It's not the quantity, but timing

Study says no video games on school nights:

According to Dr. Iman Sharif, the results were clear-cut. "On weekdays, the more they watched, the worse they did," said Dr. Sharif. Weekends were another matter, with gaming and TV watching habits showing little or no effect on academic performance, as long as the kids spent no more than four hours per day in front of the console or TV. "They could watch a lot on weekends, and it didn't seem to correlate with doing worse in school," noted Dr. Sharif.

The study was using self-reporting by kids, which has its problems, but is OK in this case, I think. The key information they did not gather was the timing of game-playing and TV watching.

On schooldays, the only time they can do this is late in the evening, after homework and dinner and sports and everything else have been done. Exposure to light from the screens, as well as the emotional involvement (perhaps raised adrenaline?) phase-delays the kids' already delayed circadian clocks. Instead of getting 9 hours of sleep, they get 5 or 6. Of course they perform miserably at school and the athletic field, feel lousy and misbehave - they are chronically sleep-deprived.

On weekends, kids are likely to play and watch in the morning or early afternoon, which does not affect the phase of their sleep-wake cycle.

I let my kids play games first thing when they come home from school. They do homework later - it gradually puts them to sleep so they are not sleep deprived.

Hat-tip: Ed Cone.