Abstract We investigate random walks driven by an asymmetric dual-choice game inspired by the rock-paper-scissors game. In this game, Player A selects either paper or scissors, while Player B chooses either rock or scissors. The random walk is generated by the game’s outcomes: a player advances by + 1 step upon winning, while the opponent regresses by − 1 step. Each player follows a strategy categorized as either a mixed strategy or an adaptive strategy optimized via a genetic algorithm (GA) that analyzes the opponent’s past hands. According to game theory, this game possesses a mixed strategy Nash equilibrium. We show that the game-driven random walk exhibits a biased random walk with correlated steps when one player employs a GA-based strategy and the opponent follows a mixed strategy. We find that when the opponent plays according to the probabilities at the Nash equilibrium, the step correlations disappear. These findings highlight the interplay between strategic adaptation and Nash equilibrium in game-driven random walks.