Many Players
Fair division with three or more players is considerably more complex than the two player case.
Proportional division is the easiest and the article describes some procedures which can be applied with any number of players. Finding the minimum number of cuts needed is an interesting mathematical problem.
Envy-free division was first solved for the 3 player case in 1960 independently by John Selfridge of Northern Illinois University and John Horton Conway at Cambridge University. The best algorithm uses at most 5 cuts.
The Brams-Taylor procedure was the first cake-cutting procedure for four or more players that produced an envy-free division of cake for any number of persons and was published by Steven Brams and Alan Taylor in 1995. This number of cuts that might be required by this procedure is unbounded. A bounded moving knife procedure for 4 players was found in 1997.
There are no discrete algorithms for an exact division even for two players, a moving knife procedure is the best that can be done. There are no exact division algorithms for 3 or more players but there are 'near exact' algorithms which are also envy-free and can achieve any desired degree of accuracy.
A generalization of the surplus procedure called the equitable procedure (EP) achieves a form of equitability. Equitability and envy-freeness can be incompatible for 3 or more players.
Read more about this topic: Fair Division
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