Bargaining with Habit Formation - Duke...

Bargaining with Habit Formation

Bahar Leventoglu∗

Duke UniversityJune 2012

Abstract

Habit formation is a well-documented behavioral regularity in psy-chology and economics, however its implications on bargaining out-comes have so far been overlooked. I study an otherwise standardRubinstein bargaining model with habit-forming players. In equilib-rium, a player can strategically exploit his opponent’s habit formingbehavior via unilateral transfers off the equilibrium path to generateendogenous costs and gain bargaining leverage at no cost to himselfon the equilibrium path. Uncertainty about habit formation may leadto a delay in bargaining.

1 Introduction

Habit formation is a well documented behavioral regularity in psychologyand behavioral economics (Camerer and Loewenstein 2004). Accordingly,human beings form habits for consumption and their current satisfactionlevel tends to be highly correlated with their past consumption level. Habitformation has been incorporated extensively to a number of research pro-grams in economics, finance, international conflict, management science andsocial psychology among others.1 However, the formal bargaining literature

∗Thanks go to Alexandre Debs, Robert Powell and Huseyin Yildirim for helpful com-ments and suggestions.

1For example, habit formation has been used to account for the consumption data inthe US as well as other countries (Ferson and Constantinides 1991, Braun et al. 1993),

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has so far overlooked its implications in bargaining. In this paper, I exam-ine the role of habit formation in bargaining, and in particular, I explorehow players can exploit habit formation to generate bargaining leverage innegotiations.I introduce habit formation into an otherwise standard bargaining model

(Rubinstein 1982).2 Two players collectively receive a flow of v units ofconsumption good over an infinite time-horizon. The players can consume thegood only if they mutually agree to share it. The players discount their futurepayoffs. They make offers in an alternating fashion in the beginning of everyperiod. If a player’s proposal is accepted, the two players share v accordinglyforever. If a proposal is rejected, v of the current period perishes. I extendthis standard model as follows: When an offer is rejected, player 1 can makea unilateral transfer from his other resources to player 2, which player 2 mayaccept or reject. In the benchmark case, I model player 2 as an individual thatforms habits for consumption and her current satisfaction levels tend to behighly correlated with her past consumption levels (Camerer and Loewenstein2004). In particular, player 2 uses her consumption from the previous periodas a reference point, and pays a cost if her current consumption falls belowher past consumption level (Rozen 2010). Her cost is a linear function of thedifference between current and previous consumption levels. I refer to thecoeffi cient of this linear function as the cost coeffi cient.In the subgame perfect equilibrium of the game, player 1 gains bargaining

leverage by exploiting player 2’s habit forming behavior to generate endoge-nous costs for player 2. Specifically, as long as offers are rejected, whichhappens only off the equilibrium path, player 1 alternates between makinga unilateral transfer one period, which player 2 accepts, and no transfer thefollowing period. Since player 2 pays a cost only when she consumes lessthan what she has consumed in the previous period, the alternating schemeof unilateral transfers lowers player 2’s continuation payoff off the equilib-rium path when an offer is rejected. In turn, she accepts lower offers onthe equilibrium path. This bargaining leverage comes to player 1 at no costsince he exploits player 2’s habit formation by unilateral transfers only off

and some notable asset pricing anomalies such as the equity premium puzzle (Abel 1990,Constantinides 1990, Campbell and Cochrane 1999). Scholars in growth economics (e.g.Carroll, Overland, and Weil 1997) and monetary economics (e.g. Fuhrer 2000) have alsoutilized rational models of habit formation for their explanatory and predictive power.

2See Osborne and Rubinstein (1990) for an extensive review of the literature on bar-gaining models.

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the equilibrium path without actually making any transfer on the equilib-rium path. In addition, a higher cost coeffi cient benefits player 1. Player 2cannot commit to not accepting transfers by player 1, because momentarybenefit of the transfer exceeds the gains in bargaining leverage that player2 would gain by refusing the transfer. These qualitative features still holdwhen both players exhibit habit forming behavior or when the proposer ischosen randomly every period.The model predictions survive in the presence of asymmetric information,

however, an equilibrium delay may emerge. When player 1 is uncertain aboutwhether player 2 exhibits habit forming behavior or not, if player 2 makesthe first offer, a low cost type player 2 can credibly signal her type to player1 in a separating equilibrium by facing a risk of delay in negotiations. Sucha separating equilibrium is possible only if the higher cost coeffi cient is nottoo high. Otherwise, the low cost and the high cost types pool by makingthe offer that the high cost type would make in the complete informationgame. A higher cost coeffi cient benefits player 1 by increasing his expectedpayoff. It also increases the likelihood of a delay, which only harms player2 of low cost type. The potential delay disappears when player 1 makes thefirst offer. More importantly, player 1 continues to exploit player 2’s habitformation without actually making any transfers in the equilibrium of theincomplete information game. That is, the threat of the unilateral transferis suffi cient to generate bargaining leverage for player 1.Closest to my work is the literature on bargaining games in which players

can endogenously determine disagreement payoffs (Haller and Holden 1990,Fernandez and Glazer 1991, Avery and Zemsky 1994, Busch and Wen 1995).In the standard Rubinstein model, disagreement payoffs are fixed and thereis a unique perfect equilibrium which is effi cient and has stationary equilib-rium offers. In contrast, when disagreement payoffs are endogenous, theremay exist multiple equilibria and some of these equilibria may be ineffi cient,explaining wasteful phenomena such as strikes, delay in bargaining and wars.The driving force behind these results is that actions that are taken in thestage game that determines the disagreement payoffs may determine equilib-ria that are going to be played in future. This allows for multiple equilibria,some of which are ineffi cient.My model is similar to these earlier work in the sense of disagreement

payoffs being endogenous. However, in contrast, a player’s past consumptionlevel is a payoff relevant state of the game in my model. This has nontrivialimplications on model predictions. For example, Haller and Holden (1990)

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and Fernandez and Glazer (1991) study wage bargaining between a firm anda union. The union may strike in case of disagreement, which is costly forboth the firm and the union workers. In this model, not striking is a Nashequilibrium of the stage game and therefore the unique equilibrium of thestandard Rubinstein game without a strike option is an equilibrium of thegame with the strike option (Busch and Wen 1995, Corollary 1). In contrast,for a region of parameters, there is no equilibrium in my model in whichplayer 1 does not make any unilateral transfer on and offthe equilibrium path.Therefore the unique equilibrium of the standard Rubinstein game withouta transfer option is not an equilibrium in my model even though no transferis the unique Nash equilibrium of the stage game. This is because a transfertoday changes preferences over consumption in future via habit formationand player 1 finds it optimal to manipulate player 2’s future preferences.This is not solely an equilibrium phenomenon as in the earlier work, in whichactions today do not change fundamentals in future —in particular, in thesemodels, players continue to have the same preferences regardless of whatactions have been played in the past, however, players’ past actions maydetermine equilibria that are going to be played in future.Finally, the prospect theory of international conflict (Levy 1996, 1997a,

1997b; also see Berejikian 2004) hypothesizes that political leaders of ad-versary states behave differently when they are bargaining over gains thanwhen they are bargaining over loses (Levy 1996). Although this is close to mywork in terms of exploring non-standard preference patterns in bargaining,my work does not rest on prospect theory: First, players’payoff functionsare weakly concave everywhere in my model,3 and second, the insights thatare derived from my model are fundamentally different than that come fromprospect theory.I introduce the complete information bargaining model with habit forma-

tion in the next section and discuss the equilibrium in Section 3. I considertwo extensions of the benchmark model in Section 4. I first consider the casewhen both players exhibit habit forming behavior, and then the case whenthe proposer is selected randomly every period. In Section 5, I study the casewhen player 1 is uncertain whether player 2 exhibits habit forming behavioror not. I defer all the technical analysis to the appendix.

3Prospect theory (Kahneman and Tversky 1979) postulates that an individual evaluatesalternatives with respect to a reference point and assigns value to gains and losses withrespect to the reference rather than to final assets. The value function is generally concavefor gains, convex for losses and steeper for losses than for gains.

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2 Bargaining with habit forming players

The benchmark model is an extension of Rubinstein (1982) bargaining model:There are two players 1 (he) and 2 (she). The players bargain over a flow ofv units of a perishable consumption good that they can share and consumeonly if they mutually agree to do so. The players make offers in an alternatingand deterministic order. The player that is selected to make an offer makes aproposal (v−x, x) where x is player 2’s share and v−x is player 1’s share. If anagreement is reached, the players consume their shares of the flow thereafter.When a player rejects a proposal, he makes the next proposal. Before thenext proposal, player 1 offers to make a unilateral transfer of y ≥ 0 to player2, which costs player 1 ψy, ψ ≥ 0. Then player 2 decides whether to acceptor reject the transfer. She consumes y if she accepts the transfer. The gameproceeds to the next period and continues until one player accepts the other’sproposal.Player i discounts future payoffs by δi ∈ [0, 1). Let zit denote player i’s

consumption in period t and yt be player 1’s unilateral transfer in period t.Player 1’s per-period payoff is z1t−ψyt, which is his consumption minus thecost of unilateral transfer in the given period. Then his lifetime utility isgiven by ∑

t

δt1(z1t − ψyt)

Player 2’s lifetime utility is given by∑t

δt2 [z2t − φ[z2,t−1 − z2t]+]

where

[z2,t−1 − z2t]+ =

{z2,t−1 − z2t if z2,t−1 > z2t0 otherwise

φ[z2,t−1 − z2t]+ captures player 2’s cost from her habit for past consump-tion. If player 2’s current consumption is at least as much as her consumptionin previous period, i.e. z2t ≥ z2,t−1, then player 2 does not face any addi-tional cost, and her per-period payoff is her current consumption. If hercurrent consumption is less than her consumption in the previous period, i.e.z2t < z2,t−1, then player 2 pays a cost of φ(z2,t−1 − z2t) and her per-periodpayoff is her current consumption minus the cost. This cost increases lin-early with the difference between consumption levels in current and previous

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periods. The marginal cost coeffi cient φ ≥ 0 measures how costly it is for theplayer when there is a gap between the current and past consumption levels.The higher φ is, the greater costs player 2 pays. The only part of the costthat is exogenous is φ, and later I discuss how this parameter can vary, forexample by consumption type, regime type of states or sanction type whenthe model is applied to international bargaining and sanctions.The benchmark model captures scenarios in which player 1’s income from

other resources is large enough in comparison to unilateral transfers he couldmake so that he does not suffer any cost from habit forming behavior.I discuss the equilibrium of the benchmark model in the next section.

Then I show that the qualitative features of this complete information gameis robust to various changes in modelling assumptions on habit forming be-havior of player 1, the bargaining protocol and the information structure.

3 Equilibrium

A strategy profile of offers, unilateral transfers and acceptance/rejection de-cisions is a subgame perfect equilibrium if it is a Nash equilibrium in everysubgame.Formally, xit ∈ [0, v] is player i’s offer for player 2’s share in period t;

yit ≥ 0 is player 1’s unilateral transfer when player i’s offer of xit is rejected;ait(x) ∈ {accept, reject} is player i’s decision to accept or reject an offer ofx made by the other player in period t; Tt(y) ∈ {accept, reject} is player 2’sdecision to accept or reject a transfer of y made by player 1 in period tI use the following convention: t represents a period that player 1 makes

an offer. So I refer to 1’s offer periods as ...t − 2, t, t + 2, ... and 2’s offerperiods as ..., t− 1, t+ 1, t+ 3...4

Player 1’s strategy is denoted by σ1 = (x1t, y1t, y2t, a1t)t=1,2,..., player 2’sstrategy is denoted by σ2 = (x2t, a2t, Tt)t=1,2,.... A player makes an offer oran acceptance/rejection decision every other period. One can set xit andait arbitrarily in periods that are not relevant for player i. This will notcause any confusion in the analysis. A strategy profile (σ1, σ2) is a subgameperfect equilibrium if the continuation of (σ1, σ2) forms a Nash equilibriumat every subgame. An equilibrium is Markov perfect if strategies dependonly on payoff relevant state of the game, which is player 2’s previous period

4In general, let 1 make offers at t = τ + 2n, n = 0, 1, 2, ... and 2 make offers at t+ 1 ort− 1, where τ = 1 if 1 makes the first offer and τ = 0 if 2 makes the first offer.

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consumption. I focus on the time-invariant Markov perfect equilibrium ofthe game in which strategies do not depend on time.When there is no habit formation, the model reduces to the standard

Rubinstein bargaining game, which has a unique and effi cient equilibriumwith stationary offers of

x∗1 =δ2(1− δ1)1− δ1δ2

v and x∗2 =1− δ1

1− δ1δ2v

Let

φ∗ =(1− δ1)ψδ1(1− δ2)

.

The following proposition characterizes optimal unilateral transfers whenplayer 2 accepts all unilateral transfers. I defer all the proofs to the Ap-pendix.

Proposition 1 Suppose that player 2 commits to accepting all unilateraltransfers. In a subgame perfect equilibrium, player 1’s unilateral transferwhen 1 rejects 2’s offer is given by

y2,t+1 =

{0 if φ ≤ φ∗

x1,t+2 if φ > φ∗

Since no transfer is always an option for player 1 and y2,t+1 > 0 whenφ > φ∗, it must be the case that player 1 benefits from this transfer. Inother words, player 1 gains bargaining leverage by exploiting player 2’s habitforming behavior, which in turn hurts player 2. However player 2 has theoption of rejecting a transfer. The next proposition states that player 2 willaccept y2,t+1 = x1,t+2 in any subgame perfect equilibrium when φ > φ∗.

