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Wine version: 1.7.20
Distribution: Xubuntu x64
Distribution version: 14.04 (Trusty)
Graphics card: Nvidia, GeForce GT 740M
Graphics card driver: 331.38 (using nvidia-prime)
Comments: Steam loads fine, BioShock Infinite downloads and upon first run it installs VirtualC and DirectX/D3D. Upon running the game it doesn't load and it gives a small window that says "Message" and an "OK" button.

When I click OK Steam kicks back to the question window asking if I want to run BioShock or the benchmarking tool.

The output of the log file from the time of running Steam to when BioShock fails to start:

[06/18/14 10:39:25] - Running wine-1.7.20 Steam.exe (Working directory : /home/rsty/.PlayOnLinux/wineprefix/Steam/drive_c/Program Files/Steam)
[0618/103926:ERROR:resource_bundle.cc(417)] Failed to load C:\Program Files\Steam\bin\chrome.pak
Some features may not be available.
[0618/103926:ERROR:network_change_notifier_win.cc(126)] WSALookupServiceBegin failed with: 8
Setting breakpad minidump AppID = 8870
Steam_SetMinidumpSteamID:  Caching Steam ID:  76561197961426402 [API loaded no]

Display settings have been left to defaults with video memory set to match what my card has. Every once in a while when tweaking a few things the game does seem to run but no window is visible and no audio from the game is heard. I will try to replicate that situation and post the output of the log for that case as well.

I am also running Compiz as my WM and I've had experiences where minimized windows were not visible when restored so I tried switching to XFWM for a couple tests and the results were no different.

its dx10+ only..so no point trying it... wine doesnt support that.

theres hope that Aspyr will port it to linux at some point. but nothing is confirmed