You can see this "game" in Pride and Prejudice when you look at the various women and the reasons they each marry whom they marry. Charlotte plays the game by marrying the first eligible prospect who proposes, and probably even seeks his attention, because marriage to Collins is way better than being destitute. Elizabeth plays the game a little differently. She turns down an amazing proposal from an incredibly wealthy man in the hopes that she can marry for true love eventually. She doesn't realize at the time that she actually will be the big winner of the game when she marries that wealthy man for true love!
During the time period, that was probably accurate. Women could not find work like they can today. The only way for a woman to be supported was through her family and marriage. A woman would begin to feel like she was a burden to her family if she was not married by a certain age. Society expected a young woman to marry and have children. A woman who did not would be looked down upon. Independence for a woman in Jane Austen's time was more about running your own household rather than having your own career.
Certainly that was true in Austen's own time, when marriage, especially among the upper classes, was seen as a way to achieve economic stability. Austen's novels are full of women who are encouraged to marry a man with good economic prospects, and the reverse was certainly true of men, who could certainly improve their prospects by marrying a wealthy man. The obvious problem here, the tension between marrying for love and marrying for economic reasons, is one of the major themes of her work. For women with few economic prospects, marriage was one avenue to social mobility. If it was a game, it was a high-stakes game.