Proposition 2 Suppose that φ > φ∗. In any subgame perfect equilibrium,player 2 accepts y2,t+1 = x1,t+2.

That y2,t+1 = xh1 > 0 and player 2 accepts the transfer is a striking result.This result differentiates my model from earlier work that studies bargainingwith endogenously determined disagreement payoffs (Haller and Holden 1990,Fernandez and Glazer 1991, Avery and Zemsky 1994, Busch and Wen 1995).My model is similar to these earlier work in the sense of disagreement payoffsbeing endogenous. However, in contrast, a player’s past consumption level

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is a payoff relevant state of the game in my model. This has nontrivial andnovel implications on model predictions. For example, Haller and Holden(1990) and Fernandez and Glazer (1991) study wage bargaining between afirm and a union. The union may strike in case of disagreement, which iscostly for both the firm and the union workers. In that model, not striking isa Nash equilibrium of the stage game and therefore the unique equilibriumof the standard Rubinstein game without a strike option is an equilibriumof the game with the strike option (Busch and Wen 1995, Corollary 1). Incontrast, if φ > φ∗, there is no equilibrium in my model in which player1 does not make any unilateral transfer on and off the equilibrium path.Therefore the unique equilibrium of the standard Rubinstein game without atransfer option is not an equilibrium when φ > φ∗ in my model even thoughno transfer is the unique Nash equilibrium of the stage game.I discuss the time-invariant Markov perfect equilibrium next. Define the

following:

xh1 =δ2

1− δ1δ2 + (1− δ2)φ− δ2(1− δ1)ψ(1− δ1)v,

xh2 =1 + (1− δ2)φ

1− δ1δ2 + (1− δ2)φ− δ2(1− δ1)ψ(1− δ1)v,

yh1 = 0 and yh2 = xh1

The following proposition summarizes the Markov perfect equilibrium ofthe game with habit formation.

Proposition 3 (i) (Low Cost) If φ ≤ φ∗, in the time-invariant Markovperfect equilibrium of the game, player 1 does not make any unilateral transferwhen an offer is rejected; if player 1 made a unilateral transfer of y afterrejecting 2’s offer in the previous period, then player 1 offers x1(y), where

x1(y) =

{−(1− δ2)φy + δ2x

∗2 if y ≤ x∗1

1+(1−δ2)φx∗1

1+(1−δ2)φ otherwise,

he accepts any offer smaller than or equal to x∗2 and rejects any other offer;player 2 offers x∗2, accepts any offer greater than or equal to x1(y) and rejectsany other offer. Player 2 accepts any transfer by player 1. x1(0) = x∗1 sothat the habit formation has no effect on offers on the equilibrium path.(ii) (High Cost) If φ > φ∗, in the time-invariant Markov perfect equi-

librium of the game, player 1 does not make any unilateral transfer when 2

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rejects his offer and he makes a unilateral transfer of xh1 when he rejects 2’soffer; if player 1 made a unilateral transfer of y after rejecting 2’s offer inthe previous period, then player 1 offers xh1(y), where

xh1(y) =

{−(1− δ2)φy + δ2x

h2 if y ≤ xh1

xh1 otherwise,

he accepts any offer smaller than or equal to xh2 and rejects any other offer;player 2 offers xh2 , accepts any offer greater than or equal to x

h1(y) and rejects

any other offer. Player 2 accepts any transfer by player 1.

When φ ≤ φ∗, there is no unilateral transfer on the equilibrium pathand x1(0) = x∗1. That is, the players offer x

∗1 and x

∗2 in equilibrium, so habit

formation does not have any effect on equilibrium behavior.When φ > φ∗, player 2’s habit formation changes the equilibrium behavior

of both players. On the equilibrium path, y2 = xh1 and xh1(x

h1) = xh1 so that

the players offer xh1 and xh2 in equilibrium. The first interesting observation

is that player 1 takes advantage of player 2’s habit formation to decrease theequilibrium offers for player 2:

xh1 < x∗1 and xh2 < x∗2

This bargaining leverage comes to player 1 at no additional cost since theoffers are accepted immediately in equilibrium and player 1 never makes aunilateral transfer on the equilibrium path.The timing of 1’s unilateral transfer off the equilibrium path reveals how

player 1 takes advantage of player 2’s habit forming behavior. First, player1 alternates between making a unilateral transfer one period and no transferthe following period. This creates a cost for player 2 in periods with nounilateral transfer and decreases her continuation payoff when an offer isrejected off the equilibrium path. This outcome is also predicted when theproposer is selected randomly every period (see Section 4.2).Second, the timing of the unilateral transfer off the equilibrium path

matters. Player 1 can make a unilateral transfer either after rejecting player2’s offer or after player 2 rejects his offer. It is optimal for Player 1 to makethe unilateral transfer after he rejects player 2’s offer. Intuitively, player 2collects a larger share of surplus when her offer is accepted. Player 1 canrecover some of this surplus by threatening player 2 with a rejection. Afterrejecting player 2’s offer, he can unilaterally transfer the amount that he will

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propose next period. This transfer will create costs for player 2 in the nextperiod if she rejects it, therefore she will accept smaller offers next period.Then player 1’s continuation payoffwill be higher, since he can induce player2 to accept smaller offers next period. Anticipating player 1’s actions off theequilibrium path and his continuation payoff, player 2 rationally makes andaccepts smaller offers in comparison to the case without habit formation.Player 1’s bargaining leverage gain can be demonstrated more clearly at

the threshold φ = φ∗. Habit formation has no impact on the equilibriumbehavior when φ ≤ φ∗. Substituting φ = φ∗ in xhi , the limit of x

hi as φ

approaches to φ∗ from above become

xh,lim1 =δ1δ2

(1− δ1δ2)(δ1 + (1− δ1)ψ)(1− δ1)v and

xh,lim2 =1− δ1

1− δ1δ2v

ψ > 0 impliesxh,lim1 < x∗1 and x

h,lim2 = x∗2

The jump from xh,lim1 to x∗1 is surprising. Although player 1 cannot utilizehabit formation when φ ≤ φ∗, he can utilize it when φ is just slightly aboveφ∗. This discontinuity shows the strategic opportunity player 1 gains fromplayer 2’s habit formation off the equilibrium path.The closed form solution yields intuitive comparative statics. dxi

dφ< 0 and

dxidψ

> 0 for both i = 1, 2 when φ > φ∗.5 In other words, both players decreasetheir equilibrium offers for player 2’s share when the cost from habits, φ,increases or the cost of unilateral transfer, ψ, decreases. Also dφ∗

dψ> 0 and

∂xhi 2

∂ψ∂φ> 0 for both i = 1, 2 when φ > φ∗.6 In words, if player 1’s cost of

unilateral transfers gets larger, it becomes more diffi cult for him to exploitplayer 2’s habit forming behavior.This completes the discussion of the benchmark model. Next, I discuss

the extensions of the model.

5 ∂xh1

∂φ = − (1−δ2)δ2(1−δ1)v[1−δ1δ2+(1−δ2)φ−δ2(1−δ1)ψ]2 < 0 and

∂xh2∂φ = − δ2(1−δ2)(δ1+(1−δ1)ψ)(1−δ1)v

[1−δ1δ2+(1−δ2)φ−δ2(1−δ1)ψ]2 < 0.

6 dφ∗

dψ = (1−δ1)ψδ1(1−δ2) > 0,

∂xh21∂ψ∂φ = − 2δ2(1−δ1)

1−δ1δ2+(1−δ2)φ−δ2(1−δ1)ψ∂xh1∂φ > 0 and ∂xh22

∂ψ∂φ =[1+δ1δ2+(1−δ2)φ+δ2(1−δ1)ψ](1−δ2)δ2(1−δ1)2v

[1−δ1δ2+(1−δ2)φ−δ2(1−δ1)ψ]3 > 0.

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4 Extensions of the benchmark model

The benchmark model predicts how and when player 1 can exploit player2’s habit forming behavior in bargaining. The qualitative features of thesepredictions are robust to several extensions. I will discuss below the casewhere player 1 also exhibits habit forming behavior, as well as the case wherethe proposer is selected randomly every period. In Section 5, I study the casewhere player 1 is uncertain about whether player 2 exhibits habit formingbehavior or not. In order to simplify the analysis, I will assume that player2 accepts player 1’s transfers, so I will not study her decision to accept orreject a transfer in the appendix.

4.1 Two-sided Habit Formation

Set ψ = 1. That is, player 1’s consumption falls by the amount of his uni-lateral transfer to 2. In addition, assume that player 1 also pays a cost dueto habit formation whenever the consumption level falls below that in theprevious period. If he consumes z1t, his lifetime utility is given by∑

t

δt1 [z1t − γ[z1,t−1 − z1t]+]

where

[z1,t−1 − z1t]+ =

{z1,t−1 − z1t if z1,t−1 > z1t0 otherwise

γ ≥ 0, and γ[z1,t−1 − z1t]+ captures player 1’s cost from his habit for pastconsumption.Define

φ∗∗ =(1− δ1)(1 + γ)

δ1(1− δ2),

xhh1 =δ2(1− δ1)

(1− δ2)(1 + φ)− γδ2(1− δ1)v,

xhh2 =(1− δ1) (1 + (1− δ2)φ)

(1− δ2)(1 + φ)− γδ2(1− δ1)v,

yhh1 = 0 and

yhh2 =

{xhh1 if φ > φ∗∗

0 if φ ≤ φ∗∗

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The following summarizes the equilibrium:

Proposition 4 (i) (Low Cost) If φ ≤ φ∗∗, in the time-invariant Markovperfect equilibrium of the game, player 1 does not make any unilateral transferwhen an offer is rejected; if player 1 made a unilateral transfer of y afterrejecting 2’s offer in the previous period, then player 1 offers x1(y), where

x1(y) =

{−(1− δ2)φy + δ2x

∗2 if y ≤ x∗1

1+(1−δ2)φx∗1

1+(1−δ2)φ otherwise,

he accepts any offer smaller than or equal to x∗2 and rejects any other offer;player 2 offers x∗2, accepts any offer greater than or equal to x1(y) and rejectsany other offer. x1(0) = x∗1 so that habit formation has no effect on offerson the equilibrium path.(ii) (High Cost) If φ > φ∗∗, in the time-invariant Markov perfect equi-

librium of the game, player 1 does not make any unilateral transfer when 2rejects his offer and he makes a unilateral transfer of xhh1 when he rejects 2’soffer; if player 1 made a unilateral transfer of y after rejecting 2’s offer inthe previous period, then player 1 offers xhh1 (y), where

xhh1 (y) =

{−(1− δ2)φy + δ2x

hh2 if y ≤ xhh1

xhh1 otherwise,

he accepts any offer smaller than or equal to xhh2 and rejects any other offer;player 2 offers xhh2 , accepts any offer greater than or equal to x

hh1 (y) and

rejects any other offer.

The expression of φ∗∗ is very similar to the expression of φ∗.When player1 makes a unilateral transfer of y in the original model, his cost is ψy; in thisnew setting, the cost has two components: the direct cost y is the decrease inhis consumption, and γy is the cost thanks to habit formation. φ∗∗ becomesequal to φ∗ when ψ = 1 and γ = 0.

xhh1 (xhh1 ) = xhh1 so that the players offer xhh1 and xhh2 on the equilibriumpath. An increase in player 1’s cost coeffi cient γ increases xhhi , benefitingplayer 2 and harming player 1.

