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The local navigation of pedestrians amid a crowd is generally believed to involve no anticipation beyond (at best) the avoidance of the most imminent collisions. We show that current models rooted in this belief fail to reproduce some key features experimentally evidenced when a dense static crowd is crossed by an intruder. We identify the missing ingredient as the pedestrians' ability to plan their motion well beyond the next interaction, whence they may accept to move towards denser regions for a short time. To account for this effect, we introduce a minimal model based on mean-field game theory, which proves remarkably successful in replicating the aforementioned observations as well as other daily-life situations involving collective behaviour in dense crowds, such as partial metro boarding. This demonstrates the ability of game approaches to capture the anticipatory effects at play in operational crowd dynamics.