This paper explores the question of whether it's possible to discover a well-defined property of game systems that corresponds to what game designers and players mean by the term ``depth.'' We propose a measurable property of a game's formal system, which we call `d', that corresponds to the capacity of a game to absorb dedicated problem-solving attention and allow for sustained, long-term learning. To define this property we develop a formal model that measures how susceptible a game is to partial solutions under conditions of steadily increasing computational resources. We then sketch out several directions for using the model to investigate questions about the structural properties of games that produce these effects.

Game balancing is the fine-tuning phase in which a functioning game is adjusted to be deep, fair, and interesting. Balancing is difficult and time-consuming, as designers must repeatedly tweak parameters, and run lengthy playtests to evaluate the effects of these changes. If designers could receive immediate feedback on their designs, they could explore a vast space of variations, and select only the most promising games for playtesting. Such automated design feedback has been difficult to achieve, as there is no mathematical formulation of game balance that unifies many of its forms. We argue for a formulation in which carefully restricted agents are played against standard agents. We develop this restricted-play balance framework, and evaluate its utility by building a tool capable of calculating measures of balance for a large family of games. By applying this tool to an educational card game, we demonstrate how the framework and tool allow designers to rapidly evaluate and iterate on the balance of their games.