The British government, concerned about the number of venomous cobras in Delhi, offered a bounty for every dead cobra. Initially, this was a successful strategy; large numbers of snakes were killed for the reward. Eventually, however, enterprising people began to breed cobras for the income.
> When the government became aware of this, the reward program was scrapped. When cobra breeders set their now-worthless snakes free, the wild cobra population further increased.
French officials wanted a lifestyle more like home, including flush toilets.
So they built a sewer system. It was a perfect breeding ground for rats.
Now, Hanoi had a rat problem.
They hired exterminators, but it wasn't enough. So they offered a bounty to anyone who would kill a rat and bring in the severed tail as proof. It was much easier for them to dispose of a collection of rat tails instead of entire rat carcasses.
The locals quickly figured out that you could remove a rat's tail without killing it, and send it back into the sewers to breed more rats with fresh young tails.
https://www.atlasobscura.com/articles/hanoi-rat-massacre-190...
https://en.wikipedia.org/wiki/Great_Hanoi_Rat_Massacre
Here is an article that ties together the rat massacre, the cobra effect, factory workers' gloves, overtime pay, and heat-based bonuses:
Note that I'm not saying they won't, just that it's a bad idea in a new and exciting way.
I suspect if you time-box the reward that'll fix the cobra effect problem. Announce that you'll only pay for dead rats for ~4 months and at least in the final month people should realize that its time to kill the golden goose and provide all the rats they have instead of breeding for more.