If you are livestreaming audio only, you may need performance licensing. If you are live streaming audio and video, you may also need a synchronization license. Consult with an entertainment attorney.
No, that is not a viable ruse to get around the copyright law. You would be broadcasting copyrighted music and none of the fair use exceptions apply. You need a license to do this. See a copyright
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In many (most) video games you can shut off the music or turn its volume down to zero. That would avoid the problem. Otherwise, music belongs to the owner, and all You may also want to clear your actions with the video game owner, since it appears you might end up streaming their proprietary content.
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