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4.2 Random offers

Suppose that, at the beginning of each period, player i is selected to makean offer with probability πi, where π1 + π2 = 1. πi may represent player i’srelative bargaining power.When there is no habit formation, i.e. φ = 0, there will be no unilateral

transfer in equilibrium and equilibrium offers are given by

xr1 =δ2π2

1− δ2π1 − δ1π2(1− δ1)v

xr2 =1− δ2π1

1− δ2π1 − δ1π2(1− δ1)v

Consider φ > 0. In the time-invariant Markov perfect equilibrium, whenoffers are rejected off the equilibrium path, player 1 makes a positive unilat-eral transfer every other period. Player 1’s optimal transfers do not dependon past actions, but equilibrium offers depend on player 1’s unilateral trans-fer in the previous period. Let s ∈ {0, 1} denote the state of the game atthe beginning of a period. Then s = 0 if player 1 was prescribed to makeno unilateral transfer in the previous period, and s = 1 if he was prescribedto make a transfer in the previous period. Denote by xi0(y) the offer thatplayer i makes if i is chosen to make an offer, s = 0, and he made a transferof y in the previous period. Similarly, denote by xi1(y) the offer that playeri makes if i is chosen to make an offer and s = 1, and he made a transfer ofy in the previous period.Define

φ∗∗∗ =1− δ1

δ1π1(1− δ2)and let (x11, x21, x10, x20) solve the following linear equation system,

x11 =δ2

1 + (1− δ2)φ(π1x10 + π2x20)

x21 = (1− δ1)v + δ1 (π1x10 + π2x20)

x10 = (1− δ2)x11 + δ2 (π1x11 + π2x21)

x20 = (1− δ1)(v + x11) + δ1 (π1x11 + π2x21)

Also

x11(y) =

{−(1− δ2)φy + δ2 (π1x10 + π2x20) if y ≤ x11x11 otherwise

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and

x10(y) =

x10 if y ≤ x11x10 − φ(y − x11) if x11 < y ≤ xx otherwise

where x = x10+φ(1−δ2)x111+(1−δ2)φ < x10. Notice that x10(0) = x10 and x11(y) = x11,

which are player 1’s offers on the equilibrium path when he did not deviatefrom his equilibrium transfer in the previous period. Player 2’s equilibriumoffers are x20 and x21 independently of past actions.The following summarizes the equilibrium:

Proposition 5 (i) (Low Cost) If φ ≤ φ∗∗∗, in the time-invariant Markovperfect equilibrium of the game, player 1 does not make any unilateral transferwhen an offer is rejected; if player 1 made a unilateral transfer of y in theprevious period, then player 1 offers xr1(y), where

xr1(y) =

−(1− δ2)φy + δ2 (π1xr1 + π2x

r2) if y ≤

δ2(π1xr1+π2xr2)1+(1−δ2)φ

δ2(π1xr1+π2xr2)1+(1−δ2)φ otherwise,

he accepts any offer smaller than or equal to xr2 and rejects any other offer;player 2 offers xr2, accepts any offer greater than or equal to x

r1(y) and rejects

any other offer. xr1(0) = xr1 so that habit formation has no effect on offerson the equilibrium path.(ii) (High Cost) If φ > φ∗∗∗, in the time-invariant Markov perfect equi-

librium of the game,

(0) if s = 0, i.e. player 1’s equilibrium strategy in the previous period wasnot to make a unilateral transfer, and he made a transfer of y ≥ 0,player 1 offers x10(y), accepts any offer smaller than or equal to x20and rejects any other offer; makes a unilateral transfer of x11 if anoffer is rejected; player 2 offers x20, accepts any offer greater than orequal to x10(y) and rejects any other offer.

(y) if s = 1, i.e. player 1’s equilibrium strategy in the previous period wasto make a unilateral transfer of x11, and he made a transfer of y ≥ 0,player 1 offers x11(y), accepts any offer smaller than or equal to x21and rejects any other offer; makes no unilateral transfer when an offeris rejected; player 2 offers x21, accepts any offer greater than or equalto x11(y) and rejects any other offer.

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As I mentioned before, these equilibrium predictions have important im-plications for international bargaining. In particular, the alternating patternof unilateral transfers in equilibrium implies that sanctions will be effective ifthe target state has not been subject to sanctions yet and a temporary easeof sanctions will be more effective than constant sanctioning if the targetstate has been sanctioned for some time and has adapted,Player 1’s equilibrium offers are smaller than player 2’s offers in both

states, i.e. x1s < x2s for s ∈ {0, 1}, and x11 is the smallest offer madein equilibrium. Comparative statics analysis yields intuitive results. Anincrease in π1, φ or δ1 and a decrease in δ2 decreases every xis, increasingplayer 1’s payoff in equilibrium.

5 Incomplete Information

Consider the benchmark model with ψ = 1. Suppose that player 1 doesnot know if player 2 exhibits habit formation or not. Let φ ∈ {φl = 0, φh},φ∗ < φh. If φ = φl = 0, then player 2 does not exhibit habit formation. Ifφ = φh, then player 2 exhibits habit formation. Player 2’s type is her privateinformation. It is common knowledge that she exhibits habit formation withprobability θ, i.e. Pr(φ = φh) = θ. The outcome is predicted by BayesianNash equilibrium.Under complete information, if φ = φl, then yi = 0 for i = 1, 2 and the

equilibrium offers are given by

x∗1 =δ2(1− δ1)1− δ1δ2

v and x∗2 =1− δ1

1− δ1δ2v

and if φ = φh, the equilibrium offers are given by

xh1 =δ2(1− δ1)

(1− δ2)(1 + φh)v and xh2 =

1 + (1− δ2)φh(1− δ2)(1 + φh)

(1− δ1)v,

in that case, y1 = 0 and y2 = xh1 .Assume that player 2 makes the first offer. When different types of player

2 make different offers in a separating equilibrium, the continuation gamereduces to a complete information game. Define

φ =δ2(1− δ1δ2)

(1− δ2)(δ22 − 1 + δ1δ2)− 1 > φ∗ and α =

x∗2 − xh2x∗2 − δ2x∗1

15

The following proposition summarizes the Bayesian equilibrium whenplayer 2 makes the first offer.

Proposition 6 Assume that player 2 makes the first offer.(i) If δ1 <

1−δ22δ2

or φh ∈ [φ∗, φ], the following is a separating equilibrium:Type φl offers x∗2, which player 1 accepts with probability 1−α. Type φh offersxh2 < x∗2, which player 1 accepts with probability 1. Player 2 rejects any otheroffer bigger than xh2 , accepts any other offer smaller than xh2 . If player 2offers x∗2, then player 1 updates his belief to φ = φl, otherwise he updateshis belief to φ = φh and the players play the equilibrium of the associatedcomplete information games. The equilibrium payoff of both types of player2 is xh2 .(ii) If δ1 >

1−δ22δ2

and φh > φ, there does not exist any separating equilib-rium. The following is a pooling equilibrium: Both types of player 2 offer xh2 .Player 1 rejects offers x > xh2 and accept offers x ≤ xh2 . If player 2 offersx 6= xh2 , then player 1 updates his belief to φ = φh and plays according to theequilibrium of the associated complete information game. The equilibriumpayoff of both types of player 2 is xh2 .

Since xh2 is a decreasing function of φh, α increases with φh. That is, thelikelihood of a delay when player 1 receives an offer of x∗2 increases. However,since player 1 is indifferent between accepting and rejecting x∗2, this increasein the likelihood of delay does not decrease his expected payoff. In contrast,since xh2 decreases with φh , his payoff of v−xh2 increases. Type φl’s expectedpayoff decreases because of the delay.Consider the game in which player 1 makes the first offer. There is no sep-

arating equilibrium of this game in which different types of player 2 separatethemselves by their acceptance/rejection decision. This is because player 2of type φh can achieve a higher payoff by imitating φl, after which the gameturns into a complete information game with φ = φl and φh collects thehighest payoff she can. Therefore, the continuation game after a potentialrejection is an incomplete information game in which player 2 makes the of-fer. By Proposition (6), the equilibrium payoff of both types of player 2 isxh2 whether they play a separating or pooling equilibrium in the continuationincomplete information game. Then the payoff from rejecting player 1’s offeris δ2xh2 for both types of player 2, so that player 1 offers δ2x

h2 and both types

accept in equilibrium. The following proposition summarizes the Bayesianequilibrium when player 1 makes the first offer.

16

Proposition 7 Assume that player 1 makes the first offer. Player 1 offersδ2x

h2 . Both types of player 2 accept offers x ≥ δ2x

h2 and reject offers x < δ2x

h2 .

If an offer is rejected, player 1 updates his belief to Pr(φ = φh) = 1, and theplayers play the equilibrium strategies of the complete information game withφ = φh.

If player 1 knew that φ = φh, he would have offered xh1 < δ2xh2 . Thus,

player 2 of type φh collects an information rent of R(φh) = δ2xh2 − xh1 . In the

complete information game, player 2 of type φh is hurt by an increase in φh,since xh1 and x

h2 are decreasing functions of φh. She is also hurt by an increase

in φh in the incomplete information game for the same reason. However, bothR(φh) and R(φh)/x

h1 are increasing functions of φh. That is, player 1 pays her

a larger information rent in both absolute and relative values as φh increases.

6 Conclusion

In this paper, I examine the role of habit formation in bargaining, and showhow players can exploit their opponent’s habit formation to generate endoge-nous costs for the opponent. Introduction of habit formation brings in newstrategic tools and incentives. In the subgame perfect equilibrium of the com-plete information game, a player can exploit the habit forming behavior ofhis opponent off the equilibrium path. This increases his equilibrium payoffin comparison to the case with no habit formation. Since unilateral transfersare made only off the equilibrium path, this increase in payoff comes for freeto the player. The qualitative features of the models are robust to severalextensions. Introduction of informational asymmetry may cause delay innegotiations without changing the qualitative predictions.These findings have direct implications in international bargaining. In

particular, the carrot-stick approach is a preferred policy in internationalnegotiations. The conventional wisdom is that offering a combination ofrewards and punishments may be effective in getting one’s opponent to makeconcessions in bargaining. My findings suggest that, even a “free carrot”may sometimes improve the hand of a negotiating country if the citizens ofthe opponent country exhibit habit forming behavior.The international relations literature mostly studies such problems in

isolation that is, independent of other possible coercive tools, for example,the actual use of force (e.g. see Jonge Oudraat 2000 for a criticism of the

17

sanctions literature). It is quite possible that such policies can work moreeffectively when they are a part of a comprehensive strategy. That includesthe use of force as an outside option. I leave this problem for future research.

18

References

[1] Abel, Andrew B. 1990. "Asset Prices under Habit Formation and Catch-ing up with the Joneses." American Economic Review. 80(2): 38-42.

[2] Avery, Christopher and Peter B. Zemsky (1994), “Money Burning andMultiple Equilibria in Bargaining,”Games and Economic Behavior, 7,154-168.

[3] Busch, Lutz-Alexander and Quan Wen. 1995. “Perfect Equilibria in aNegotiation Model.”Econometrica 63(3): 545-565.

[4] Berejikian, Jeffrey D. 2004. International Relations Under Risk: Fram-ing State Choice. SUNY Press, Albany, NY.

[5] Braun, Phillip A., Constantinides, George M. and Wayne E. Ferson.1993. "Time Nonseparability of Aggregate Consumption: InternationalEvidence." European Economic Review. 37(5): 897-920.

[6] Campbell, John Y. and J. H. Cochrane. 1999. "By Force of Habit: AConsumption-Based Explanation of Aggregate Stock Market Behavior."Journal of Political Economy. 107(2): 205-51.

[7] Carroll, Christopher D., Overland, Jody, and David N. Weil. 1997."Comparison Utility in a Growth Model." Journal of Economic Growth.2(4): 339-67.

[8] Camerer, C., and G. Loewenstein. 2004. "Behavioral Economics: Past,Present, and Future" in Advances in Behavioral Economics, ed. by C.Camerer, G. Loewenstein, and M. Rabin. Princeton University Press.

[9] Constantinides, George M. 1990. "Habit Formation: A Resolution of theEquity Premium Puzzle." Journal of Political Economy. 98(3): 519-43.

[10] Fernandez, Rachel and Jacob Glazer. 1990. "Striking for a Bargain Be-tween Two Completely Informed Agents," American Economic Review81: 240-252.

[11] Ferson, Wayne E. and George M. Constantinides. 1991. "Habit persis-tence and durability in aggregate consumption: Empirical tests." Jour-nal of Financial Economics. 29(2): 199-240.

19

[12] Fuhrer, Jeffrey C. 2000. "Habit Formation in Consumption and Its Im-plications for Monetary-Policy Models." American Economic Review.90(3): 367-390.

[13] Haller, Hans Hermann and Steinar Holden. 1990. "A Letter to the Editoron Wage Bargaining." Journal of Economic Theory 52: 232-236.

[14] Jonge Oudraat, Chantal de. 2000. "Making Economic Sanctions Work."Survival 42(3): 105-127.

[15] Kahneman, Daniel and Amos Tversky. 1979. "Prospect theory: Ananalysis of decisions under risk." Econometrica. 47: 313-327.

[16] Levy, Jack S. 1996. "Loss Aversion, Framing and Bargaining: The Im-plications of Prospect Theory for International Conflict." InternationalPolitical Science Review 17(2): 179-195.

[17] Levy, Jack S. 1997a. "Prospect Theory and the Cognitive-Rational De-bate." In Nehemia Geva and Alex Mintz, eds., Decisionmaking on Warand Peace: The Cognitive-Rational Debate. Boulder, CO: Lynne Rien-ner, pp 33-50.

[18] Levy, Jack S. 1997b. "Prospect Theory, Rational Choice, and Interna-tional Relations." International Studies Quarterly. 41(1): 87-112.

[19] Osborne, Martin J. and Ariel Rubinstein. 1990. Bargaining and Markets.Academic Press.

[20] Pape, Robert. 1997. "Why Economic Sanctions do not Work." Interna-tional Security 2:90-136.

[21] Rozen, Kareen. 2010. "Foundations of Intrinsic Habit Formation."Econometrica. 78(4): 1341—1373.

[22] Rubinstein Ariel. 1982. "Perfect Equilibrium in a Bargaining Model."Econometrica. 50(1): 97-110.

