Abstract: Bargaining is the process in which players try to reach an agreement through alternating offers. Bargaining process usually takes a lot of time. The main problem faced by players in bargaining is that players reach an agreement on how to cooperate before truly cooperating and achieving results. On the one hand, each player hopes to reach a consensus agreement. On the other hand, each player hopes to reach an agreement that is as beneficial as possible to itself. Therefore, there are conflicts among players in the bargaining process-each player tries to reach an agreement that is as beneficial as possible to itself. As a result, after paying a high price, both players are likely to reach an agreement or fail to do so. The cost (frictions) incurred by players in the bargaining process comes from two facts: bargaining is time-consuming, while time is valuable to players. Therefore, reaching a consensus agreement between both players will pay a price (i.e., time cost).
In the Rubinstein bargaining game, the time cost of a player is reflected by the discount rate, whose impacts on subgame perfect equilibrium are investigated. It is worth noting that expected utility theory dominates the analysis of bargaining theory, despite much evidence that it fails to characterize or predict adequately human behavior. Experimental studies on economics and psychology have demonstrated that players tend to exhibit irrational behavior. It means that expected utility theory cannot accurately describe such behavior. Therefore, KAHNEMAN and TVERSKY (1992) proposed prospect theory, which is an alternative to expected utility theory. Prospect theory can explain the phenomenon that players cannot follow the principle of maximizing expected utility. Prospect theory is a modification of expected utility theory that deviate in the following three aspects: (1)Different from expected utility theory that evaluates the final wealth, prospect theory evaluates the outcomes with respect to a reference level. Deriving gains and losses are regarded as prospects. (2)The marginal utility in benefits is smaller than in losses. That is, losses loom larger than gains. (3)Probability weighting: small probabilities are overestimated, while other probabilities are underestimated. Prospect theory and its variants are currently the most commonly used behavioral decision-making models. Therefore, our work incorporates prospect theory into the classic Rubinstein bargaining game, and explores the impact of loss aversion and probability weighting in prospect theory on the subgame perfect equilibrium.
It is worth noting that some works have explored the application of prospect theory in game theory. But they mainly focus on the impact of prospect theory on static matrix games and dynamic matrix games. A few scholars have explored the impact of prospect theory on bargaining games, and they mainly focus on the impact of loss aversion behavior of players on the alternating-offer bargaining game. However, those works do not consider the weighting of risk probability by players when there is a risk of bargaining breakdown. Different from the extant literatures, we introduce prospect theory into Rubinstein bargaining games, not only analyzing the impact of loss aversion behavior on the subgame perfect equilibrium, but also exploring the influence of weighted risk probability of players.
This paper reconsiders the Rubinstein bargaining game, where players have prospect theory preferences, and their proposals depend on bargaining history. We construct a PT-subgame perfect equilibrium and prove the existence and uniqueness of the PT-subgame perfect equilibrium. The findings are shown as follows: (1)When the reference points of two players are high, the equilibrium share of players will not be related to loss aversion behavior, but only will depend on the probability weighted behavior of players; when the reference points of two players are low, the equilibrium share of players will depend on both their loss aversion behavior and probability weighted behavior. (2)If the probability of bargaining continuing is high, probability weighting is advantageous to the first-player with loss aversion behavior; otherwise, whether probability weighting is advantageous to the first-player with loss aversion behavior depends on the weight coefficient. (3)Only when the degree of loss aversion for the first-player is higher than that of the competitor and the probability of bargaining continuing is sufficiently high, prospect theory preference may have adverse effects on the first-player; otherwise, prospect theory preference is advantageous to the first-player.

Key words: Rubinstein bargaining, prospect theory, referent point, PT-subgame perfect equilibrium

摘要： 期望效用理论不能准确地刻画或预测人类的行为,KAHNEMAN和TVERSKY(1992)提出的前景理论成为刻画或预测人类行为最有效的一种方法。为此,将前景理论引入Rubinstein谈判博弈,构建PT-子博弈完美均衡,并证明存在性及其唯一性,探究前景理论偏好对谈判博弈的影响。研究发现:(1)当两个局中人的参考点较高时,局中人的均衡份额与损失规避行为无关,仅取决于局中人的概率加权行为;当两个局中人的参考点较低时,局中人的均衡份额不仅与损失规避行为有关,而且取决于局中人的概率加权行为。(2)如果谈判继续的概率较大,概率加权对具有损失规避行为的先行动者有利,否则,概率加权对具有损失规避行为的先行动者是否有利取决于权系数。(3)只有当先行动者的损失规避程度高于对手且谈判继续的概率δ足够高时,前景理论偏好可能会对先行动者产生不利影响;否则,前景理论偏好对先行动者有利。

关键词: Rubinstein谈判, 前景理论, 参考点, PT-子博弈完美均衡