# Subgame Perfect Equilibrium in the Rubinstein Bargaining Game with Loss Aversion

Authors: Zhongwei Feng; Chunqiao Tan
Journal: Complexity
 (2019)

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## Abstract

Rubinstein bargaining game is extended to incorporate loss aversion, where the initial reference points are not zero. Under the assumption that the highest rejected proposal of the opponent last periods is regarded as the associated reference point, we investigate the effect of loss aversion and initial reference points on subgame perfect equilibrium. Firstly, a subgame perfect equilibrium is constructed. And its uniqueness is shown. Furthermore, we analyze this equilibrium with respect to initial reference points, loss aversion coefficients, and discount factor. It is shown that one benefits from his opponent’s loss aversion coefficient and his own initial reference point and is hurt by loss aversion coefficient of himself and the opponent’s initial reference point. Moreover, it is found that, for a player who has a higher level of loss aversion than the other, although this player has a higher initial reference point than the opponent, this player can(not) obtain a high share of the pie if the level of loss aversion of this player is sufficiently low (high). Finally, a relation with asymmetric Nash bargaining is established, where player’s bargaining power is negatively related to his own loss aversion and the initial reference point of the other and positively related to loss aversion of the opponent and his own initial reference point.


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## Body

## 1. Introduction

A large number of experimental literature pieces on bargaining explore the nature of agreements and disagreements and the dynamics of bargaining. There are two critical conclusions: firstly, in real bargaining problem, bargaining is a gradual process and the agreements can be reached after many periods. Secondly, there is a strictly positive probability of disagreement. For the classical bargaining problem of dividing a pie, whose size is one unit, between two bargainers, Rubinstein [1] assumed that preferences of bargainers are time dependent. In many bargaining situations, however, the assumption may be violated and the share finally obtained by a bargainer may depend on the history of alternating offers made so far. In particular, the phenomenon of loss aversion in bargaining problems is pointed out by Driesen et al. [2] as follows: a share of x% is evaluated less if a share of y% with y>x has been within reach at an earlier stage of the game.Kahneman and Tversky [3] first proposed loss aversion. As the most striking result of the investigation of reference-dependent utility functions, loss aversion is applied to lots of applications with fixed reference point [4, 5]. For the situation of loss aversion where the reference points are fixed, we can regard it as a special case of risk aversion. Roth [6] investigated the impact of risk aversion on the classical Rubinstein alternating offers bargaining model in the context of full rationality of bargainers. However, in many applications, it is likely that a loss depends on history of the bargaining. That is, the reference points are endogenous [7]. Shalev [8] considered objective discount and loss aversion and obtained the unique subgame perfect equilibrium (SPE) of the Rubinstein bargaining with the transformed discount factors. Compte and Jehiel [9] assumed that a new bargaining phase begins at a fixed cost if a breakdown of the bargaining occurs. In each new bargaining phase, the reference points can be adjusted and the highest offer received over the process of bargaining and the first mover is chosen from the two agents at random with probability 1/2. Li [10] assumed that a player would rather reject any share that is less than the highest offer in the past and found a unique subgame perfect equilibrium. Schwartz and Wen [11] assumed that a proposal of a bargainer made to the other cannot be less than a proposal made to that player. Hyndman [12] assumed that a bargainer with reference dependent preferences prefers his current reference point to impasse to consuming below their current reference points. Closer to this paper is Driesen et al. [2]; they investigated the impact of loss aversion on the Rubinstein bargaining game based on the assumption that the initial reference points are zero and one’s reference point in the process of bargaining is regarded as the highest rejected offer of his opponent last rounds of bargaining. Although it is reasonable that the reference points at the beginning of bargaining are zero in a lot of instances, it may be essential that the reference points at the beginning of bargaining are not zero in others. For example, a player may transfer his expectations derived from previous opponents when he enters into a new bargaining situation with another player. Thus, how to investigate the impact of loss aversion and the initial reference points on the classical Rubinstein bargaining game is a valuable topic and also the objective of this paper.We adopt the model of loss aversion proposed by Shalev [13]. In Shalev’s model, a player’s preference is modeled by the following elements: basic utility function of decision-maker, loss aversion coefficient, and reference point. The outcomes that are less than some reference point are regarded as losses. And the corresponding values of utilities are scaled down by loss aversion parameter. A number of applications are consistent with Shalev’s model of loss version. The basic assumption of Shalev’s model is that the loss aversion coefficient is regarded as a constant parameter, which makes the model be easily used. The loss aversion coefficient is constant in the following two different aspects: first, for the utility of an outcome that is less than the reference outcome, it can be obtained from the basic utility by subtracting a disutility, which is obtained from the size of the loss multiplied by a parameter — loss aversion coefficient. Second, the parameter is constant across different reference points; that is, the loss aversion coefficient does not depend on reference point [14].In this paper, we extend the analysis of the Rubinstein bargaining game to incorporate loss aversion and reference dependence, where the initial reference points are not zero. We assume that a bargainer’s reference point in the bargaining process is equal to the highest rejected offer of his opponent that is higher than his own initial reference point, since it can be regarded as the share that could have been obtained so far. A simple modification of the Rubinstein bargaining game can be transformed into a new bargaining game with loss aversion and reference dependence through changed reference points, which depends on the history of bargaining. In our model, subgames depend not only on the initial reference points, but also on the impact the history of bargaining has on preferences, which leads to much more complications to analyzing the characterization of SPE. On the other hand, for Rubinstein bargaining game with loss aversion and reference dependence, where the initial reference points are not zero and the discount factor is regarded as the probability of entering a new phase of bargaining after rejecting a proposal of a player, we construct the unique subgame perfect equilibrium (SPE), and its features are shown. Finally, we analyze the impact of loss aversion coefficients and the discounting factor (or the probability of continuation) on subgame perfect equilibrium.The remainder of the present paper is organized as follows. After preliminaries in Section2, we define the SPE in the Rubinstein bargaining model with loss aversion, construct a SPE, and concern uniqueness of the SPE in Section 3. In Section 4, we discuss convergence of the subgame perfect equilibrium for the probability of continuation. In Section 5, conclusion is given.


## 2. Preliminaries

### 2.1. Rubinstein Bargaining Model

Player 1 and player 2 have to reach an agreement on how to divide one unit of a pie. The set of all possible partitions is denoted by(1)Z≕z1,z2∈R2∣z1+z2=1,z1,z2≥0.Bargaining occurs at times t∈T=1,2,…. For simplicity, the set of odd moments is denoted by Todd≔1,3,…… and the set of even moments is denoted by Teven≔2,4,……. At odd moments, player 1 offers (z1,z2)∈Z and this proposal is accepted (Y) or rejected (N) by player 2. At even moments, for players 1 and 2, their roles are reversed. We assume that the history of bargaining is common knowledge; that is, one knows all previous offers at any moment t∈T, including his own and those of his opponent. If players accept the proposal (z1,z2), then player i (i=1,2) obtains zi and the bargaining ends. If a proposal offered by player i is rejected by his opponent, then the bargaining continues to a new phase with probability 0<δ<1 and ends in probability 1-δ. For the latter case, it means that the bargaining game ends in disagreement; that is, players obtain the shares (r10,r20), where r10 and r20 represent the initial reference points of players.For each odd moment, a strategyf played by player 1 specifies a proposal in Z that depends on the history of the bargaining so far; on the other hand, for each even moment, a decision Y orN is made, where this decision not only depends on the proposal at the current phase but also on the history of bargaining. Similarly, player 2 plays a strategy g, with the roles for moments t∈Todd and t∈Teven are reversed.At timet∈T, the history of the bargaining is denoted by ht, which is defined as a vector of proposals of bargainers. Specifically, ht≔(z1,z2,…,zt), where zs∈Z for all s≤t. Furthermore, at time t∈T, all possible histories ht are denoted by Ht in the bargaining game. That is, Ht≔∏s=1tZ, let H0≔(h0), where h0 is an empty history.LetF be strategy set of player 1, denoted by sequences of functions ftt∈T where for t=1: ft∈Z, for t>1 and t∈Todd: ft: Ht-1→Z, and for t∈Teven: ft: Ht→Y,N, and letG be strategies of player 2, denoted by sequences of functions gtt∈T where for t∈Todd: gt: Ht→Y,N and for t∈Teven: gt: Ht-1→Z. An agreement path is denoted by (ht,a), which is a history ht∈Ht ending in agreement at time t. All time t agreement paths are denoted by At≔(ht,a)∣ht∈Ht. The set A≔⋃t∈TAt contains all histories that end in agreement. Similarly, a disagreement path is denoted as (ht,d), which means that a history ht∈Ht ends in disagreement at time t. Dt≔(ht,d)∣ht∈Ht contains all time t disagreement paths. D≔⋃t∈TDt contains all histories ending in disagreement. All objects of (ht,c) are denoted by the set Ct; that is, histories do not end at momentt. On the other hand, we define H∞≔(z1,z2,…)∣zt∈Zforallt∈T. The elements of H∞ are defined as infinite paths. Therefore, the set that contains all paths of the bargaining game can be denoted by H_≔H∞∪A∪D. Note that a set of paths in H_ is determined by a strategy profile (f,g)∈F×G. In particular, if an agreement at timet is reached when players play (f,g), the set of paths associated with (f,g) not only contains t-1 paths in the set D but also contains one in the set A. If an agreement is never reached when (f,g) is played, then the set only contains paths in D.The functionξi≔H_∖H∞→ri0,1 is introduced, which is used to specify the share that player i(i=1,2) obtains for each finite path in H_. Specifically, for all ht∈Ht, ht=(z1,z2,…,zt), we have that ξi(ht,a)≔zit and ξi(ht,d)≔ri0, where ri0 is the initial reference point of player i.


