This paper explicitly models investor behaviour in financial markets allowing for traits linked to a notion of imperfect rationality. We study an extreme form of posterior overconfidence where some risk neutral investors overestimate the precision of their private information. They compete in market orders with another group of informed traders who have rational expectations. The participation of overconfident traders in the market leads to higher transactions volume, larger depth, more volatile and more informative prices. More importantly, such traders may make higher expected profits than rational ones and may even earn more than if they switched to rational behaviour. Their unconscious commitment to aggressive trading offers them a `first mover advantage'. I consider an extension with risk averse market makers and find that the nature of results depends on whether exogenous noise trading exists.