ K. This implies:

! ! ! Kz j Kz j zi λj ≥ ≥K ≥ 1 − (1 − z j )b 1 − (1 − Kz j )b 1 − (1 − zi )b ! ! zj 1 1 zi ≤ − ln = Ti T j = − ln λj 1 − (1 − z j )b λi 1 − (1 − zi )b

The first inequality on the first line is directly implied, the second follows because (1 − b)(1 − z j ) + z j = (1 − Kz j )(1 − b) + Kz j − (K − 1)z j b ≤ (1 − Kz j )(1 − b) + Kz j , the third because Kz j ≥ zi . The second line stating T j ≤ T i is then simply a rearrangement of the inequality of the first and final term on line one. The bound immediately λi λ

also ensures that z j j

−1

≥ K and so in the Baseline equilibrium we must also have T j ≤ T i .

Let z j ≤ min{z0 , z00 , 12 }, then the payoff to player i in the Baseline equilibrium is ui (1 − α j ). To ensure Qi > 0 we must certainly have ui (mi ) > ui (1 − α j ) and hence Ui∗c > ui (1 − α j ). In the Baseline equilibrium j’s payoff is λ

− λi

U Bj := u j (αi ) − (u j (α j ) − u j (1 − αi ))zi z j j . We need to compare this to her payoff in a P1 equilibrium: U ∗c j

zj = (1 − zi )bu j (m j ) + (1 − b(1 − zi ))ui (α j ) − (ui (αi ) − ui (1 − α j ))zi 1 − (1 − z j )b

!− λλi

j

Notice that B U ∗c j − Uj

zi

 !− λλi   − λi j  zj  λj  =(1 − zi )b + (ui (αi ) − ui (1 − α j )) z j −  zi 1 − (1 − z j )b  !− λλi  λ u j (1 − mi ) − ui (α j )  − λi  j  2  j  ≥K + (ui (αi ) − ui (1 − α j ))z j 1 −  zj 2−b u j (1 − mi ) − ui (α j )

where the second line follows from (1 − zi )b is equivalent to 2 − b ≥ 2(1 − (1 − z j )b)).

u j (1−mi )−ui (α j ) zi

≥K

u j (1−mi )−ui (α j ) zj

and

zj 1−(1−z j )b

2 ≥ z j 2−b when z j ≤

1 2

(the

λ

The final expression on the second line explodes as z j → 0 because



 i 2 − λj 2−b

< 1. We previously established that

u j (1−mi )−ui (α j ) K zj

the limit of as z j → 0 is finite. This implies that there exists z > 0 such for z j ≤ z we have a P1 B ∗c B  equilibrium with mi = mi and U ∗c j > U j and U i > U i . QED. Proof of Proposition 5. Suppose this were not true, then there must exist some sequence of games (ri , ui , αi , zni , mn , bn ) with zn1 → 1 and a sequence of P1 equilibria in each (I suppress the subscript n in what follows for simpler notation). I first claim that any (sub)sequence of these P1 equilibria must satisfy lim T ∗ = 0. This follows immediately   z1 from the fact that T ∗ ≤ T 1 = − λ11 ln (1−z1 )(1−b)+z → 0. 1 R ∗ Notice that s 0, for all sufficiently large n we need mi > αi −ε for i = 1, 2, in order to have Qi ≥ 0. Choosing ε = α1 +α2 2 −1 we have m1 + m2 > 1, a contradiction.  R Proof of Proposition 6. Consider the expression for s
∗

(ui (αi ) − ui (1 − α j )) +

z j ui (1 − α j ) (1 − e−ri T ) − (1 − b)ri e−ri T 1 − zj (1 − b)2

29

∗

dT ∗ db

It is clear that

dT ∗ db

< 0 (agents have larger reputations at the start of the war of attrition phase for larger b). The

result will therefore be proved if we can establish that 1, below.

dH cj (0) db

≥ 0 when λ j ≥ λi . This is exactly the claim of Lemma 

Lemma 1. Rational agent j’s time zero concession in a P1 equilibrium is increasing in b if λ j ≥ λi and decreasing otherwise. Proof. Recall that = min{0, V} where W(b, θ) := 1 − the following partial derivatives: H cj (0)

z¯ j z¯iX −1 1−¯z j


}, z¯ j (b) :=

zj 1−(1−z j )b

< 1 and X :=

λj λi .

