The New York Times' fine psychology reporter Benedict Carey considers the evidence, such as there is, in "Shooting in the Dark. [2]"
My guess would be that, all else being equal, violent video games do encourage violence, but what's not equal is that video games are such a massive time sink that they don't leave much of the day free to get off the couch and into trouble. On balance, they make the world less violent (if more pudgy).
I was paid to read mass murderer Anders Breivik's manifesto a couple of years ago. My impression is that his addiction to online strategy games such as World of Warcraft played a role in his desire to be a player in the real world, to win at 21st Century European history by setting into motion a series of events he had calculated would make him the architect of the new Europe of the second half of this century.
Does that mean World of Warcraft is a bad thing? Perhaps, but what if World of Warcraft had been around to absorb the energies of Marx, Lenin, Trotsky, Stalin, Hitler, Mao, and Pol Pot, to deflect their ambitions to foresee and control history into a cyber fantasy world?