Incentive-Compatible Division Procedures?

So, most people should be familiar with the fail-safe way for 2 people to divide an object:

- The first person cuts the object into 2 pieces

- The second person chooses the object they wish to take first.

Since the second person is going to choose the largest available piece, the first person has a strong incentive to make an even cut, as if it is un-even, they will get less of the final product.

So, I got around to thinking about this problem in the case of N people. Here is a procedure that I think is incentive-compatible for 3 people:

- The first person makes 2 pieces, designating one as a 'small' piece and one as a 'large' piece

- The second person gets 2 choices. They may:

- Take the 'small' piece for themselves. The large piece is then divided using the 2-person division procedure between the first and third person

- Give the 'small' piece to the first person, and split the large piece with the 3rd person - using the same procedure used for 2 people.

So, here's a bunch of somewhat open-ended questions.

- Does my procedure for 3 people work? Can you prove it, preferably somewhat elegantly? If it does not work, show how it can be broken.

- The procedure for 3 people build recursively on the 2 person procedure. If this procedure works... can we construct a general procedure for N-people that also works recursively?

What do you all think?