20

A Equilibrium Analysis

In equilibrium, each player chooses an optimal action at every subgame. Iwill assume that a player accepts an offer when he is indifferent betweenaccepting and rejecting it given the continuation equilibrium. If player j isbetter off by accepting player i’s offer, then player i can increase his/herpayoff by slightly decreasing player j’s share, which player j will continue toaccept. Therefore player i’s optimal offer makes player j indifferent betweenaccepting and rejecting i’s offer. Consider an equilibrium at which offers andtransfers are accepted. Given 1’s strategy for transfers, yit, the conditionsfor the equilibrium offers that are accepted are given by

x1t1− δ2

− φ [y2,t−1 − x1t]+ = (y1t − φ [y2,t−1 − y1t]+)

+ δ2

[x2,t+11− δ2

− φ [y1t − x2,t+1]+]

(1)

v − x2,t+11− δ1

= −ψy2,t+1 +δ1

1− δ1(v − x1,t+2) (2)

In the first equation, the left hand side is player 2’s payoff if she accepts x1t.x1t1−δ2 is her lifetime utility from consuming x1t forever and −φ [y2,t−1 − x1t]+is the audience cost that she pays in period t. Since there is no change inthe consumption level thereafter, there is no further cost after period t. Theright hand side is her payoff if she rejects the offer x1t. In that case, sheconsumes y1t, the unilateral transfer by player 1 and pays the audience costof φ [y2,t−1 − y1t]+ in period t, and then her offer of x2,t+1 will be accepted inperiod t+ 1, which she will consume thereafter, and she will pay a one-periodcost of −φ [y1t − x2,t+1]+ in period t+ 1.In the second equation, the left hand side is player 1’s payoff if he accepts

2’s offer x2,t+1 in period t+1 and the right hand side is his payoff if he rejectsit. In that case, player 1’s offer of x1,t+2 will be accepted in period t+ 2, andhe will pay a one-period unilateral transfer cost, ψy2,t+1.To simplify the expressions, multiply both sides of (1) by (1 − δ2) and

multiply both sides of (2) by (1 − δ1). I will use the following equations inthe analysis.

21

x1t − (1− δ2)φ [y2,t−1 − x1t]+ = (1− δ2)[y1t − φ [y2,t−1 − y1t]+

]+ δ2

[x2,t+1 − (1− δ2)φ [y1t − x2,t+1]+

](3)

v − x2,t+1 = −(1− δ1)ψy2,t+1 + δ1(v − x1,t+2) (4)

I will solve for a time-invariant Markov perfect equilibrium with

xit = xi and yit = yi for all t

A.1 Equilibrium with no habit formation

Suppose that φ = 0. In that case subgame perfection requires y1 = y2 = 0and conditions 3 and 4 for the equilibrium offers reduce to

x∗1 = δ2x∗2

v − x∗2 = δ1(v − x∗1)

which yields

x∗1 =δ2(1− δ1)1− δ1δ2

v and x∗2 =1− δ1

1− δ1δ2v

A.2 Equilibrium with habit formation

When player 2 exhibits habit forming behavior, player 1 may find it optimalto take a unilateral action to gain bargaining leverage in the negotiations.Next, I will solve for 1’s optimal unilateral transfers under the assumptionthat player 2 commits to accepting all transfers.

A.2.1 Optimal choice for unilateral transfer yi when player 2 ac-cepts

Assume that player 2 commits to accepting all transfers. First consider player1’s decision for a unilateral transfer when 1 rejects 2’s offer.

Proposition 8 In a subgame perfect equilibrium, player 1 does not makeany unilateral transfer when his offer is rejected. That is, y1t = 0 for all t inevery subgame perfect equilibrium.

22

Proof. Consider the node at which 2 rejects 1’s offer x1t in period t. Atthis node of the game, 1’s continuation payoff as a function of y1t is given by(after multiplying by (1− δ1))

ur1 = −(1− δ1)ψy1t + δ1(v − x2,t+1)

where x2,t+1 is determined by

v − x2,t+1 = −(1− δ1)ψy2,t+1 + δ1(v − x1,t+2)

Then ∂x2,t+1∂y1t

= 0 so that

dur1

dy1t= −(1− δ1)ψ < 0

which implies thaty1t = 0

in equilibrium. That is, if 2 rejects 1’s offer, it is optimal for 1 not to makeany unilateral transfer in that period.Next consider the unilateral transfer by 1 after 1 rejects 2’s offer.

Proposition 9 Assume that player 2 commits to accepting all transfers. Ina subgame perfect equilibrium, player 1’s unilateral transfer when 1 rejects2’s offer is given by

y2,t+1 =

{0 if φ ≤ φ∗

x1,t+2 if φ > φ∗

where

φ∗ =(1− δ1)ψδ1(1− δ2)

Proof. Consider the node at which 1 rejects 2’s offer x2,t+1 in period t + 1.At that node of the game, 1’s continuation payoff as a function of y2,t+1 isgiven by (after multiplying by (1− δ1))

ur2 = −(1− δ1)ψy2,t+1 + δ1(v − x1,t+2)


x1,t+2 − (1− δ2)φ [y2,t+1 − x1,t+2]+ = (1− δ2)[y1,t+2 − φ [y2,t+1 − y1,t+2]+

]+ δ2

[x2,t+3 − (1− δ2)φ [y1,t+2 − x2,t+3]+

]23

Substituting y1t = 0 for all t from Proposition 8, I obtain

x1,t+2 − (1− δ2)φ [y2,t+1 − x1,t+2]+ = −(1− δ2)φy2,t+1 + δ2x2,t+3 (5)

If y2,t+1 > x1,t+2 then this equation reduces to

(1 + (1− δ2)φ)x1,t+2 = δ2x2,t+3

so thatdx1,t+2dy2,t+1

= 0

anddur2

dy2,t+1= −(1− δ1)ψ < 0

This implies that 1’s utility increases by an decrease in y2,t+1 when y2,t+1 >x1,t+2 so that y2,t+1 ≤ x1,t+2 in equilibrium. Then equation (5) reduces to

x1,t+2 = −(1− δ2)φy2,t+1 + δ2x2,t+3

so thatdx1,t+2dy2,t+1

= −(1− δ2)φ

which impliesdur2

dy2,t+1= −(1− δ1)ψ + δ1(1− δ2)φ

Then

dur12dy2,t+1

< 0 if φ < φ∗

= 0 if φ = φ∗

> 0 if φ > φ∗

where

φ∗ =(1− δ1)ψδ1(1− δ2)

When y2,t+1 ≤ x1,t+2, this implies that 1’s utility increases with y2,t+1 only ifφ > φ∗, that is, if the cost coeffi cient is suffi ciently large. Then in equilibrium,

y2,t+1 =

{0 if φ ≤ φ∗

x1,t+2 if φ > φ∗

Next I will prove that it is optimal for player 2 to accept player 1’s transfer.

24

A.2.2 Player 2’s optimal action to accept or reject a transfer

Consider a continuation game after player 2 rejects player 1’s offer of x1,t.Suppose that player 1 makes a transfer of y1t ≥ 0. Since player 2 will makethe next offer and x2,t+1 is independent of y1t, it is optimal for player 2 toaccept y1t.Next consider player 2’s decision to accept or reject y2,t+1 = x1,t+2 after

player 1 rejects her offer of x2,t+1. Then [y2,t+1 − x1,t+2]+ = 0. If she acceptsy2,t+1, then x1,t+2 is given by

x1,t+2 = (1−δ2)[y1,t+2 − φ [y2,t+1 − y1,t+2]+

]+δ2

[x2,t+3 − (1− δ2)φ [y1,t+2 − x2,t+3]+

]Since y1,t+2 = 0 in any subgame perfect equilibrium, this yields

x1,t+2 = −(1− δ2)φy2,t+1 + δ2[x2,t+3 − (1− δ2)φ [y1,t+2 − x2,t+3]+

]Then player 2’s payoff from accepting y2,t+1 is

(1− δ2)[y2,t+1 − φ [y1t − y2,t+1]+

]+ δ2x1,t+2 (6)

If she rejects the transfer, then player 1’s offer the next period is given by

x′1,t+2 = (1− δ2)y1,t+2 + δ2[x2,t+3 − (1− δ2)φ [y1,t+2 − x2,t+3]+

]= δ2

[x2,t+3 − (1− δ2)φ [y1,t+2 − x2,t+3]+

]and her payoff from rejecting y2,t+1 is

−(1− δ2)φy1t + δ2x′1,t+2

Player 2 accept the transfer if (6) is greater than or equal to (??). equiv-alently, [

y2,t+1 − φ [y1t − y2,t+1]+]− δ2φy2,t+1 + φy1t ≥ 0 (7)

Player 1 might have deviated in the past, so y1t is not necessarily equalto zero. If y1t > y2,t+1, (7) is equivalent to

1 + (1− δ2)φ ≥ 0

which holds, so player 2 accepts y2,t+1. If y1t ≤ y2,t+1, (7) becomes

(1− δ2φ)y2,t+1 + φy1t ≥ 0

25

y1t ≤ y2,t+1 implies

(1− δ2φ)y2,t+1 + φy1t ≥ (1− δ2φ)y1t + φy1t

= (1 + (1− δ2)φ)y1t ≥ 0

so that (7) holds and player 2 accepts y2,t+1. This completes the proof ofProposition 2.Next I will solve for the equilibrium offers.

A.2.3 Optimal offers with low cost: φ ≤ φ∗

On the equilibrium path of a time-invariant Markov perfect equilibrium, xit =xhi and yit = yhi for all t. Substituting y

h1 = yh2 = 0 from propositions 8 and

9, conditions 3 and 4 for the equilibrium offers reduce to

xh1 = δ2xh2

v − xh2 = δ1(v − xh1)

which yieldsxh1 = x∗1 and x

h2 = x∗2

Consider a period following y2,t−1, which may be different than his equi-librium transfer of zero. By propositions 8 and 9, y1t′ = y2,t′+1 = 0 and areindependent of past decisions for all t′ ≥ t. Also x2,t+1 and x1,t+2 are indepen-dent of y2,t−1, because player 1 will choose y1t = 0 and the game will revertto its time-invariant Markov perfect equilibrium path after t. So x2,t+1 = x∗2and x1,t+2 = x∗1. Substitute these in (3) and obtain

x1t − (1− δ2)φ [y2,t−1 − x1t]+ = −(1− δ2)φy2,t−1 + δ2x∗2

If y2,t−1 ≤ x1t, [y2,t−1 − x1t]+ = 0 so that

x1t = −(1− δ2)φy2,t−1 + δ2x∗2

and y2,t−1 ≤ x1t becomes equivalent to

y2,t−1 ≤ −(1− δ2)φy2,t−1 + δ2x∗2 ⇔

y2,t−1 ≤δ2x∗2

1 + (1− δ2)φ≡ x (8)

26

If y2,t−1 > x1t, (3) becomes

x1t − (1− δ2)φ(y2,t−1 − x1t) = −(1− δ2)φy2,t−1 + δ2x∗2 ⇒

x1t = x

So player 1’s optimal offer in a time-invariant Markov perfect equilibrium isgiven by x1(yt−1) where

x1(y) =

{−(1− δ2)φy + δ2x

∗2 if y ≤ x

x otherwise

Notice that x1(0) = δ2x∗2 = x∗1.

Consider a period following y1t, which may be different than his equilib-rium transfer of zero. Since y2,t+1 and x1,t+2 are independent of y1t, x2,t+1 isindependent of y1t in (4) so that x2,t+1 = x∗2.

A.2.4 Optimal offers with high cost: φ > φ∗

On the equilibrium path of a time-invariant Markov perfect equilibrium, xit =xhi and yit = yhi for all t. Substituting y

h1 = 0 and yh2 = xh1 from propositions

8 and 9, conditions 3 and 4 for the equilibrium offers reduce to

xh1 = −(1− δ2)φxh1 + δ2xh2

v − xh2 = −(1− δ1)ψxh1 + δ1(v − xh1)

which yields

xh1 =δ2

1− δ1δ2 + (1− δ2)φ− δ2(1− δ1)ψ(1− δ1)v and

xh2 =1 + (1− δ2)φ

1− δ1δ2 + (1− δ2)φ− δ2(1− δ1)ψ(1− δ1)v

φ > φ∗ implies that (1− δ2)φ > δ2(1− δ1)ψ so xh1 > 0 and xh2 > 0.Consider a period following y2,t−1, which may be different than his equi-

librium transfer of xh1 . x2,t+1 and x1,t+2 are independent of y2,t−1, becauseplayer 1 will choose y1t = 0 by propositions 8 and the game will revert to itstime-invariant Markov perfect equilibrium path after t. So x2,t+1 = xh2 andx1,t+2 = xh1 . Also by propositions 8 and 9, y1t′ = 0 and y2,t′+1 = x1,t′+2 = xh1

27

and are independent of past decisions for all t′ ≥ t. Substitute these in (3)and obtain

x1t − (1− δ2)φ [y2,t−1 − x1t]+ = −(1− δ2)φy2,t−1 + δ2xh2


x1t = −(1− δ2)φy2,t−1 + δ2xh2


y2,t−1 ≤ −(1− δ2)φy2,t−1 + δ2xh2 ⇔

y2,t−1 ≤δ2x

h2

1 + (1− δ2)φ= xh1


x1t − (1− δ2)φ(y2,t−1 − x1t) = −(1− δ2)φy2,t−1 + δ2xh2 ⇒

x1t = xh1

So player 1’s optimal offer in a time-invariant Markov perfect equilibrium isgiven by xh1(yt−1) where

xh1(y) =

{−(1− δ2)φy + δ2x

h2 if y ≤ xh1

xh1 otherwise

Notice that xh1(xh1) = xh1 .

Consider a period following y1t, which may be different than his equilib-rium transfer of zero. Since y2,t+1 and x1,t+2 are independent of y1t, x2,t+1 isindependent of y1t in (4) so that x2,t+1 = xh2 .Next, given the offers and players’decisions to accept and reject offers, I

will solve for player 2 optimal action when player 1 makes an offer.

