As I see it we have two VERY separate issues here.
Can I play games on my work computer on off time?
This question you've covered yourself generally saying they don't take issue with this. We could get into debates on whether you should or shouldn't but to be fair this behavior is actually encouraged in some businesses so we'll just assume it's all on the up and up.
Can I install non-work related third party software on my work computer?
Maybe, but generally it's a REALLY bad idea... Some companies won't make a big stink about this, others will, but even if yours doesn't I would advise against it.
Why shouldn't I if they are okay with it? Well several reasons, the first is you can't necessarily predict or control what that program might do to a computer. Steam is a perfect example here. Last fall (2013) Steam pushed out a really bad driver update for select NVIDIA devices, any computers who received this update became unstable and in many cases would leave you with a black screen on boot. This required people to go into safe mode and uninstall the offending driver and or do a rollback to prior to installing it... Now that mistake would waste quite a bit of your productive time plus likely involve IT, that won't make IT or your boss happy...
Admittedly that was a one time thing... the problem is almost everything out there has security and stability holes in it. Steam lets you load games based on tons of frameworks all of which have their own issues... XNA, Unity, JAVA, WCF, Flash, etc. When IT oversees the installation of programs they have a basic inventory of what's on their network. This lets them mitigate potential issues proactively, and in the event new attack vectors become known they can mitigate them reactively. They can only do this effectively when they know what's on their network, since Steam is effectively a program to download other programs and their dependencies it's really hard to know what ultimately ends up on your computer. (this is hard to deal with for IT and will not make them happy)
Not to mention you're putting your personal account on a company computer... that's just dangerous stuff should things go sour. (this is more dangerous in the hassle sense, but I've seen steam accounts get banned due to someone hijacking the account)
What can I do?
Throw steam on a personal laptop! If the people over top of you are all on board with you gaming on off time by all means do it. But keep that gaming from bleeding into work time in the means of causing computer stability and performance issues. In this case as well if something does go wrong you can keep working as if nothing happened than deal with whatever happened on your own time.
If you don't have a personal laptop that is able to play Civ, perhaps you could pick up less resource intense titles for your "at work free time" (besides I've personally found that when I was allowed to game at work during lunch quick games were far more desirable at work, games like civ you barely get enough time to get much out of over a lunch break.)