  Nearly all simulation-based games have environment parameters that affect
incentives in the interaction but are not explicitly incorporated into the game
model. To understand the impact of these parameters on strategic incentives,
typical game-theoretic analysis involves selecting a small set of
representative values, and constructing and analyzing separate game models for
each value. We introduce a novel technique to learn a single model representing
a family of closely related games that differ in the number of symmetric
players or other ordinal environment parameters. Prior work trains a
multi-headed neural network to output mixed-strategy deviation payoffs, which
can be used to compute symmetric $\varepsilon$-Nash equilibria. We extend this
work by making environment parameters into input dimensions of the regressor,
enabling a single model to learn patterns which generalize across the parameter
space. For continuous and discrete parameters, our results show that these
generalized models outperform existing approaches, achieving better accuracy
with far less data. This technique makes thorough analysis of the parameter
space more tractable, and promotes analyses that capture relationships between
parameters and incentives.