A.3 Player 2’s optimal action to accept or reject atransfer

Consider a continuation game after player 2 rejects player 1’s offer of x1,t.Suppose that player 1 makes a transfer of y1t ≥ 0. Since player 2 will makethe next offer and x2,t+1 is independent of y1t, it is optimal for player 2 toaccept y1t. I summarize this in the proposition.

28

Proposition 10 In any subgame perfect equilibrium, it is optimal for player2 to accept any y1t.

Next consider player 2’s decision to accept or reject y2,t+1 after player 1rejects her offer of x2,t+1.

A.3.1 Low cost: φ ≤ φ∗

Suppose that player 1 makes a transfer of y2,t+1 ≥ 0. If player 2 accepts thetransfer, her continuation payoff is

(1− δ2)[y2,t+1 − φ [y1t − y2,t+1]+

]+ δ2

[x1,t+2 − (1− δ2)φ [y2,t+1 − x1,t+2]+

](9)

where

x1,t+2 = x1(y2,t+1) =

{−(1− δ2)φy2,t+1 + δ2x

∗2 if y2,t+1 ≤ x

x otherwise

If she rejects the transfer, then her continuation payoff is

−(1− δ2)φy1t + δ2x′1,t+2 (10)

wherex′1,t+2 = x1(0) = δ2x

∗2

Player 2 accepts the transfer if and only if (9) is greater than or equal to(10). Suppose that y2,t+1 ≤ x. Then

y2,t+1 − x1,t+2 = [1 + (1− δ2)φ] y2,t+1 − δ2x∗2≤ [1 + (1− δ2)φ] x− δ2x∗2 = 0

so (9) becomes

(1− δ2)[y2,t+1 − φ [y1t − y2,t+1]+

]+ δ2x1,t+2

= (1− δ2)[y2,t+1 − φ [y1t − y2,t+1]+

]+ δ2 [−(1− δ2)φy2,t+1 + δ2x

∗2]

Then (9) ≥ (10) is equivalent to

(1−δ2)[y2,t+1 − φ [y1t − y2,t+1]+

]+δ2 [−(1− δ2)φy2,t+1 + δ2x

∗2] ≥ −(1−δ2)φy1t+δ22x∗2

which is equivalent to

(1− δ2)[y2,t+1 − φ [y1t − y2,t+1]+

]− δ2(1− δ2)φy2,t+1 ≥ −(1− δ2)φy1t (11)

29

If y1t > y2,t+1, (11) becomes equivalent to

1 + (1− δ2)φ ≥ 0

so that player 2 accepts y2,t+1. If y1t ≤ y2,t+1, (11) becomes equivalent to

(1− δ2φ)y2,t+1 + φy1t ≥ 0


(1− δ2φ)y2,t+1 + φy1t ≥ (1− δ2φ)y1t + φy1t

= (1 + (1− δ2)φ)y1t ≥ 0

so that player 2 accepts y2,t+1.Now suppose that y2,t+1 > x. Then x1,t+2 = x and

y2,t+1 − x1,t+2 = y2,t+1 − x > 0

so (9) becomes

(1− δ2)[y2,t+1 − φ [y1t − y2,t+1]+

]+ δ2 [x− (1− δ2)φ(y2,t+1 − x)]

= (1− δ2)[y2,t+1 − φ [y1t − y2,t+1]+

]− δ2(1− δ2)φy2,t+1 + δ22x

∗2


(1− δ2)[y2,t+1 − φ [y1t − y2,t+1]+

]− δ2(1− δ2)φy2,t+1 ≥ −(1− δ2)φy1t (12)


1 + (1− δ2)φ ≥ 0


(1− δ2φ)y2,t+1 + φy1t ≥ 0


(1− δ2φ)y2,t+1 + φy1t ≥ (1− δ2φ)y1t + φy1t

= (1 + (1− δ2)φ)y1t ≥ 0

so that player 2 accepts y2,t+1. I summarize these results in the followingproposition.

Proposition 11 If φ ≤ φ∗ players’offers and their decisions to accept offersare given as in Proposition 3(i), then it is optimal for player 2 to accept anyy2,t+1.

30

A.3.2 High: φ > φ∗

Suppose that player 1 makes a transfer of y2,t+1 ≥ 0. If player 2 accepts thetransfer, her continuation payoff is

(1− δ2)[y2,t+1 − φ [y1t − y2,t+1]+

]+ δ2

[x1,t+2 − (1− δ2)φ [y2,t+1 − x1,t+2]+

](13)

where

x1,t+2 = xh1(y) =

{−(1− δ2)φy2,t+1 + δ2x

h2 if y2,t+1 ≤ xh1

xh1 otherwise

If she rejects the transfer, then her continuation payoff is

−(1− δ2)φy1t + δ2x′1,t+2 (14)

wherex′1,t+2 = xh1(0) = δ2x

h2

Player 2 accepts the transfer if and only if (13) is greater than or equalto (14).Suppose that y2,t+1 ≤ xh1 . Then

y2,t+1 − x1,t+2 = [1 + (1− δ2)φ] y2,t+1 − δ2xh2≤ [1 + (1− δ2)φ]xh1 − δ2x∗2 = 0

so (13) becomes

(1− δ2)[y2,t+1 − φ [y1t − y2,t+1]+

]+ δ2x1,t+2

= (1− δ2)[y2,t+1 − φ [y1t − y2,t+1]+

]+ δ2

[−(1− δ2)φy2,t+1 + δ2x

h2

]Then (13) ≥ (14) becomes equivalent to

(1−δ2)[y2,t+1 − φ [y1t − y2,t+1]+

]+δ2

[−(1− δ2)φy2,t+1 + δ2x

h2

]≥ −(1−δ2)φy1t+δ22xh2

which is equivalent to

(1− δ2)[y2,t+1 − φ [y1t − y2,t+1]+

]− δ2(1− δ2)φy2,t+1 ≥ −(1− δ2)φy1t (15)


1 + (1− δ2)φ ≥ 0

31


(1− δ2φ)y2,t+1 + φy1t ≥ 0


(1− δ2φ)y2,t+1 + φy1t ≥ (1− δ2φ)y1t + φy1t

= (1 + (1− δ2)φ)y1t ≥ 0

so that player 2 accepts y2,t+1.Now suppose that y2,t+1 > xh1 . Then x1,t+2 = xh1 and

y2,t+1 − x1,t+2 = y2,t+1 − xh1 > 0

so (13) becomes

(1− δ2)[y2,t+1 − φ [y1t − y2,t+1]+

]+ δ2

[xh1 − (1− δ2)φ(y2,t+1 − xh1)

]= (1− δ2)

[y2,t+1 − φ [y1t − y2,t+1]+

]− δ2(1− δ2)φy2,t+1 + δ22x

h2


(1− δ2)[y2,t+1 − φ [y1t − y2,t+1]+

]− δ2(1− δ2)φy2,t+1 ≥ −(1− δ2)φy1t (16)


1 + (1− δ2)φ ≥ 0


(1− δ2φ)y2,t+1 + φy1t ≥ 0


(1− δ2φ)y2,t+1 + φy1t ≥ (1− δ2φ)y1t + φy1t

= (1 + (1− δ2)φ)y1t ≥ 0

so that player 2 accepts y2,t+1. I summarize these results in the followingproposition.

Proposition 12 If φ > φ∗ players’offers and their decisions to accept offersare given as in Proposition 3(ii), then it is optimal for player 2 to accept anyy2,t+1.

32

B Two-sided Habit Formation

Let w be player 1’s income from other resources. I will normalize w to zeroand allow for negative consumption. Although I use the normalization w = 0,I will set up the equations by referring to w, because it is easier this way. Asit will become evident, w will drop out of the analysis so normalization willnot play any role.Given 1’s strategy for unilateral transfers, yit, I will derive the conditions

for the equilibrium offers next. The condition for 1’s offer is the same asbefore:

x1t − (1− δ2)φ [y2,t−1 − x1t]+ = (1− δ2)[y1t − φ [y2,t−1 − y1t]+

]+ δ2

[x2,t+1 − (1− δ2)φ [y1t − x2,t+1]+

](17)

Consider player 2’s offer x2,t+1. If 1 accepts the offer, he will consumew and v − x2,t+1 forever, which will generate a lifetime payoff of w+v−x2,t+11−δ1 .Player 1’s previous period consumption is (w − y1t), which is less than hiscurrent consumption (w + v − x2,t+1) so that he does not pay any cost. Ifhe rejects the offer, he makes a unilateral transfer of y2,t+1 to player 2, sohis current consumption becomes w − y2,t+1, for which he pays a one-periodcost of γ [(w − y1t)− (w − y2,t+1)]+ = γ [y2,t+1 − y1t]+ . His offer of x1,t+2 isaccepted next period, which generates a lifetime payoff of w+v−x1,t+2

1−δ1 fromthe next period on. Since w − y2,t+1 ≤ w + v − x1,t+2, he does not pay anyadditional cost in period t+2. Then the condition for 2’s offer becomes (aftermultiplying both sides by 1− δ1)

w + v − x2,t+1 = (1− δ1)((w − y2,t+1)− γ [y2,t+1 − y1t]+

)+ δ1(w + v − x1,t+2)

which reduces to

v − x2,t+1 = −(1− δ1)(y2,t+1 + γ [y2,t+1 − y1t]+

)+ δ1(v − x1,t+2) (18)

When φ = 0, subgame perfection requires y1 = y2 = 0 and conditions 17and 18 for the equilibrium offers yield x∗1 and x

∗2. The analysis of the case

with habit formation follows the similar steps in Section A.

33

B.1 Optimal choice for yiI will solve for y1t and y2,t+1 simultaneously. Consider any subgame perfectequilibrium. y1t is the amount of unilateral transfer by 1 after 2 rejects 1’soffer of x1t in period t. At that node of the game, 1’s continuation payoff asa function of y1t is given by (after multiplying by (1− δ1))

ur1 = (1− δ1)((w − y1t)− γ [y1t − y2,t−1]+

)+ δ1(w + v − x2,t+1) (19)


v − x2,t+1 = −(1− δ1)(y2,t+1 + γ [y2,t+1 − y1t]+

)+ δ1(v − x1,t+2) (20)

Now consider the node at which 1 rejects 2’s offer. At that node ofthe game, 1’s continuation payoff as a function of y2,t+1 is given by (aftermultiplying by (1− δ1))

ur2 = (1− δ1)((w − y2,t+1)− γ [y2,t+1 − y1t]+

)+ δ1(w + v − x1,t+2) (21)


x1,t+2 − (1− δ2)φ [y2,t+1 − x1,t+2]+ = (1− δ2)[y1,t+2 − φ [y2,t+1 − y1,t+2]+

]+ δ2

[x2,t+3 − (1− δ2)φ [y1,t+2 − x2,t+3]+

](22)

Suppose that y2,t+1 < y1t. Then γ [y2,t+1 − y1t]+ = 0, ur2 in (21) becomesindependent of yt, and y2,t+1 that maximizes ur2 becomes independent of y1tso that ∂y2,t+1

∂y1t= 0. Then dx2,t+1

dy1t= 0 from (20) so that

dur1

dy1t=

{−(1− δ1) if y1t ≤ y2,t−1−(1− δ1)(1 + γ) if y1t > y2,t−1

that is, dur1

dy1t< 0, which implies that y1t = 0. But y2,t+1 < y1t = 0 cannot

hold so y2,t+1 ≥ y1t must hold in equilibrium. Then (19), (20), (21) and (22)become

ur1 = (1− δ1)(w − y1t) + δ1(w + v − x2,t+1)v − x2,t+1 = −(1− δ1) (y2,t+1 + γ(y2,t+1 − y1t)) + δ1(v − x1,t+2) (23)

ur2 = (1− δ1) ((w − y2,t+1)− γ(y2,t+1 − y1t)) + δ1(w + v − x1,t+2)

34

x1,t+2 − (1− δ2)φ [y2,t+1 − x1,t+2]+ = (1− δ2)(y1,t+2 − φ [y2,t+1 − y1,t+2]+

)+ δ2

[x2,t+3 − (1− δ2)φ [y1,t+2 − x2,t+3]+

](24)

dx2,t+1dy1t

= (1− δ1)γ from (23) so that dur1

dx2,t+1= −(1− δ1)− δ1(1− δ1)γ < 0.

This implies yt1 = 0 for all t. So I will substitute y1,t+2 = 0 below.Consider the case y2,t+1 > x1,t+2. Since y2,t+1 ≥ y1,t+2 = 0, [y2,t+1 − y1,t+2]+ =

y2,t+1 and x1,t+2 becomes independent of y2,t+1 in (24). So dur2

dy2,t+1= −(1 −

δ1)(1 + γ) < 0 which implies that a decrease in y2,t+1 increases ur2 soy2,t+1 > x1,t+2 cannot hold. This implies that y2,t+1 ≤ x1,t+2 must holdin equilibrium. Then (24) becomes

x1,t+2 = (1−δ2)(y1,t+2 − φ [y2,t+1 − y1,t+2]+

)+δ2

[x2,t+3 − (1− δ2)φ [y1,t+2 − x2,t+3]+

]Since y1,t+2 = 0, this equality reduces to

x1,t+2 = −(1− δ2)φy2,t+1 + δ2x2,t+3

Then dx1,t+2dy2,t+1

= −(1− δ2)φ and dur2

dy2,t+1= −(1− δ1)(1 +γ) + δ1(1− δ2)φ. Define

φ∗∗ =(1− δ1)(1 + γ)

δ1(1− δ2)

Then dur2

dy2,t+1> 0 if and only if φ > φ∗∗ so

y2,t+1 =

{x1,t+2 if φ > φ∗∗

0 if φ ≤ φ∗∗

The expression of φ∗∗ is very similar to the expression of φ∗. When 1 makesa unilateral transfer of y in the original model, his cost is ψy; in this newsetting, the cost has two components, the direct cost y, and γy, which is thecost thanks to habit formation.I summarize these results in the following proposition.

