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Seems like after you have the solution for 2 and 3 people, the N-solution is just to split up in teams. Say you have 8 people, split in to two teams of 4. Team 1 (With 4 people) now represents "Person A" in the 2 person scenario, so Team 1 gets to cut, Team 2 gets to choose. Now Team 1 and Team 2 repeat the process internally for their halves of the pizza, and so forth.

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That's good for most cases, but I don't think it works for groups of prime cardinality. In addition, the problem often allows each participant to have a different value function (e.g., one person wants lots of pepperoni but the others prefer mushrooms). In such a variant, a team might have problems agreeing on which piece is most desirable.

Four person game: A1 says to B1 and B2, "Hey, give me a bigger piece and I'll make the cut in your favor". This doesn't solve collusion.

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