We conjecture that the thought processes used to solve dominant strategy games and mixed strategy games are quite distinct. Two-person games with dominant strategies can be treated as simple decision problems that involve no assessment of one's partner. By contrast, two-person games with mixed strategies require that one think about one's partner. We measure differences in electroencephalogram (EEG) activity while a human subject is playing two-person games. We time-lock the EEG to a common event and use the average across many trials and subjects to find an Event Related Potential (ERP) associated with the common event. The ERP is the brain's response to events—in this case our different games. Our findings lend modest support for the idea that subjects respond to types of games differently.