Abstract
Following a novel approach, where the emphasis is on configuration spaces and equivariant topology, we prove several results addressing the envy-free division problem in the presence of an unpredictable (secretive, non-cooperative) 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 should be able to share 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 need 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. We emphasize that in both settings, some players may prefer to choose degenerate pieces of the cake. Moreover, the players 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.
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