On amending the Maskin's theorem by using complex numbers

The Maskin's theorem is a fundamental work in the theory of mechanism design. A recent work [Wu, Quantum mechanism helps agents combat ``bad'' social choice rules. \emph{Intl. J. of Quantum Information} 9 (2011) 615-623] shows that when an additional condition is satisfied, the Maskin's theorem will not hold if agents use quantum strategies. Inspired by the quantum mechanism, in this paper, we will propose an algorithmic mechanism which uses complex numbers. We show by an example that a Pareto-efficient social choice rule that is not monotonic may be Nash implemented by using the algorithmic mechanism. This result is positive not only to the agents, but also to the designer if the designer wishes to maximize the total social surplus