We found our data in the outputs of their models but who can do anything about it...
If the crawlers refuse to voluntarily respect your robots.txt, then you are well within your rights to poison their data.
Sue for $180,000 per infringement which should be calculated for each illegal API call.
A person should be able to write in a terms of use or license page on their website that says "do not include any content from this website in your AI training data. if you do you will be billed $100 billion dollars." And it should be enforceable. It just turns out that nerds like to say "oh that would be too hard or too expensive, so we're going to ignore it."
I looked into this a bit (not a lawyer) and it seems that robots.txt isn't legally binding to either party, but this seems to have two major implications for AI agents (and crawlers/scrapers in general).
First, even if the robots.txt says you can crawl the site, that isn't a copyright grant of any kind or permission to copy/use that data outside of the permissions granted by the TOS.
Second, ignoring the robots.txt while also pirating the site contents could point to bad-faith and makes a much stronger case for double-damage penalties due to willful infringement.
If the site TOS doesn't explicitly grant an AI agent rights to copy out the site content AND the AI agent is ignoring the robots.txt at the same time, it seems a lot more likely that there's a strong copyright infringement case against the agent owner.
Unauthorized access, system damage, and maybe even extortion all apply here.