It sounds like you're taking these photos to preserve memories ... this makes me think you're probably not zooming in on the tiniest details (aka "pixel peeping") to critique the quality of the gear.
Canon's new EF 70-200mm f/2.8L IS USM II isn't just a good 70-200... it's probably the best-in-class 70-200 available in the industry. But keep in mind the original 70-200 f/2.8 (the version I) was extremely good. The new version isn't much sharper in the center but it is a bit sharper in the corners. Also it's image stabilization performance is improved. But it's about $2100 for the new version and you can still get the original version at a significantly reduced price of about $1350. ($750 less) If you want the lottery then the version II is the lens you want - no question. But at a $750 premium... this would really need to be a lens you plan to use heavily to justify the price.
For indoor games (or even outdoor games if played at night under field lighting) the lighting isn't particularly bright and you'll want to be able to use reasonably high shutter speeds to reduce/eliminate effects of motion blur. For that... you probably want a lens that can collect as much light as possible such as an f/2.8 version.
Be mindful that you're going to lose some depth of field if you shoot at f/2.8 ... at 200mm & f/2.8, a lens focused to a subject 50' away will only have about 2' in focus. Suppose this is basketball... if you want BOTH the ball AND the player in focus at the same time, they could easily be more than 2' apart (even if they have control of the ball). In comparison to an f/4 lens... the DoF at f/4 is just shy of 3' (2.86' according to the calculator at DOFmaster.com). f/4 is giving you about 30% more DoF as compared to f/2.8. So this gain in light which results in you being able to halve the time the shutter is open does come with a trade-off.
Canon's lens (even the original EF 70-200mm f/2.8L IS USM) is better than the Sigma & Tamron versions and a new copy is only about $50 more (in my opinion it's worth the extra $50).
B&H Photo lists the Tamron & Sigma versions for $1299 and the Canon is $1349.
Sigma has made the industry take notice lately ... going from a 3rd party lens maker that offers "the lens you settle for" vs. the "the lens you want" because the "the lens you want" is too expensive. Their latest lenses are hugely improved and some of their latest lenses are "the lens you want". But their 70-200 is not one of those latest lenses that are getting everyone's attention.
If you had an abundance of light and could shoot with very fast shutter speeds (1/1000th sec or faster) then I'd say you could get a non-IS lens. But I expect you'll find the lighting is not adequate and you may be forced to shoot at slow-is shutter speeds possibly 1/250th sec or slower) and you're probably going to want IS to compensate for camera movement. This is even more true if you go with an f/4 lens instead of an f/2.8 lens -- where f/4 is collecting only half as much light as f/2.8.