On Hierarchies of Fairness Notions in Cake Cutting: From Proportionality to Super Envy-Freeness

We consider the classic cake-cutting problem of producing fair allocations for $n$ agents, in the Robertson–Webb query model. In this model, it is known that: (i) proportional allocations can be computed using $O(n \log n)$ queries, and this is optimal for deterministic protocols; (ii) envy-free allocations (a subset of proportional allocations) can be computed using $O\left( n^{n^{n^{n^{n^{n}}}}} \right)$ queries, and the best known lower bound is $\Omega(n^2)$; (iii) perfect allocations (a subset of envy-free allocations) cannot be computed using a bounded (in $n$) number of queries. In this work, we introduce two hierarchies of new fairness notions: \newnotioninverse \,(\newnotioninverseabbrev) and \newnotionlinear \,(\newnotionlinearabbrev). An allocation is \newnotioninverseabbrev-$k$ if the allocation is complete and, for any subset of agents $S$ of size at most $k$, every agent $i \in S$ believes the value of all pieces allocated to agents in $S$ to be at least $\frac{1}{n-|S|+1}$, making the union of all pieces allocated to agents not in $S$ at most $\frac{n-|S|}{n-|S|+1}$; for \newnotionlinearabbrev-$k$ allocations, these bounds become $\frac{|S|}{n}$ and $\frac{n-|S|}{n}$, respectively. Intuitively, these notions of fairness ask that, for every agent $i$, the collective value (from the perspective of agent $i$) that a group of agents receives is limited. If the group includes $i$, its value is lower-bounded, and if the group excludes $i$, it is upper-bounded, thus providing the agent some protection against the formation of coalitions. Our hierarchies bridge the gap between proportionality, envy-freeness, and super envy-freeness. \newnotioninverseabbrev-$k$ and \newnotionlinearabbrev-$k$ coincide with proportionality for $k=1$. For all $k \leq n$, \newnotioninverseabbrev-$k$ allocations are a superset of envy-free allocations (i.e., easier to find). On the other hand, for $k \in [2, \lceil n/2 \rceil - 1]$, \newnotionlinearabbrev-$k$ allocations are incomparable to envy-free allocations. For $k \geq \lceil n/2 \rceil$, \newnotionlinearabbrev-$k$ allocations are a subset of envy-free allocations (i.e., harder to find), while \newnotionlinearabbrev-$n$ coincides with super envy-freeness: the value of each agent for their piece is at least $1/n$, and their value for the piece allocated to any other agent is at most $1/n$. We prove that \newnotioninverseabbrev-$n$ allocations can be computed using $O(n^4)$ queries in the Robertson–Webb model. On the flip side, finding \newnotioninverseabbrev-$2$ (and therefore all \newnotioninverseabbrev-$k$ for $k \geq 2$) allocations requires $\Omega(n^2)$ queries, while \newnotionlinearabbrev-$2$ (and therefore all \newnotionlinearabbrev-$k$ for $k \geq 2$) allocations cannot be computed using a bounded (in $n$) number of queries. Our results reveal that envy-free allocations occupy a curious middle ground, between a computationally impossible notion of fairness, \newnotionlinearabbrev-$\lceil n/2 \rceil$, and a computationally ``easy'' notion, \newnotioninverseabbrev-$n$.