### 2.2. Loss Aversion

Kahneman and Tversky [3] first proposed loss aversion; its central assumption is that gains are smaller than losses. For instance, the decrease in utility of loss of 10 dollars if one has 100 dollars is larger than the increase in utility of gain of 10 dollars if one has 90 dollars. Shalev [13] proposed another model of loss aversion to measure this. In Shalev’s model, loss aversion of each player i(i=1,2) is characterized by loss aversion coefficient λi, where λi is nonnegative. And let ri be a reference point of player i. The utility function is given by the following transformation:(2)wizi,λi,ri=ziifzi≥rizi-λiri-ziifzi<rior, equivalently,(3)wizi,λi,ri=1+λizi-λimax⁡ri,ziIf the outcomes that are less than reference point are regarded as losses, then the values of utilities are scaled down by λi. If the values of the payoffs are higher than that of the reference point, then the payoffs are left unchanged.In a number of applications, the reference points are usually given exogenously, which sidesteps the important question of the significance of the reference points. Thus, the fact that players’ reference points are endogenous is the motivation for this paper.


## 2.1. Rubinstein Bargaining Model

Player 1 and player 2 have to reach an agreement on how to divide one unit of a pie. The set of all possible partitions is denoted by(1)Z≕z1,z2∈R2∣z1+z2=1,z1,z2≥0.Bargaining occurs at times t∈T=1,2,…. For simplicity, the set of odd moments is denoted by Todd≔1,3,…… and the set of even moments is denoted by Teven≔2,4,……. At odd moments, player 1 offers (z1,z2)∈Z and this proposal is accepted (Y) or rejected (N) by player 2. At even moments, for players 1 and 2, their roles are reversed. We assume that the history of bargaining is common knowledge; that is, one knows all previous offers at any moment t∈T, including his own and those of his opponent. If players accept the proposal (z1,z2), then player i (i=1,2) obtains zi and the bargaining ends. If a proposal offered by player i is rejected by his opponent, then the bargaining continues to a new phase with probability 0<δ<1 and ends in probability 1-δ. For the latter case, it means that the bargaining game ends in disagreement; that is, players obtain the shares (r10,r20), where r10 and r20 represent the initial reference points of players.For each odd moment, a strategyf played by player 1 specifies a proposal in Z that depends on the history of the bargaining so far; on the other hand, for each even moment, a decision Y orN is made, where this decision not only depends on the proposal at the current phase but also on the history of bargaining. Similarly, player 2 plays a strategy g, with the roles for moments t∈Todd and t∈Teven are reversed.At timet∈T, the history of the bargaining is denoted by ht, which is defined as a vector of proposals of bargainers. Specifically, ht≔(z1,z2,…,zt), where zs∈Z for all s≤t. Furthermore, at time t∈T, all possible histories ht are denoted by Ht in the bargaining game. That is, Ht≔∏s=1tZ, let H0≔(h0), where h0 is an empty history.LetF be strategy set of player 1, denoted by sequences of functions ftt∈T where for t=1: ft∈Z, for t>1 and t∈Todd: ft: Ht-1→Z, and for t∈Teven: ft: Ht→Y,N, and letG be strategies of player 2, denoted by sequences of functions gtt∈T where for t∈Todd: gt: Ht→Y,N and for t∈Teven: gt: Ht-1→Z. An agreement path is denoted by (ht,a), which is a history ht∈Ht ending in agreement at time t. All time t agreement paths are denoted by At≔(ht,a)∣ht∈Ht. The set A≔⋃t∈TAt contains all histories that end in agreement. Similarly, a disagreement path is denoted as (ht,d), which means that a history ht∈Ht ends in disagreement at time t. Dt≔(ht,d)∣ht∈Ht contains all time t disagreement paths. D≔⋃t∈TDt contains all histories ending in disagreement. All objects of (ht,c) are denoted by the set Ct; that is, histories do not end at momentt. On the other hand, we define H∞≔(z1,z2,…)∣zt∈Zforallt∈T. The elements of H∞ are defined as infinite paths. Therefore, the set that contains all paths of the bargaining game can be denoted by H_≔H∞∪A∪D. Note that a set of paths in H_ is determined by a strategy profile (f,g)∈F×G. In particular, if an agreement at timet is reached when players play (f,g), the set of paths associated with (f,g) not only contains t-1 paths in the set D but also contains one in the set A. If an agreement is never reached when (f,g) is played, then the set only contains paths in D.The functionξi≔H_∖H∞→ri0,1 is introduced, which is used to specify the share that player i(i=1,2) obtains for each finite path in H_. Specifically, for all ht∈Ht, ht=(z1,z2,…,zt), we have that ξi(ht,a)≔zit and ξi(ht,d)≔ri0, where ri0 is the initial reference point of player i.


## 2.2. Loss Aversion

Kahneman and Tversky [3] first proposed loss aversion; its central assumption is that gains are smaller than losses. For instance, the decrease in utility of loss of 10 dollars if one has 100 dollars is larger than the increase in utility of gain of 10 dollars if one has 90 dollars. Shalev [13] proposed another model of loss aversion to measure this. In Shalev’s model, loss aversion of each player i(i=1,2) is characterized by loss aversion coefficient λi, where λi is nonnegative. And let ri be a reference point of player i. The utility function is given by the following transformation:(2)wizi,λi,ri=ziifzi≥rizi-λiri-ziifzi<rior, equivalently,(3)wizi,λi,ri=1+λizi-λimax⁡ri,ziIf the outcomes that are less than reference point are regarded as losses, then the values of utilities are scaled down by λi. If the values of the payoffs are higher than that of the reference point, then the payoffs are left unchanged.In a number of applications, the reference points are usually given exogenously, which sidesteps the important question of the significance of the reference points. Thus, the fact that players’ reference points are endogenous is the motivation for this paper.


## 3. Equilibrium in the Rubinstein Bargaining Game with Loss Aversion

At timet, all the proposals made to a player by his opponent so far, possibly including the proposal at time t, specify all the shares that this player could have obtained up to the current time t. Thus, the maximum of those shares can be regarded as the reference point of this player, since the maximum of those shares represents what this player could have obtained: shares below this reference point represent losses and their utilities are evaluated by (3).For real bargaining problems, a player may transfer his expectations derived from previous opponents when he enters into a new bargaining situation with another player. For such situations, it is more appropriate that the initial reference points of players are not equal to zero. However, if playeri starts the bargaining by offering an equal split (1/2, 1/2) to his opponent, there is a risk that — if breakdown occurs and a new bargaining phrase starts — the reference point of his opponent switches to a new value that it is larger than the initial value 1/2 [9]. Thus, let r10 and r20 be the initial reference points of players 1 and 2, where r10,r20∈(0,1/2). At any moment t≥1, the reference point of player 1 is r1t=max⁡r10,z1s∣s=2,4,……≤t and the reference point of player 2 is r2t=max⁡r20,z2s∣s=1,3,……≤t, if z1,z2,…,zt∈Z are the offers made up to timet. Thus, the Rubinstein bargaining game that is extended to incorporate loss aversion and reference dependence is not history independent anymore.For playeri, the utility functions for agreement paths and disagreement paths are defined as ui(ht,a)=wi(ξi(ht,a),ri(ht),λi,ri0) and ui(ht,d)=wi(ξi(ht,d),ri(ht),λi,ri0), respectively. In H∞, the utility evaluation of player i is defined as ui=-λi for all h∈H∞, which means that the utility of perpetual disagreement is −λi.LetUi:F×G→R be the expected utility function and the strategy profile (f,g)∈F×G be played from the moment t∈T, where t is the moment up until the history is known. Then (f∣ht,g∣ht) is played at moment t+1 and Ui(f∣ht,g∣ht) is defined as the expected utility of player i at time t if (f,g)∈F×G is played.Definition 1.
The strategy profile(f,g) is called a SPE if, for every t∈T and every ht∈Ht, it satisfies the following two conditions:(4)U1f∣ht,g∣ht≥U1f~∣ht,g∣ht∀f~and(5)U2f∣ht,g∣ht≥U2f∣ht,g~∣ht∀g~.