Proposition 13 In a subgame perfect equilibrium, player 1’s optimal uni-lateral transfers are y1t = 0 and

y2,t+1 =

{x1,t+2 if φ > φ∗∗

0 if φ ≤ φ∗∗

35

B.2 Optimal offers with low costs: φ ≤ φ∗∗

On the equilibrium path of a time-invariant Markov perfect equilibrium, xit =xhhi and yit = yhhi for all t. Substituting yhh1 = yhh2 = 0 from Propositions 13,conditions 17 and 18 for the equilibrium offers reduce to

xhh1 = δ2xhh2

v − xhh2 = δ1(v − xh1h)

which yieldsxhh1 = x∗1 and x

hh2 = x∗2

Consider a period following y2,t−1, which may be different than his equi-librium transfer of zero. x2,t+1 and x1,t+2 are independent of y2,t−1, becauseplayer 1 will choose y1t = 0 by Proposition 13 and the game will revert to itstime-invariant Markov perfect equilibrium path after t. So x2,t+1 = x∗2 andx1,t+2 = x∗1. Also By Proposition 13, y1t′ = 0 and are independent of pastdecisions for all t′ ≥ t. Substitute these in (17) and (18) to obtain

x1t − (1− δ2)φ [y2,t−1 − x1t]+ = −(1− δ2)φy2,t−1 + δ2x∗2

v − x∗2 = δ1(v − x∗1)

If y2,t−1 ≤ x1t, then [y2,t−1 − x1t]+ = 0, so the first equation becomes

x1t = −(1− δ2)φy2,t−1 + δ2x∗2

and y2,t−1 ≤ x1t becomes equivalent to y2,t−1 ≤ x, where x is given by (8).If y2,t−1 > x1t, (17) becomes

x1t − (1− δ2)φ(y2,t−1 − x1t) = −(1− δ2)φy2,t−1 + δ2x∗2 ⇒

x1t = x

So player 1’s optimal offer in a time-invariant Markov perfect equilibrium isgiven by x1(yt−1) where

x1(y) =

{−(1− δ2)φy + δ2x

∗2 if y ≤ x

x otherwise

Notice that x1(0) = δ2x∗2 = x∗1.

Consider a period following y1t, which may be different than his equilib-rium transfer of zero. Since y2,t+1 and x1,t+2 are independent of y1t, x2,t+1 isindependent of y1t in (18) so that x2,t+1 = x∗2.

36

B.3 Optimal offers with high cost: φ > φ∗∗

On the equilibrium path of the time-invariant Markov perfect equilibrium,xit = xhhi and yit = yhhi . By Proposition 13, y1t = 0 and y2,t+1 = x1,t+2 sothat yhh1 = 0 and yhh2 = xhh1 . Substituting these in (17) and (18), the offerson the equilibrium path are given by

x1hh = −(1− δ2)φxhh1 + δ2x

hh2

v − xhh2 = −(1− δ1)(1 + γ)xhh1 + δ1(v − xhh1 )

which yields

xhh1 =δ2(1− δ1)

(1− δ2)(1 + φ)− γδ2(1− δ1)v and

xhh2 =(1− δ1) (1 + (1− δ2)φ)

(1− δ2)(1 + φ)− γδ2(1− δ1)v

Consider a period following y2,t−1, which may be different than his equi-librium transfer of xhh1 . x2,t+1 and x1,t+2 are independent of y2,t−1, becauseplayer 1 will choose y1t = 0 by Proposition 13 and the game will revert to itstime-invariant Markov perfect equilibrium path after t. So x2,t+1 = xhh2 andx1,t+2 = xhh1 . Also By Proposition 13, y1t′ = 0 and y2,t′+1 = x1,t′+2 = xhh1 andare independent of past decisions for all t′ ≥ t. Substitute these in (17) andobtain

x1t − (1− δ2)φ [y2,t−1 − x1t]+ = −(1− δ2)φy2,t−1 + δ2xhh2


x1t = −(1− δ2)φy2,t−1 + δ2xhh2


y2,t−1 ≤ −(1− δ2)φy2,t−1 + δ2xhh2 ⇔

y2,t−1 ≤δ2x

hh2

1 + (1− δ2)φ= xhh1


x1t − (1− δ2)φ(y2,t−1 − x1t) = −(1− δ2)φy2,t−1 + δ2xhh2 ⇒

x1t = xhh1

37

So player 1’s optimal offer in a time-invariant Markov perfect equilibrium isgiven by xhh1 (yt−1) where

xhh1 (y) =

{−(1− δ2)φy + δ2x

hh2 if y ≤ xhh1

xhh1 otherwise

Notice that xhh1 (xhh1 ) = xhh1 .Consider a period following y1t, which may be different than his equilib-

rium transfer of zero. Since y2,t+1 and x1,t+2 are independent of y1t, x2,t+1 isindependent of y1t in (18) so that x2,t+1 = xhh2 .

C The model with randomly selected pro-posers

I will solve for a time-invariant Markov perfect equilibrium at which offers areaccepted immediately and they depend only on player 2’s past consumptionlevel off the equilibrium path.Consider a subgame perfect equilibrium, not necessarily time-invariant

Markov perfect, in which all offers are accepted. Consider period t andsuppose that player 1 made a transfer of yt−1 in the previous period, whichmay be different than his equilibrium transfer. Optimality requires thateach player’s offer makes the other player indifferent between accepting andrejecting it. So the optimal offers in period t are determined by

x1t − (1− δ2)φ [yt−1 − x1t]+ = (1− δ2)[yt − φ [yt−1 − yt]+

](25)

+ δ2

(π1(x1,t+1 − (1− δ2)φ [yt − x1,t+1]+)

+π2(x2,t+1 − (1− δ2)φ [yt − x2,t+1]+)

)and

v − x2,t = −(1− δ1)yt + δ1 [v − (π1x1,t+1 + π2x2,t+1)] (26)

The right hand sides of (25) and (26) take into account that player i willmake the offer xi,t+1 with probability πi in period t+ 1.First consider the case φ = 0. Subgame perfection requires that yt = 0

for all t. Then (25) and (26) reduce to

x1 = δ2 (π1x1 + π2x2)

v − x2 = δ1 [v − (π1x1 + π2x2)]

38

Let xr1 and xr2 be the equilibrium offers when φ = 0, which solve the equation

system above. Then

xr1 =δ2π2

1− δ2π1 − δ1π2(1− δ1)v

xr2 =1− δ2π1

1− δ2π1 − δ1π2(1− δ1)v

I will solve for the case φ > 0 next.

C.1 Optimal choice for ytWhen an offer is rejected off the equilibrium path in period t, the offers inperiod t+ 1 are determined by

x1,t+1 − (1− δ2)φ [yt − x1,t+1]+ = (1− δ2)[yt+1 − φ [yt − yt+1]+

](27)

+ δ2

(π1(x1,t+2 − (1− δ2)φ [yt+1 − x1,t+2]+)

+π2(x2,t+2 − (1− δ2)φ [yt+1 − x2,t+2]+)

)and

v − x2,t+1 = −(1− δ1)yt+1 + δ1 [v − (π1x1,t+2 + π2x2,t+2)] (28)

Player 1’s decision problem after a rejection in period t is given by

maxyt

ur1 = −(1− δ1)yt + δ1 [v − (π1x1,t+1 + π2x2,t+1)]

subject to (27) and (28)

Past decisions do not enter into this optimization problem so yt is indepen-dent of yt′ for all t′ < t. Therefore, the following analysis also applies tothe off-equilibrium nodes that follow a period in which player 1 makes aunilateral transfer that is different than his equilibrium transfer level.Suppose that yt > x1,t+1. If yt > yt+1, then (27) becomes

x1,t+1 =1

1 + (1− δ2)φ[(1− δ2)(1 + φ)yt+1

+ δ2

(π1(x1,t+2 − (1− δ2)φ [yt+1 − x1,t+2]+)

+π2(x2,t+2 − (1− δ2)φ [yt+1 − x2,t+2]+)

)]

39

so thatdx1,t+1dyt

= 0

and the first order derivative of ur1 with respect to yt is given by

dur1dyt

= −(1− δ1) < 0

so yt > yt+1 cannot hold at the optimal solution. Then yt > x1,t+1 impliesyt ≤ yt+1 and (27) becomes

x1,t+1 =1

1 + (1− δ2)φ[(1− δ2)(φyt + yt+1)

+ δ2

(π1(x1,t+2 − (1− δ2)φ [yt+1 − x1,t+2]+)

+π2(x2,t+2 − (1− δ2)φ [yt+1 − x2,t+2]+)

)]

so thatdx1,t+1dyt

=(1− δ2)φ

1 + (1− δ2)φand the first order derivative of the objective with respect to yt is given by

dur1dyt

= −(1− δ1)− δ1π1(1− δ2)φ

1 + (1− δ2)φ< 0

so this case cannot hold in equilibrium either.Therefore yt ≤ x1,t+1 holds in equilibrium. If yt ≥ yt+1, then (27) becomes

x1,t+1 = (1− δ2) [yt+1 − φ(yt − yt+1)]

+ δ2

(π1(x1,t+2 − (1− δ2)φ [yt+1 − x1,t+2]+)

+π2(x2,t+2 − (1− δ2)φ [yt+1 − x2,t+2]+)

)so that

dx1,t+1dyt

= −(1− δ2)φ

and the first order derivative of the objective with respect to yt is given by

dur1dyt

= −(1− δ1) + δ1π1(1− δ2)φ

If dur1

dyt> 0, that is

φ > φ∗∗∗ =1− δ1

δ1π1(1− δ2)

40

then yt = x1,t+1 at the optimal solution. yt = 0 if φ ≤ φ∗∗∗.If yt ≤ yt+1, then (27) becomes

x1,t+1 = (1− δ2)yt+1

+ δ2

(π1(x1,t+2 − (1− δ2)φ [yt+1 − x1,t+2]+)

+π2(x2,t+2 − (1− δ2)φ [yt+1 − x2,t+2]+)

)so that

dx1,t+1dyt

= 0

and the first order derivative of the objective with respect to yt is given by

dur1dyt

= −(1− δ1) < 0

so that yt = 0 at the optimal solution. I summarize these results in thefollowing proposition:

Proposition 14 In a subgame perfect equilibrium, for all t, yt is independentof actual transfers and offers made in periods t′ < t. Moreover,(i) If φ ≤ φ∗∗∗, then yt = 0.(ii) If φ > φ∗∗∗, then if the equilibrium transfer in the next period is zero,

i.e. is yt+1 = 0, then the equilibrium transfer in period t is yt = x1,t+1,otherwise yt = 0.

Next, I will solve for the optimal offers on the time-invariant Markovperfect equilibrium path. Later I will solve for the optimal offers off theequilibrium path.Optimal offers on the time-invariant Markov perfect equilibrium pathIf φ ≤ φ∗∗∗, then substitute yt = 0 for all t in (25) and (26) and obtain

x1t = xr1 and x2t = xr2.Let φ > φ∗∗∗. By Proposition 14, yt is either zero or a positive number.

