... therefore I am.
Oops, sorry, wrong philosopher! (just kidding)
I think what Wittgenstein was trying to convey with the concept of games was something very subtle and difficult to understand. You must realize that after thinking about his subject quite a lot, he eventually realized that there was nothing he could say about it. He found the ultimate reason to be silent, and he did remain silent for several years. But of course one can only be silent for so long.
In any case, as I understand it the problem has to do with language. How can you explain language if you need a language to understand any explanation of anything? Isn't an explanation of language simply beside the point? Well, yes and no. As much as it is a fact that you already understand a language before you can understand any explanation of it, it's also a fact that we are unaware of some of its more subtle aspects. Those are the ones Wittgenstein was trying to talk about, and this is where the concept of games comes in.
The important point about games, when it comes to linguistic analysis, is not that someone wins, but that there are rules whose sole purpose is to prevents them from winning. This is certainly true of any game and any language. It's very easy to win a game if you're not constrained by rules while your opponents are. So winning isn't important, it's winning by the rules that really matters. Likewise for language, you can always prove any assertion "true" if you don't abide by the linguistic conventions associated with the particular subject you're talking about (just read Alan and Dick to know what I mean)
Taking the analogy an extra step, it's interesting to look at communication as a sort of game. The game's goal is to convince your opponents that what you are saying is true. The game's rules are completely arbitrary, but if you don't follow them nobody cares if you win. In human communication, whomever cheats always loses. But of course some people enjoy cheating as it gives them the illusion that they have won but are not recognized (and some people take this very seriously!)
So, to comment on your assertion, math is in fact a competition. If you prove a theorem true you win, and every mathematician will acknowledge your feat as long as they do not think you have broken any rule of the game.
Hope this helps.