Envy-free division in the presence of a dragon

The following problem comes from mathematical economics and is solved via methods of equivariant topology and combinatorial geometry. $N$ friends wish to divide a cake. Each of them has his individual preferences (one likes whipped cream roses, one chooses the biggest tile, the other one wishes to lose weight and takes the smallest tile). The cake should be cut into $N$ tiles, and the tiles should be allocated to the friends in such a way that none of them envies the others. Is this always possible? There are two scenarios with a dragon: (1) once the cake is cut, a dragon comes ad unpredictably takes one of the tiles, (2) once the cake is cut, a dragon comes ad unpredictably swallows one of the friends.