Evolution, lateralization, cognition and fighting

What could be the evolutionary advantage of having two brain hemispheres? 

Lateralization is a fundamental property of brain and behaviour throughout the animal kingdom. However, its benefits in terms of Darwinian function remains somewhat elusive.  What could be the evolutionary advantage of having two brain hemispheres that are each dominant for different functions? A potential benefit is increasing cognitive performance. Ton Groothuis will show results of a test with human subjects in the lab that for the first time demonstrate improved cognitive skills by subjects with a stronger task division between both hemispheres.
Another potential benefit is related to handedness. Left-handedness is claimed to be associated with disease prevalence but still exists as a clear minority in all human societies. This suggests it has benefits as well, otherwise selection would have get rid of it.  The “fighting hypothesis” attempts to explain this benefit, stating that left-handers have a surprise effect in fights as long as they are in the minority. Indeed, combat sports, such as boxing and tennis, show a surprisingly overrepresentation of left-handers relative to non combat sports and the general population. To test this hypothesis more directly, Groothuis and his group went to a pre-industrial society in the highlands of Papua Indonesia, until recently involved in frequent combat.

Ton Groothuis is full professor and head of the research group Behavioural Biology, which participates in the Centre for Behaviour and Neuroscience at the University of Groningen. In his research he integrates mechanistic and evolutionary explanations of behaviour, using a wide array of animal species, including humans. One of his research lines is the developmental plasticity and evolution of lateralization of brain and behaviour.

Interesting links
Bionieuws (2005) Waarom we twee hersenhelften hebben
Intermediair (2012) Linkshandig? Meer kans op een sportcarrière