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Entrapment occurs when police engage in behavior that would induce an ordinarily law-abiding citizen to commit a crime. You're not police, so it's only entrapment if you're found to have been acting on the police's behalf. Since you contacted them, and they're not really working with you, that's a gray area. If you installed a device that linked to your cell phone (and I have seen them), and tracked it yourself, or watched and followed the person, that wouldn't be entrapment at all. However, even if you call police and ask them to find the item using the GPS, I think the DA will be able to successfully overcome an entrapment claim. Police do sometimes leave unlocked vehicles (or decoy cars) to track car thieves and that's not entrapment, because most people don't just take unlocked cars.
If the bike were locked up, that tends to negate a claim that you did something that would cause a law-abiding person to break the law. (Of course, most of us wouldn't take a bike that wasn't locked up, either).
Once you know who is taking the items, there may also be a civil suit against the person if you are able to figure out who it is and can prove that all the thefts
The primary downsides are, as you said, the time involved, the cost of the device, and losing whatever item is taken. There's also the risk of retaliation by the thief.
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