I-Cut-You-Choose with more than 2 players

I stumbled across this video today (someone posted it in the new BGDL+ forums) talking about I-Cut-You-Choose with more than 2 players. It's not the process I'm using in my newest design, an I-Cut-You-Choose Worker Placement game, but I thought it would be really good to keep track of it in case I'd like to review it later:

The basic idea being that P1 splits something into 3 groups, P2 decides one thing they DON'T want, and trims one of the other two so they're "equal," then P3 chooses one thing... P2 gets the thing they trimmed (if it's still available), or the other, "equivalent" thing (if it's not), and P1 gets the remaining thing, and everyone should be happy... then they similarly divide up the trimmed piece.

This process would work for any number of players, but it looks like it gets pretty cumbersome pretty quickly as you add people. A pragmatic solution might be to discard the "trimmed" bit, or do something else with it (add it to the next group to be split?) - then it still should be the case that each player is happy with what they get, and it doesn't get too iterative.

For example, P1 takes 13 cube and splits them into 3 piles of 3, 4, and 6 cubes. Note that while these aren't even with respect to total number of cubes, they might be even (or close to even) with respect to total value of cubes. Then, P2 decides they don't want the 3-cube pile, and trims 1 cube off the 6-cube pile (making it a 5-cube pile instead) -- the trimmed cube is discarded, or maybe used later in some other part of the game. P3 get to pick any of the three piles, maybe they choose the 3-cube pile, because those specific cubes are really good for their position. Then P2 gets the 5-cube pile that they had trimmed, and P1 gets the remaining 4-cube pile.

P1 should be happy, because they (theoretically) split the cubes such that they'd like any pile, P2 should be happy, because they (again, theoretically) trimmed the 6-cube pile so they'd be equally happy with the 4-cube or the 5-cube pile, and P3 should be happy, because they got 1st pick of all the piles.

I'll note that "everyone being happy" is the goal when splitting a cake, but in a game there might be more of an aspect of "getting away with something." If the cubes have different (non-obvious) values to different players, then things like P3 in that example taking the 3-cube pile over the other two are reasonable, and there's an opportunity to make piles that are not so much "even" -- tempt someone into leaving you what you want by making a pile that's attractive to them! That's the crux of the ICYC mechanism in games, I think.