Figure 1: A guitar model featuring a couple different wood materials: quilted maple on the body and walnut on the neck. The image is rendered using our comprehensive, volumetric, procedural model of wood. We simulate most of the significant wood features: growth rings, pores, rays, and growth distortions. Furthermore, our model can produce the anisotropic specular highlight arising from reflection from the subsurface fiber structure, as seen in the quilted maple figure. The fiber directions are automatically derived from the growth distortions. Model by Nikos Natsios. Inset: A photograph of real quilted maple.

A key challenge in game design is achieving balance between the strategies available to the players. Traditionally this has been done through playtesting, with its difficult requirements of time, labor, and interpretation of results. To make it quicker and easier to balance games, we propose a game-theoretic approach that automatically balances strategies based on a mathematical model of the game. Specifically, we model the balance problem as modifying a zero-sum game, using one variable per strategy, so that every strategy has an incentive to be employed. We begin with a special case where these variables affect player payoffs multiplicatively, and show that the simple Sinkhorn-Knopp algorithm can be used to balance the game. We then proceed to analyze the more general case where the variables have a monotonic effect on payoffs, and show that it is amenable to standard optimization methods. We give examples inspired by well-known game series including Pokemon and Warhammer 40,000.

Mechanics involving the roll of multiple dice---a "dice pool"---commonly appear in tabletop board games and role-playing games. Existing general-purpose dice pool probability calculators resort to exhaustive enumeration of all possible sorted sequences of rolls, which can quickly become computationally intractable. We propose a dynamic programming algorithm that can efficiently compute probabilities for a wide variety of dice pool mechanics while limiting the need for bespoke optimization. We also present Icepool, a pure Python implementation of the algorithm combined with a library of common dice operations.