### 3.1. Constructing Equilibrium

In this subsection, we construct a SPE for the Rubinstein bargaining game with loss aversion and reference dependence, where the initial reference points are not zero. Players’ strategies in the SPE are stationary Markov strategies: both proposals and decisions of acceptance or rejection depend only on the initial reference points and the current reference points. The SPE in this paper still satisfy the following two characteristics that share with SPE in the classical bargaining game proposed by Rubinstein: (i) every proposal in equilibrium is immediately accepted; and (ii) for the decision of acceptance or rejection, players are always indifferent in equilibrium. In our model, a SPE is constructed based on the assumption that a player’s proposal should make the other one indifferent between this proposal and his own proposal in the next phrase.Att∈Todd, player 1 offers x∈Z. We assume that player 2 offers y∈Z at moment t+1 and this proposal will be accepted by his opponent if the proposal x is rejected. Let r20 be the initial reference point of player 2 and let r2 be a reference point of player 2 at time t-1. If x is accepted by player 2, then we have(6)1+λ2x2-r20-λ2max⁡r2-r20,x2-r20≥δ1+λ2y2-r20-λ2max⁡y2-r20,max⁡r2-r20,x2-r20-1-δλ2max⁡r2-r20,x2-r20which means that player 2 should estimate the proposal x at moment t at least as high as the proposal of himself y at the time t+1 after rejecting x. Similarly, we can give another inequality at even moments as follows:(7)1+λ1y1-r10-λ1max⁡r1-r10,y1-r10≥δ1+λ1x1-r10-λ1max⁡x1-r10,max⁡r1-r10,y1-r10-1-δλ1max⁡r1-r10,y1-r10The equilibrium can be constructed by assuming the inequalities (6) and (7) to be equalities. Let μi=1+λi1-δ for i=1,2. It follows from (6) with equality that we can obtain the following three cases:(I)r2>y2>x2:δ(y2-r20)=(x2-r20).(II)y2≥r2>x2:δ(y2-r20)=(1+λ2)(x2-r20)-δλ2(r2-r20).(III)y2>x2≥r2:δ(y2-r20)=μ2(x2-r20).Similarly, we can obtain the following three cases from (7) with equality(1)r1>x1>y1:δ(x1-r10)=(y1-r10).(2)x1≥r1>y1:δ(x1-r10)=(1+λ1)(y1-r10)-δλ1(r1-r10).(3)x1>y1≥r1:δ(x1-r10)=μ1(y1-r10).For reference points of players, we can obtain a partition of[ri0,1](i=1,2) of all possible pairs (r1,r2) into nine sets by combining these equations (see Figure 1). In Figure 1, these sets are denoted by X1,I,…,X3,III,X4, where X4 represents the set of the initial reference points.Figure 1
The partitionX1,I,……,X3,III,X4, where X4 represents the set of the initial reference points.Therefore, the nine sets are formally described. All associated equilibrium proposals are given as follows.① Region 1, I(8)X1,I=r1,r2,ri∈ri0,1∣r1>11+δ+δr101+δ-r201+δ,r2>11+δ-r101+δ+δr201+δThe equilibrium proposals inX1,I are shown as follows:(9)x1,I=11+δ+δr101+δ-r201+δ,δ1+δ-δr101+δ+r201+δ;y1,I=δ1+δ+r101+δ-δr201+δ,11+δ-r101+δ+δr201+δ② Region 1, III(10)X1,III=r1,r2,ri∈ri0,1∣r1>1+λ21+λ2+δ+δr101+λ2+δ-1+λ2r201+λ2+δ,r2≤δ1+λ2+δ-δr101+λ2+δ+1+λ2r201+λ2+δThe equilibrium proposals inX1,III are shown as follows:(11)x1,III=1+λ21+λ2+δ+δr101+λ2+δ-1+λ2r201+λ2+δ,δ1+λ2+δ-δr101+λ2+δ+1+λ2r201+λ2+δ;y1,III=δ1+λ21+λ2+δ+μ2r101+λ2+δ-δ1+λ2r201+λ2+δ,1+λ21-δ1+λ2+δ-μ2r101+λ2+δ+δ1+λ2r201+λ2+δ③ Region 3, I(12)X3,I=r1,r2,ri∈ri0,1∣r1≤δ1+λ1+δ+1+λ1r101+λ1+δ-δr201+λ1+δ,r2>1+λ11+λ1+δ-1+λ1r101+λ1+δ+δr201+λ1+δThe equilibrium proposals inX3,I are shown as follows:(13)x3,I=1+λ11-δ1+λ1+δ+δ1+λ1r101+λ1+δ-1+λ11-δr201+λ1+δ,δ1+λ11+λ1+δ-δ1+λ1r101+λ1+δ+1+λ11-δr201+λ1+δ;y3,I=δ1+λ1+δ+1+λ1r101+λ1+δ-δr201+λ1+δ,1+λ11+λ1+δ-1+λ1r101+λ1+δ+δr201+λ1+δ④ Region 3, III(14)X3,III=r1,r2,ri∈ri0,1∣r1≤δμ2-δμ1μ2-δ2+μ2r10μ1-δμ1μ2-δ2-δr20μ2-δμ1μ2-δ2,r2≤δμ1-δμ1μ2-δ2-δr10μ1-δμ1μ2-δ2+μ1r20μ2-δμ1μ2-δ2The equilibrium proposals inX3,III are shown as follows:(15)x3,III=μ1μ2-δμ1μ2-δ2+δr10μ1-δμ1μ2-δ2-μ1r20μ2-δμ1μ2-δ2,δμ1-δμ1μ2-δ2-δr10μ1-δμ1μ2-δ2+μ1r20μ2-δμ1μ2-δ2;y3,III=δμ2-δμ1μ2-δ2+μ2r10μ1-δμ1μ2-δ2-δr20μ2-δμ1μ2-δ2,μ2μ1-δμ1μ2-δ2-μ2r10μ1-δμ1μ2-δ2+δr20μ2-δμ1μ2-δ2⑤ Region 1, II(16)X1,II=r1,r2,ri∈ri0,1∣r1>μ2-δ+δλ21-r21+λ2-δ2+δ1-δr101+λ2-δ2-1+λ21-δr201+λ2-δ2,δ1+λ2+δ-δr101+λ2+δ+1+λ2r201+λ2+δ<r2≤11+δ-r101+δ+δr201+δ.The equilibrium proposals in X1,II are shown as follows:(17)x1,II=μ2-δ+δλ21-r21+λ2-δ2+δ1-δr101+λ2-δ2-1+λ21-δr201+λ2-δ2,δ1-δ+δλ2r21+λ2-δ2-δ1-δr101+λ2-δ2+1+λ21-δr201+λ2-δ2;y1,II=δμ2-δ+δ2λ21-r21+λ2-δ2+1+λ21-δr101+λ2-δ2-δ1+λ21-δr201+λ2-δ2,μ2-δ+δ2λ2r21+λ2-δ2-1+λ21-δr101+λ2-δ2+δ1+λ21-δr201+λ2-δ2⑥ Region 3, II(18)X3,II=r1,r2,ri∈ri0,1∣r1≤δμ2-δ+λ2δ1-r2μ11+λ2-δ2+1+λ2μ1-δr10μ11+λ2-δ2-δμ2-δr20μ11+λ2-δ2,δμ1-δμ1μ2-δ2-δr10μ1-δμ1μ2-δ2+μ1r20μ2-δμ1μ2-δ2<r2<1+λ11+λ1+δ-1+λ1r101+λ1+δ+δr201+λ1+δ.The equilibrium proposals inX3,II are shown as follows:(19)x3,II=μ1μ2-δ+λ2δ1-r2μ11+λ2-δ2+δμ1-δr10μ11+λ2-δ2-μ1μ2-δr20μ11+λ2-δ2,δμ1-δ+λ2r2μ1μ11+λ2-δ2-δμ1-δr10μ11+λ2-δ2+μ1μ2-δr20μ11+λ2-δ2y3,II=δμ2-δ+λ2δ1-r2μ11+λ2-δ2+1+λ2μ1-δr10μ11+λ2-δ2-δμ2-δr20μ11+λ2-δ2,1+λ2μ1-δ+λ2δ2r2μ11+λ2-δ2-1+λ2μ1-δr10μ11+λ2-δ2+δμ2-δr20μ11+λ2-δ2⑦ Region 2, I(20)X2,I=r1,r2,ri∈ri0,1∣δ1+λ1+δ+1+λ1r101+λ1+δ-δr201+λ1+δ<r1≤11+δ+δr101+δ-r201+δ,r2>μ1-δ+δλ11-r11+λ1-δ2-1-δ1+λ1r101+λ1-δ2-δ1-δr201+λ1-δ2.The equilibrium proposals inX2,I are shown as follows:(21)x2,I=μ1-δ+δ2λ1r11+λ1-δ2+δ1-δ1+λ1r101+λ1-δ2-1+λ11-δr201+λ1-δ2,δμ1-δ+δ2λ11-r11+λ1-δ2-δ1-δ1+λ1r101+λ1-δ2+1+λ11-δr201+λ1-δ2;y2,I=δ1-δ+δλ1r11+λ1-δ2+1-δ1+λ1r101+λ1-δ2-δ1-δr201+λ1-δ2,μ1-δ+δλ11-r11+λ1-δ2-1-δ1+λ1r101+λ1-δ2+δ1-δr201+λ1-δ2.⑧ Region 2, III(22)X2,III=r1,r2,ri∈ri0,1∣δμ2-δμ1μ2-δ2+μ2r10μ1-δμ1μ2-δ2-δr20μ2-δμ1μ2-δ2<r1≤1+λ21+λ2+δ+δr101+λ2+δ-1+λ2r201+λ2+δ,r2≤δμ1-δ+λ1δ1-r1μ21+λ1-δ2-δμ1-δr10μ21+λ1-δ2+μ2-δ1+λ1r20μ21+λ1-δ2.The equilibrium proposals inX2,III are shown as follows:(23)x2,III=μ2-δ1+λ1+δ2λ1r1μ21+λ1-δ2+δμ1-δr10μ21+λ1-δ2-μ2-δ1+λ1r20μ21+λ1-δ2,δμ1-δ+λ1δ1-r1μ21+λ1-δ2-δμ1-δr10μ21+λ1-δ2+μ2-δ1+λ1r20μ21+λ1-δ2;y2,III=δμ2-δ+λ1r1μ2μ21+λ1-δ2+μ2μ1-δr10μ21+λ1-δ2-δμ2-δr20μ21+λ1-δ2,μ2μ1-δ+λ1δ1-r1μ21+λ1-δ2-μ2μ1-δr10μ21+λ1-δ2+δμ2-δr20μ21+λ1-δ2⑨ Region 2, IIFor the setX2,II, its boundaries are described by the neighboring sets’ boundaries. The equilibrium proposals in X2,II are shown as follows:(24)x2,II=1+λ1μ2-δ+δλ21-r2+δ2λ1r11+λ11+λ2-δ2+δ1-δ1+λ1r101+λ11+λ2-δ2-1-δ1+λ11+λ2r201+λ11+λ2-δ2,δμ1-δ+λ1δ1-r1+λ21+λ1r21+λ11+λ2-δ2-δ1-δ1+λ1r101+λ11+λ2-δ2+1-δ1+λ11+λ2r201+λ11+λ2-δ2;y2,II=δμ2-δ+λ2δ1-r2+λ11+λ2r11+λ11+λ2-δ2+1-δ1+λ11+λ2r101+λ11+λ2-δ2-δ1-δ1+λ2r201+λ11+λ2-δ2,1+λ2μ1-δ+λ1δ1-r1+δ2λ2r21+λ11+λ2-δ2-1-δ1+λ11+λ2r101+λ11+λ2-δ2+δ1-δ1+λ2r201+λ11+λ2-δ2.In the setX1,I, the equilibrium proposals are independent of loss aversion coefficients. If r10=r20=0, we can obtain the classical Rubinstein equilibrium proposals x=(1/(1+δ),δ/(1+δ)); y=(δ/(1+δ),1/(1+δ)). Moreover, if r10=r20=0, we can obtain the equilibrium proposals of Driesen et al. [2] in the sets X1,I,…,X3,III, respectively.The equilibrium proposals in the setsX1,III, X3,I, and X3,III depend on the initial reference points but not onr1 andr2. The equilibrium proposals in the sets X1,II and X3,II depend on the initial reference points and the referent pointsr2 but not on the referent points r1. In the sets X2,I and X2,III, the associated equilibrium proposals depend on the initial reference points and player 1’s referent pointsr1 but not on player 2’s referent pointsr2. In the set X2,II, the associated equilibrium proposals not only depend on the initial reference points but also the referent pointsr1 andr2.


