Answer: Some compare this to owning a vending machine, which many poskim (Shemiras Shabbos Kehilchasa 20:70) allow one to keep running over Shabbos. As one hasn’t predetermined the sale, they aren’t considered to be actively doing a melacha. One doing so should declare that the money earned will not be acquired until after Shabbos.
R’ Akiva Eiger forbids one from pre-arranging an acquisition to take place on Shabbos, however. Accordingly, bidding for an item on eBay that is due to end on Shabbos may be prohibited.
In truth, the issue is far more complicated, as while winning a bid commits one to buy, no transaction has taken place until payment has been received. Some have dismissed the possible prohibition arguing that the time of transaction is not until the credit card payment has cleared. That poses other problems, as according to that argument, one would be forbidden to shop on Thursday, if they knew the transaction won’t clear until Shabbos! Websites that are set up to automatically send a software download or coupon, etc. would be particularly problematic.
Certainly, this is a most complex matter that needs to be investigated further by today’s poskim . (Perhaps they will need to create innovative solutions similar to Heter Iska.)