The Gift Exchange Problem

TL;DR

This paper analyzes the gift exchange problem, deriving formulas and recurrences to count the number of game play sequences given the number of players, gifts, and steal limits.

Contribution

It provides new mathematical formulas and recurrences for counting the possible game sequences in the gift exchange problem.

Findings

Derived explicit formulas for the number of game sequences.

Established recurrence relations for different parameters.

Identified open questions for further research.

Abstract

The aim of this paper is to solve the "gift exchange" problem: you are one of n players, and there are n wrapped gifts on display; when your turn comes, you can either choose any of the remaining wrapped gifts, or you can "steal" a gift from someone who has already unwrapped it, subject to the restriction that no gift can be stolen more than a total of S times. The problem is to determine the number of ways that the game can be played out, for given values of S and n. Several recurrences and explicit formulas are given for these numbers, although some open questions remain.