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Honestly, you probably don’t even need to check them. If they’re Steam Deck Verified, they’ll probably just work. If they’re Steam Deck Playable, they’ll probably just work (most common issues are controls or small text, neither is an issue on desktop/laptop).
You honestly only need to check protondb if you have an issue, or maybe if you’re buying a game that’s “Unsupported” or something. 9/10 times, just clicking Play will work fine if it’s a single player game.
You mean deck verified shows on the Steam shop so you don’t have to check on protondb? I don’t use the deck so I didn’t follow this much.
Yes, Deck Verified on Steam means Valve has verified that it works on the Deck. If it works on the Deck, it’ll work on Linux, because the Deck runs Linux and it’s basically the same software stack.
Protondb is a community project where people submit reports about what works and what doesn’t. It’s a good resource, but only really needed if something isn’t working right out of the box.