Individuals often aim to reverse undesired outcomes in interactions with automated systems, like loan denials, through system-recommended actions (recourse) or manipulation actions (e.g., misreporting feature values). While providing recourse benefits users and enhances system utility, it also increases transparency, enabling more strategic exploitation by individuals, especially when groups share information. We show that this tension could potentially lead systems to strategically withhold recourse, challenging assumptions about universal recourse provision in current literature. We propose a framework to investigate the interplay of transparency, recourse, and manipulation and demonstrate that rational utility-maximizing systems frequently withhold recourse, leading to decreased population utility, particularly impacting sensitive groups. To mitigate these effects, we explore the role of recourse subsidies, finding them effective in increasing the provision of recourse actions by rational systems.