Growing evidence suggests that the brain uses an attention schema, or a simplified model of attention, to help control what it attends to. One proposed benefit of this model is to allow agents to model the attention states of other agents, and thus predict and interact with other agents. The effects of an attention schema may be examined in artificial agents. Although attention mechanisms in artificial agents are different from in biological brains, there may be some principles in common. In both cases, select features or representations are emphasized for better performance. Here, using neural networks with transformer attention mechanisms, we asked whether the addition of an attention schema affected the ability of agents to make judgements about and cooperate with each other. First, we found that an agent with an attention schema is better at categorizing the attention states of other agents (higher accuracy). Second, an agent with an attention schema develops a pattern of attention that is easier for other agents to categorize. Third, in a joint task where two agents must predict each other to paint a scene together, adding an attention schema improves performance. Finally, the performance improvements are not caused by a general increase in network complexity. Instead, improvement is specific to tasks involving judging, categorizing, or predicting the attention of other agents. These results support the hypothesis that an attention schema has computational properties beneficial to mutual interpretability and interactive behavior. We speculate that the same principles might pertain to biological attention and attention schemas in people.