Preferences (Partial Pre-Orders) on Complex Numbers -- in View of Possible Use in Quantum Econometrics

Abstract

In economic application, it is desirable to find an optimal solution -- i.e., a solution which is preferable to any other possible solution. Traditionally, the state of an economic system has been described by real-valued quantities such as profit, unemployment level, etc. For such quantities, preferences correspond to natural order between real numbers: all things being equal, the more profit the better, and the smaller unemployment, the better. Lately, it turned out that to adequately describe economic phenomena, it is often convenient to use complex numbers. From this viewpoint, a natural question is: what are possible orders on complex numbers? In this paper, we show that the only possible orders are orders on real numbers.

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