A fundamental characteristic common to both human vision and natural language is their compositional nature. Yet, despite the performance gains contributed by large vision and language pretraining, recent investigations find that most - if not all - our state-of-the-art vision-language models struggle at compositionality. They are unable to distinguish between images of "a girl in white facing a man in black" and "a girl in black facing a man in white". Moreover, prior work suggests that compositionality doesn't arise with scale: larger model sizes or training data don't help. This paper develops a new iterated training algorithm that incentivizes compositionality. We draw on decades of cognitive science research that identifies cultural transmission - the need to teach a new generation - as a necessary inductive prior that incentivizes humans to develop compositional languages. Specifically, we reframe vision-language contrastive learning as the Lewis Signaling Game between a vision agent and language agent, and operationalize cultural transmission by iteratively resetting one of the agent's weights during training. After every iteration, this training paradigm induces representations that become "easier to learn", a property of compositional languages: e.g. our model trained in CC3M and CC12M improves standard CLIP by 4.7%, 4.0% respectfully in the SugarCrepe benchmark.