Having just read a Slashdot article "Violent Games Credited With Reducing Crime Levels" on this topic that suggests that violent video games reduce violent crime, I thought it worthwhile to post it to this interesting thread.
The Slashdot article cites the paper "Understanding the Effects of Violent Video Games on Violent Crime", dated April 7, 2011, the abstract for which is (my emphasis):
Psychological studies invariably find a positive relationship between violent video game play and aggression. However, these studies cannot account for either aggressive effects of alternative activities video game playing substitutes for or the possible selection of relatively violent people into playing violent video games. That is, they lack external validity. We investigate the relationship between the prevalence of violent video games and violent crimes. Our results are consistent with two opposing effects. First, they support the behavioral effects as in the psychological studies. Second, they suggest a larger voluntary incapacitation effect in which playing either violent or non-violent games decrease crimes. Overall, violent video games lead to decreases in violent crime.
The answer, based on this study, is: Violent video games cause or correlate with an increase in violent behaviour. However, it's worth noting that this violent behaviour is offset by the distraction and release of aggression through playing of the (violent or non-violent) video games, resulting in an overall reduction in violent crime.
While this study purports to answer the question, a lack of consensus is likely to remain on the topic. I think this study is, however, an important piece.
— EDIT —
I would also add a reference to the article No link found between movie, video game violence and societal violence (2014), which is summarized:
Christopher Ferguson (Stetson University) published his findings in the Journal of Communication. Ferguson conducted two studies that raised the question if whether the incidence of violence in media correlates with actual violence rates in society. The first study looked at movie violence and homicide rates between 1920 and 2005. The second study looked at videogame violence consumption and its relationship to youth violence rates from 1996-2011. He found that societal consumption of media violence is not predictive of increased violence rates in society.
and
"Entertainment Software Ratings Board ratings were used to estimate the violent content of the most popular video games for the years 1996-2011. These estimates of societal video game violence consumption were correlated against federal data on youth violence rates during the same years. Violent video game consumption was strongly correlated with declines in youth violence. However, it was concluded that such a correlation is most likely due to chance and does not indicate video games caused the decline in youth violence. ... Previous studies have focused on laboratory experiments and aggression as a response to movie and videogame violence, but this does not match well with real-life exposure.
It may be worth noting that this appears to be the first article by Christopher Ferguson on the topic.