Quick Answer: Every player brings a wrapped gift to a central pile. Players draw random numbers to determine turn order. On your turn, you can either open a new wrapped gift from the pile, OR steal an unwrapped gift from a previous player. A gift is "dead" (safe) after it has been stolen 3 times.
How to Determine Turn Order
The entire game hinges on turn order. Going first is terrible; going last is incredibly powerful. Use a digital Random Number Generator (1 to N players) or a custom spinner to strictly enforce a random, mathematically fair distribution of turns.
The Standard "3 Steal" Rule
Without limits, a highly desirable gift (like an espresso maker) will be stolen back and forth for three hours, ruining the party. You MUST implement the 3-Steal Cap. Once a specific item is stolen from someone three times, it freezes. The third thief owns it permanently.
Mathematical Strategy
If you draw a number in the early/middle of the game, DO NOT STEAL. Open a new gift. Stealing early leaves more mystery gifts in the pile for late-game players to snatch. Your best mathematical move is to open a new gift, hope someone steals it from you late in the game, which then triggers your ability to counter-steal when most of the board is revealed.