What is the term for the arbitrary rule that automatically places the children of a union between members of different socioeconomic groups in the less-privileged group?

The correct term is hypodescent.

Explanation:

The anthropological term hypodescent describes a system where the people of mixed races are automatically put into the category that is considered to be socially inferior.  This is not a system that acknowledges mixed race categories or degrees of racial mixing. In this notion, any racial mixing dilutes the purity of the racial categories. It is a system that tends to discourage racial mixing and intermarriage. We have the example of this kind of system in American history, which is the notion of the one-drop rule. In this framework, one drop of black ancestry made you black by the dominant social standards. This over simplifies the notions of race and it is a very discriminatory part of our history that has perpetrated systemic racism.  

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