Steam – If I buy Arma 2 through Steam will I be able to install it on multiple computers

In my own experience you can simple copy paste the entire Steam/Valve folder and run it from it's new destination.

Hence, yes you can just copy the entire game's folder with the updates to the other computer

And I just stumbled upon a Super User question about this. Quoting John T:

I do this all the time it is fine to do. Any files that do not match are usually re-downloaded from Steam, hence why it will still update a bit after a fresh copy. If you do not want all the sounds and maps which take up a lot of space, you can simply take the .gcf files for each game from Program Files/Steam/SteamApps, these are the biggest files in the update which will save the most time.

Steam also has a "Backup" feature you can use. The files it creates can be moved from PC to PC and restored on any one.

You cannot log onto the same Steam account in multiple places or it will kick other computers off. Additionally, I think you are unable to 'double buy' a game (though if you got the same game different ways—in a pack and a la carte, for example—you might) on the same account unless it's a gift.

However, you may be able to launch Battlefield outside of Steam (look in the steamapps folder for the executable) however, and work from there. As you suggest, playing online with the same copy may not work, but a LAN game may. Using one copy on multiple computers simultaneously is a little grey though.