### 3.2. Subgame Perfect Equilibrium and Its Uniqueness

To find a SPE, the strategiesf^ and g^, which are the strategies of players 1 and 2, are defined according to the sets Xω and the proposals xω and yω, where ω∈1,I,…,3,III. At any time t∈Todd, for player 1, take the (unique) Xω containing reference point (r1,r2) for any (r1,r2) with r1≥r10, r2≥r20: then the corresponding proposal xω is made by player 1. At any time t∈Teven and for any (r1,r2) with r1≥r10, r2≥r20, take again the relevant set Xω: then a proposal z is accepted by player 1 if and only if z1≥y1ω. Similarly, the strategy g^ for player 2 can be defined.Theorem 2.
The strategy profile(f^,g^) is a SPE; the outcome of the game is(25)x3,III=μ1μ2-δμ1μ2-δ2+δr10μ1-δμ1μ2-δ2-μ1r20μ2-δμ1μ2-δ2,δμ1-δμ1μ2-δ2-δr10μ1-δμ1μ2-δ2+μ1r20μ2-δμ1μ2-δ2.Proof.
See AppendixA.Obviously, the outcome in Theorem2 depends on the initial reference points of players but not on the reference pointsr1 andr2. It is interesting to note that it is a SPE introduced by Driesen et al. [2] if r10=r20=0; i.e., (μ1(μ2-δ)/(μ1μ2-δ2),δ(μ1-δ)/(μ1μ2-δ2)) is the SPE in Driesen et al.’s [2] result, which is independent of the reference points.For the situation where players have the same level of loss aversion, i.e.,λ≔λ1=λ2>0, another interesting observation is that there exist the following cases.(i) The initial reference points are equal but not zero, i.e.,r≔r10=r20>0. In this case, the outcome is(26)μ-μ-δrμ+δ,δ+μ-δrμ+δ,whereμ≔μ1=μ2.Sinceμ-δ=(1+λ)(1-δ)>0, we have δ/(μ+δ)+μ-δr/(μ+δ)>δ/(μ+δ), and δ/(μ+δ) is player 2’s share in Driesen et al.’s [2] outcome. Thus, player 2 (the player who does not start proposing) benefits from the initial reference points.Similarly, sinceμ/(μ+δ)-μ-δr/(μ+δ)<μ/(μ+δ), and μ/(μ+δ) is payoff of player 1 in Driesen et al.’s [2] outcome, player 1 suffers loss because of existing initial reference points.(ii) The initial reference points of players are not equal; there exist the following three different cases.Ifδr10=μr20, then the outcome of the game is (μ/(μ+δ),δ/(μ+δ)), which is the outcome of Driesen et al.’s [2] game. This implies that players do not benefit from the initial reference points compared to Driesen et al.’s [2] case.Ifδr10<μr20, then the outcome is ((μ+δr10-μr20)/(μ+δ),(δ-δr10+μr20)/(μ+δ)), which means that μ/(μ+δ)>(μ+δr10-μr20)/(μ+δ); (δ-δr10+μr20)/(μ+δ)>δ/(μ+δ). Compared to Driesen et al.’s [2] case, player 1, who starts proposing, has a disadvantage and player 2, who does not start proposing, has an advantage. In other words, player 2 (the player who does not start proposing) benefits from the initial reference points and player 1, who starts proposing, suffers loss because of existing initial reference points.Ifδr10>μr20, then μ/(μ+δ)<(μ+δr10-μr20)/(μ+δ) and (δ-δr10+μr20)/(μ+δ)<δ/(μ+δ). Player 1, who starts proposing, has an advantage and player 2, who does not start proposing, has a disadvantage compared to Driesen et al.’s [2] case. In other words, player 1, who starts proposing, benefits from his own initial reference points and player 2 (the player who does not start proposing) suffers loss because of existing initial reference points.Finally, it is important to note that proposals can never be below the reference points on the equilibrium path. For example, if a proposal of player 1 would be below his own reference points, then player 1 has made a higher proposal last phases and so he would improve his payoff by accepting the higher proposal.Now, we show that the subgame perfect equilibrium is unique. For the strategy profile(f,g), it satisfies the following three conditions:(I) The strategiesf and g are stationary Markov strategies. At each time t∈Todd, the proposal prescribed by f does not depend on time but on the reference points at timet and the initial reference point, and at each time t∈Teven, the Y/N decision prescribed byf depends on the proposal of player 2, the reference points at timet, and the initial reference point. Similarly, the strategy g for player 2 can be described.(II) Immediate acceptance: According tof, player 1 makes any proposal that is accepted by his opponent according to g, and conversely.(III) Indifference between acceptance and rejection: For a proposal made by player 1, his opponent is indifferent between accepting this proposal or rejecting it according to the strategy profile(f,g), and conversely.An interesting observation is that above three conditions are satisfied by the SPE in the Rubinstein bargaining.Theorem 3.
The pair of strategy(f^,g^) is the unique SPE, which satisfies the conditions: (I), (II), and (III).Proof.
See AppendixB.The condition (I) implies that the equilibrium strategies are history-dependent despite the impact this play has on reference points of the two players. Nevertheless, it does not mean that bargainers are limited to stationary Markov strategies. In fact, the condition (II) must be satisfied by any SPE in some subgames; i.e., the reference points in these subgames are higher than the (equilibrium) payoff. Condition (III) requires for a proposal made by a player that his opponent is indifferent between accepting or rejecting this proposal.


