"Walk/Don't Walk" street sign wired into bathroom light
A 2 ways switch should totally do it:


wire the common of the switch with the line cable
wire one of the switch ways to the bathroom light and the don't walk light
wire the other switch way to the walk light
wire all lights second wire to the neutral cable
if your walk/don't walk sign has a metal frame, wire it to the protective earth cable


Just ensure that: 


the bathroom light and don't walk light are connected in parallel, not in series
your 2 ways switch is graded for the sum of your two lights (look at the number of watts on the bulbs)


You may want to add a second switch somewhere so the walk sign won't stay on 24/7 while the gaming room is empty (perhaps wiring everything to the main light switch off the gaming room?)

EDIT: I should add that, since it implies live electric current, YOU SHOULD HAVE A FULLY QUALIFIED PROFESSIONAL CHECKING YOUR DESIGN BEFORE YOU DO ANYTHING, and CHECKING WHAT YOU'VE DONE BEFORE ANY CONNECTION TO YOUR POWER GRID! (depending where you live, this may be mandatory by law; if it's not, you should still do it)

Or in other words, you do as you want, but I'm not responsible if anything goes wrong after you followed instruction from "some guy on the internet".