Since a lot of the plot of MindGames is caused by the play of the Great Game, an explanation would probably be helpful:
First and foremost, The Great Game is NOT like an athletic contest where two sides try to score a goal.
It's more a symphony of violence where the conductor builds to a crescendo. The whole symphony is part of the Great Game, not just the big bang at the end. 'Points' are judged on how well the violence has been orchestrated, its strength, its 'beauty'. 'Points' go to the Starborn's reputation among its peers.
When more than one Starborn is playing, they're not opponents but rather friendly rivals. Picture two brothers on safari, each trying to outdo the other in spectacular kills - until one kills the last living white rhino. Or, the same two brothers in college, each trying to bed a more beautiful blond - until one of them beds the Homecoming Queen. Occasionally, a Starborn might make his rival's play a little more "challenging" - like telling the Homecoming Queen that his brother has gonorrhea right before their Big Date - but it's all in good fun.
The Starborn want to see ever-larger battles all over the Solar System. Each battle is a movement in the symphony, building to the eventual crescendo - the destruction of all space-traveling life. Once mankind has again been confined to his ancient homeworld, the round will be considered over.