## 3.1. Constructing Equilibrium

In this subsection, we construct a SPE for the Rubinstein bargaining game with loss aversion and reference dependence, where the initial reference points are not zero. Players’ strategies in the SPE are stationary Markov strategies: both proposals and decisions of acceptance or rejection depend only on the initial reference points and the current reference points. The SPE in this paper still satisfy the following two characteristics that share with SPE in the classical bargaining game proposed by Rubinstein: (i) every proposal in equilibrium is immediately accepted; and (ii) for the decision of acceptance or rejection, players are always indifferent in equilibrium. In our model, a SPE is constructed based on the assumption that a player’s proposal should make the other one indifferent between this proposal and his own proposal in the next phrase.Att∈Todd, player 1 offers x∈Z. We assume that player 2 offers y∈Z at moment t+1 and this proposal will be accepted by his opponent if the proposal x is rejected. Let r20 be the initial reference point of player 2 and let r2 be a reference point of player 2 at time t-1. If x is accepted by player 2, then we have(6)1+λ2x2-r20-λ2max⁡r2-r20,x2-r20≥δ1+λ2y2-r20-λ2max⁡y2-r20,max⁡r2-r20,x2-r20-1-δλ2max⁡r2-r20,x2-r20which means that player 2 should estimate the proposal x at moment t at least as high as the proposal of himself y at the time t+1 after rejecting x. Similarly, we can give another inequality at even moments as follows:(7)1+λ1y1-r10-λ1max⁡r1-r10,y1-r10≥δ1+λ1x1-r10-λ1max⁡x1-r10,max⁡r1-r10,y1-r10-1-δλ1max⁡r1-r10,y1-r10The equilibrium can be constructed by assuming the inequalities (6) and (7) to be equalities. Let μi=1+λi1-δ for i=1,2. It follows from (6) with equality that we can obtain the following three cases:(I)r2>y2>x2:δ(y2-r20)=(x2-r20).(II)y2≥r2>x2:δ(y2-r20)=(1+λ2)(x2-r20)-δλ2(r2-r20).(III)y2>x2≥r2:δ(y2-r20)=μ2(x2-r20).Similarly, we can obtain the following three cases from (7) with equality(1)r1>x1>y1:δ(x1-r10)=(y1-r10).(2)x1≥r1>y1:δ(x1-r10)=(1+λ1)(y1-r10)-δλ1(r1-r10).(3)x1>y1≥r1:δ(x1-r10)=μ1(y1-r10).For reference points of players, we can obtain a partition of[ri0,1](i=1,2) of all possible pairs (r1,r2) into nine sets by combining these equations (see Figure 1). In Figure 1, these sets are denoted by X1,I,…,X3,III,X4, where X4 represents the set of the initial reference points.Figure 1
The partitionX1,I,……,X3,III,X4, where X4 represents the set of the initial reference points.Therefore, the nine sets are formally described. All associated equilibrium proposals are given as follows.① Region 1, I(8)X1,I=r1,r2,ri∈ri0,1∣r1>11+δ+δr101+δ-r201+δ,r2>11+δ-r101+δ+δr201+δThe equilibrium proposals inX1,I are shown as follows:(9)x1,I=11+δ+δr101+δ-r201+δ,δ1+δ-δr101+δ+r201+δ;y1,I=δ1+δ+r101+δ-δr201+δ,11+δ-r101+δ+δr201+δ② Region 1, III(10)X1,III=r1,r2,ri∈ri0,1∣r1>1+λ21+λ2+δ+δr101+λ2+δ-1+λ2r201+λ2+δ,r2≤δ1+λ2+δ-δr101+λ2+δ+1+λ2r201+λ2+δThe equilibrium proposals inX1,III are shown as follows:(11)x1,III=1+λ21+λ2+δ+δr101+λ2+δ-1+λ2r201+λ2+δ,δ1+λ2+δ-δr101+λ2+δ+1+λ2r201+λ2+δ;y1,III=δ1+λ21+λ2+δ+μ2r101+λ2+δ-δ1+λ2r201+λ2+δ,1+λ21-δ1+λ2+δ-μ2r101+λ2+δ+δ1+λ2r201+λ2+δ③ Region 3, I(12)X3,I=r1,r2,ri∈ri0,1∣r1≤δ1+λ1+δ+1+λ1r101+λ1+δ-δr201+λ1+δ,r2>1+λ11+λ1+δ-1+λ1r101+λ1+δ+δr201+λ1+δThe equilibrium proposals inX3,I are shown as follows:(13)x3,I=1+λ11-δ1+λ1+δ+δ1+λ1r101+λ1+δ-1+λ11-δr201+λ1+δ,δ1+λ11+λ1+δ-δ1+λ1r101+λ1+δ+1+λ11-δr201+λ1+δ;y3,I=δ1+λ1+δ+1+λ1r101+λ1+δ-δr201+λ1+δ,1+λ11+λ1+δ-1+λ1r101+λ1+δ+δr201+λ1+δ④ Region 3, III(14)X3,III=r1,r2,ri∈ri0,1∣r1≤δμ2-δμ1μ2-δ2+μ2r10μ1-δμ1μ2-δ2-δr20μ2-δμ1μ2-δ2,r2≤δμ1-δμ1μ2-δ2-δr10μ1-δμ1μ2-δ2+μ1r20μ2-δμ1μ2-δ2The equilibrium proposals inX3,III are shown as follows:(15)x3,III=μ1μ2-δμ1μ2-δ2+δr10μ1-δμ1μ2-δ2-μ1r20μ2-δμ1μ2-δ2,δμ1-δμ1μ2-δ2-δr10μ1-δμ1μ2-δ2+μ1r20μ2-δμ1μ2-δ2;y3,III=δμ2-δμ1μ2-δ2+μ2r10μ1-δμ1μ2-δ2-δr20μ2-δμ1μ2-δ2,μ2μ1-δμ1μ2-δ2-μ2r10μ1-δμ1μ2-δ2+δr20μ2-δμ1μ2-δ2⑤ Region 1, II(16)X1,II=r1,r2,ri∈ri0,1∣r1>μ2-δ+δλ21-r21+λ2-δ2+δ1-δr101+λ2-δ2-1+λ21-δr201+λ2-δ2,δ1+λ2+δ-δr101+λ2+δ+1+λ2r201+λ2+δ<r2≤11+δ-r101+δ+δr201+δ.The equilibrium proposals in X1,II are shown as follows:(17)x1,II=μ2-δ+δλ21-r21+λ2-δ2+δ1-δr101+λ2-δ2-1+λ21-δr201+λ2-δ2,δ1-δ+δλ2r21+λ2-δ2-δ1-δr101+λ2-δ2+1+λ21-δr201+λ2-δ2;y1,II=δμ2-δ+δ2λ21-r21+λ2-δ2+1+λ21-δr101+λ2-δ2-δ1+λ21-δr201+λ2-δ2,μ2-δ+δ2λ2r21+λ2-δ2-1+λ21-δr101+λ2-δ2+δ1+λ21-δr201+λ2-δ2⑥ Region 3, II(18)X3,II=r1,r2,ri∈ri0,1∣r1≤δμ2-δ+λ2δ1-r2μ11+λ2-δ2+1+λ2μ1-δr10μ11+λ2-δ2-δμ2-δr20μ11+λ2-δ2,δμ1-δμ1μ2-δ2-δr10μ1-δμ1μ2-δ2+μ1r20μ2-δμ1μ2-δ2<r2<1+λ11+λ1+δ-1+λ1r101+λ1+δ+δr201+λ1+δ.The equilibrium proposals inX3,II are shown as follows:(19)x3,II=μ1μ2-δ+λ2δ1-r2μ11+λ2-δ2+δμ1-δr10μ11+λ2-δ2-μ1μ2-δr20μ11+λ2-δ2,δμ1-δ+λ2r2μ1μ11+λ2-δ2-δμ1-δr10μ11+λ2-δ2+μ1μ2-δr20μ11+λ2-δ2y3,II=δμ2-δ+λ2δ1-r2μ11+λ2-δ2+1+λ2μ1-δr10μ11+λ2-δ2-δμ2-δr20μ11+λ2-δ2,1+λ2μ1-δ+λ2δ2r2μ11+λ2-δ2-1+λ2μ1-δr10μ11+λ2-δ2+δμ2-δr20μ11+λ2-δ2⑦ Region 2, I(20)X2,I=r1,r2,ri∈ri0,1∣δ1+λ1+δ+1+λ1r101+λ1+δ-δr201+λ1+δ<r1≤11+δ+δr101+δ-r201+δ,r2>μ1-δ+δλ11-r11+λ1-δ2-1-δ1+λ1r101+λ1-δ2-δ1-δr201+λ1-δ2.The equilibrium proposals inX2,I are shown as follows:(21)x2,I=μ1-δ+δ2λ1r11+λ1-δ2+δ1-δ1+λ1r101+λ1-δ2-1+λ11-δr201+λ1-δ2,δμ1-δ+δ2λ11-r11+λ1-δ2-δ1-δ1+λ1r101+λ1-δ2+1+λ11-δr201+λ1-δ2;y2,I=δ1-δ+δλ1r11+λ1-δ2+1-δ1+λ1r101+λ1-δ2-δ1-δr201+λ1-δ2,μ1-δ+δλ11-r11+λ1-δ2-1-δ1+λ1r101+λ1-δ2+δ1-δr201+λ1-δ2.⑧ Region 2, III(22)X2,III=r1,r2,ri∈ri0,1∣δμ2-δμ1μ2-δ2+μ2r10μ1-δμ1μ2-δ2-δr20μ2-δμ1μ2-δ2<r1≤1+λ21+λ2+δ+δr101+λ2+δ-1+λ2r201+λ2+δ,r2≤δμ1-δ+λ1δ1-r1μ21+λ1-δ2-δμ1-δr10μ21+λ1-δ2+μ2-δ1+λ1r20μ21+λ1-δ2.The equilibrium proposals inX2,III are shown as follows:(23)x2,III=μ2-δ1+λ1+δ2λ1r1μ21+λ1-δ2+δμ1-δr10μ21+λ1-δ2-μ2-δ1+λ1r20μ21+λ1-δ2,δμ1-δ+λ1δ1-r1μ21+λ1-δ2-δμ1-δr10μ21+λ1-δ2+μ2-δ1+λ1r20μ21+λ1-δ2;y2,III=δμ2-δ+λ1r1μ2μ21+λ1-δ2+μ2μ1-δr10μ21+λ1-δ2-δμ2-δr20μ21+λ1-δ2,μ2μ1-δ+λ1δ1-r1μ21+λ1-δ2-μ2μ1-δr10μ21+λ1-δ2+δμ2-δr20μ21+λ1-δ2⑨ Region 2, IIFor the setX2,II, its boundaries are described by the neighboring sets’ boundaries. The equilibrium proposals in X2,II are shown as follows:(24)x2,II=1+λ1μ2-δ+δλ21-r2+δ2λ1r11+λ11+λ2-δ2+δ1-δ1+λ1r101+λ11+λ2-δ2-1-δ1+λ11+λ2r201+λ11+λ2-δ2,δμ1-δ+λ1δ1-r1+λ21+λ1r21+λ11+λ2-δ2-δ1-δ1+λ1r101+λ11+λ2-δ2+1-δ1+λ11+λ2r201+λ11+λ2-δ2;y2,II=δμ2-δ+λ2δ1-r2+λ11+λ2r11+λ11+λ2-δ2+1-δ1+λ11+λ2r101+λ11+λ2-δ2-δ1-δ1+λ2r201+λ11+λ2-δ2,1+λ2μ1-δ+λ1δ1-r1+δ2λ2r21+λ11+λ2-δ2-1-δ1+λ11+λ2r101+λ11+λ2-δ2+δ1-δ1+λ2r201+λ11+λ2-δ2.In the setX1,I, the equilibrium proposals are independent of loss aversion coefficients. If r10=r20=0, we can obtain the classical Rubinstein equilibrium proposals x=(1/(1+δ),δ/(1+δ)); y=(δ/(1+δ),1/(1+δ)). Moreover, if r10=r20=0, we can obtain the equilibrium proposals of Driesen et al. [2] in the sets X1,I,…,X3,III, respectively.The equilibrium proposals in the setsX1,III, X3,I, and X3,III depend on the initial reference points but not onr1 andr2. The equilibrium proposals in the sets X1,II and X3,II depend on the initial reference points and the referent pointsr2 but not on the referent points r1. In the sets X2,I and X2,III, the associated equilibrium proposals depend on the initial reference points and player 1’s referent pointsr1 but not on player 2’s referent pointsr2. In the set X2,II, the associated equilibrium proposals not only depend on the initial reference points but also the referent pointsr1 andr2.


