Homework Policy - Extension with Partial Solutions for Partial Credit
I give weekly assignments and post the full solutions immediately after the assignment is due.  However, I also give partial credit (50%) to any assignments turned in after its due date but before the final exam.  

I have seen a handful of students utilize this system.  Typically they are the ones who normally complete assignments but missed the deadline due to extraneous circumstances.  I consider these use-cases to be exactly in the spirit of my policy.  Occasionally, I will have a student who submits virtually all of the assignments in the last week of the term.  I see this as an unintended consequence of the policy (I would actually prefer to discourage this kind of procrastination).

There are a few downsides from an instructor's point of view:


There is a higher organizational load than if you enforce a strict deadline.  I find the marginal cost of grading a late assignment to be surprisingly high.  If assignments came any more frequently than weekly, I am confident my system would become onerous, and the one you propose would be even worse.
There is a risk that an honest (but poor) on-time effort by a student could be surpassed by a dishonest late effort by another student.  I haven't seen this but it is a worry in the back of my mind.  I try to avoid this by actively encouraging students who do poorly (worse than 50%) to re-submit after looking at the solutions.