Fantasy Flight made a board game called 'Orcs' that plays upon this factor greatly. Best done with four players. Each player is a horde of orcs. You all serve the dark lord. On each round you assemble your armies into marching ranks, of mixed clans to assault the four good-guy strongholds.
What happens is you have troops that reflect rock-paper-scissors and three special trump card troops. You go in rounds putting two guys down (side by side) with ONE face up, behind whoever else is there. Now, the dark lord only rewards the strongest clan who partakes in a fight, so before you assault the good guys, starting with the guys in the far back of the march, you can challenge the rank in front of you. You compare your rock-paper-scissors factors, or if you are the same type, whoever has the higher number. If you defeat both you can plow through even more guys. If you both lose one, the non-beaten guys stay to fight. If you lose both, everybody laughs at you. I believe the setup is Wolf Riders > Infantry > Archers > Wolf Riders. Archers and wolves have values 1 to 3, infantry has values 1 to 4.
There are three trumps. The Champion, who will automatically win the fight against the good guys. He doesn't fight fellow troops, and fellow troops cannot drop him. He counts for zero points in the main fight however. The Shaman is great. If receiving a challenge, he diverts that challenge to the next rank up, even if it's the attackers own troops. If the Shaman is on the attack, he doubles the value of the other unit in his rank. Shaman is zero points in the main fight. The Scout is never deployed to the table, you announce him and peek at a face down token. All trumps are once per game.
The game is played in four rounds, and if 3 of the 4 good guy strongholds fall, orcs win, otherwise EVERYBODY LOSES.
So the idea is you knock off a few guys in hopes that he is somewhat below you, but you pray that the collective clan infighting doesn't get to the point where you don't have enough forces to fight the good guys. The game is kept fresh with a big mess of random event tokens that do varying things each round.