Tight Lower Bounds for Unequal Division

TL;DR

This paper establishes tight lower bounds on the number of cuts needed for unfair cake division in ratio (a:b), showing that the minimal cuts grow at least as fast as the double logarithm of (a+b).

Contribution

It proves new lower bounds matching known upper bounds for the number of cuts in unfair division, advancing understanding of the problem's complexity.

Findings

Lower bound: f(a,b) ≥ lg(lg(a+b)) for all a,b.

Upper bound: f(a,b) ≤ 1 + lg(lg(a+b)) for infinitely many (a,b).

Bounds are tight up to an additive constant.

Abstract

Alice and Bob want to cut a cake; however, in contrast to the usual problems of fair division, they want to cut it unfairly. More precisely, they want to cut it in ratio $(a:b)$. (We can assume gcd(a,b)=1.) Let f(a,b) be the number of cuts will this take (assuming both act in their own self interest). It is known that f(a,b) \le \ceil{lg(a+b)}. We show that (1) for all a,b, f(a,b) \ge lg(lg(a+b)) + (2) for an infinite number of (a,b), f(a,b) \le 1+lg(lg(a+b).