nach oben

Theory and Decision

Erschienen in:

17.02.2018

Implementing egalitarianism in a class of Nash demand games

verfasst von: Emin Karagözoğlu, Shiran Rachmilevitch

Erschienen in: Theory and Decision | Ausgabe 3-4/2018

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

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.

Vorheriger Artikel Dictatorship on top-circular domains

Nächster Artikel Why do young women marry old men?

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

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.

Abreu, D., & Pearce, D. (2015). A dynamic reinterpretation of Nash bargaining with endogenous threats. Econometrica, 83, 1641–1655.CrossRef

Anbarcı, N., & Boyd, J. H. (2011). Nash demand game and the Kalai–Smorodinsky solution. Games and Economic Behavior, 71, 14–22.CrossRef

Anbarcı, N. (2001). Divide-the-dollar game revisited. Theory and Decision, 50, 295–304.CrossRef

Ashlagi, I., Karagözoğlu, E., & Klaus, B. (2012). A non-cooperative support for equal division in estate division problems. Mathematical Social Sciences, 63, 228–233.CrossRef

Binmore, K. (1987). Nash bargaining theory (II). In K. Binmore & P. Dasgupta (Eds.), The economics of bargaining. Cambridge: Basil Blackwell.

Brams, S. J., & Taylor, A. D. (1994). Divide the dollar: Three solutions and extensions. Theory and Decision, 37, 211–231.CrossRef

Carlsson, H. (1991). A bargaining model where parties make errors. Econometrica, 59, 1487–1496.CrossRef

Cetemen, E. D., & Karagözoğlu, E. (2014). Implementing equal division with an ultimatum threat. Theory and Decision, 77, 223–236.CrossRef

Howard, J. V. (1992). A social choice rule and its implementation in perfect equilibrium. Journal of Economic Theory, 56, 142–159.CrossRef

Myerson, R. B. (1978). Refinements of the Nash equilibrium concept. International Journal of Game Theory, 7, 73–80.CrossRef

Nash, J. F. (1953). Two person cooperative games. Econometrica, 21, 128–140.CrossRef

Rachmilevitch, S. (2017). Punishing greediness in divide-the-dollar games. Theory and Decision, 82, 341–351.CrossRef

Rubinstein, A., Safra, Z., & Thomson, W. (1992). On the interpretation of the Nash bargaining solution and its extension to non-expected utility preferences. Econometrica, 60, 1171–1186.CrossRef

Titel: Implementing egalitarianism in a class of Nash demand games
verfasst von: Emin Karagözoğlu
Shiran Rachmilevitch
Publikationsdatum: 17.02.2018
Verlag: Springer US
Erschienen in: Theory and Decision / Ausgabe 3-4/2018
Print ISSN: 0040-5833
Elektronische ISSN: 1573-7187
DOI: https://doi.org/10.1007/s11238-018-9656-x

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Weitere Artikel der Ausgabe 3-4/2018

Moral judgments, gender, and antisocial preferences: an experimental study

Stable coalition structures in symmetric majority games: a coincidence between myopia and farsightedness

An exploration of third parties’ preference for compensation over punishment: six experimental demonstrations

Ambiguous life expectancy and the demand for annuities

Dictatorship on top-circular domains

The strength of sensitivity to ambiguity