  Game theory is central to the understanding of competitive interactions
arising in many fields, from the social and physical sciences to economics.
Recently, as the definition of information is generalized to include entangled
quantum systems, quantum game theory has emerged as a framework for
understanding the competitive flow of quantum information. Up till now only
two-player quantum games have been demonstrated. Here we report the first
experiment that implements a four-player quantum Minority game over tunable
four-partite entangled states encoded in the polarization of single photons.
Experimental application of appropriate quantum player strategies give
equilibrium payoff values well above those achievable in the classical game.
These results are in excellent quantitative agreement with our theoretical
analysis of the symmetric Pareto optimal strategies. Our result demonstrate for
the first time how non-trivial equilibria can arise in a competitive situation
involving quantum agents and pave the way for a range of quantum transaction
applications.
