Subgame Perfect Equilibrium in the Rubinstein Bargaining Game with Loss Aversion

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.


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 % is evaluated less if a share of % with  >  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 2 Complexity 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 decisionmaker, 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 Section 2, 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.

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 For the latter case, it means that the bargaining game ends in disagreement; that is, players obtain the shares ( 0 1 ,  0 2 ), where  0 1 and  0 2 represent the initial reference points of players.For each odd moment, a strategy  played by player 1 specifies a proposal in  that depends on the history of the bargaining so far; on the other hand, for each even moment, a decision  or N 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 , with the roles for moments  ∈  odd and  ∈  even are reversed.
At time  ∈ , the history of the bargaining is denoted by ℎ  , which is defined as a vector of proposals of bargainers.Specifically, ℎ  fl ( 1 ,  2 , . . .,   ), where   ∈  for all  ≤ .Furthermore, at time  ∈ , all possible histories ℎ  are denoted by   in the bargaining game.That is,   fl ∏  =1 , let  0 fl (ℎ 0 ), where ℎ 0 is an empty history.
Let  be strategy set of player 1, denoted by sequences of functions (  ) ∈ where for  = 1:   ∈ , for  > 1 and  ∈   :   :  -1 → , and for  ∈  V :   :   → {, }, and let G be strategies of player 2, denoted by sequences of functions (  ) ∈ where for  ∈   :   :   → {, } and for  ∈  V :   :  -1 → .An agreement path is denoted by (ℎ  , ), which is a history ℎ  ∈   ending in agreement at time .All time  agreement paths are denoted by   fl {(ℎ  , ) | ℎ  ∈   }.The set A fl ⋃ ∈   contains all histories that end in agreement.Similarly, a disagreement path is denoted as (ℎ  , ), which means that a history ℎ  ∈   ends in disagreement at time .  fl {(ℎ  , ) | ℎ  ∈   } contains all time  disagreement paths. fl ⋃ ∈   contains all histories ending in disagreement.All objects of (ℎ  , ) are denoted by the set   ; that is, histories do not end at moment t.On the other hand, we define  ∞ fl {( 1 ,  2 , . ..) |   ∈  for all  ∈ }.The elements of  ∞ are defined as infinite paths.Therefore, the set that contains all paths of the bargaining game can be denoted by  fl  ∞ ∪  ∪ .Note that a set of paths in  is determined by a strategy profile (, ) ∈  × .In particular, if an agreement at time t is reached when players play (, ), the set of paths associated with (, ) not only contains  − 1 paths in the set  but also contains one in the set .If an agreement is never reached when (, ) is played, then the set only contains paths in .
The function   fl \ ∞ → [ 0  , 1] is introduced, which is used to specify the share that player  ( = 1, 2) obtains for each finite path in .Specifically, for all ℎ  ∈   , ℎ  = ( 1 , or, equivalently, If the outcomes that are less than reference point are regarded as losses, then the values of utilities are scaled down by   .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.

