Abstract
We analyze a bargaining model which is a generalization of the model of Rubinstein (Econometrica 50(1):97–109, 1982) from the viewpoint of the process of how a proposer is decided in each period. In our model, a player’s probability to be a proposer depends on the history of proposers and players divide a pie of size 1. We derive a subgame perfect equilibrium (SPE) and analyze how its SPE payoffs are related to the process. In the bilateral model, there is a unique SPE. In the n-player model, although SPE may not be unique, a Markov perfect equilibrium (MPE) similar to the SPE in the bilateral model exists. In the case where the discount factor is sufficiently large, if the ratio of opportunities to be a proposer converges to some value, players divide the pie according to the ratio of this convergent value under these equilibria. This result implies that although our process has less regularity than a Markov process, the same result as in the model that uses a Markov process holds. In addition to these results, we show that the limit of the SPE (or the MPE) payoffs coincides with the asymmetric Nash bargaining solution weighted by the convergent values of the ratio of the opportunities to be a proposer.
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Notes
Note that, even if some player’s probability to be a proposer increases gradually over time as in our example of price negotiation, her probability to be a proposer converges to some constant value since the value of the probability to be a proposer is bounded. In this sense, we can consider that the negotiation calms down and the proposal ratio stays in some value in the long run even in the situation that some player’s probability to be a proposer increases gradually over time.
Let A be the order \(1 \rightarrow 2\) where player 1 proposes first and player 2 proposes second. Also, let B be the order \(2 \rightarrow 1\). Now, for example, consider the deterministic process \(A B B A A A B B B B \cdots \). In this process, the proposal ratio during 2 periods is 1 : 1. Therefore, this process satisfies the condition in Theorem 4. However, for all \(k \in \mathbb {N}\), this process is not represented by a Markov process in which probabilities depend on the previous k periods. Although this process is a very extreme example, even if we set A and B arbitrarily, such orders satisfy our condition. This fact implies that Theorem 4 can analyze not only simple situations which are represented by a Markov process but also more complex situations.
In Proposition 2, we have to assume the condition of convergence for all \(\pi \in \bigcup _{t \in \mathbb {N}} N^{t - 1}\). If this condition does not hold, after some history, the limit of the division which is proposed in the unique SPE may not exist. Even if we assume such a condition, we can still consider the process where players’ probabilities to be a proposer depends on the periods and the probabilities converge to some values in the limit. Therefore, this condition is more general than assuming a Markov process.
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Acknowledgements
I am grateful to Ryo Kawasaki, two anonymous referees, an anonymous associate editor, and the editor for extremely valuable comments. I am also grateful to Keisuke Bando, Toshiyuki Hirai, Shin Kishimoto, Toshiji Miyakawa, Shigeo Muto, Tadashi Sekiguchi, Satoru Takahashi, and participants at Game Theory Workshop 2017 held at The University of Electro-Communications, The 2017 Spring National Conference of The Operations Research Society of Japan held at Okinawaken Shichouson Jichikaikan, East Asian Game Theory Conference 2017 held at The National University of Singapore, and European Meeting on Game Theory 2018 held at University of Bayreuth for helpful comments and suggestions. This work was supported by JSPS KAKENHI Grant Number JP18J20162.
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Hanato, S. Equilibrium payoffs and proposal ratios in bargaining models. Int J Game Theory 49, 463–494 (2020). https://doi.org/10.1007/s00182-019-00698-w
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DOI: https://doi.org/10.1007/s00182-019-00698-w