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The issue is that they ban from their website (UI interface) but they do not ban from the game/blockchain. They could ban the bots from the website but if the bots don't interact with the site (which to my knowledge they don't), that wouldn't do any good.

I would imagine it is not just a ban from the website ui. It is most likely a server API. The game isn't played just by blockchain transactions. Whether the bot is using the website or not, it is going through the Splinterlands servers. Think about when the game goes down for maintenance and is inaccessible, they aren't bringing the entire blockchain down, just the game. A ban essentially does this, isolated to an account. So if they actually detect botting from an account and wanted to ban it, they definitely could. They just don't want to.

Yes, since they are the only ones processing the custom transaction related to Splinterlands from the Hive blockchain they could block them.

I mean to clarify and say currently the way they ban is by not allowing banned users to access the official Splinterlands website.

I don't believe they have ever banned someone from the API. Even though from a technological standpoint they could, I don't believe they do that and it would be a large change in the way they operate/enforce policy if they did.

Is there a source that actually confirms this? Seems... unlikely to be done that way from a technical standpoint. As a software engineer, it makes zero sense to me that they would ban an account from a website ui and not just interacting with the server as a whole.