A compositional game to fairly divide homogeneous cake

TL;DR

This paper introduces a new game-theoretic model for fairly dividing a homogeneous cake among multiple players, achieving a Nash equilibrium with contiguous pieces and linear complexity, addressing fairness issues in traditional cake-cutting mechanisms.

Contribution

The paper proposes a novel compositional game for homogeneous cake division that guarantees fairness, contiguous pieces, and linear complexity, improving upon naive methods.

Findings

Nash equilibrium with equal division and contiguous pieces.

Naive composition leads to exponential unfairness and high Price of Anarchy.

The Biggest Player rule achieves decentralized fairness with linear complexity.

Abstract

The central question in the game theory of cake-cutting is how to fairly distribute a finite resource among multiple players. Most research has focused on how to do this for a heterogeneous cake in a situation where the players do not have access to each other's valuation function, but I argue that even sharing homogeneous cake can have interesting mechanism design. Here, I introduce a new game, based on the compositional structure of iterated cake-cutting, that in the case of a homogeneous cake has a Nash equilibrium where each of $n$ players gets $1/n$ of the cake. Furthermore, the equilibrium distribution is the result of just $n-1$ cuts, so each player gets a contiguous piece of cake. Naive composition of the `I cut you choose' rule leads to an exponentially unfair cake distribution with a Gini-coefficient that approaches 1, and suffers from a high Price of Anarchy. This cost is completely eliminated by the proposed \textit{Biggest Player} rule for composition which achieves decentralised and asynchronous fairness at linear Robertson-Webb complexity. After introducing the game, proving the fairness of the equilibrium, and analysing the incentive structure, the game is implemented in Haskell and the Open Game engine to make the compositional structure explicit.