When engaging in social interaction, people rely on their ability to reason about unobservable mental content of others, which includes goals, intentions, and beliefs. This so-called theory of mind ability allows them to more easily understand, predict, and influence the behavior of others. People even use their theory of mind to reason about the theory of mind of others, which allows them to understand sentences like Alice believes that Bob does not know about the surprise party'. But while the use of higher orders of theory of mind is apparent in many social interactions, empirical evidence so far suggests that people do not use this ability spontaneously when playing strategic games, even when doing so would be highly beneficial. In this paper, we attempt to encourage participants to engage in higher-order theory of mind reasoning by letting them play a game against computational agents. Since previous research suggests that competitive games may encourage the use of theory of mind, we investigate a particular competitive game, the Mod game, which can be seen as a much larger variant of the well-known rock-paper-scissors game. By using a combination of computational agents and Bayesian model selection, we simultaneously determine to what extent people make use of higher-order theory of mind reasoning, as well as to what extent computational agents can encourage the use of higher-order theory of mind in their human opponents. Our results show that participants who play the Mod game against computational theory of mind agents adjust their level of theory of mind reasoning to that of their computer opponent. Earlier experiments with other strategic games show that participants only engage in low orders of theory of mind reasoning. Surprisingly, we find that participants who knowingly play against second- and third-order theory of mind agents apply up to fourth-order theory of mind themselves, and achieve higher scores as a result.