Let yt ∈ {0, yr} in the time-invariant Markov perfect equilibrium. If yt+1 = 0,then yt = yr. If yt+1 = yr, then yt = 0.Define the state of the game, s ∈ {0, 1}, as follows: s = 0 if yt+1 = 0,

or equivalently yt−1 = 0, and s = 1 if yt+1 = yr, or equivalently yt−1 = yr.s = 0 indicates that no unilateral transfer was supposed to be made in theprevious period and no unilateral transfer will be made in equilibrium the

41

next period. Therefore player 1 will make a unilateral transfer if an offer isrejected when s = 0.Equilibrium offers will depend on s and on the actual transfer made by

player 1 in the previous period, which may differ from his equilibrium trans-fer. Let xis be player i’s offer on the equilibrium path when the state of thegame is given by s and player 1 makes his equilibrium transfer of yt ∈ {0, yr}whenever an offer is rejected.Suppose that yt = yr. Then it must be the case that yt−1 = yt+1 = 0, i.e.

the current state of the game is s = 0, so x1,t = x10, x2,t = x20, x1,t+1 = x11and x2,t+1 = x21. Then yt−1 = 0 and yt = x11 by Proposition 14. Substitutethese values in (25) and (26) and obtain

x10 = (1− δ2)x11+ δ2

(π1x11 + π2(x21 − (1− δ2)φ [x11 − x21]+)

)(29)

v − x20 = −(1− δ1)x11 + δ1 [v − (π1x11 + π2x21)] (30)

Now suppose that yt = 0. Then it must be the case that yt−1 = yt+1 = yr,i.e. the current state of the game is s = 1, so x1,t = x11, x2,t = x21, x1,t+1 =x10 and x2,t+1 = x20. Then yt−1 = x11 by Proposition 14. Substitute thesevalues in (25) and (26) and obtain

x11 = −(1− δ2)φx11 + δ2 (π1x10 + π2x20) (31)

v − x21 = δ1 [v − (π1x10 + π2x20)] (32)

Next I will show that x11 < x21 so that [x11 − x21]+ = 0 in (29). From(31) obtain

x11 =δ2

1 + (1− δ2)φ(π1x10 + π2x20)

and from (32)x21 = (1− δ1)v + δ1 (π1x10 + π2x20)

so that x11 < x21 is equivalent to

δ21 + (1− δ2)φ

(π1x10 + π2x20) < (1− δ1)v + δ1 (π1x10 + π2x20)⇔

[δ2

1 + (1− δ2)φ− δ1

](π1x10 + π2x20) < (1− δ1)v

42

which holds if

δ21 + (1− δ2)φ

− δ1 < 0⇔

δ2 − δ1δ1(1− δ2)

< φ

Check thatδ2 − δ1δ1(1− δ2)

< φ∗∗∗ =1− δ1

δ1π1(1− δ2)< φ

so that x11 < x21 and therefore [x11 − x21]+ = 0. Then equations (29), (30),(31) and (32) simplify to

x10 = (1− δ2)x11 + δ2 (π1x11 + π2x21) (33)

v − x20 = −(1− δ1)x11 + δ1 [v − (π1x11 + π2x21)]

x11 = −(1− δ2)φx11 + δ2 (π1x10 + π2x20)

v − x21 = δ1 [v − (π1x10 + π2x20)]

equivalently

x10 = (1− δ2)x11 + δ2 (π1x11 + π2x21) (34)

x20 = (1− δ1)(v + x11) + δ1 (π1x11 + π2x21)

x11 =δ2

1 + (1− δ2)φ(π1x10 + π2x20)

x21 = (1− δ1)v + δ1 (π1x10 + π2x20)

This characterizes the equilibrium offers and transfers. Since yt is indepen-dent of past decisions, Proposition 14 also characterizes yt offthe equilibrium.Next, I will characterize the optimal offers off the equilibrium.

C.2 Optimal offers with low cost: φ ≤ φ∗∗∗

By Proposition 14, yt = 0 for all t independently of past decisions. Supposethat player 1 made a unilateral transfer of yt−1 ≥ 0 in the previous period,which may be different than his equilibrium offer of zero. Since yt = 0, thegame will revert to its time-invariant Markov perfect equilibrium path aftert+ 1 on, so that x1t′ = xr1 and x2t′ = xr2 for all t

′ > t. Substitute these in (25)and (26), and obtain

43

x1,t − (1− δ2)φ [yt−1 − x1,t]+ = −(1− δ2)φyt−1 + δ2 (π1xr1 + π2x

r2) (35)

andv − x2,t = δ1 [v − (π1x

∗1 + π2x

∗2)]

so that x2,t = xr2 for all yt−1.If yt−1 ≤ x1,t, then [yt−1 − x1,t]+ = 0 and (35) becomes

x1,t = −(1− δ2)φyt−1 + δ2 (π1xr1 + π2x

r2)

Then yt−1 ≤ x1,t becomes equivalent to

yt−1 ≤ −(1− δ2)φyt−1 + δ2 (π1xr1 + π2x

r2)⇔

yt−1 ≤δ2 (π1x

r1 + π2x

r2)

1 + (1− δ2)φ≡ x

If yt−1 > x1,t, (35) becomes

x1,t − (1− δ2)(φyt−1 − x1,t) = −(1− δ2)φyt−1 + δ2 (π1xr1 + π2x

r2)⇒

x1,t = x

So player 1’s optimal offer is xr1(yt−1), where

xr1(y) =

{−(1− δ2)φy + δ2 (π1x

r1 + π2x

r2) if y ≤ x

x otherwise

Notice that xr1(0) = xr1.

C.3 Optimal offers with high cost: φ > φ∗∗∗

By Proposition 14, yt ∈ {0, x11} for all t independently of past decisions.Suppose that player 1 made a unilateral transfer of yt−1 in the previousperiod, which may be different than his equilibrium offer. Let the state ofthe game be st ∈ {0, 1} in period t and st+1 6= st be the state of the gamein period t+ 1. Since yt only depends on st but not past decisions, the gamewill revert to its time-invariant Markov perfect equilibrium path after t + 1on, so that x1,t+1 = x1st+1 and x2,t+1 = x2st+1 .

44

First consider x2,t, which is determined by (26):

v − x2,t = −(1− δ1)yt + δ1 [v − (π1x1,t+1 + π2x2,t+1)]

Since yt, x1,t+1 and x2,t+1 are independent of yt−1, this equation implies thatx2,t = x2s for all yt−1.Next consider x1,t. Define

∆20 = δ2 (π1x10 + π2x20) ,

∆21 = δ2 (π1x11 + π2x21)

Let st = 1. Then yt = 0, st+1 = 0 so that x1,t+1 = x10 and x2,t+1 = x20.Also [yt−1 − yt]+ = yt−1. Substitute these values in (25). If yt−1 ≤ x1,t then[yt−1 − x1,t]+ = 0 and (25) becomes

x1,t = −(1− δ2)φyt−1 + ∆20

Then yt−1 ≤ x1,t becomes equivalent to yt−1 ≤ x11.If yt−1 > x1,t, (25) becomes

x1,t − (1− δ2)φ(yt−1 − x1,t) = −(1− δ2)φyt−1 + δ2(π1x10 + π2x20)⇒x1,t = x11

So player 1’s optimal offer is x11(yt−1), where

x11(y) =

{−(1− δ2)φy + ∆20 if y ≤ x11x11 otherwise

Next consider st = 0. Then yt = x11, st+1 = 1 so that x1,t+1 = x11and x2,t+1 = x21. Substitute these values in (25). If yt−1 ≤ yt = x11 then[yt−1 − yt]+ = 0. Suppose also that yt−1 ≤ x1,t. Then (25) becomes

x1,t = (1− δ2)x11 + ∆21 = x10

So yt−1 ≤ x1,t becomes equivalent to yt−1 ≤ x10. Since x11 < x10, yt−1 ≤x11 < x10 so that yt−1 ≤ x1,t holds.If yt = x11 < yt−1 ≤ x10, then [yt−1 − yt]+ = yt−1−x11. Suppose also that

yt−1 ≤ x1,t. Then (25) becomes

x1,t = (1− δ2) [(1 + φ)x11 − φyt−1] + ∆21

= x10 − φ(1− δ2)(yt−1 − x11)

45

and yt−1 ≤ x1,t becomes equivalent to

yt−1 ≤ x10 − φ(1− δ2)(yt−1 − x11)⇔

yt−1 ≤x10 + φ(1− δ2)x11

1 + (1− δ2)φ≡ x

If yt−1 > x1,t, (25) becomes

x1,t − (1− δ2)φ(yt−1 − x1,t) = (1− δ2) [(1 + φ)x11 − φyt−1] + ∆2y ⇒x1,t = x.

x ≤ x10 is equivalent to x11 ≤ x10. I will show in Section C.4 that x11 < x10,so x < x10.Then player 1’s optimal offer is x10(yt−1), where

x10(y) =

x10 if y ≤ x11x10 − φ(y − x11) if x11 < y ≤ xx otherwise

C.4 Some Comparisons

I will make some observations before proceeding with the comparative staticsanalysis.First, I will show that player 1’s offers are smaller than player 2’s offers

in both states. I have already showed that x11 < x21. Obtain x10 < x20 asfollows:

x10 < x20 ⇔(1− δ2)x11 + δ2 (π1x11 + π2x21) < (1− δ1)(v + x11) + δ1 (π1x11 + π2x21)⇔(δ1 − δ2) [x11 − (π1x11 + π2x21)] < (1− δ1)v

Since δ1−δ2 < 1−δ2, the last inequality above holds if x11−(π1x11 + π2x21) <v. Check

x11 − (π1x11 + π2x21) < x11 ≤ v,

so that x10 < x20.

46

Next I will show that x11 < x10 so that x11 is the smallest offer made onthe equilibrium path. Use the equation system (33) to obtain

x10 − x11 = (1− δ2)(1 + φ)x11 + δ2 (π1(x11 − x10) + π2(x21 − x20))

Use the equation system (34) to obtain

x20 − x21 =(1− δ1)x11 − δ1π1(x10 − x11)

1 + δ1π2

Substitute this in the previous equation and obtain(1 + δ2π1 −

δ1π1δ2π21 + δ1π2

)(x10 − x11) =

[(1− δ2)(1 + φ)− (1− δ1)δ2π2

1 + δ1π2

]x11

Check that

1 + δ2π1 −δ1π1δ2π21 + δ1π2

> 0⇔

(1 + δ2π1)(1 + δ1π2) > δ1π1δ2π2 ⇔1 + δ2π1 + δ1π2 > 0

which is true. Next check that

(1− δ2)(1 + φ)− (1− δ1)δ2π21 + δ1π2

> 0

holds if

(1− δ2)−(1− δ1)δ2π2

1 + δ1π2> 0⇔

(1− δ2)(1 + δ1π2) > (1− δ1)δ2π2 ⇔1− δ2 + δ1π2 − δ1δ2π2 > δ2π2 − δ1δ2π2 ⇔

1− δ2 + δ1π2 > δ2π2 ⇔1− δ2(1− π2) + δ1π2 > 0

which holds. So since x11 > 0, I obtain x10 − x11 > 0, that is, x11 < x10.

47

C.5 Comparative Statics

Solving for xis is cumbersome. So I will perform comparative statics analysiswithout giving closed form solutions of xis. It is trivial to show that xis > 0for all i ∈ {1, 2}, s ∈ {0, y}. Also, consider any linear equation system

X = AX + b

where X a K × 1 vector of unknowns, A = [aij] is K × K and b = [bk] isK × 1. Assume that aij ≥ 0 for all i, j. Then it is easy to prove that (i) ifbk ≥ 0 for all k, then xk ≥ 0 for all k, and (ii) if bk ≤ 0 for all k, then xk ≤ 0for all k. I will use (i) and (ii) repeatedly below.Comparative Statics on Bargaining Power, π1Let π1 = π. Then π2 = 1 − π. Let x′is denote the partial derivative of

xis with respect to π. Take the total differential of the equation system (34)with respect to π and obtain

x′11 =δ2

1 + (1− δ2)φ[(x10 − x20) + πx′10 + (1− π)x′20]

x′21 = δ1 [(x10 − x20) + πx′10 + (1− π)x′20]

x′10 = (1− δ2)x′11 + δ2 [(x11 − x21) + πx′11 + (1− π)x′21]

x′20 = (1− δ1)x′11 + δ1 [(x11 − x21) + πx′11 + (1− π)x′21]

Let X = [x′11, x′21, x

′10, x

′20]

t and write this equation system in the form ofX = AX + b. Then aij ≥ 0 for all i, j. Also x10 − x20 < 0 and x11 − x21 < 0imply that bk ≤ 0 for all k. So by (ii) above, x′is < 0 for every i = 1, 2 ands = 0, y. That is, an increase in 1’s relative bargaining power increases 1’spayoff and decreases 2’s payoff in equilibrium.Comparative Statics on Habit formation, φLet x′is denote the partial derivative of xis with respect to φ. Take the

total differential of the equation system (34) with respect to φ and obtain

x′11 = − δ2(1− δ2)[1 + (1− δ2)φ]2

(πx10 + (1− π)x20) +δ2

1 + (1− δ2)φ(πx′10 + (1− π)x′20)

x′21 = δ1 (πx′10 + (1− π)x′20)

x′10 = (1− δ2)x′11 + δ2 (πx′11 + (1− π)x′21)

x′20 = (1− δ1)x′11 + δ1 (πx′11 + (1− π)x′21)

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Similarly, − δ2(1−δ2)[1+(1−δ2)φ]2

(πx10 + (1− π)x20) < 0, so by (ii) above, x′is < 0 forevery i = 1, 2 and s = 0, y. That is, as 2’s habit formation gets stronger, 1’spayoff increases in equilibrium.Comparative Statics on 2’s patience, δ2Let x′is denote the partial derivative of xis with respect to δ2. Take the

total differential of the equation system (34) with respect to δ2 to obtain

x′11 =1 + φ

[1 + (1− δ2)φ]2(πx10 + (1− π)x20) +

δ21 + (1− δ2)φ

(πx′10 + (1− π)x′20)

x′21 = δ1 (πx′10 + (1− π)x′20)

x′10 = (1− π)(x21 − x11) + (1− δ2)x′11 + δ2 (πx′11 + (1− π)x′21)

x′20 = (1− δ1)x′11 + δ1 (πx′11 + (1− π)x′21)

Since 1+φ

[1+(1−δ2)φ]2(πx10 + (1− π)x20) > 0 and (1 − π)(x21 − x11) > 0, by (i)

above, x′is > 0 for every i = 1, 2 and s = 0, y. That is, an increase in 2’sdiscount rate increases 2’s payoff in equilibrium.Comparative Statics on 1’s patience, δ1Let x′is denote the partial derivative of xis with respect to δ1. Take the

total differential of the equation system (34) with respect to δ1 to obtain

x′11 =δ2

1 + (1− δ2)φ(πx′10 + (1− π)x′20)

x′21 = (πx10 + (1− π)x20 − v) + δ1 (πx′10 + (1− π)x′20)

x′10 = (1− δ2)x′11 + δ2 (πx′11 + (1− π)x′21)

x′20 = (1− π)(x21 − x11)− v + (1− δ1)x′11 + δ1 (πx′11 + (1− π)x′21)

Since (πx10 + (1− π)x20 − v) < 0 and (1 − π)(x21 − x11) − v < 0, by (ii)above, x′is < 0 for every i = 1, 2 and s = 0, y. That is, an increase in 1’sdiscount rate increases 1’s payoff in equilibrium.

