# Deterministic, Strategyproof, and Fair Cake Cutting

## TL;DR

This paper investigates the limitations of deterministic, strategyproof, and fair mechanisms in cake cutting, revealing several impossibility results and proposing improvements to existing algorithms under specific conditions.

## Contribution

It provides new impossibility results for strategyproof and approximately proportional cake cutting mechanisms and offers a modified algorithm with better incentive properties for small agent groups.

## Key findings

- No strategyproof, approximately proportional, non-wasteful mechanisms for piecewise constant valuations.
- Contiguous allocations cannot be strategyproof and approximately proportional.
- Modified Even-Paz algorithm improves incentive compatibility for two or three agents.

## Abstract

We study the classic cake cutting problem from a mechanism design perspective, in particular focusing on deterministic mechanisms that are strategyproof and fair. We begin by looking at mechanisms that are non-wasteful and primarily show that for even the restricted class of piecewise constant valuations there exists no direct-revelation mechanism that is strategyproof and even approximately proportional. Subsequently, we remove the non-wasteful constraint and show another impossibility result stating that there is no strategyproof and approximately proportional direct-revelation mechanism that outputs contiguous allocations, again, for even the restricted class of piecewise constant valuations. In addition to the above results, we also present some negative results when considering an approximate notion of strategyproofness, show a connection between direct-revelation mechanisms and mechanisms in the Robertson-Webb model when agents have piecewise constant valuations, and finally also present a (minor) modification to the well-known Even-Paz algorithm that has better incentive-compatible properties for the cases when there are two or three agents.