In episode 1 of Bet, titled 'Game On,' the main game played is called 'Skirmish.' It involves each player being dealt 7 cards, with the game consisting of 7 rounds. In each round, players play one card, and the high card wins. Yumeko, the protagonist, uses a clever strategy to ensure her opponent, Mary, does not cheat.

Yumeko ensures Mary doesn't cheat by playing her cards blind after initially betting 10,000 dollars. She also uses a mirror on her mother's poker chip to observe any hints Ryan might be giving Mary, thus preventing cheating.

Yumeko participates in the game as part of her strategy to climb the ranks of the gambling leaderboard at St. Dominic's Prep. Her ultimate goal is to confront a man named Ray, a parent of one of the school's legacy students, by becoming a top gambler.

During the game, Yumeko offers to donate half of her winnings to Ryan if she wins, which would set him free from his 'pet' status. This offer is a strategic move to motivate Ryan and also to challenge Mary.

Yumeko's character influences the dynamics of the game by her bold and unpredictable moves. She not only challenges Mary but also uses her intelligence and strategic thinking to outmaneuver her opponent, showcasing her determination and skill as a gambler.