Equilibrium in the Rubinstein Bargaining Game with Loss Aversion
At time , all the proposals made to a player by his opponent so far, possibly including the proposal at time , specify all the shares that this player could have obtained up to the current time .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 player  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  0 1 and  0 2 be the initial reference points of players 1 and 2, where  0 1 ,  0 2 ∈ (0, 1/2).At any moment  ≥ 1, the reference point of player 1 is For player , the utility functions for agreement paths and disagreement paths are defined as   (ℎ  , ) =   (  (ℎ  , ),   (ℎ  ),   ,  0  ) and   (ℎ  , ) =   (  (ℎ  , ),   (ℎ  ),   ,  0  ), respectively.In  ∞ , the utility evaluation of player  is defined as   = −  for all ℎ ∈  ∞ , which means that the utility of perpetual disagreement is −  .
Let   :  ×  →  be the expected utility function and the strategy profile (, ) ∈  ×  be played from the moment  ∈ , where  is the moment up until the history is known.Then ( | ℎ  ,  | ℎ  ) is played at moment  + 1 and   ( | ℎ  ,  | ℎ  ) is defined as the expected utility of player  at time  if (, ) ∈  ×  is played.
Definition 1.The strategy profile (, ) is called a SPE if, for every  ∈  and every ℎ  ∈   , it satisfies the following two conditions: and 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.At  ∈   , player 1 offers  ∈ .We assume that player 2 offers  ∈  at moment +1 and this proposal will be accepted by his opponent if the proposal  is rejected.Let  0 2 be the initial reference point of player 2 and let  2 be a reference point of player 2 at time  − 1.If  is accepted by player 2, then we have which means that player 2 should estimate the proposal  at moment  at least as high as the proposal of himself  at the time  + 1 after rejecting .Similarly, we can give another inequality at even moments as follows: The equilibrium can be constructed by assuming the inequalities ( 6) and ( 7) to be equalities.Let   = 1 +   (1 − ) for  = 1, 2. It follows from (6) with equality that we can obtain the following three cases: ).Similarly, we can obtain the following three cases from (7) with equality (1) ).For reference points of players, we can obtain a partition of [ 0  , 1] ( = 1, 2) of all possible pairs ( 1 ,  2 ) into nine sets by combining these equations (see Figure 1).In Figure 1, these sets are denoted by  1,I , . . .,  3,III ,  4 , where  4 represents the set of the initial reference points.
Therefore, the nine sets are formally described.All associated equilibrium proposals are given as follows.
A Region 1, I The equilibrium proposals in  1,I are shown as follows: B Region 1, III The equilibrium proposals in  1,III are shown as follows: C Region 3, I The equilibrium proposals in  3,I are shown as follows: The equilibrium proposals in  3,III are shown as follows: The equilibrium proposals in  1,II are shown as follows: The equilibrium proposals in  3,II are shown as follows: The equilibrium proposals in  2,I are shown as follows: Complexity 7 The equilibrium proposals in  2,III are shown as follows: For the set  2,II , its boundaries are described by the neighboring sets' boundaries.The equilibrium proposals in  2,II are shown as follows: In the set  1,I , the equilibrium proposals are independent of loss aversion coefficients.If  0 1 =  0 2 = 0, we can obtain the classical Rubinstein equilibrium proposals  = (1/(1 + ), /(1 + ));  = (/(1 + ), 1/(1 + )).Moreover, if  0 1 =  0 2 = 0, we can obtain the equilibrium proposals of Driesen et al. [2] in the sets  1,I , . . .,  3,III , respectively.
The equilibrium proposals in the sets  1,III ,  3,I , and  3,III depend on the initial reference points but not on r 1 and r 2 .The equilibrium proposals in the sets  1,II and  3,II depend on the initial reference points and the referent points r 2 but not on the referent points  1 .In the sets  2,I and  2,III , the associated equilibrium proposals depend on the initial reference points and player 1's referent points r 1 but not on player 2's referent points r 2 .In the set  2,II , the associated equilibrium proposals not only depend on the initial reference points but also the referent points r 1 and r 2 .

Subgame Perfect Equilibrium and Its
Uniqueness.To find a SPE, the strategies f and ĝ, which are the strategies of players 1 and 2, are defined according to the sets   and the proposals   and   , where  ∈ {1, I, . . ., 3, III}.At any time  ∈   , for player 1, take the (unique)   containing reference point (r 1 , r 2 ) for any (r 1 , r 2 ) with  1 ≥ Obviously, the outcome in Theorem 2 depends on the initial reference points of players but not on the reference points r 1 and r 2 .It is interesting to note that it is a SPE introduced by Driesen et al. [2] if  0 1 =  0 2 = 0; i.e., ( [2] result, which is independent of the reference points.
For the situation where players have the same level of loss aversion, i.e.,  fl  1 =  2 > 0, another interesting observation is that there exist the following cases.
(ii) The initial reference points of players are not equal; there exist the following three different cases.
If  0 1 =  0 2 , then the outcome of the game is (/( + ), /( + )), 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.
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 (, ), it satisfies the following three conditions: (I) The strategies  and  are stationary Markov strategies.At each time  ∈   , the proposal prescribed by  does not depend on time but on the reference points at time t and the initial reference point, and at each time  ∈  V , the / decision prescribed by f depends on the proposal of player 2, the reference points at time t, and the initial reference point.Similarly, the strategy  for player 2 can be described.
(II) Immediate acceptance: According to , player 1 makes any proposal that is accepted by his opponent according to , 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 (, ), and conversely.
An interesting observation is that above three conditions are satisfied by the SPE in the Rubinstein bargaining.

eorem . The pair of strategy ( f, ĝ) is the unique SPE, which satisfies the conditions: (I), (II), and (III).
Proof.See Appendix B.
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.

