My Grandfather has a few acres in the Azores, which he raises some chickens and grows some fruit and vegetables. Over the past 15 years or so he has had the problem (i believe the whole island may have this problem) of someone stealing crops just before harvest and some tools here and there. The thief is never consistent, but its always a let down.
No you don't want to use a game camera if he is moving at night the flash of the camera will alert him too it. The best deterrent I have found is a loud barking dog. Some well place rock salt in the back side works wonders too.
I ended up making my own infra-red security camera from a raspberry pi (little cheap credit card size computer- £25), an infra red camera (r-pi make one, £16), some IR LEDs (£4) and a battery pack (£10)- which all worked out far cheaper than me buying one, but obviously required more effort setting it up.
R-pi has some software called motion-mmal that will monitor the camera and start recording when motion is seen. Infra-red means the thief (or in my case, wildlife) can't see the 'light' or the camera- no flashes or anything). Very minimal setup (plug camera in.. install software- put camera in position)
My battery pack is quite small, it only lasts overnight (It also have a solar panel that cost as much as the rest of the project put together- that charges it everyday)
I've used a couple of different Moultrie cameras. The no-glow infrared ones don't make any noticeable flash at all, and the low-glow are unnoticeable when placed correctly, meaning higher than eye level and pointing down. Very easy to use and the batteries last a long time.
I know someone who had one in their backyard, caught a known intruder, pressed charges, and the photos were enough evidence to get the intruder locked up. I use mine to get pictures of deer and bears though.