If First are going to compete with Lothian, then first of, this might be good for the passengers as it might result in lower fares, however there is a chunk of LB passengers who have ridacards, so would not use First services. Just need to look at when First ran the 44 service, to see what happened there. First could 'flood' Edinburgh with 150 buses (questions from others on the vehicle locations, etc..) but it would not really make a dent in Edinburgh even if they did. That may cover a few key routes but not them all. 26 for example needs around 35 buses, 25 with 21 buses. If First took the 22, 25, 26, 34 - those 4 alone would need peak vehicle of 99 buses. Add on the 11 and 16 and it is 146 buses (so they'd have 4 spare for breakdowns). To have any serious effect, First would need to pump 450-500 buses in to Edinburgh and would need to stick at it with much lower fares, and a high reliable service with buses on par if not better in frequency to Lothian. Hard to do on some routes.
If First were to select key routes, and time the buses carefully and lower prices v LB then maybe they'd get some passengers over but not anything to sing home about as, as I said, there is a chunk of passengers with ridacards so they are locked in to the Lothian ecosystem. This is why I am shocked LB have not gave the option to those for LCB to have a ridacard option yet, to lock them in to the ecosystem.
Plus, if First were to do this, and start a full on bus war, Lothian would just go harder in to WL at a faster speed and First would lose in the end.