Packing Rectangles: A Cake Sharing Puzzle

Abstract

In the quest to solve a mathematical conundrum based on a two-player cake allocation game, apparently unsolved since 2004, this paper presents an in-depth analysis of each player’s challenges and opportunities.

At the beginning of the research stands a conjecture about the subgame perfect Nash equilibrium, whose implications are discussed in some depth. Although a full proof cannot be presented here, the work establishes several equilibrium properties and lies the foundation for an algorithmic implementation of the player’s task. Understanding of the game dynamics is further fuelled by the analysis of several game variants, obtained either by slightly altering the game’s objective or by restricting the action space of the opponents. While two of these variants may be solved completely, the third version may be translated into a lower bound for the player’s minimal return.