I think that people run White Elephant parties differently. I explain how the one I attend works.
LOL
Let's say GaySpeak is having a White Elephant. Myself, Bowyn, Jay, Airborn, Wakeboarder, SacredCake and Almac are all attending.
We each go through our house and find something we no longer want. The uglier and tackier the better. Broken is ok too! LOL You wrap the item up as if it was new, and bring it to the party. Numbers are drawn, whoever gets #1 goes first. We'll say that Airborn is #1, and she takes a gift and opens it. I'm #2. I can choose a gift from the pile, or "steal" Airborn's gift. If I steal Airborn's opened gift, she then picks a second gift and opens it. It continues on until all the gifts are opened, and then because Airborn went first and didn't have the opportunity to "steal", she then can review all the opened gifts and opt to keep hers, or make one last steal.
LOL
It's actually very funny, many laughs happen over how tacky/ugly the gifts are, and then when everyone leaves and gets home, they generally throw out the gift they got, and no one feels badly about it. LOL
Last year, my gift that I wrapped as new was two identical Cowboy Mermen Christmas ornaments.
LOL