First, I shall formally define a White Elephant Gift Exchange.
The game is made up of n players, who each bring a gift with a value v1, v2, ... vn-1, vn between 0 and 1, inclusive, distributed randomly uniformly.
Players go in an arbitrary order chosen before the game begins. Each present begins wrapped, which conceals its value. Each player knows the value of his or her gift, or more formally, Player i knows the value of vi for all 1 ≤ i ≤ n.
When it is a player's turn they must do one of the following:
- Open one of the still-wrapped presents, publicly revealing its value to all players. It then becomes the turn of the next player in line.
- Steal an opened present (note the rule below) from another player. It then becomes that player's turn, but they may not steal from the player who stole from them immediately before.
Each present can be stolen k times. After that point, it is considered "safe" and may no longer be stolen by any player.
When all presents have been unwrapped, the game is over and players may not steal presents any more.
So the question then becomes "After the game begins, what is the best strategy for maximizing the expected value of your gift at the end of the game? Which position in line has the highest expected gift value if all players use the optimal strategy?"
A side question would be "If each player was able to choose the value of the gift they bring before the game begins, what is the optimal value of gift to bring to the exchange?" I suspect the optimal value of the gift you bring is 0, but that doesn't make for a very interesting game. I suppose it could be a Nash Equilibrium, but I doubt it.