The gateway is the device allowing your local LAN to communicate with the external world. Assuming your gaming adapter is a LAN extension device (sounds like it), then your gateway should be 192.168.1.1.

If the device is NOT is LAN extension, than you have a conflict as you may have two touching networks with the same subnet. (but it does sound like the purpose of your wireless gamign adapter is the correct one for your needs, so 192.168.1.1).

See Mojo's signature