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Dynamic Games and Applications

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11-07-2020

Bargaining, Reference Points, and Limited Influence

Authors: Tarık Kara, Emin Karagözoğlu, Elif Özcan-Tok

Published in: Dynamic Games and Applications | Issue 2/2021

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Abstract

We study the emergence of reference points in a bilateral, infinite horizon, alternating offers bargaining game. Players’ preferences exhibit reference dependence, and their current offers have the potential to influence each other’s future reference points. However, this influence is limited in that it expires in a finite number of periods. We first construct a subgame perfect equilibrium that involves an immediate agreement and study its properties. Later, we also show the existence of an equilibrium where agreement is reached with delay. We show that expiration lengths and initial reference points play a crucial role for the existence of this equilibrium. For instance, we show that equilibrium with a delayed agreement does not exist when the initial reference point is (0, 0). Finally, we provide comparative static analyses on model parameters, compare two variations of our model, and compare our findings with those of the closest paper to ours, Driesen et al. (Math Soc Sci 64:103–118, 2012).

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Ashenfelter and Bloom [2], Blount et al. [7], Kristensen and Gärling [37], Bohnet and Zeckhauser [8], Gächter and Riedl [18, 19], Gimpel [20], Bartling and Schmidt [3], Herweg and Schmidt [24], Fehr et al. [16], Karagözoğlu and Riedl [33], Bolton and Karagözoğlu [9] and Karagözoğlu and Kocher [32] are only some of these studies, all of which reported that reference points —in the form of existing contracts, reservation prices, expired contracts, historical contractual conditions, informal agreements, (fairness) norms—significantly influence the whole bargaining process and the negotiated agreement.

The reader is referred to Benartzi and Thaler [6], Shalev [44, 45], Compte and Jehiel [10], Falk and Knell [15], Köszegi and Rabin [34‐36], Gimpel [20], Li [38], Vartiainen [47], Abeler et al. [1], Baucells et al. [5], Giorgi and Post [21], Hyndman [26], Driesen et al. [12], Sarver [43], Roels and Su [41], Karagözoğlu and Keskin [31] and Karagözoğlu et al. [30] among others.

In the bargaining game \(\Gamma \), (i) the pie is desirable, (ii) breakdown is the worst outcome, (iii) \(u_{i}^{t}(x,r)>u_{i}^{t+1}(x,r)\) for each t, x and r, and (iv) \(u_{i}\) is continuous. The game is stationary in the sense that player i’s preference between his share from the division x at period t and his share from division y at period \(t+1\) is independent of t when period t is player i’s turn to make an offer. Note that the reference point of player i at period \(t+1\) is equal to his reference point at period t, i.e., \(r_{i}^{t+1}=r_{i}^{t}\). These properties of \(\Gamma \) allow us to use the expected payoff at period \(t+1\) as the continuation payoff of the game (see [40, pp. 33–34 and 74–75]).

See [17] for a similar modeling assumption. Note that our assumption is weaker than their endogenous commitment assumption.

Note that the deviation (from the strategies of the corresponding immediate-agreement equilibrium) that brings about the delayed-agreement equilibrium is a bilateral one, and as such, the immediate-agreement equilibrium and the delayed-agreement equilibrium coexist.

The reader is referred to “Appendix C” for the mathematical derivations of the results presented here.

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Title: Bargaining, Reference Points, and Limited Influence
Authors: Tarık Kara
Emin Karagözoğlu
Elif Özcan-Tok
Publication date: 11-07-2020
Publisher: Springer US
Published in: Dynamic Games and Applications / Issue 2/2021
Print ISSN: 2153-0785
Electronic ISSN: 2153-0793
DOI: https://doi.org/10.1007/s13235-020-00359-8

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