'Psychic games' is a bit of an overloaded phrase and can refer to a number of things. Let's mention four possible interpretations:
The first interpretation refers to quizzes and similar articles which can be found in popular magazines. These kind of psychic games generally don't have much to offer except maybe some entertainment value.
The second interpretation refers to small tests, short practice session or on-the-spot questions a psychic can set for himself in order to continuously train without too much active effort. These psychic games are useful for developing a structured approach for gaining insight.
The third interpretation refers to games as systems. A game (as in 'board game') is generally a system of balance in which its composing elements interact with each other. If you pay attention, playing such games helps develop a sense of coherence and balance. Conversely, more complex systems can be approached with a playful, hands-on attitude which furthers understanding.
The fourth interpretation refers to the emotional power-games or dramas humans like to act out, either consciously or subconsciously.
A decent approach to psychic games in general is to ignore the first kind, train the second, understand the third, and observe the fourth until it gets too boring.