I would imagine that it depends on the amount of the credit being offered, and thus their potential risk.
If the principal is low, they may do no, or one, credit pull, but as the principal, and thus the risk, rises, they are probably more apt to want to check for things reported to one CRA that might not have been reported to another. They pay fees to pull.
As for whether they do a soft or hard pull, unless their inquiry is related only to a promotional offer for credit that you did not initiate, they get the same information with an SP as with an HP. So as for benefit to them, it doesnt matter.
The real issue, in my opinion, between soft and hard inquiries is what is the difference in what is required by the creditor to get an inquiry coded as soft?
If their inquiry is actually one that qualifies as a hard pull (i.e., you initiated the request for new or additional credit), how do they get it coded in a manner that precludes its appearance in a consumer's credit report? Do they provide a fake reason for the pull, or are their codes that say "for consumer initiated request for credit, but dont post for others to see"? What is their incentive to do that, when either gives them your full CR?
I have never seen an adequate explanation of how credtiors get hard pull items coded as soft pulls, so the answer to that is a mystery to me.
