Distributive vs. Procedural Justice. The vast majority of research in fair division can be situated within distributive justice5, which is primarily concerned with socially fair outcomes. However, the human judgement of decisions is often impacted by perceived fairness of the procedures. In other words, the interaction between distributive fairness and procedural justice (Tyler and Allan Lind 2002) in allocation of resourcesparticularly when dealing with approximate fairness axiomsrequires an in-depth investigation.

Let us revisit the instance given in Example 1: both circled and underlined allocations satisfy EF1; however, the circled allocation may be perceived to be fairer as it is the outcome of a fair procedure (in this case the Round-Robin algorithm): agents pick their favorite items among the remaining items one by one according to the 1, 2, 3, 1 ordering. More importantly, note that the underlined allocation is theoretically more desirable as it results in higher social welfare (it is simultaneously EF1 and Pareto optimal). In the context of allocating a divisible resource, procedures that are envy-free are perceived to be fairer by participants than those that are proportional (but necessarily not envy-free) (Kyropoulou, Ortega, and Segal-Halevi 2022). 5Distributive justice has a long history in philosophy within Rawls theory of justice in distribution of social goods. We refer the readers to the seminal works of Adams (1963) and Rawls (1971).

These findings imply that participantsall things equal can correctly identify (and prefer) procedures that provide stronger fairness guarantees. Yet, the comparison between fair procedures and fair outcomes remains an interesting open problem. Are envy-free outcomes through centralized algorithms perceived to be fairer compared to fair procedures that only guarantee weaker notions of fairness (e.g. proportionality)? 4 Uncertainty and Temporal Effects Thus far we mainly highlighted challenges in devising and evaluating axioms of fairness in isolated, static, settings. Yet, human judgement is often impacted by uncertainty of the outcomes or procedures as well as dynamic or temporal nature of parameters or decisions.

Fair Lotteries. Randomization has emerged as an alternative approach to approximate fairness notions such as EF1. The goal is often to achieve fairness ex-ante by allowing agents to participate in fair lotteries. While recent efforts investigated theoretical boundaries of achieving ex-ante envyfreeness and ex post EF1 solutions (Freeman, Shah, and Vaish 2020; Aziz 2020), the interaction between the two approach and how they impact the perception of fairness remain mainly open.