Is there an expression for asking two people to perform a task, but neither do it, assuming that the other will?

"Fell through the cracks"

to not be noticed or dealt with.

• Little details often fall through the cracks.
• Too many young people slip through the cracks in the health system.

[Cambridge English Dictionary]

The almost identical English phrase is falling between two stools which, as the Cambrige dictionary entry under the link shows, is idiomatic and carries an almost identical meaning.

It is used, however, to describe many other situations as well: for example someone who tries to follow two different career paths and achieves success in neither; a business which tries to do two incompatible things and does neither of them well or a film which tries to be both comic and serious but fails to convince on either count.

Given the long history of common development of the French and English languages it is possible that the two idioms have the same origin.

Everybody’s business is nobody’s business Here ‘business’ means ‘duty’ or ‘task’. When nobody is directly responsible for doing a thing, nobody does it, because everybody thinks somebody else is or will be doing it. http://proverbhunter.com/everybodys-business-is-nobodys-business/