In daily life, we often copy the gestures and expressions of those we communicate with, but recent evidence shows that such mimicry has a physiological counterpart: interaction elicits linkage, which is a concordance between the biological signals of those involved. To find out how the type of social interaction affects linkage, pairs of participants played a turn-based computer game in which the level of competition was systematically varied between cooperation and competition. Linkage in the beta and gamma frequency bands was observed in the EEG, especially when the participants played directly against each other. Emotional expression, measured using facial EMG, reflected this pattern, with the most competitive condition showing enhanced linkage over the facial muscle-regions involved in smiling. These effects were found to be related to self-reported social presence: linkage in positive emotional expression was associated with self-reported shared negative feelings. The observed effects confirmed the hypothesis that the social context affected the degree to which participants had similar reactions to their environment and consequently showed similar patterns of brain activity. We discuss the functional resemblance between linkage, as an indicator of a shared physiology and affect, and the well-known mirror neuron system, and how they relate to social functions like empathy.