In this work, we question the equity of a batch-matching on horizon policy managed by a private Autonomous Mobility on Demand operator. We check the sensitivity of policy rules and horizon length on AMoD operator's profit and equity for customers. We propose a naive pricing scheme the transportation authority could implement to improve the system's equity. This work is the first step toward comparing a bi-level optimization problem (where profit and equity objectives stand on different levels) and a multi-objective fair optimization problem (where a public stakeholder operates AMoD).