At first sight the question seems impossible. There is definitely not enough information to decide who plays who and when but, believe it or not, there in enough information to answer the particular question of who lost in the ninth game.

Firstly lets see how many games there were altogether. To do this we an simply add together the number of games played by each individual, and divide that total by two, since there are two players in each game. This tells us that there were 11 games in total. Because of the way they have of rotating players, even if a player loses, they will be back at the table three games later. The key now is to look at Billie, who has played the least games. If we assume they lost every game they played they would sill play roughly a third of the games. Say Billie played in game 1, they would play again in game 4, game 7 and game 10, but that is too many games. Similarly if they played and lost in game 2, they would reappear for game 5, game 8 and game 11. Still too many. The only possibility is that Billie played and lost in game 3, game 6 and game 9.

Billie lost in the ninth game.