The eyes are a rich channel for non-verbal communication in our daily interactions. We propose social gaze interaction as a game mechanic to enhance user interactions with virtual characters. We develop a game from the ground-up in which characters are designed to be reactive to the player's gaze in social ways, such as getting annoyed when the player seems distracted or changing their dialogue depending on the player's apparent focus of attention. Results from a qualitative user study provide insights about how social gaze interaction is intuitive for users, elicits deep feelings of immersion, and highlight the players' self-consciousness of their own eye movements through their strong reactions to the characters.