D Incomplete Information with Alternatingoffers

I will compute the separating and pooling Bayesian equilibria of the game.

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D.1 Player 2 makes the first offer

In this case, if player 1 knows that φ = φl, then yi = 0 for i = 1, 2 and theequilibrium offers are given by

x∗1 =δ2(1− δ1)1− δ1δ2

v and x∗2 =1− δ1

1− δ1δ2v

If he knows that φ = φh, then the equilibrium offers are given by

xh1 =δ2(1− δ1)

(1− δ2)(1 + φh)v and xh2 =

1 + (1− δ2)φh(1− δ2)(1 + φh)

(1− δ1)v

in this case, y1 = 0 and y2 = xh1 .

Separating EquilibriumSuppose that there is a separating equilibrium in which type φl makes an

offer of xl and type φh makes an offer of xh 6= xl. If both offers are acceptedimmediately, the type with the lower offer would benefit by making the offerof the other type. So player 1 has to reject at least one of the offers withpositive probability. Suppose he rejects xl with probability αl and xh withprobability αh.After observing xτ , player 1 updates his belief to φ = φτ , τ ∈ {l, h},

the game turns into a complete information game and equilibrium offers andunilateral transfers are given as above in the continuation game.Then type φh does not imitate type φl if and only if

(1−αh)xh +αh((1− δ2)y2 + δ2xh1) = (1−αh)xh +αhxh1 ≥ (1−αl)xl +αlδ2x

∗1

And type φl does not imitate type φh if and only if

(1−αl)xl +αlδ2x∗1 ≥ (1−αh)xh +αh((1− δ2)y2 + δ2x

h1) = (1−αh)xh +αhxh1

Note that I have substituted y2 = xh1 above. These two inequalities implythat

(1− αl)xl + αlδ2x∗1 = (1− αh)xh + αhxh1 (36)

If αl ∈ (0, 1), then player 1 is indifferent between accepting and rejectingxl and it must be the case that

v − xl = δ1(v − x∗1)⇒ xl = x∗2

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Then (36) becomes

(1− αl)x∗2 + αlδ2x∗1 = (1− αh)xh + αhxh1 (37)

Player 1’s payoff from receiving the offer of xh is

(1− αh)(v − xh) + αh[−(1− δ1)y2 + δ1(v − xh1)]

If xh satisfies

v − xh > −(1− δ1)y2 + δ1(v − xh1)

then αh = 0 is optimal for 1 so that xh is not optimal for type φh, becauseif she offers xh + ε and even if player 1 updates her beliefs to φ = φh afterobserving this offer, it is optimal for player 1 to accept this offer. So

v − xh = −(1− δ1)y2 + δ1(v − xh1) (38)

must be satisfied in such equilibrium. Then substituting y2 = xh1 ,

xh = (1− δ1)v + xh1 = xh2

Then (37) implies

(1− αl)x∗2 + αlδ2x∗1 = (1− αh)xh2 + αhxh1

which solves for αl as

αl =x∗2 − (1− αh)xh2 − αhxh1

x∗2 − δ2x∗1(39)

Next I will check if αl ∈ [0, 1]. Since xh2 > xh1 , αl achieves its minimum when

αh = 0, then αl becomes

αlmin =x∗2 − xh2x∗2 − δ2x∗1

(40)

αlmin ≥ 0 if and only if x∗2 ≥ xh2 , which is equivalent to φ ≥ φ∗, which holds.So αl ≥ 0.

αlmin ≤ 1 if and only if δ2x∗1 ≤ xh2 . This last inequality is equivalent to

(δ22 − 1 + δ1δ2)(1− δ2)φ ≤ (1− δ1δ2)− δ22(1− δ2)

51

Since 1−δ1δ2 > 1−δ2 > δ22(1−δ2), the right hand is positive. If δ22−1+δ1δ2 <

0, equivalently δ1 ≤ 1−δ22δ2, then this inequality is satisfied for all values of φ.

Otherwise, it is satisfied if and only if φ ≤ φ where

φ =δ2(1− δ1δ2)

(1− δ2)(δ22 − 1 + δ1δ2)− 1 > φ∗.

If δ1 ≤ 1−δ22δ2

or φ∗ < φ ≤ φ, then the following is a separating Bayesianequilibrium. Type φl offers x∗2, which player one accepts with probability1 − αlmin. Type φh offers x

h2 < x∗2, which player 1 accepts with probability

1. Player 2 rejects any other offer bigger than xh2 , accepts any other offersmaller than xh2 . If player 2 offers x

∗2, then player 1 updates his belief to

φ = φl, otherwise he updates his belief to φ = φh and the players play theequilibrium of the associated complete information games. The equilibriumpayoff of both types of player 2 is xh2 .Next I will show that αh > 0 cannot hold in such separating equilibrium

if δ1 ≤ 1−δ22δ2

or φ∗ < φ ≤ φ. Suppose that αh > 0. If type φh offers xh2 − εinstead, for small ε > 0, it is optimal for player 1 to accept her offer because(i) if he updates his belief so that φ = φh, his continuation payoff fromrejecting it is −(1− δ1)y2 + δ1(v− xh1), which is less than v− xh2 + ε by (38).Given that player 1 will accept the offer of xh2 − ε with probability 1, typeφh’s payoff from offering xh2 − ε is xh2 − ε, and her payoff from offering xh2 is

(1− αh)xh2 + αh((1− δ2)y2 + δ2xh1) = (1− αh)xh2 + αhxh1

Since (1−αh)xh2+αhxh1 < xh2 for αh > 0, there exists small enough ε such that

(1−αh)xh2 +αhxh1 < xh2 − ε, which contradicts with the optimality of offeringxh2 . There is no such separating equilibrium with αh > 0 when δ1 ≤ 1−δ22

δ2or

φ∗ < φ ≤ φ.

Pooling EquilibriumSuppose that δ22 − 1 + δ1δ2 > 0 and φ > φ. No αh and αl pair satisfies

(39), because αl > 1 even when αh minimizes αl in (40). So no separatingequilibrium with αl ∈ (0, 1) exists in this case. Alternatively, consider αh ∈(0, 1), then player 1 is indifferent between accepting and rejecting xh and itmust be the case that

v − xh = −(1− δ1)y2 + δ1(v − xh1)⇒ xh = xh2

52

Then (36) becomes

(1− αl)xl + αlδ2x∗1 = (1− αh)xh2 + αhxh1 (41)

Player 1’s payoff from receiving the offer of xl is

(1− αl)(v − xl) + αlδ1(v − x∗1)

Suppose that xl satisfies

(v − xl) < δ1(v − x∗1)

then optimality for player 1 requires αl = 1, so (41) becomes

δ2x∗1 = (1− αh)xh2 + αhxh1

which implies

αh =xh2 − δ2x∗1xh2 − xh1

< 0

because δ2x∗1 > xh2 in this case. So (v − xl) ≥ δ1(v − x∗1) must hold.If xl satisfies

(v − xl) > δ1(v − x∗1)then αl = 0 is optimal. Then from (41),

xl = (1− αh)xh2 + αhxh1

which implies that xl < xh2 for αh > 0 since xh2 > xh1 . If type φh offers

xh2 − ε instead, for small ε > 0, it is optimal for player 1 to accept her offerbecause (i) if he updates his belief so that φ = φh, his continuation payofffrom rejecting it is −(1− δ1)y2+ δ1(v−xh1), which is less than v−xh2 + ε, thepayoff from accepting the offer; (ii) if he updates his belief so that φ = φl, hiscontinuation payoff from rejecting it is δ1(v−x∗1), which is less than v−xh2+εbecause δ2x∗1 > xh2 in this case and δ1 < 1. Given that player 1 will acceptthe offer of xh2 − ε with probability 1, type φh’s payoff from offering xh2 − ε isxh2 − ε, and her payoff from offering xh2 is

(1− αh)xh2 + αh((1− δ2)y2 + δ2xh1) = (1− αh)xh2 + αhxh1

Since (1 − αh)xh2 + αhxh1 < xh2 for αh > 0, there exists small enough ε such

that (1 − αh)xh2 + αhxh1 < xh2 − ε, which contradicts with the optimality ofoffering xh2 . So x

l satisfies (v − xl) ≤ δ1(v − x∗1).

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So xl must satisfy(v − xl) = δ1(v − x∗1)

which implies xl = x∗2. Then from (41)

(1− αl)x∗2 + αlδ2x∗1 = (1− αh)xh2 + αhxh1

This inequality cannot hold since δ2x∗1 > xh2 in this case and x∗2 > xh2 > xh1 .

So no such separating equilibrium exists either.I will show that the following constitutes a pooling equilibrium in this

case: Both types of player 2 offer xh2 . Player 1 rejects offers x > xh2 andaccept offers x ≤ xh2 . If player 2 offers x 6= xh2 , then player 1 updates hisbelief to φ = φh and plays according to the equilibrium of the associatedcomplete information game.Since xh2 is the lowest offer player 1 can secure in any equilibrium, it

is optimal for him to accepts x ≤ xh2 . Consider a continuation game aftera rejection. In this case, player 1 believes that φ = φh and there will beno updating of his degenerate beliefs. So it is optimal for him to play hisequilibrium strategy of the associated complete information game. Sincethere will be no updating in beliefs, it is optimal for both types of player 2to offer xh2 and accept any offer x ≥ xh1 .Given player 1’s strategy in the continuation game, both types of player

2 will achieve the payoff of −(1 − δ1)y2 + δ1(v − xh1) = xh2 after a rejection.Given that player 1 rejects any offer x > xh2 , then it is optimal for both typesto offer xh2 , which player 1 accepts with probability 1. Since x > xh2 is anoff equilibrium offer, player 1 can set his belief to φ = φh after receiving anyoffer of x > xh2 .The equilibrium payoff of both types of player 2 is xh2 in the separating

and pooling equilibria. Player 1’s payoff in the separating equilibrium isgiven by

θ(v − xh2) + (1− θ)[αlδ1(v − x∗1) + (1− αl)(v − x∗2)

]and his payoff in the pooling equilibrium is v − xh2 .

D.2 When Player 1 makes the offer

x∗1 is the highest offer player 1 makes in any equilibrium. So both types ofplayer 2 accept any offer x ≥ x∗1. Also x

h1 is the lowest offer that player 1

makes in any equilibrium.

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Consider a separating equilibrium in which player 1 offers x ∈ [xh1 , x∗1].

Since the continuation game becomes a complete information game andδ2x∗2 = x∗1, type φl rejects any offer x < x∗1 and offers x

∗2 the next period,

which player 1 accepts. Then type φh must be accepting x < x∗1 in the sep-arating equilibrium. But she can achieve δ2x∗2 = x∗1 > x by imitating typeφl. So x < x∗1 cannot be a separating equilibrium offer. Since both types ac-cept x∗1, x

∗1 cannot be a separating equilibrium offer either. So no separating

equilibrium exists.Consider a pooling equilibrium in which player 1 offers x ∈ [xh1 , x

∗1]. Sup-

pose that type φτ rejects x with probability βτ , τ ∈ {l, h}. After a rejection,player 1 updates his beliefs according to Bayes rule to

θ′ = Pr(φ = φh) =θβh

θβh + (1− θ)βl.

Suppose that βh > 0. Then θ′ ∈ (0, 1). Since player 2 makes the offer thenext period, the continuation game has either a separating equilibrium or apooling equilibrium. In both cases, the payoff of both types of player 2 isequal to xh2 (see the conclusion of the previous section). Then both typesaccept any offer x ≥ δ2x

h2 in the first period. So player 1 offers δ2x

h2 in such

a equilibrium. Since player 2 of both types is indifferent between acceptingand rejecting the offer, they can reject it with any probability. Since rejectionis costly for player 1, player 1 can avoid rejection by offering δ2xh2 + ε. Forδ2x

h2 to be optimal, it must be that βh = βl = 0. Since any rejection is off

equilibrium in this case, player 1 can set his belief to φ = φh. Given that hewill believe that φ = φh after a rejection, the game will turn into a completeinformation game with φ = φh, so both types of player 2 can achieve at mostxh2 in the continuation game after rejection. So it is optimal for them toaccept player 1’s offer of δ2xh2 in the first period.In this case, the information rent type φh collects is

R(φh) = δ2xh2 − xh1 =

δ2(1− δ2)(1− δ1)φh(1− δ2)(1 + φh)

v > 0

and

R′(φh) =R(φh)

1 + φh> 0

and (R(φh)

xh1

)′= ((1− δ2)φh)′ = 1− δ2 > 0

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