There's a really interesting question about incentivization behind this. Everybody knows that there are too many dogs, and as a result many die in shelters because they aren't adopted. Yet many folks continue to buy dogs from breeders, particularly in America where they actually sell dogs in stores!
So what is the best solution so that exactly the right amount of dogs exist and none are killed. This is innovative, but I suspect training doesn't scale. Perhaps make it illegal to sell dogs until we're out of strays? Maybe a tax deduction for adopting a dog, or a tax penalty on buying a dog? Or an anti-breeding campaign (like the anti-fur one which has been very successful)?
Professional breeders are not the problem. It is not in breeders' interests to flood the market and collapse prices.
The problem is irresponsible pet owners who fail to neuter and spay their pets.
Santa Cruz County, California, where I used to live, enacted in 1995 a law requiring neutering and spaying of pets. (A breeder's licensed can be purchased for something like $100 annually, IIRC.) The numbers of animals in the shelters dropped dramatically.
Efforts to pass a similar bill at the state level failed, however.
However, I'll note that all shelter neuter and spay, so if it irresponsible pet owners who are responsible, than it is limited to those owners who did not get their dogs from a shelter.
The cost may still be a little too far out, but what about requiring breeders to register the dog's DNA sequence in some database. If the dog is found as a stray, a large fine is imposed on the breeder or the most recently registered owner. The fines could go towards programs for neutering and finding homes for strays. If particular breeders are frequently coming up as strays, they could loose their breeder license, etc. Of course to play my own devil's advocate, the risk here is that the owners who have lost interest in keeping a pet kill the animals instead of releasing them.
The point of this is not to make people adopt abandoned dogs because they can drive, it is to make people more likely to adopt abandon dogs by removing the conception that they are stupid/uncontrollable.
I think the real issue with overpopulating dogs is that they are treated as a commodity, so eccess stock is almost inevitable. Of course, every species that ever existed is always in a state of overpopulation, or exponential growth.
So what is the best solution so that exactly the right amount of dogs exist and none are killed. This is innovative, but I suspect training doesn't scale. Perhaps make it illegal to sell dogs until we're out of strays? Maybe a tax deduction for adopting a dog, or a tax penalty on buying a dog? Or an anti-breeding campaign (like the anti-fur one which has been very successful)?
I'd love to hear some cool ideas here.