## 3.2. Subgame Perfect Equilibrium and Its Uniqueness

To find a SPE, the strategiesf^ and g^, which are the strategies of players 1 and 2, are defined according to the sets Xω and the proposals xω and yω, where ω∈1,I,…,3,III. At any time t∈Todd, for player 1, take the (unique) Xω containing reference point (r1,r2) for any (r1,r2) with r1≥r10, r2≥r20: then the corresponding proposal xω is made by player 1. At any time t∈Teven and for any (r1,r2) with r1≥r10, r2≥r20, take again the relevant set Xω: then a proposal z is accepted by player 1 if and only if z1≥y1ω. Similarly, the strategy g^ for player 2 can be defined.Theorem 2.
The strategy profile(f^,g^) is a SPE; the outcome of the game is(25)x3,III=μ1μ2-δμ1μ2-δ2+δr10μ1-δμ1μ2-δ2-μ1r20μ2-δμ1μ2-δ2,δμ1-δμ1μ2-δ2-δr10μ1-δμ1μ2-δ2+μ1r20μ2-δμ1μ2-δ2.Proof.
See AppendixA.Obviously, the outcome in Theorem2 depends on the initial reference points of players but not on the reference pointsr1 andr2. It is interesting to note that it is a SPE introduced by Driesen et al. [2] if r10=r20=0; i.e., (μ1(μ2-δ)/(μ1μ2-δ2),δ(μ1-δ)/(μ1μ2-δ2)) is the SPE in Driesen et al.’s [2] result, which is independent of the reference points.For the situation where players have the same level of loss aversion, i.e.,λ≔λ1=λ2>0, another interesting observation is that there exist the following cases.(i) The initial reference points are equal but not zero, i.e.,r≔r10=r20>0. In this case, the outcome is(26)μ-μ-δrμ+δ,δ+μ-δrμ+δ,whereμ≔μ1=μ2.Sinceμ-δ=(1+λ)(1-δ)>0, we have δ/(μ+δ)+μ-δr/(μ+δ)>δ/(μ+δ), and δ/(μ+δ) is player 2’s share in Driesen et al.’s [2] outcome. Thus, player 2 (the player who does not start proposing) benefits from the initial reference points.Similarly, sinceμ/(μ+δ)-μ-δr/(μ+δ)<μ/(μ+δ), and μ/(μ+δ) is payoff of player 1 in Driesen et al.’s [2] outcome, player 1 suffers loss because of existing initial reference points.(ii) The initial reference points of players are not equal; there exist the following three different cases.Ifδr10=μr20, then the outcome of the game is (μ/(μ+δ),δ/(μ+δ)), which is the outcome of Driesen et al.’s [2] game. This implies that players do not benefit from the initial reference points compared to Driesen et al.’s [2] case.Ifδr10<μr20, then the outcome is ((μ+δr10-μr20)/(μ+δ),(δ-δr10+μr20)/(μ+δ)), which means that μ/(μ+δ)>(μ+δr10-μr20)/(μ+δ); (δ-δr10+μr20)/(μ+δ)>δ/(μ+δ). Compared to Driesen et al.’s [2] case, player 1, who starts proposing, has a disadvantage and player 2, who does not start proposing, has an advantage. In other words, player 2 (the player who does not start proposing) benefits from the initial reference points and player 1, who starts proposing, suffers loss because of existing initial reference points.Ifδr10>μr20, then μ/(μ+δ)<(μ+δr10-μr20)/(μ+δ) and (δ-δr10+μr20)/(μ+δ)<δ/(μ+δ). Player 1, who starts proposing, has an advantage and player 2, who does not start proposing, has a disadvantage compared to Driesen et al.’s [2] case. In other words, player 1, who starts proposing, benefits from his own initial reference points and player 2 (the player who does not start proposing) suffers loss because of existing initial reference points.Finally, it is important to note that proposals can never be below the reference points on the equilibrium path. For example, if a proposal of player 1 would be below his own reference points, then player 1 has made a higher proposal last phases and so he would improve his payoff by accepting the higher proposal.Now, we show that the subgame perfect equilibrium is unique. For the strategy profile(f,g), it satisfies the following three conditions:(I) The strategiesf and g are stationary Markov strategies. At each time t∈Todd, the proposal prescribed by f does not depend on time but on the reference points at timet and the initial reference point, and at each time t∈Teven, the Y/N decision prescribed byf depends on the proposal of player 2, the reference points at timet, and the initial reference point. Similarly, the strategy g for player 2 can be described.(II) Immediate acceptance: According tof, player 1 makes any proposal that is accepted by his opponent according to g, and conversely.(III) Indifference between acceptance and rejection: For a proposal made by player 1, his opponent is indifferent between accepting this proposal or rejecting it according to the strategy profile(f,g), and conversely.An interesting observation is that above three conditions are satisfied by the SPE in the Rubinstein bargaining.Theorem 3.
The pair of strategy(f^,g^) is the unique SPE, which satisfies the conditions: (I), (II), and (III).Proof.
See AppendixB.The condition (I) implies that the equilibrium strategies are history-dependent despite the impact this play has on reference points of the two players. Nevertheless, it does not mean that bargainers are limited to stationary Markov strategies. In fact, the condition (II) must be satisfied by any SPE in some subgames; i.e., the reference points in these subgames are higher than the (equilibrium) payoff. Condition (III) requires for a proposal made by a player that his opponent is indifferent between accepting or rejecting this proposal.


## 4. Analysis of the Equilibrium

Here, we discuss the impact of loss aversion coefficients on the SPE(f^,g^) and investigate the SPE (f^,g^) with respect to the discount factor (or the probability of continuation of game) δ. Then, we analyze what happens when δ tends to 1 and discuss what happens for different continuation probabilities. Finally, we investigate what happens when the time lapse between proposals goes to zero.Since the setX3,III is the relevant set at the beginning of bargaining game, we focus on this set. In fact, the comparative statics results are similar in subgames. For the strategy profile(27)x3,III=μ1μ2-δμ1μ2-δ2+δr10μ1-δμ1μ2-δ2-μ1r20μ2-δμ1μ2-δ2,δμ1-δμ1μ2-δ2-δr10μ1-δμ1μ2-δ2+μ1r20μ2-δμ1μ2-δ2,we restrict ourselves to the analysis of player 1, since what player 1 gains is what his opponent losses. By differentiating with respect to λ1 and λ2, we have(28)dx13,IIIdλ1=-δ21-δ21+λ21-r10-r20μ1μ2-δ22<0,dx13,IIIdλ2=δμ11-δ21+λ11-r10-r20μ1μ2-δ22>0.Thus, a player is hurt by loss aversion of himself and benefits from his opponent’s at given initial reference points.Example 4.
Consider that players 1 and 2 with loss aversion bargain over a pie, whose size is one. The outcomes of the bargaining are shown inTheorem 1. Let δ = 0.6, r10=0.1, and r20=0.2. Figures 2 and 3 show player 1’s equilibrium payoff with respect to loss aversion coefficients λ1 and λ2, respectively.Figure 2
The changes of equilibrium share of player 1 asλ1.Figure 3
The changes of equilibrium share of player 1 asλ2.From Figure2, it follows that player 1’s equilibrium share is decreasing as λ1. That is, player 1 is hurt by his own loss aversion. In Figure 3, player 1’s equilibrium share is increasing as λ2; i.e., player 1 benefits from player 2’s loss aversion.By differentiating with respect tor10 and r20, we have(29)dx13,IIIdr10=δμ1-δμ1μ2-δ2,dx13,IIIdr20=-μ1μ2-δμ1μ2-δ2Thus, a player benefits from his initial reference point and is hurt by the reference point of the opponent.


