Sure, that's possible. But now you're asking customers to make a buying decision under pressure, for something they'll only be able to use within a narrow time window. Although this is still economically rational in many cases, it seems to me that the whole point of Groupon is that it promotes impulsive and slightly irrational shopping - a deal that's 'too good to pass up,' even though you may not really want or need the product or service that's on sale. If you attach too many conditions to said deal, then the utility of the purchase itself is subject to the same analysis.

It's a cool idea from an engineering standpoint, but I'm not sure about whether it has the fun factor.

People make buying decisions under pressure when deep discounts are involved (auctions). And I don't think it is as hard to do as you seem think it is. Especially once a customer is familiar with the process.