According to these log files, your XNA Game Studio 3.1 installation is failing due to the same issue described in
this Connect bug. As a workaround, you should be able to go to the Programs and Features control panel, uninstall the item named Microsoft XNA Framework Redistributable 3.0, then re-run XNA Game Studio 3.1 setup. Can you please give that a try and let us know if you encounter any additional installation problems?
Thanks!