Consider

  z¯ j dW −X X(1 − z¯i )¯z−X = i − z¯i + 1 db (1 − z¯ j )(1 − b)   z ¯ ∂2 V j == (1 − z¯i )¯z−X zi )) + z¯−X zi ) i (1 − Xln(¯ i ln(¯ dbdX (1 − z¯ j )(1 − b) dW d2 W Notice that dW db is continuous in all variables and db X=1 = 0. To establish the result we show that dbdX > 0 dW whenever dW = 0, which implies a strict single crossing property for dW z−X = i db . Note that db = 0 implies (1 − z¯i )¯  db 1 −X X z¯i − 1 . Hence define: Y(X) := Xln(¯zi ) + z¯−X i −1=

X(1 − z¯ j )(1 − b) d2 W dW =0 dbdX db z¯ j

dY Notice in turn that dX = −ln(¯zi )(¯z−X i − 1) > 0, where the strict inequality follows from z¯i < 1. Noting that Y|X=0 = 0 and X > 0 we are done. 

Proof of Proposition 7. Let zˆ = max{z1 , z2 } and γˆ = max{γ1 , γ2 }. Define G using the linear ODE, g(t) =   zˆ γˆ (1 − G(t)) + (1−ˆz)(1−b) for t ∈ [0, T ∗ ] with the boundary condition G(0) = 0 and T ∗ defined by G(T ∗ ) = 1. This G ensures that an agent who has accepted mediation will find it optimal to concede only at T ∗ because it U c (t,σ ) necessarily implies i dt j ≥ 0 on [0, T ∗ ). As argued above in the text, to establish the existence of a D1 equilibrium, it merely remains to show that this   ∗ u (1−α ) zˆ G implies e−ri T ≤ iui (αi )j . Solving for G in closed form we get G(t) = 1 + (1−ˆz)(1−b) (1 − e−ˆγt ) for t ≤ T ∗ . For ∗ zˆ 1 ) u1 (1−α2 ) ˆ = max{ u2u(1−α , u1 (α1 ) } ∈ (0, 1), G(T ∗ ) = 1 we must then have e−ˆγT = (1−ˆz)(1−b)+ˆ z . Let rˆ = min{r1 , r2 } > 0 and y 2 (α2 ) rˆ   ∗ γ ˆ zˆ then we must certainly have an equilibrium if yˆ ≥ e−ˆrT = (1−ˆz)(1−b)+ˆ z . Notice that that the right hand side of this equation is continuously increasing in zˆ and converges to zero as zˆ → 0. Hence, defining z as the unique value   γrˆˆ z at which (1−z)(1−b)+z = yˆ we are done.  Proof of Proposition 8. Suppose this were not true, then there must exist some sequence of games (ri , ui , αi , zni , m0,n , m∞,n , bn , Gn ) with zn1 → 1 and a sequence of D1 equilibria in each (I suppress the subscript n in what follows for simpler notation). I first claim that any (sub)sequence of these D1 equilibria must satisfy lim T ∗ = 0. Suppose not, and so c ∗ there is a (sub)sequence with lim T ∗ > 0, and hence lim e−r2 T < 1. In any D1 equilibrium Uic (T ∗ , σ1 ) ≤ U i := ∗ ∗ (1 − z j )ui (mi ) + z j e−ri T ui (1 − α j ). If lim T ∗ > 0, then lim U 2 = e−r2 lim T u2 (1 − α1 ) < u2 (1 − α1 ), which is a contradiction, as agent 2 could do better by conceding at time 0 instead.   ∗ Next, notice that agent i can always obtain supt∈[05 ,∞) Uin (t, σ j ) ≥ U ni := e−ri T (1 − z j )ui (αi ) + ui (1 − α j )z j by not c   ∗ U i −U ni compromising and conceding an instant after T ∗ . Next notice that 1−z = ui (mi ) − eri T ui (αi ) , which converges j to ui (mi ) − ui (αi ) as T ∗ → 0. Hence, for any ε > 0, we need mi > αi − ε for i = 1, 2, for all sufficiently large n for agent i to be willing to compromise. Choosing ε = α1 +α2 2 −1 , we must then have m1 + m2 > 1, a contradiction. 

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Proof of Proposition 9. Keep T ∗ := min{− λ11 ln(z1 ), − λ12 ln(z2 )}, 1 − G j (t) = For t ∈ (0, T ∗ ) consider the continuous function qi (t) =

zj λ j (T ∗ −t) 1−z j (e

− 1).

