Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [CHTW] R. Cleve, P. Hoyer, B. Toner and J. Watrous, ``Consequences and Limits of Nonlocal Strategies'', Proc. of the 19th Annual Conference on Computational Complexity (2004), 236-249.
  • [DL] G. Dahl and S. Landsburg, ``Quantum Strategies'', preprint, available at http://www. landsburg.com/dahlcurrent.pdf.
  • [EW] Jens Eisert and Martin Wilkens, Quantum games, J. Modern Opt. 47 (2000), no. 14-15, 2543–2556. Seminar on Fundamentals of Quantum Optics, V (Kühtai, 2000). MR 1801549, 10.1080/095003400750039537
  • [EWL] Jens Eisert, Martin Wilkens, and Maciej Lewenstein, Quantum games and quantum strategies, Phys. Rev. Lett. 83 (1999), no. 15, 3077–3080. MR 1720182, 10.1103/PhysRevLett.83.3077
  • [I] S. Landsburg, ``Intertwining'', working paper available at http://www.landsburg.com/ twining4.pdf.
  • [NE] S. Landsburg, ``Nash Equilibrium in Quantum Games'', RCER Working Paper #524, Rochester Center for Economic Research, February 2006.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 91A05, 81P45

Retrieve articles in all journals with MSC (2010): 91A05, 81P45


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
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