The theory of games provides a mathematical formalization of strategic choices, which have been studied in both economics and neuroscience, and more recently has become the focus of neuroeconomics experiments with human and non-human actors. This paper reviews the results from a number of game experiments that establish a unitary system for forming subjective expected utility maps in the brain, and acting on these maps to produce choices. Social situations require the brain to build an understanding of the other person using neuronal mechanisms that share affective and intentional mental states. These systems allow subjects to better predict other players' choices, and allow them to modify their subjective utility maps to value pro-social strategies. New results for a trust game are presented, which show that the trust relationship includes systems common to both trusting and trustworthy behaviour, but they also show that the relative temporal positions of first and second players require computations unique to that role.