they remained quiet, the piece they receive is larger in their eyes than Left, so they also don't envy the shouter. The shouter receives Left, which is equal to the piece they could receive by remaining silent and larger than the third piece, hence the shouter does not envy any of the quieters. Following this strategy each person gets the largest or one of the largest pieces by their own valuation and therefore the division is envy-free. The same analysis shows that the division is envy-free even in the somewhat degenerate case when there are two shouters, and the leftmost piece is given to any of them. Dividing a 'bad' cake The moving-knives procedure can be adapted for chore division - dividing a cake with a negative value. See also The Fair pie-cutting procedure provides a simpler solution to the same problem, using only 3 rotating knives, when the cake is a 1-dimensional circle ("pie"), The Robertson–Webb rotating-knife procedure provides an even simpler solution, using only 1 rotating knife, when the cake is 2-dimensional. Moving-knife procedure References Fair division protocols Cake-cutting