Is the game, "go," a proper noun? What about "checkers" or "chess"?

The game of Go is...

or

The game of go is...

Apparently the International Go Federation capitalizes it. Its dictionary entry doesn't appear to be (from what I have seen). It seems to fit the definition of a proper noun as it isn't preceded by an article (such as "the" or "an").

So which is it?

Also, where does that put checkers, chess, basketball, football, etc. They are never (or at least not regularly) capitalized, yet they seem to have the same characteristics.

I predict the argument is going to be that Monopoly is capitalized because it is a brand and go should not be capitalized since it is not a brand and it is not referring to a specific instance of anything. Is that correct?

I'd say that Go is often capitalised simply because go is already a very common English word, so most sentences about the game would become significantly harder to parse if it weren't somehow differentiated typographically. Personally, I prefer italics to capitalisation, where possible. The names of games are often capitalised regardless of whether they're trademarks, seemingly in arbitrary fashion based on what looks good or feels right.

The OED does not capitalize the name of the game, but its quotes do all have capitals. A third option would be not capitalized but italicised, because it is a foreign word.

I suspect that the argument goes that names may lose their capital once they are "felt" to have turned into regular words - a very vague criterion, of course; but compare how compound nouns lose their spaces and then their hyphens, which is equally difficult to define.