### 4.1. Convergence of the Subgame Perfect Equilibrium for Continuation Probability

We investigate convergence of the subgame perfect equilibrium in the following two different aspects:(1) Convergence of the subgame perfect equilibrium for a commonδIn this subsection, we analyze what happens to the SPE whenδ goes to 1.(30)limδ→1⁡x13,III=1+λ21-r20+1+λ1r102+λ1+λ2,limδ→1⁡x23,III=1+λ11-r10+1+λ2r202+λ1+λ2.An interesting observation is that the limit equilibrium proposals forδ tending to 1 are equal to the limit equilibrium proposals in the result obtained by Driesen et al. [2] if the initial reference points are equal to zero.We can repeat this for all subgames. In the limit forδ tending to 1, the nine sets of Figure 1 and the limit equilibrium proposals are shown in Figures 4, 5, 6, and 7 for the case where λ2>λ1.Figure 4
The equilibrium partitions forδ→1, with λ2>λ1 and r≔r10=r20.Figure 5
The equilibrium partitions forδ→1, with λ2>λ1 and r10>r20.Figure 6
The equilibrium partitions forδ→1, with r10<r20 and λ1<λ2<λ1(1-2r10)+2(r20-r10)/(1-2r20).Figure 7
The equilibrium partitions forδ→1, with r10<r20 and λ1(1-2r10)+2(r20-r10)/(1-2r20)<λ2.In Figure4, the nine sets of Figure 1 and the limit equilibrium proposals are shown in the limit for δ tending to 1, for the case where λ2>λ1 and r≔r10=r20. If r=0, all of regions in Figure 4 are consistent with that of Figure 3 obtained in Driesen et al. [2]. Moreover, the limit outcome is (0.5, 0.5) in X1,I when δ→1. If r≠0, the limit outcomes of player 2 in some sets are higher than that of player 2 in Driesen et al.’s [2] outcomes. In X1,I, the limit outcome is an equal share (0.5, 0.5) when δ→1. The limit equilibrium outcome in X1,II and X3,II is (1-r2,r2), while it is (r1,1-r1) in X2,I and X2,III, which are also the limit equilibrium outcome in Driesen et al.’s [2] outcomes. The limit equilibrium outcome is ((1+λ1r)/(2+λ1),(1+λ1-λ1r)/(2+λ1)) in X3,I and the limit equilibrium outcome is ((1+λ2-λ2r)/(2+λ2),(1+λ2r)/(2+λ2)) in X1,III. The set X2,II becomes the line r1+r2=1 where r1∈((1+λ1r)/(2+λ1),(1+λ2-λ2r)/(2+λ2)).In Figure5, the nine sets of Figure 1 and the limit equilibrium proposals are shown in the limit for δ tending to 1, for the case where λ2>λ1 and r10>r20. In the set of X1,I, the limit equilibrium partition is 1+r10-r20/2,1-r10+r20/2. Since r10>r20, player 1 benefits from the reference points compared to player 1’s share in Driesen et al.’s [2] outcomes. The limit equilibrium proposal is 1-r2,r2 in X1,II and X3,II, while the limit equilibrium outcome is r1,1-r1 in X2,I and X2,III, which are also the limit equilibrium partition in Driesen et al.’s [2] outcomes. The limit equilibrium proposal in X3,I is (1+λ1r10-r20/2+λ1, 1+λ1-1+λ1r10+r20/2+λ1), where player 1 benefits from his own initial reference point compared to player 1’s share in Driesen et al.’s [2] outcomes. And in X1,III, it is ((1+λ2+r10-(1+λ2)r20)/2+λ2, 1-r10+1+λ2r20/2+λ2), where player 1 benefits from his own initial reference point compared to player 1’s share in Driesen et al.’s [2] outcome if r10>1+λ2r20, and player 2 benefits from his own initial reference point compared to player 2’s share in Driesen et al.’s [2] outcome if r10<1+λ2r20. The set X2,II becomes the line r1+r2=1, where r1∈1+λ1r10-r20/2+λ1,1+λ2+r10-1+λ2r20/2+λ2. In X3,III, the limit equilibrium partition is (1+λ2+1+λ1r10-1+λ2r20/2+λ1+λ2, (1+λ1-1+λ1r10+1+λ2r20)/(2+λ1+λ2)), where player 1 benefits from the initial reference point r10 compared to player 1’s share in Driesen et al.’s [2] outcome.In Figure6, the nine sets of Figure 1 and the limit equilibrium proposals are shown in the limit for δ tending to 1, for the case where r10<r20 and λ1<λ2<λ11-2r10+2r20-r10/1-2r20. In X1,I, the limit equilibrium partition is 1+r10-r20/2,1-r10+r20/2. Since r10<r20, player 2 benefits from the reference points compared to player 2’s share in Driesen et al.’s [2] outcome. The limit equilibrium outcome in X1,II and X3,II is 1-r2,r2, while it is r1,1-r1 in X2,I and X2,III, which are also the limit equilibrium outcome in Driesen et al.’s outcomes [2]. The limit equilibrium partition in X3,I is 1+λ1r10-r20/2+λ1,1+λ1-1+λ1r10+r20/2+λ1, where player 1 benefits from his own initial reference point compared to player 1’s share in Driesen et al.’s [2] outcomes if r10<r20<λ1r10. And in X1,III, it is (1+λ2+r10-1+λ2r20/2+λ2, (1-r10+(1+λ2)r20)/(2+λ2)), where the share of player 1 is lower than that of player 1 in Driesen et al.’s [2] outcomes, because r10<r20. The set X2,II becomes the line r1+r2=1 where r1∈((1+λ1r10-r20)/(2+λ1), (1+λ2+r10-(1+λ2)r20)/(2+λ2)). In X3,III, the limit equilibrium partition is ((1+λ2+1+λ1r10-1+λ2r20)/(2+λ1+λ2), (1+λ1-1+λ1r10+1+λ2r20)/(2+λ1+λ2)). Since λ1<λ2<λ1(1-2r10)+2(r20-r10)/(1-2r20), player 2’s share assigned by the limit equilibrium partition is higher than that of player 1. That is, although player 2 has a higher loss aversion coefficient than that of player 1, player 2 can obtain a high share of the pie since this player has a high initial reference point.In Figure7, the nine sets of Figure 1 and the limit equilibrium proposals are shown in the limit for δ tending to 1, for the case where r10<r20 and λ1(1-2r10)+2(r20-r10)/(1-2r20)<λ2. In X3,III, the limit equilibrium outcome is(31)1+λ2+1+λ1r10-1+λ2r202+λ1+λ2,1+λ1-1+λ1r10+1+λ2r202+λ1+λ2.Since λ1(1-2r10)+2(r20-r10)/(1-2r20)<λ2, player 2’s share assigned by the limit equilibrium partition is lower than that of player 1. That is, although player 2 has a higher initial reference point than that of player 1, player 2 cannot obtain a high share of the pie because of higher loss aversion level of himself. The analysis of the limit equilibrium partition in other sets is similar to that of the limit equilibrium partition in Figure 6.


### 4.2. Convergence of the Subgame Perfect Equilibrium forδ1≠δ2

Consider the situation where each playeri has his own continuation probability δi(i=1,2). δi is interpreted as the probability of the bargaining occurs at time t+1 if player i rejected his opponent’s proposal at time t. It follows from inequalities (6) and (7) that(32)1+λ2x2-r20-λ2max⁡r2-r20,x2-r20≥δ21+λ2y2-r20-λ2max⁡y2-r20,max⁡r2-r20,x2-r20-1-δ2λ2max⁡r2-r20,x2-r20and(33)1+λ1y1-r10-λ1max⁡r1-r10,y1-r10≥δ11+λ1x1-r10-λ1max⁡x1-r10,max⁡r1-r10,y1-r10-1-δ1λ1max⁡r1-r10,y1-r10In particular, we can obtain the unique SPE by assuming the inequalities (32) and (33) are equalities, which satisfies conditions (I), (II), and (III).We further generalize the SPE if there exists a time lapseΔ between proposals. Moreover, after the last proposal was rejected by player i,i=1,2, the waiting time for breakdown of the game is a probability distribution function. We assume that the waiting time is exponentially distributed with parameter βi, which is the survival rate. After a proposal was rejected by player i, the probability that the bargaining game continues is denoted by δiΔ, where δi=exp⁡-1/βi. Since the reference points in X3,III are the relevant at the beginning of bargaining game, we restrict ourselves to analyzing this case. The outcomes are(34)x3,III=μ1μ2-δ2Δμ1μ2-δ1Δδ2Δ+δ2Δr10μ1-δ1Δμ1μ2-δ1Δδ2Δ-μ1r20μ2-δ2Δμ1μ2-δ1Δδ2Δ,δ2Δμ1-δ1Δμ1μ2-δ1Δδ2Δ-δ2Δr10μ1-δ1Δμ1μ2-δ1Δδ2Δ+μ1r20μ2-δ2Δμ1μ2-δ1Δδ2Δand(35)y3,III=δ1Δμ2-δ2Δμ1μ2-δ1Δδ2Δ+μ2r10μ1-δ1Δμ1μ2-δ1Δδ2Δ-δ1Δr20μ2-δ2Δμ1μ2-δ1Δδ2Δ,μ2μ1-δ1Δμ1μ2-δ1Δδ2Δ-μ2r10μ1-δ1Δμ1μ2-δ1Δδ2Δ+δ1Δr20μ2-δ2Δμ1μ2-δ1Δδ2Δ,where μi=1+λi1-δiΔ for i=1,2. For Δ tending to 0, we can derive (36)limΔ→0⁡x3,III=1+λ21-r20ln⁡δ2+r101+λ1ln⁡δ11+λ1ln⁡δ1+1+λ2ln⁡δ2,1+λ11-r10ln⁡δ1+r201+λ2ln⁡δ21+λ1ln⁡δ1+1+λ2ln⁡δ2=limΔ→0⁡y3,III.Note that this is an asymmetric Nash bargaining solution, as shown by Harsanyi and Selten (1972) and Kalai (1977). That is, this is the solution to the following optimization problem(37)maxz∈Z⁡z1αz21-αwhere α is defined as bargaining power of player 1 and α=(1+λ21-r20ln⁡δ2+r101+λ1ln⁡δ1)/(1+λ1ln⁡δ1+1+λ2ln⁡δ2).It is easy to check thatα is negatively related to λ1 and positively related to λ2. Obviously, α depends on the initial reference points of players, where α is increasing as r10 and decreasing as r20.


