This relationship is commonly referred to as the Rock-Scissors-Paper relationship. This classic childrens game is summarized as: rock defeats scissors, scissors defeat paper, paper defeats rock. Each element defeats and is defeated by an element.
This sort of relationship is commonly seen in strategy games where certain units have strengths or weaknesses against others. We note that intransitive relationships can involve more than 3 elements. However, increasing the number of elements involved can lead to a combinatorial explosion in the number of relationships involved. (Rollings and Morris 2000)
In this vertical scrolling shmup has only two varieties of enemies, white and black. The player can switch between firing white and black bullets, which allows the player to kill enemies faster since white enemies are weak to black and black enemies are weak to white.
The concept of intransitive relationships is very apparent in the first series of the Nintendo Game Boy Pokemon games. The main character captures/collects a team of element oriented "pokemon" and battles enemies in a turn based battle system. Players gain strategic advantages by using pokemon with certain elemental characteristics against enemies with contrasting abilities. For example a pokemon which is considered a "water type" pokemon is given advantage over a foe "fire type" pokemon, and thus its attacks are "super-effective" resulting in up to double the normal point damage done. The fire pokemon, in turn, are particularly effective against bug and ice pokemon.
In Halo 2 certain weapons are more effectice against certain opponents. This is a weak example because it is situational, it is entirely possible to kill anyone with any weapon if you get them in the right situation.
Rollings, A. and D. Morris (2000). Game Architecture and Design. Scottsdale, Arizona, Coriolis.