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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Nash equilibria in quantum games


Author: Steven E. Landsburg
Journal: Proc. Amer. Math. Soc. 139 (2011), 4423-4434
MSC (2010): Primary 91A05, 81P45
Published electronically: April 19, 2011
MathSciNet review: 2823088
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Abstract: For any two-by-two game $ \mathbf{G}$, we define a new two-player game $ \mathbf{G}^Q$. The definition is motivated by a vision of players in game $ \mathbf{G}$ communicating via quantum technology according to the protocol introduced by J. Eisert and M. Wilkins.

In the game $ \mathbf{G}^Q$, each player's (mixed) strategy set consists of the set of all probability distributions on the 3-sphere $ \mathbf{S}^3$. Nash equilibria in the game can be difficult to compute.

Our main theorems classify all possible mixed-strategy equilibria. First, we show that up to a suitable definition of equivalence, any strategy that arises in equilibrium is supported on at most four points; then we show that those four points must lie in one of a small number of allowable geometric configurations.


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Additional Information

Steven E. Landsburg
Affiliation: Department of Economics, University of Rochester, Rochester, New York 14627

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-10838-4
PII: S 0002-9939(2011)10838-4
Received by editor(s): October 24, 2009
Received by editor(s) in revised form: October 17, 2010
Published electronically: April 19, 2011
Communicated by: Richard C. Bradley
Article copyright: © Copyright 2011 Steven E. Landsburg