Meanwhile the landlord down the street that was able to order his own credit, criminal, and eviction reports on the spot swoops up all the good tenants.  

@Justice Cannieis the 401k with the current or past employer?

If it is a reasonable rent, I would have the guy stay, as long as he passed the background (criminal/credit/eviction) check.  

A couple (with three children) would like to move in, but the wife stated that she would be the primary applicant and would not provide any information other than income for her husband (no background/criminal check or credit score).

Could the Decline recommendation be based on criminal or eviction history? 

If your state has online criminal records, I'd look him up first to make sure you're not dealing with a violent convicted felon or anything.

@Kevin Perk , we left our prior screening because it was only credit, and the new one is nationwide criminal plus credit for a lower price.

Was he screened prior to move in for criminal history?

But I don't know if any of those screening companies actually check with previous landlords either - I suspect they only pull credit and criminal and verify employment.The best approach, depending on your # of units and time available, might be a hybrid - do the quick and easy checks yourself, and then only if someone passes those, turn it over to a screening company to complete the rest.For example, in my state it's free to do a criminal check - one of the few things my state gets right btw :) So I'm usually checking that within a half hour of getting the application.

They might be afraid if it's a scam or criminal.