Christmas Gift Exchange Games

Abstract

The Christmas gift exchange is a popular party game played around Christmas. Each participant brings a Christmas present to the party, and a random ordering of the participants, according to which they will choose gifts, is announced. When a participant’s turn comes, she can either open a new gift with unknown value, or steal an already opened gift with known value from someone before her in the ordering; in the second case, the person whose gift was stolen gets to make the same choice. A gift cannot be stolen more than once.

We model the gift exchange as a sequential game of perfect information and characterize its equilibria, showing that each player plays a threshold strategy in the subgame perfect equilibrium of the game. We compute the expected utility of players as a function of the position in the random ordering; the first player’s utility is vanishingly small relative to every other player. We then analyze a different version of the game, also played in practice, where the first player is allowed an extra turn after all presents have been opened—we show that all players have equal utility in the equilibrium of this game. Finally, we analyze the equilibria of two variants of this game, one where a gift can be stolen more than once, and another where players have complete information about the value of the gift.