People model other people’s mental states in order to understand and predict their behavior. Sometimes they model what others think about them as well: “He thinks that I intend to stop.” Such second-order theory of mind is needed to navigate some social situations, for example, to make optimal decisions in turn-taking games. Adults sometimes find this very difficult. Sometimes they make decisions that do not fit their predictions about the other player. However, the main bottleneck for decision makers is to take a second-order perspective required to make a correct opponent model. We report a methodical investigation into supporting factors that help adults do better. We presented subjects with two-player, three-turn games in which optimal decisions required second-order theory of mind (Hedden and Zhang, 2002). We applied three “scaffolds” that, theoretically, should facilitate second-order perspective-taking: 1) stepwise training, from simple one-person games to games requiring second-order theory of mind; 2) prompting subjects to predict the opponent’s next decision before making their own decision; and 3) a realistic visual task representation. The performance of subjects in the eight resulting combinations shows that stepwise training, but not the other two scaffolds, improves subjects’ second-order opponent models and thereby their own decisions.

An arbitrary number of competitors are presented with independent Poisson streams of offers consisting of independent and identically distributed random variables having the uniform distribution on [0, 1]. The players each wish to accept a single offer before a known time limit is reached and each aim to maximize the expected value of their offer. Rejected offers may not be recalled, but they are passed on to the other players according to a known transition matrix. This paper finds equilibrium points for two such games, and demonstrates a two-player game with an equilibrium point under which the player with the faster stream of offers has a lower expected reward than his opponent.

An optimal linear control (4) is deduced for the symmetric team control problem specified by relations (1)–(3). The generating function (6) for the optimal operator is in general irrational if there is more than one agent (controller), indicating that the recursions associated with effective state estimation are no longer of finite order, as in the single-agent case.