AI Will Not Start a Nuclear War, but Humans Might

Metal Gear Solid: Peace Walker is about this idea. Set during the Cold War, the theory is that no human will choose nuclear retaliation if attacked, since it means mutual annihilation. However, American nuclear doctrine at the time assumes the threat of retaliation is the only thing deterring an attack. They attempt to solve this by building an AI which will retaliate, but are baffled when it too refuses to return fire.