In this paper, we propose a competitive game in which a player wearing an augmented reality (AR) head-mounted display (HMD) and a player not wearing an HMD share not only a virtual environment but also the structure of a physical environment. Through the proposed game, we explore the interaction between players in an online multiplayer game using an AR HMD, which is enjoyable and has a high social presence. For this exploration, we created a game design that actively utilizes a physical environment and the asymmetry between players wearing and not wearing an HMD. We implemented the designed game and conducted a user study (n=14) to evaluate the game using the Game Experience Questionnaire and an our own questionnaire. The results revealed that the players had a highly positive affect toward the game and showed a high social presence. We also obtained insights into how to make the game more interesting and to increase social presence of players.