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Social Choice and Welfare

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01-09-2015

Bargaining with split-the-difference arbitration

Author: Kang Rong

Published in: Social Choice and Welfare | Issue 2/2015

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Abstract

We analyze an alternating-offer model in which an arbitrator uses the split-the-difference arbitration rule to determine the outcome if both players’ offers are rejected by the opponents. We find that the usual chilling effect of split-the-difference arbitration arises only when the discount factor is sufficiently large. When the discount factor is sufficiently small, players tend to reach agreement immediately. When the discount factor is in the middle range, delayed agreements might arise. We also find that as long as players are not excessively impatient, then the player who makes the first offer obtains an equilibrium payoff that is not greater than his opponent.

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Split-the-difference arbitration is widely used in the real world. For example, NHL salary disputes are required to be resolved by arbitration, and in practice, the arbitrator usually splits the difference between the disputed offers (see, for example, http://nypost.com/2009/08/03/rangers-near-z-day/).

In this “middle” range, immediate-agreement equilibrium may also arise.

Although the symmetric arbitration solution and the split-the-difference arbitration rule coincide with each other when the Pareto frontier is linear, Rong (2012a) focuses on the case where both players are sufficiently patient, while this paper studies the general case where the discount factor can be any number between 0 and 1.

We will further illustrate this point in section 3.2.

\((x_{1}^{*}(\delta ),f(x_{1}^{*}(\delta )))\) is such that Player 2 is indifferent between option \(A\) and option \(R_{c}\). \((x_{2}^{*}(\delta ),f(x_{2}^{*}(\delta )))\) is such that Player 2 is indifferent between option \(A\) and option \(R_{e}\). \((x_{3}^{*}(\delta ),f(x_{3}^{*}(\delta )))\) is such that Player 2 is indifferent between option \(R_{c}\) and option \(R_{e}\).

Use the fact that \(x_{1}^{*} \rightarrow 1\) as \(\delta \rightarrow 0\).

The offer \((0,1)\) is also not strictly inside the acceptance region or the weak/strong rejection region. However, \((0,1)\) cannot be Player 1’s equilibrium offer because \((0,1)\) is strictly dominated by the offer \((1,0)\) and thus Player 1 will never make the offer \((0,1)\) in equilibrium.

Actually, if Player 1 makes the offer \((x_{2}^{*},f(x_{2}^{*}))\), then Player 2 is indifferent between acceptance and rejection. However, using tie-breaking rule 1, Player 2 chooses to accept the offer \((x_{2}^{*},f(x_{2}^{*}))\).

Actually, Player 2 is indifferent between making the counteroffer \(\left( \frac{\delta }{2-\delta }x_{3}^{*},1-\frac{\delta }{2-\delta }x_{3}^{*}\right) \) and making the extreme offer. Using tie-breaking rule 2, Player 2 makes the counteroffer \(\left( \frac{\delta }{2-\delta }x_{3}^{*},1-\frac{\delta }{2-\delta }x_{3}^{*}\right) \) which Player 1 accepts.

For the case where \(\delta \in [0.763,0.781)\), the graphic explanation is similar.

Notice that Player 1’s payoff obtained from the Rubinstein equilibrium is strictly decreasing in \(\delta \) for \(\delta \in (0,1)\).

See also Farber (1981) for a similar analysis in the setting where the two players make simultaneous offers.

When \(x_{1}=x_{2}^{*}\), Player 1 is indifferent between option \(A\) and option \(R_{e}\).

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Title: Bargaining with split-the-difference arbitration
Author: Kang Rong
Publication date: 01-09-2015
Publisher: Springer Berlin Heidelberg
Published in: Social Choice and Welfare / Issue 2/2015
Print ISSN: 0176-1714
Electronic ISSN: 1432-217X
DOI: https://doi.org/10.1007/s00355-015-0896-1

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