Analysis of the Equilibrium
Here, we discuss the impact of loss aversion coefficients on the SPE ( f, ĝ) and investigate the SPE ( f, ĝ) with respect to the discount factor (or the probability of continuation of game) .Then, we analyze what happens when  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 set  3,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 we restrict ourselves to the analysis of player 1, since what player 1 gains is what his opponent losses.By differentiating with respect to  1 and  2 , we have Thus, a player is hurt by loss aversion of himself and benefits from his opponent's at given initial reference points.From Figure 2, it follows that player 1's equilibrium share is decreasing as  1 .That is, player 1 is hurt by his own loss aversion.In Figure 3, player 1's equilibrium share is increasing as  2 ; i.e., player 1 benefits from player 2's loss aversion.
By differentiating with respect to  0 1 and  0 2 , we have Thus, a player benefits from his initial reference point and is hurt by the reference point of the opponent.

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 common In this subsection, we analyze what happens to the SPE when  goes to 1.
(30) An interesting observation is that the limit equilibrium proposals for  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 for  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  2 >  1 .
In Figure 4, the nine sets of Figure 1 and the limit equilibrium proposals are shown in the limit for  tending to 1, for the case where  2 >  1 and  fl  0 1 =  0 2 .If  = 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  1,I when  → 1.If  ̸ = 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  1,I , the limit 10 Complexity  outcome is an equal share (0.5, 0.5) when  → 1.The limit equilibrium outcome in  1,II and  3,II is (1 −  2 ,  2 ), while it is ( 1 , 1 −  1 ) in  2,I and  2,III , which are also the limit equilibrium outcome in Driesen et al. 's [2] outcomes.The limit equilibrium outcome is ) in  3,I and the limit equilibrium outcome is In Figure 5, the nine sets of Figure 1 and the limit equilibrium proposals are shown in the limit for  tending to 1, for the case where  2 >  1 and  0 1 >  0 2 .In the set of  1,I , the limit equilibrium partition is , 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 −  2 ,  ) in  1,II and  3,II , while the limit equilibrium outcome is ( 1 , 1 −  1 ) in  2,I and  2,III , which are also the limit equilibrium partition in Driesen et al. 's [2] outcomes.The limit equilibrium proposal in  3,I is 2 )/(2 +  1 )), 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  1,III , it is , where player 1 benefits from his own initial reference point compared to player 1's share in Driesen et al. 's [2] outcome if  0 1 > (1 +  2 ) 0 2 , and player 2 benefits from his own initial reference point compared to player 2's share in Driesen et al. 's [2] the limit equilibrium partition is ((1 , where player 1 benefits from the initial reference point  0 1 compared to player 1's share in Driesen et al. 's [2] outcome.
In Figure 6, the nine sets of Figure 1 and the limit equilibrium proposals are shown in the limit for  tending to 1, for the case where  0 1 <  0 2 and In  1,I , the limit equilibrium partition is , 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  1,II and  3,II is (1 −  2 ,  2 ), while it is ( 1 , 1 −  1 ) in  2,I and  2,III , which are also the limit equilibrium outcome in Driesen et al. 's outcomes [2].The limit equilibrium partition in  3,I is ((1 + )).In  3,III , the limit equilibrium partition is ((1 , 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 Figure 7, the nine sets of Figure 1 and the limit equilibrium proposals are shown in the limit for  tending to 1, for the case where  0 1 <  0 2 and  1 (1 − 2 0 1 ) + 2( 0 2 −  0 1 )/(1 − 2 0 2 ) <  2 .In  3,III , the limit equilibrium outcome is ) . (31) Since  1 (1 − 2 0 1 ) + 2( 0 2 −  0 1 )/(1 − 2 0 2 ) <  2 , 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.

