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Once Origin is installed you can launch it directly from its installer. Start the newly added "game" ie: the Origin installer and install it. In Steam, choose the "Add non-Steam Game" command and select OriginThinSetup.exe from where ever you placed it. It doesn't matter where you put it but your Downloads folder is a good option. Transfer OriginThinSetup.exe to your Linux machine. (Alternatively, Redditor GGG_246 informs these are also available at winehq) You need to do this because viewing the site from a Linux machine doesn't give the Windows download option. On a Windows machine, download OriginThinSetup.exe from their site. There might be other better ways, in which case please let me know. This post explains the best way I found to set it up. Unfortunately there doesn't seem to be a way to have it automatically shutdown when you close the game so that needs to be done manually. Luckily there's a way to launch Origin and then get it to run a specific game - which I'll cover below. It's like EA's version of Steam but crappier - especially for lounge room gaming machines that don't have a mouse or keyboard since it doesn't support controllers. Unfortunately the only way to download, install and run most EA games is via their launcher Origin. "Why do this when EA Games already install OriginThinSetup." This is specifically for games that aren't on Steam, in this case NFS The Run. For me, everything else I play is in Steam, it works fine and I just wanted to add this one non-Steam game.
I personally haven't looked into Lutris, but have only heard good things about it. "Why not just use Lutris?" No reason except that I just like using Steam as the launcher for this lounge-room based, controller-only gaming machine.Update: since posting this article there's a been some discussion on Reddit about why this is necessary: Since this was the first time I'd tried a non-Steam game under Proton I knew there'd be some tinkering around but I got it working in the end. The Run isn't listed on ProtonDB so there was a chance it wouldn't be playable but at just $5 there wasn't much to lose.
When I saw the PC version on sale I couldn't resist grabbing a copy to see if I could get it to work on my new Linux gaming machine (which I wrote about here). One of my favourite PS3 games was Need for Speed: The Run.