|InterJournal Complex Systems, 2029
|Manuscript Number: |
Submission Date: 20080226
|Quantum Computing and The Nash Bargaining Problem|
At ICCS, 2006, we discussed the schema proposed by Bernardo Huberman and Tadd Hogg (“Quantum Solution of Coordination Problems”, http://arxiv.org/PS_cache/quant-ph/pdf/0306/0306112v1.pdf). We characterized this schema as a “Realizable Quantum Nash Equilibrium” an important distinction between their proposed mechanism applying quantum entanglement to actual games and the more theoretical, general treatment of quantum Nash equilibria as discussed by Meyer et al., or the general formulation of the quantum game form discussed by Landsberg (“Quantum Game Theory”, Notices of the American Mathematical Society, Volume 51, No. 4, April, 2004). The most obvious application of Huberman and Hogg’s mechanism was to generate a quantum Nash equilibrium in a simple game (one much studied in chaos and complexity theory for its deceptively complex dynamics) of rock-paper-scissors, producing a new quantum Nash equilibrium of .66, or twice the value of the classical Nash equilibrium. The more general solution is described in Cheon and Tsutsui, who map the new values of quantum Nash equilibria into a projective plane perpendicular to the traditional solution basin (“Game Theory Formulated on Hilbert Space, http://arxiv.org/PS_cache/quant-ph/pdf/0605/0605134v1.pdf). However, that analysis contains an implicit assumption about the distribution of the payoffs between the coordinated players which is, in fact, deserving of a more extended, explicit analysis. and a treatment of this new type of equilibrium as a Nash bargaining problem. In this paper we draw on John Nash’s work “The Bargaining Problem” (Econometrica, Vol. 18, No. 2, April, 1950) to extend the analysis of Huberman and Hogg’s quantum coordination schema to Nash bargaining problems.
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