Convergence of the Subgame Perfect Equilibrium for
Consider the situation where each player  has his own continuation probability   ( = 1, 2).  is interpreted as the probability of the bargaining occurs at time  + 1 if player  rejected his opponent's proposal at time .It follows from inequalities ( 6) and ( 7) that and In 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 lapse Δ between proposals.Moreover, after the last proposal was rejected by player ,  = 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   , which is the survival rate.After a proposal was rejected by player , the probability that the bargaining game continues is denoted by  Δ  , where   = exp(−1/  ).Since the reference points in  3,III are the relevant at the beginning of bargaining game, we restrict ourselves to analyzing this case.The outcomes are and where   = 1 +   (1 −  Δ  ) for  = 1, 2. For Δ tending to 0, we can derive 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 max where  is defined as bargaining power of player 1 and It is easy to check that  is negatively related to  1 and positively related to  2 .Obviously,  depends on the initial reference points of players, where  is increasing as  0 1 and decreasing as  0 2 .

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.

A. Proof of Theorem 2
The one-deviation property is used to prove Theorem 2. According to this property, the sufficient condition that a strategy profile (, ) ∈  ×  is a SPE is no one can improve his own payoffs by deviating unilaterally only once.
The one-deviation property, as Hendon et al. [15] pointed out, holds in infinite-horizon extensive-form games.These games are continuous at infinity.In order to define the continuity at infinity, we have the following: For any  > 0, there exists a number  ∈  such that if, for (, ), (  ,   ) ∈  × , we have (  ,   ) = (  ,   ) for all  ≤ , then |  (, ) −   (  ,   )| < .Lemma A. .The bargaining game, where bargainers are loss averse and their initial reference points are not zero, is continuous at infinity.Proof.Let  > 0, and let the strategy profiles (, ), (  ,   ) ∈  ×  satisfy (  ,   ) = (  ,   ) for all  ≤ , where  > max =1,2 log  /(1+  ).For two such strategy profiles, if player  obtains the whole pie from a strategy profile at time  + 1, while the other one results in perpetual disagreement, we have the following.
For the former, player  would obtain   =   + (1 − ) ∑  =1  −1   (ℎ  , ).For the latter he would obtain It follows from this equality and   −   ≥ 0 that Thus, the game is continuous at infinity.
By Lemma A.1, the one-deviation property can be used.
Proof of Theorem 2. The sufficient condition that the strategy profile ( f, ĝ) is a SPE is that no one can improve his share by deviating unilaterally at one point in time.The utility of share that player 1 obtains according to the following strategy f is denoted by  * 1 .Let ℎ -1 ∈  -1 ; that is, ℎ -1 is a history continuing to time .We assume that ℎ -1 satisfies ( 1 (ℎ -1 ),  2 (ℎ -1 )) ∈   with  ∈ Ω, and ℎ  = (ℎ  −1 , ) with  ∈ .If  ∈  odd ( even ), then the proposal  is made by player 1 (6).If  is rejected, then the bargaining continues with probability  or ends in disagreement with probability 1 − .If the bargaining continues to moment  + 1, it ends in accepting the proposal at  + 1, since the strategy profile ( f, ĝ) is prevalent.
To present that the strategy f is the best response to the strategy ĝ, we distinguish the following two cases:  ∈   and  ∈  V .For each case, the following three subcases are considered: (

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 in Theorem 1.Let  = 0.6,  0 1 = 0.1, and  0 2 = 0.2.Figures2 and 3show player 1's equilibrium payoff with respect to loss aversion coefficients  1 and  2 , respectively.

1 Figure 2 :Figure 3 :
Figure 2: The changes of equilibrium share of player 1 as  1 .
01 ,  2 ≥  0 2 : then the corresponding proposal   is made by player 1.At any time  ∈  V and for any ( 1 ,  2 ) with  1 ≥  0 1 ,  2 ≥  0 2 , take again the relevant set   : then a proposal  is accepted by player 1 if and only if  1 ≥   1 .Similarly, the strategy ĝ for player 2 can be defined.