О делении без зависти в присутствии дракона

Доклад по совместной работе с Р. Живалевичем https://arxiv.org/abs/2112.12969.
We prove several results addressing the envy-free division problem in the presence of an unpredictable (secretive) player, called the "dragon". There are two basic scenarios.
1. There are r−1 players and a dragon. Once the "cake" is divided into r parts, the dragon makes his choice and grabs one of the pieces. After that the players want to divide the remaining pieces in an envy-free fashion.
2. There are r+1 players who divide the cake into r pieces. A ferocious dragon comes and swallows one of the players. The players want to cut the cake in advance in such a way that no matter who is the unlucky player swallowed by the dragon, the remaining players can share the tiles in an envy-free manner.
In both settings the players are allowed to choose degenerate pieces of the cake. Moreover, they construct in advance both a cut of the cake and a "decision tree", allowing them to minimize the uncertainty of what pieces can be given to each of the players.