In Ohio I know of two types of permits. One isthat any farmer can get if he complains about crop damage. Most of the time the DNR will hand out a few of these permits to any farmer that ask if he can show any damage. I think they almost always hand out a few just to make the farmer happy, usually only a couple. The second is one where they come out and walk the farmers fields over real good, do some math and actually find out how much damage he has and then if it is severe enough they will issue permits that can be used anytime of the year. I know of one guy in town that has a real honey hole and is infested with deer like crazy and he gets to hunt all year long! But I have never heard of night hunting being allowed.

EDIT: Wanted to add after reading the post after mine that in Ohio its only does on the permit for crop damage permits. The only time I have ever seen bucks shot with special permits is in urban areas where hunting is generally not allowed and the DNR has handed out special permits to archery hunters only and they have taken bucks, but have heard it is rare. Ohio is also having problems around the inner parts of Cleavland and has hired a marksman company to shoot deer, buck or doe and this also has nothing to do with farming, but public safety in the big city parks. But for open hunting country the crop damage permits are doe only.