White Elephant parties and fair division

It's holiday party season, which means that it's time for white elephant parties. For those who don't know what a white elephant party is, here's a description from Wikipedia:

Each participant supplies one wrapped gift. The gifts are placed in a central location, and participants determine in which order they will take turns selecting them. The first person opens a wrapped gift, and the turn ends. On subsequent turns, each person can open a new present or gets the choice to "steal" another person's gift. The gift cannot be stolen once the third participant touches the gift (i.e. - it is stolen for the 2nd time). When a person's gift is stolen, that person can either choose another wrapped gift to open or can steal from another player. The game is over when the last person goes and the first person goes again

I just finished my algorithms class with the "traditional" (n=2) discussion of fair division algorithms (aka cake cutting, but with actual cake). White elephant parties are also a form of fair division with indivisible goods.

A few years ago, I started wondering about the theory behind white elephant exchanges and posted a question on cstheory. The answer I got (from usul) was excellent (tl;dr: worst-case definitions of fairness don't work). But it's also a few years old, and I wonder if there are new results on this topic.