Do games rest on an unresolvable contradiction? On the one hand, they are social affairs, designed to unite people. In chess clubs, for example, people of all ages, races, creeds, etc., come together and enjoy each other's company. On the other hand, games are competitive affairs, appealing to our most raw and neanderthalic impulses to clobber our enemies. To become good we must prey on and exploit every weakness of the opponent, and to do this we must make him the enemy, else we won't be motivated.
A knife may be used by one person for farming and another person for killing. There is no contradiction here, it's just that the same thing may serve different purposes at different times. Moreover, you may play a game both for the purpose of trying to win and as a social affair, at the same time. You can badly want to beat your opponent without disliking him. When I play squash with my regular partners, I try very hard to win (with more enthusiasm than skill, as it happens); at the same time, I find the match a bonding experience.