∗

λi eλi (T −t) (eλ j (T −t) −1) ∈ (0, 1), first defined in ∗ ∗ ∗ (eλi (T −t) −1)+λi eλi (T −t) (eλ j (T −t) −1)  i 1 rule limt→T ∗ qi (t) = 2 . Hence there exists δ ∈ 0, 12 ∗

λ j (T ∗ −t)

λ je

equation 6. Notice that limt→0 qi (t) ∈ (0, 1), and by l’Hopital’s such that qi (t) ∈ [δ, 1 − δ] for all t ∈ (0, T ∗ ). For arbitrary ε > 0 define:

W ε (q) = u1 (q(1 + ε)(1 − α2 ) + (1 − qε)α1 ) − (qu1 (1 − α1 ) + (1 − q)u1 (α1 )) and notice that

d2 W ε = ((1 + ε)(1 − α2 ) − εα1 )2 u001 (q(1 + ε)(1 − α2 ) + (1 − qε)α1 ) ≤ 0 dq2

, and hence W ε (q) achieves its minimum for q ∈ [δ, 1 − δ] at some q ∈ {δ, 1 − δ}. Given u01 (1 − α2 ) > u1 (α1 ) we have u1 (q(1−α2 )+(1−q)α1 )) > qu1 (1−α1 )+(1−q)u1 (α1 ) for q ∈ (0, 1). By continuity of u1 , min{W ε (δ), W ε (1−δ)} > 0 for all ε > 0 sufficiently small, and hence W ε (q1 (t)) > 0 for all t ∈ (0, T ∗ ). For such an appropriately small ε > 0, therefore, set m1 (t) = 1 − m2 (t) = q1 (t)(1 + ε)(1 − α2 ) + (1 − q1 (t)ε)α1 for t ∈ (0, T ∗ ). We must then have ui (mi (t)) > qi (t)ui (1 − α j ) + (1 − qi (t))ui (αi ) for t ∈ (0, T ∗ ).   zj ri (1−αi ) R m i j , let g (t) = max{λ (t)(1 − G (t)) 1 − G (t) + For t ∈ (0, T ∗ ) let λm (t) = i i ui (mi (t))−ui (1−α j ) 1−z j : i ∈ {1, 2}, j , i} and R T∗ finally 1 − GR (t) = t gR (s)ds. I claim that gR (t) < g1 (t)(1 − G2 (t)) + g2 (t)(1 − G1 (t)) and hence 1 − GR (t) < (1 − G1 (t))(1 − G2 (t)) for t ∈ (0, T ∗ ). This claim can be proved as follows: ! zj 0 =g (t)(ui (αi ) − ui (1 − α j )) − ri ui (1 − α j ) 1 − G (t) + 1 − zj   i j j i =g (t)(1 − G (t))ui (1 − α j ) + g (t)(1 − G (t))ui (αi ) − gi (t)(1 − G j (t)) + g j (t)(1 − Gi (t)) ui (1 − α j ) ! zj i j − ri ui (1 − α j )(1 − G (t)) 1 − G (t) + 1 − zj !    zj i j j i i j < g (t)(1 − G (t)) + g (t)(1 − G (t)) ui (mi (t)) − ui (1 − α j ) − ri ui (1 − α j )(1 − G (t)) 1 − G (t) + 1 − zj j

j

The first line is simply the indifference condition in the Baseline equilibrium. The second line multiplies through by (1 − Gi (t)) and both adds and subtracts gi (t)(1 − G j (t))ui (1 − α j ). The third line follows because ui (mi (t)) > gi (t)(1 − G j (t))ui (1 − α j ) + g j (t)(1 − Gi (t))ui (αi ). Given that the third line is strictly positive for i = 1, 2,and that    z R R i by the definition of g (t) we have g (t) ui (mi (t)) − ui (1 − α j ) − ri ui (1 − α j )(1 − G (t)) 1 − G j (t) + 1−zj j = 0 for at least one agent i, it follows that gR (t) < gi (t)(1 − G j (t)) + g j (t)(1 − Gi (t)). Finally, we can choose mi (0) so that ui (mi (0))GR (0) > ui (αi )G j (0) for i = 1, 2. Given that GR (0) > 1−(1−Gi (0))(1− G j (0), this is certainly possible. In this case we have Uic (0, σ j ) = (1−z j )GR (0)ui (mi )+(1−(1−z j )GR (0))ui (1−α j ) > UiB , where UiB is the unchanged maximum payoff agent i can obtain if she does not compromise. Furthermore, for t ∈ (0, T ∗ ) we have: !   zj j i i 0 ≤g (t) ui (mi (t)) − ui (1 − α j ) − ri ui (1 − α j ) (1 − G (t))(1 − G (t)) + (1 − G (t)) 1 − zj ! c   dU z j i (t, σ j )
where the first inequality follows from the definition of gR (t), the second from the fact that 1−GR (t) < (1−G j (t))(1− Gi (t)) and the third from the definition of Uic (t, σ j ). Hence, it is optimal for rational agents to compromise and not concede until instructed to be the mediator. This establishes the existence of a G1 equilibrium, which has been constructed to exhibit strictly less delay and strictly higher payoffs for each rational agent. 

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