Nope

If the "prize" was from gambling, yes, the losses would be deductible on Schedule A.

Otherwise, no.

But the fact the business that issued the 1099 calls it a "prize" makes it very questionable if it was gambling.

I had a client who spent hundreds in postage every year entering sweepstakes.  He would occasionally win a toaster.  Oh, maybe that was for opening a bank account. I once had a client who had 11 IRA's and 11 toasters.  But that's not the question.  We deducted the postage as a miscellaneous deduction.  Then they limited it with the 2% haircut.  Then they suspended it altogether.  

But what you have is not gambling, or it would have been reported on a W-2G.

I have made an offset to the "suggested retail price" of a prize won. e.g. Refrigerator with a MFSRP of $1,000 that I can produce an ad showing it on sale at $800, then I report the $1,000 with another "other income" of "adjustment to 1099-MISC" actual value of a -$200.