We show how the fixed point problem for a special type of correspondence R which satisfies a factorisation property can be handled by considering an associated more simple fixed point problem for a correspondence B with domain typically a subset of R. In addition we analyse the fixed point problem for B under additional conditions on R that guarantee that B is at most singleton-valued. In fact we generalize, improve and make more conceptual a game theoretic technique developed by Selten and Szidarovszky.