Multiplayer quantum games

Physical Review A - Atomic, Molecular, and Optical Physics

Volumn 64, Issue 3, 2001, Pages 4-

  Benjamin, Simon C ^a   Hayden, Patrick M ^a  

^a UNIVERSITY OF OXFORD   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84861415972     PISSN: 10502947     EISSN: 10941622     Source Type: Journal    
DOI: 10.1103/PhysRevA.64.030301     Document Type: Article

References (26)

1. For an introduction see, e.g., E. Rasmusen, Games and Information (Blackwell, Oxford, 1995).
2. M. A. Nowak and K. Sigmund, Nature (London) 398, 367 (1999).
3. For a review see, e.g., C. H. Bennett and D. P. DiVincenzo, Nature (London) 404, 247 (2000).
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11. G. Collins, Sci. Am. Int. Ed. Jan. 2000.
12. J. Eisert and M. Wilkens, e-print http://xxx.lanl.gov/abs/ quant-ph/0004076.
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14. Without entanglement, the quantized form of Fig. 11(a) remains equivalent to the classical probabilistic game.
15. Adopting the general bit-flip operator (Formula presented) would simply result in a trivial rotation of the features that we discover.
16. Since we are interested in purely multipartite entanglement, we call a (pure) state maximally entangled if it is equivalent via local unitary operations to the GHZ-type state, (Formula presented).
17. Any other (Formula presented) meeting these conditions would be equivalent, via local unitary operations, to our (Formula presented), and would therefore induce equilibria corresponding to ours.
18. J. Du, e-print quant-ph/0104087.
19. S. C. Benjamin and P. M. Hayden, Phys. Rev. Lett. (to be published).
20. If the action is not drawn from (Formula presented) then the resulting state is necessarily not equivalent via local unitary operations to a GHZ state.
21. For a survey of recent work on the classical game, see http://www.unifr.ch/econophysics/minority/.
22. Interestingly, the complexity of the four player game is such that several Nash equilibria occur. Players may use criteria, such as maximum projection into the subspace of ‘classical’ moves (Formula presented), to establish a focal point, but this becomes a psychological question.
23. If we generate a pair of random classical bits, and send duplicates of these bits to each player, then Nash equilibria exist wherein the players use these bits to decide which player should be in the minority.
24. There is a focal symmetric strategy profile in this game, where each player adopts (Formula presented). However this profile is not a Nash equilibrium unless the players are actually constrained to unitary moves. One could presumably construct a similar game wherein this strategy profile does form a full Nash equilibrium—e.g., the game obtained by replacing the payoff columns in Fig. 22(b) by (Formula presented) (Formula presented) (Formula presented) (Formula presented) (Formula presented) (Formula presented) (Formula presented) (Formula presented) (top to bottom) looks promising in this respect.
25. K. Kraus, States, Effects, and Operations: Fundamental Notions of Quantum Theory, Lecture Notes in Physics Vol. 190 (Springer-Verlag, Berlin, 1983).
26. N. F. Johnson, Phys. Rev. A 63, 020302(R) (2001).