Online Fair Division for Personalized $2$-Value Instances

TL;DR

This work investigates online fair division with indivisible goods under additive valuations, focusing on personalized $2$-value instances. It introduces a tight deterministic Deferred-Priority algorithm that guarantees a per-step MMS of at least $1/(2n-1)$, with eventual improvements and a detailed phase-based analysis using priority levels. The paper further shows that very limited foresight can yield stronger EF guarantees: with foresight of length 1 for two agents, EF1 holds on every even step (EF2 temporally), and with foresight length $n-1$ for $n$ agents, EF2 holds at all steps and EF1 at round boundaries, alongside MMS guarantees of $1/n$. Finally, it connects these results to interval-restricted valuations via reductions, obtaining additive MMS and EF guarantees parameterized by the value ratio bounds. Together, these results delineate when online fairness is achievable in restricted settings and highlight the substantial gains possible with foresight.

Abstract

We study an online fair division setting, where goods arrive one at a time and there is a fixed set of $n$ agents, each of whom has an additive valuation function over the goods. Once a good appears, the value each agent has for it is revealed and it must be allocated immediately and irrevocably to one of the agents. It is known that without any assumptions about the values being severely restricted or coming from a distribution, very strong impossibility results hold in this setting. To bypass the latter, we turn our attention to instances where the valuation functions are restricted. In particular, we study personalized $2$-value instances, where there are only two possible values each agent may have for each good, possibly different across agents, and we show how to obtain worst case guarantees with respect to well-known fairness notions, such as maximin share fairness and envy-freeness up to one (or two) good(s). We suggest a deterministic algorithm that maintains a $1/(2n-1)$-MMS allocation at every time step and show that this is the best possible any deterministic algorithm can achieve if one cares about every single time step; nevertheless, eventually the allocation constructed by our algorithm becomes a $1/4$-MMS allocation. To achieve this, the algorithm implicitly maintains a fragile system of priority levels for all agents. Further, we show that, by allowing some limited access to future information, it is possible to have stronger results with less involved approaches. By knowing the values of goods for $n-1$ time steps into the future, we design a matching-based algorithm that achieves an EF$1$ allocation every $n$ time steps, while always maintaining an EF$2$ allocation. Finally, we show that our results allow us to get the first nontrivial guarantees for additive instances in which the ratio of the maximum over the minimum value an agent has for a good is bounded.

Theorems & Definitions (51)

• Definition 2.1: Personalized $2$-Value Instances
• Definition 2.2: Personalized Interval-Restricted Instances
• Definition 2.3: $\rho$-Envy-Freeness, $\rho$-EF$k$
• Definition 2.4: $\rho$-$\textrm{PROP}$, $\rho$-$\textrm{MMS}$
• Definition 2.5: Temporal Fairness
• Remark 2.6
• Theorem 3.1
• proof
• Theorem 3.2
• proof