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Theory and Decision

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01-08-2014

Implementing equal division with an ultimatum threat

Authors: Esat Doruk Cetemen, Emin Karagözoğlu

Published in: Theory and Decision | Issue 2/2014

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Abstract

We modify the payment rule of the standard divide the dollar (DD) game by introducing a second stage and thereby resolve the multiplicity problem and implement equal division of the dollar in equilibrium. In the standard DD game, if the sum of players’ demands is less than or equal to a dollar, each player receives what he demanded; if the sum of demands is greater than a dollar, all players receive zero. We modify this second part, which involves a harsh punishment. In the modified game \((D\!D^{\prime })\), if the demands are incompatible, then players have one more chance. In particular, they play an ultimatum game to avoid the excess. In the two-player version of this game, there is a unique subgame perfect Nash equilibrium in which players demand (and receive) an equal share of the dollar. We also provide an \(n\)-player extension of our mechanism. Finally, the mechanism we propose eliminates not only all pure strategy equilibria involving unequal divisions of the dollar, but also all equilibria where players mix over different demands in the first stage.

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Schelling’s (1960) argument is based on the prominence of 50–50 division. Arguments in Sugden (1986), Young (1993), Skyrms (1996), and Bolton (1997) are based on evolutionary accounts.

Equal treatment of equals: any two players with equal demands should receive the same amount. Efficiency: if the sum of demands exceeds the estate, the estate should be completely distributed. Order preservation of awards: a player with a higher demand should not receive a lower amount than a player with a lower demand.

Claims monotonicity: a player should not receive a lower amount after increasing his demand. Non-bossiness: a change in a player’s demand should not be able to change other players’ payoffs, if this change does not influence his own payoff.

Harsanyi (1977), Howard (1992), and Miyagawa (2002) are some other studies that implement the Nash bargaining solution in subgame perfect equilibrium with sequential game forms.

We thank an anonymous reviewer for bringing Malueg (2010) to our attention and suggesting an alternative proof, which also considers mixed strategies (Theorem 2).

To keep the paper simple, reader-friendly and comparable to Brams and Taylor (1994) and Anbarcı (2001), we intentionally used actions as the working horse instead of strategies.

The following arguments—explicitly or implicity—use the fact that sequential rationality implies acceptance of any offer in the second stage.

The subcase, \(d_{2}<1/2\), is not analyzed here since it is trivial and analyzed in Theorem 1.

We assume that players mix over only the Borel measurable subsets of \([0,1]\) . Also note that since players utility functions are linear they are Borel measurable.

Note that it is sufficient to check the best response of a player against a mixing opponent.

For simplicity, we assume that \(F\) is an absolutely continuous function, which implies that there exists a Lebesgue-integrable function \(f\) equal to the derivative of \(F\) almost everywhere. Moreover, this \(f\) is called the density function. Note that, alternatively \(F\) can be assumed to be continuous and have a derivative almost everywhere, which would then imply that there exists a Henstock–Kurzweil integrable \(f\) (Bartle 2001, Theorem 4.7).

Note that when \(n=2\), it is not necessary to assume \(x_{i}\in [0,d_{i}]\) since \(x_{i}>d_{i}\) is not possible.

As in Theorem 2, we assume that players mix over only the Borel measurable subsets of \([0,1]\).

As in Theorem 2, for simplicity, we assume that \(F\) is an absolutely continuous function.

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Title: Implementing equal division with an ultimatum threat
Authors: Esat Doruk Cetemen
Emin Karagözoğlu
Publication date: 01-08-2014
Publisher: Springer US
Published in: Theory and Decision / Issue 2/2014
Print ISSN: 0040-5833
Electronic ISSN: 1573-7187
DOI: https://doi.org/10.1007/s11238-013-9394-z

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