Gabriel:
You are right that "cooperation" gives the best possible outcome for the group. However, this is not the Nash Equilibrium of this game (in general). The Nash equilibrium is actually the exact opposite where both alliances compete as much as possible with each other because the situation presented is basically a "Prisoners' Dilemma" situation (a very simple to understand yet fascinating problem in game theory).
However, you did hint at a way to avoid the Prisoners' Dilemma, which is to setup a group of teams to "watch each others' backs". This, in theory (see "Iterated Prisoners' Dilemmas"), would tend to increase cooperation but in reality may not because setting up such a group of teams for such a small number of matches (only 5-7 matches per team per event) is rather difficult (not to mention preventing backstabbing from within the group itself
)
This is not to say that there won't be any rigged matches though because I am sure that situations will arise in which backstabbing is not advantageous or in which two alliances are really bound by "morality" or "loyalty" which math can't quite seem to model yet
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Just thought I'd put in my 2 cents.
Anthony.