Abstract
We add a stage to Nash’s demand game by allowing the greedier player to revise his demand if the demands are not jointly feasible. If he decides to stick to his initial demand, then the game ends and no one receives anything. If he decides to revise it down to \(1-x\), where x is his initial demand, the revised demand is implemented with certainty. The implementation probability changes linearly between these two extreme cases. We derive a condition on the feasible set under which the two-stage game has a unique subgame perfect equilibrium. In this equilibrium, there is first-stage agreement on the egalitarian demands. We also study two n-player versions of the game. In either version, if the underlying bargaining problem is “divide-the-dollar,” then equal division is sustainable in a subgame perfect equilibrium if and only if the number of players is at most four.
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Notes
An earlier result by Binmore (1987) shows that Nash’s suggestion is valid for a certain class of parametrized perturbations. An alternative approach to smoothing is to apply perturbations not to the feasible set, but to the players’ strategies. This possibility has been explored by Carlsson (1991), who added a noise component to the players’ demands.
Anbarcı (2001), Ashlagi et al. (2012), and Rachmilevitch (2017) are some other papers, which modified the “punishment clause” in DD to tackle the drawbacks mentioned above, and consequently obtained equal division of the surplus in equilibrium. Multi-stage extensions of NDG have been studied by Howard (1992) and by Anbarcı and Boyd III (2011).
There is no shortage of results in the literature that show that it matters whether the number of players is equal to or greater than 2. For example, Brams and Taylor (1994) show that in their version of DD that we described above, the equal division is dominance inducible if and only if \(n=2\).
A player’s maximal payoff in a bargaining problem is called his ideal payoff.
Rubinstein et al. (1992) study a sequential bargaining game which is similar to NDG, in which the second mover gets to chose a probability p that governs how play evolves from the third stage of the game onwards. Our probability \(\lambda \) is similar to the aforementioned p, in the sense that it is determined by one of the players.
\(\pi \)’s domain is \((\frac{1}{2},1]\): note that the combination of \(x\le \frac{1}{2}\) and \(y<x\) implies that \((x,y)\in \Delta _2\subset S\).
Note that \(\psi _S^2(\frac{x}{2})\ge 1-\frac{x}{2}\).
The above inequality is equivalent to \(\psi _S^i(\frac{x}{2})>\frac{x}{2}\); the latter holds, since \(\psi _S^i(\frac{x}{2})\ge 1-\frac{x}{2}\) and \(x\in (0,1)\).
The definition of a bargaining problem in the n-player case is a straightforward analog of the 2-player definition. Hence, for brevity, we do not repeat the details.
The reason for this can be seen in footnote 13.
More precisely: if there is a profitable deviation, then the deviation to the vector where the player asks the maximal payoff for himself and offers zero to any other player is profitable deviation; conditional on this deviation, the deviator’s payoff is independent of n.
It is easy to check that, as opposed to equal division, the first-stage demands where each player demands the entire dollar for himself are consistent with a subgame perfect equilibrium of either \(G_n^{Min}(\Delta _n)\) or \(G_n^{Av}(\Delta _n)\), for any n.
This argument relies on the specific DD structure.
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Acknowledgements
The authors thank an anonymous referee for a helpful report. Emin Karagözoğlu thanks TÜBİTAK (The Scientific and Technological Research Council of Turkey) for the post-doctoral research fellowship. Usual disclaimers apply.
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Karagözoğlu, E., Rachmilevitch, S. Implementing egalitarianism in a class of Nash demand games. Theory Decis 85, 495–508 (2018). https://doi.org/10.1007/s11238-018-9656-x
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DOI: https://doi.org/10.1007/s11238-018-9656-x