Envy-free division in the presence of a dragon

TL;DR

This paper investigates envy-free division problems involving an unpredictable 'dragon' entity, providing solutions for scenarios where the dragon's choice or presence affects fair division among players.

Contribution

It introduces novel models for envy-free division with a secretive or unpredictable participant, including decision tree strategies to handle uncertainty.

Findings

Established existence of envy-free divisions with a secretive player.

Developed algorithms for pre-structured cuts and decision trees.

Proved robustness of envy-free division under unpredictable removal or choice.

Abstract

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.