I don't think there's anything inherently wrong or worth hating as SB does about the current name; I think it just sounds weird to him because you're stipulating Greatest GAME Rivalry when everyone already knows these characters come from games. After all the prior character contests were "Great Character Battles" which is more or less the same as what he's got now but not redundant.
GameFEUDs is elegant though because it combines sponsorship and description of the contest into one term. Feud sounds a bit better - and less specific - than rivalry, which in turn sounds a bit better than adversaries.
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"Getting lucky 100% of the time... it's called PRO SKILL."
-BlAcK TuRtLe