Author Summary The ability to work out what other people are thinking is essential for effective social interactions, be they cooperative or competitive. A widely used example is cooperative hunting: large prey is difficult to catch alone, but we can circumvent this by cooperating with others. However, hunting can pit private goals to catch smaller prey that can be caught alone against mutually beneficial goals that require cooperation. Understanding how we work out optimal strategies that balance cooperation and competition has remained a central puzzle in game theory. Exploiting insights from computer science and behavioural economics, we suggest a model of ‘theory of mind’ using ‘recursive sophistication’ in which my model of your goals includes a model of your model of my goals, and so on ad infinitum. By studying experimental data in which people played a computer-based group hunting game, we show that the model offers a good account of individual decisions in this context, suggesting that such a formal ‘theory of mind’ model can cast light on how people build internal representations of other people in social interactions.