Abstract: Bargaining games have long been a central focus in game theory research due to their wide applicability in representing various economic, political and social interactions. One prominent example is the alternating offers bargaining game, initially proposed by Ariel Rubinstein in 1982. In Rubinstein’s model, two players take turns to offer terms of agreement to each other indefinitely until an agreement is reached. The players discount future payoffs, which provides an incentive to reach an agreement sooner. The analysis of subgame perfect equilibrium in Rubinstein’s model of alternating offers utilizes backward induction, which is a form of reasoning used extensively in game theory, especially in finite extensive form games. To find the subgame perfect equilibrium of the game, it is assumed that the players make decisions that maximize their own payoff, taking into account the anticipated decisions of other players. This method assumes that each player anticipates that all players will act rationally in the future and employs this anticipation to determine their current optimal action. Hence, players are assumed to be fully rational, while also assuming that they believe all other players are fully rational.
However, these assumptions may not hold in many real-world situations. Players might not be entirely rational due to cognitive limitations, lack of information, limited memory capacity, or other factors. They may also doubt that other players are fully rational. Following Herbert Simon’s introduction of bounded rationality, scholars began to explore how such cognitive limitations affect decision-making processes. Most literature to date abstracts bounded rationality as a limitation in participants’ forward-looking capabilities. Although they provide important perspectives on how bounded rationality influences decision-making processes, these theories show certain limitations when studying finite-horizon alternating bargaining games. In such games, when the number of bargaining periods is fixed, rational participants typically use backward induction to devise strategies, rather than merely observe future outcomes. Typically, participants with bounded rationality, after being able to deduce the optimal strategies for several periods through backward reasoning, may deviate from rational behavior due to changes in circumstances, computational limitations, or memory constraints, thus leading to irrational behaviors. In practical game scenarios, this manifestation of bounded rationality is usually observed when participants are able to make fully rational decisions only in smaller-scale subgames.
In light of these complexities and the limitations, our research aims to explore the dynamics of a finite-period alternating offers bargaining game, where players exhibit this specific form of bounded rationality, namely k-period backward bounded rationality. This refers to players who can only perform backward induction reasoning up to k-periods. We construct a model that captures the essence of k-period backward bounded rationality within a finite alternating offers bargaining game. A comprehensive formalization and characterization of k-period backward bounded rationality is then presented. Moreover, we turn our attention to a bargaining game that pairs a fully rational player with a player who is only rational for the final k periods of the game, where k is less than the total number of periods m. The fully rational participant is designated as player 1 and the k-period backward bounded rational player is designated as player 2. For player 2, we posit that their behavior in the initial m-k periods is characterized by unpredictability, with both proposals and acceptances distributed uniformly at random.
Our main results depend on which player is the initial proposer. Specifically, when player 2 is the initial proposer, we find that the expected utility of player 1 is always above 1/2 and increases with the discount factor. Player 2 has an expected utility that remains below 1/2. This indicates an inherent disadvantage for the player with bounded rationality when making the initial offer. On the flip side, when player 1 is the initial proposer, the expected utilities of both players trend towards 1/4 as the discount factor approaches 0. In the opposing limit, as the discount factor approaches 1 and the number of periods of irrationality m-k for the player 2 is sufficiently large, the expected utility of player 1 asymptotically approaches 1, while that of player 2 trends towards 0. Finally, the main results are validated through computer numerical simulations to ensure the accuracy of the theoretical analysis. Surprisingly, numerical simulations reveal that, in certain scenarios, the expected utility of the k-period backward bounded rational player surpasses that of the fully rational player, a phenomenon that needs further investigation.

Key words: alternating offers bargaining games, bounded rationality, backward induction, subgame perfect equilibrium

摘要： 在有限期交替议价谈判博弈中,参与者通过逆向归纳法寻找子博弈完美纳什均衡策略。这种方法假定参与者具备完美理性和无限计算能力,然而实际中,参与者的认知限制使得他们无法达到这种理想状态。为了更真实地描述现实中的谈判行为,本文引入k期逆向有限理性者的概念,这类参与者在博弈的最后k期内能够完美制定策略,但在更早的阶段则可能由于计算能力或记忆力的限制,无法完全遵循理性路径。本文分析了这类参与者在有限期交替议价谈判博弈过程中的策略,并探讨了当对手为完全理性参与者时,双方的期望效用如何受到影响。研究表明:(1)当k期逆向有限理性者首先提议时,完全理性者的期望效用始终高于k期逆向有限理性者,且随着贴现因子的增加而逐渐获得全部谈判价值; (2)当完全理性者首先提议时,其期望效用并不总是高于k期逆向有限理性者; (3)当完全理性者首先提议时,若k期逆向有限理性者的非理性时期数足够大,且贴现因子趋近于1,则完全理性者可获得全部谈判价值。通过数值算例,本研究验证了理论分析的准确性。

关键词: 交替议价谈判博弈, 有限理性, 逆向归纳法, 子博弈完美纳什均衡