Digital gaming has the potential to foster togetherness through shared, challenging, and immersive activities. In cooperative games, asymmetry of information impacts social interactions, yet its effects on player experience are not yet fully understood. To this end, we propose a framework for studying asymmetry of information in game design, expanded from previous work on asymmetry, and apply it to the design and development of a cooperative game prototype. In a study with ten pairs of players, we examined how asymmetry of information affected the player experience, through outcomes such as connectedness, communication, fun, and challenge, through a mixed-methods approach. Findings showed that players perceived differences in the distribution of information, influencing their interactions during gameplay. Future research could focus on enhancing the framework to capture the granularity of information dynamics and investigating the asymmetry of information in different scenarios other than two-player cooperative games, offering deeper insights into gaming dynamics.