The third game, can be awesome at times, may cause some burns though:
There is one aspect of enneagram personality typing, which was left out so far. Each one of us has 3 basic instincts: a Self-Preservation (SP) instinct, a Social (SO) instinct and a Sexual (SX) instinct. When each of the 9 enneagram types is further divided to account for these 3 instincts, the result comes to 27 subtypes. Understanding your subtype (not just your dominant type and your dominant instinct, but rather the interaction of the type and the instinct, which interaction is captured within the subtype) is helpful. The 3 instincts and the 27 subtypes are described in many places, here is one for example.
Just as with types, one of these instincts in each of us is dominant. It could be that SO types tend to play the Enneagram more (wanting to understand more about themselves and others around them), the SP types tend to play the Glassperlen more (wanting to strengthen their own graph of knowledge), while the SX types tend to play the game of ❤ more (craving the intensity of a one-on-one romantic relationship).