There seems to be a difference between computer games and other play:
- studies have shown that excessive computer gaming is correlated to emotional and behavioural problems
- there is no such correlation between behavioral problems and other play
(this claim might be wrong; see the comment by @deworde below)
While correlation is not causation, while there are also studies that contradict this common result, while computer gaming also has benefits, and while you can certainly control the time and games your child plays, I don't see how not letting your child play computer games might harm him or her.
It's like alcohol. If you let your child drink a sip of beer once, it certainly won't harm your child. But why do it in the first place? What benefit does it have that your child knows at an early age what beer tastes like? None. Sure, if you drink beer regularly, you might satisfy a curiosity that could otherwise lead to your child drinking beer behind your back, but if you don't drink yourself, there really is no reason why your child needs to become familiar with it.
In the same vein, if you play computer games a lot yourself, it would be strange if you forbid your child to do the same. But if you live a life with other values and habits, if you think books are great or sports or being with friends, then why would you help your child build a habit that you don't endorse? Sure, computer games are probably a small risk for a select few vulnerable kids only, but not playing computer games it no risk at all. I grew up without computers and saw my first PC when I was in my early twenties, yet I later worked as a programmer and am more "computer literate" than many younger people. You don't need to start as a child.
If my son plays computer games when he visits his friends, that's fine. Then it is a social activity, and what the kids do together is less important than how they do it and that they do something together at all. But there really is no reason why my son needs to play computer games when he is at home alone. First off, he needs to be able to deal with being alone and boredom, and if he needs gadgets to keep him from feeling lonely then he probably won't become a happy adult. Second, the best control is if you avoid situations that you have to control. No internet, no problems with your kids hacking the kid filter and watching porn. No mobile phone, no problems with your kids filming other kids beating up other kids and uploading it to YouTube. No computer games, no bleary eyed kids getting angry over having to go to bed because tomorrow is school while they haven't yet mastered the sixth level. No s**t, no problem.
For those reasons, my son is always one of the last of his peers to get technical gadgets. He has to get them eventually, or else he will be the odd kid out, but there really is no reason why he has to be the first to have them.
Essentially, your own standards should apply to your child, and your own lifestyle should be open to your child. If you play computer games, share that experience with your child on an appropriate level. If you don't play, then it is something that your child can discover on his own, when he is indepentend enough from you to manage his own affairs. Until then, share the computer-game-free experience with your child. Teach by example.
Violent Video Games
The following are meta-analyses, i.e. they summarize all (relevant) previous studies.
The evidence strongly suggests that exposure to violent video games is a causal risk factor for increased aggressive behavior, aggressive cognition, and aggressive affect and for decreased empathy and prosocial behavior.
Anderson, C. A., Shibuya, A., Ihori, N., Swing, E. L., Bushman, B. J., Sakamoto, A., ... & Saleem, M. (2010). Violent video game effects on aggression, empathy, and prosocial behavior in eastern and western countries: a meta-analytic review. Psychological bulletin, 136(2), 151.
Whereas violent video games increase aggression and aggression-related variables and decrease prosocial outcomes, prosocial video games have the opposite effects.
Greitemeyer, T., & Mügge, D. O. (2014). Video games do affect social outcomes a meta-analytic review of the effects of violent and prosocial video game play. Personality and Social Psychology Bulletin, 0146167213520459.
Pathological Video Gaming
While the relationships between video game use and negative consequences are debated, the relationships between video game addiction and negative consequences are fairly well established. ... Video game addiction was related to depression, lower academic achievement, and conduct problems, but time spent on video games was not related to any of the studied negative outcomes. ... Spending time playing video games does not involve negative consequences, but adolescents who experience problems related to video games are likely to also experience problems in other facets of life.
In other words, if you define video game addiction (VGA) as playing long hours, as most research has been doing, then there is no relation between VGA and other health problems; but if you define VGA as behavior that the individuals themselves experiences as problematic (e.g. addictive), then there is a significant correlation.
Brunborg, G., Mentzoni, R., & Frøyland, L. (2014). Is video gaming, or video game addiction, associated with depression, academic achievement, heavy episodic drinking, or conduct problems?. Journal of behavioral addictions, 3(1), 27-32.