## 4.1. Convergence of the Subgame Perfect Equilibrium for Continuation Probability

We investigate convergence of the subgame perfect equilibrium in the following two different aspects:(1) Convergence of the subgame perfect equilibrium for a commonδIn this subsection, we analyze what happens to the SPE whenδ goes to 1.(30)limδ→1⁡x13,III=1+λ21-r20+1+λ1r102+λ1+λ2,limδ→1⁡x23,III=1+λ11-r10+1+λ2r202+λ1+λ2.An interesting observation is that the limit equilibrium proposals forδ tending to 1 are equal to the limit equilibrium proposals in the result obtained by Driesen et al. [2] if the initial reference points are equal to zero.We can repeat this for all subgames. In the limit forδ tending to 1, the nine sets of Figure 1 and the limit equilibrium proposals are shown in Figures 4, 5, 6, and 7 for the case where λ2>λ1.Figure 4
The equilibrium partitions forδ→1, with λ2>λ1 and r≔r10=r20.Figure 5
The equilibrium partitions forδ→1, with λ2>λ1 and r10>r20.Figure 6
The equilibrium partitions forδ→1, with r10<r20 and λ1<λ2<λ1(1-2r10)+2(r20-r10)/(1-2r20).Figure 7
The equilibrium partitions forδ→1, with r10<r20 and λ1(1-2r10)+2(r20-r10)/(1-2r20)<λ2.In Figure4, the nine sets of Figure 1 and the limit equilibrium proposals are shown in the limit for δ tending to 1, for the case where λ2>λ1 and r≔r10=r20. If r=0, all of regions in Figure 4 are consistent with that of Figure 3 obtained in Driesen et al. [2]. Moreover, the limit outcome is (0.5, 0.5) in X1,I when δ→1. If r≠0, the limit outcomes of player 2 in some sets are higher than that of player 2 in Driesen et al.’s [2] outcomes. In X1,I, the limit outcome is an equal share (0.5, 0.5) when δ→1. The limit equilibrium outcome in X1,II and X3,II is (1-r2,r2), while it is (r1,1-r1) in X2,I and X2,III, which are also the limit equilibrium outcome in Driesen et al.’s [2] outcomes. The limit equilibrium outcome is ((1+λ1r)/(2+λ1),(1+λ1-λ1r)/(2+λ1)) in X3,I and the limit equilibrium outcome is ((1+λ2-λ2r)/(2+λ2),(1+λ2r)/(2+λ2)) in X1,III. The set X2,II becomes the line r1+r2=1 where r1∈((1+λ1r)/(2+λ1),(1+λ2-λ2r)/(2+λ2)).In Figure5, the nine sets of Figure 1 and the limit equilibrium proposals are shown in the limit for δ tending to 1, for the case where λ2>λ1 and r10>r20. In the set of X1,I, the limit equilibrium partition is 1+r10-r20/2,1-r10+r20/2. Since r10>r20, player 1 benefits from the reference points compared to player 1’s share in Driesen et al.’s [2] outcomes. The limit equilibrium proposal is 1-r2,r2 in X1,II and X3,II, while the limit equilibrium outcome is r1,1-r1 in X2,I and X2,III, which are also the limit equilibrium partition in Driesen et al.’s [2] outcomes. The limit equilibrium proposal in X3,I is (1+λ1r10-r20/2+λ1, 1+λ1-1+λ1r10+r20/2+λ1), where player 1 benefits from his own initial reference point compared to player 1’s share in Driesen et al.’s [2] outcomes. And in X1,III, it is ((1+λ2+r10-(1+λ2)r20)/2+λ2, 1-r10+1+λ2r20/2+λ2), where player 1 benefits from his own initial reference point compared to player 1’s share in Driesen et al.’s [2] outcome if r10>1+λ2r20, and player 2 benefits from his own initial reference point compared to player 2’s share in Driesen et al.’s [2] outcome if r10<1+λ2r20. The set X2,II becomes the line r1+r2=1, where r1∈1+λ1r10-r20/2+λ1,1+λ2+r10-1+λ2r20/2+λ2. In X3,III, the limit equilibrium partition is (1+λ2+1+λ1r10-1+λ2r20/2+λ1+λ2, (1+λ1-1+λ1r10+1+λ2r20)/(2+λ1+λ2)), where player 1 benefits from the initial reference point r10 compared to player 1’s share in Driesen et al.’s [2] outcome.In Figure6, the nine sets of Figure 1 and the limit equilibrium proposals are shown in the limit for δ tending to 1, for the case where r10<r20 and λ1<λ2<λ11-2r10+2r20-r10/1-2r20. In X1,I, the limit equilibrium partition is 1+r10-r20/2,1-r10+r20/2. Since r10<r20, player 2 benefits from the reference points compared to player 2’s share in Driesen et al.’s [2] outcome. The limit equilibrium outcome in X1,II and X3,II is 1-r2,r2, while it is r1,1-r1 in X2,I and X2,III, which are also the limit equilibrium outcome in Driesen et al.’s outcomes [2]. The limit equilibrium partition in X3,I is 1+λ1r10-r20/2+λ1,1+λ1-1+λ1r10+r20/2+λ1, where player 1 benefits from his own initial reference point compared to player 1’s share in Driesen et al.’s [2] outcomes if r10<r20<λ1r10. And in X1,III, it is (1+λ2+r10-1+λ2r20/2+λ2, (1-r10+(1+λ2)r20)/(2+λ2)), where the share of player 1 is lower than that of player 1 in Driesen et al.’s [2] outcomes, because r10<r20. The set X2,II becomes the line r1+r2=1 where r1∈((1+λ1r10-r20)/(2+λ1), (1+λ2+r10-(1+λ2)r20)/(2+λ2)). In X3,III, the limit equilibrium partition is ((1+λ2+1+λ1r10-1+λ2r20)/(2+λ1+λ2), (1+λ1-1+λ1r10+1+λ2r20)/(2+λ1+λ2)). Since λ1<λ2<λ1(1-2r10)+2(r20-r10)/(1-2r20), player 2’s share assigned by the limit equilibrium partition is higher than that of player 1. That is, although player 2 has a higher loss aversion coefficient than that of player 1, player 2 can obtain a high share of the pie since this player has a high initial reference point.In Figure7, the nine sets of Figure 1 and the limit equilibrium proposals are shown in the limit for δ tending to 1, for the case where r10<r20 and λ1(1-2r10)+2(r20-r10)/(1-2r20)<λ2. In X3,III, the limit equilibrium outcome is(31)1+λ2+1+λ1r10-1+λ2r202+λ1+λ2,1+λ1-1+λ1r10+1+λ2r202+λ1+λ2.Since λ1(1-2r10)+2(r20-r10)/(1-2r20)<λ2, player 2’s share assigned by the limit equilibrium partition is lower than that of player 1. That is, although player 2 has a higher initial reference point than that of player 1, player 2 cannot obtain a high share of the pie because of higher loss aversion level of himself. The analysis of the limit equilibrium partition in other sets is similar to that of the limit equilibrium partition in Figure 6.


## 4.2. Convergence of the Subgame Perfect Equilibrium forδ1≠δ2

Consider the situation where each playeri has his own continuation probability δi(i=1,2). δi is interpreted as the probability of the bargaining occurs at time t+1 if player i rejected his opponent’s proposal at time t. It follows from inequalities (6) and (7) that(32)1+λ2x2-r20-λ2max⁡r2-r20,x2-r20≥δ21+λ2y2-r20-λ2max⁡y2-r20,max⁡r2-r20,x2-r20-1-δ2λ2max⁡r2-r20,x2-r20and(33)1+λ1y1-r10-λ1max⁡r1-r10,y1-r10≥δ11+λ1x1-r10-λ1max⁡x1-r10,max⁡r1-r10,y1-r10-1-δ1λ1max⁡r1-r10,y1-r10In particular, we can obtain the unique SPE by assuming the inequalities (32) and (33) are equalities, which satisfies conditions (I), (II), and (III).We further generalize the SPE if there exists a time lapseΔ between proposals. Moreover, after the last proposal was rejected by player i,i=1,2, the waiting time for breakdown of the game is a probability distribution function. We assume that the waiting time is exponentially distributed with parameter βi, which is the survival rate. After a proposal was rejected by player i, the probability that the bargaining game continues is denoted by δiΔ, where δi=exp⁡-1/βi. Since the reference points in X3,III are the relevant at the beginning of bargaining game, we restrict ourselves to analyzing this case. The outcomes are(34)x3,III=μ1μ2-δ2Δμ1μ2-δ1Δδ2Δ+δ2Δr10μ1-δ1Δμ1μ2-δ1Δδ2Δ-μ1r20μ2-δ2Δμ1μ2-δ1Δδ2Δ,δ2Δμ1-δ1Δμ1μ2-δ1Δδ2Δ-δ2Δr10μ1-δ1Δμ1μ2-δ1Δδ2Δ+μ1r20μ2-δ2Δμ1μ2-δ1Δδ2Δand(35)y3,III=δ1Δμ2-δ2Δμ1μ2-δ1Δδ2Δ+μ2r10μ1-δ1Δμ1μ2-δ1Δδ2Δ-δ1Δr20μ2-δ2Δμ1μ2-δ1Δδ2Δ,μ2μ1-δ1Δμ1μ2-δ1Δδ2Δ-μ2r10μ1-δ1Δμ1μ2-δ1Δδ2Δ+δ1Δr20μ2-δ2Δμ1μ2-δ1Δδ2Δ,where μi=1+λi1-δiΔ for i=1,2. For Δ tending to 0, we can derive (36)limΔ→0⁡x3,III=1+λ21-r20ln⁡δ2+r101+λ1ln⁡δ11+λ1ln⁡δ1+1+λ2ln⁡δ2,1+λ11-r10ln⁡δ1+r201+λ2ln⁡δ21+λ1ln⁡δ1+1+λ2ln⁡δ2=limΔ→0⁡y3,III.Note that this is an asymmetric Nash bargaining solution, as shown by Harsanyi and Selten (1972) and Kalai (1977). That is, this is the solution to the following optimization problem(37)maxz∈Z⁡z1αz21-αwhere α is defined as bargaining power of player 1 and α=(1+λ21-r20ln⁡δ2+r101+λ1ln⁡δ1)/(1+λ1ln⁡δ1+1+λ2ln⁡δ2).It is easy to check thatα is negatively related to λ1 and positively related to λ2. Obviously, α depends on the initial reference points of players, where α is increasing as r10 and decreasing as r20.


## 5. Conclusions

A player may transfer his expectations derived from previous opponents when he enters into a new bargaining situation with another player. That is, the initial reference points in bargaining problems are not zero. In this paper, we investigate the impact of loss aversion and the initial reference points in the classical Rubinstein bargaining problem, by constructing a SPE in Rubinstein bargaining model with loss aversion and reference dependence and making a sensitivity analysis about the SPE with respect to loss aversion coefficients of bargainers. It is found that the equilibrium share of a player is negatively related to his own loss aversion and the initial reference point of the other and positively related to the opponent’s loss aversion and his own initial reference point. It is further found that the outcome converges to asymmetric Nash bargaining if the probability of breakdown tends to zero, where higher loss aversion of a player results in a higher bargaining power of the opponent and a player who has a higher initial reference point has a higher bargaining power.We introduce the unique SPE based on the following three features: stationary Markov strategies, immediate acceptance, and indifference between acceptance and rejection. It is still an open question whether the three features are necessary conditions for uniqueness of the subgame perfect equilibrium.



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