Keywords: Learning theory, Representation Learning, algorithmic game theory, Functional form games, matrix decomposition
TL;DR: A decomposition method that represents a game as a sum of planar embeddings
Abstract: The focus on equilibrium solutions in games underemphasizes the importance of understanding their overall structure. A different set of tools is needed for learning and representing the general structure of a game. In this paper we illustrate "Principle Trade-off Analysis" (PTA), a decomposition method that embeds games into a low dimensional feature space and argue that the embeddings are more revealing than previously demonstrated. Here, we develop an analogy to Principal Component Analysis (PCA). PTA represents an arbitrary two-player zero-sum game as the weighted sum of pairs of orthogonal 2D feature planes. We show that each of the feature planes represent unique strategic trade-offs (cyclic modes) and truncation of the sequence provides insightful model reduction. We demonstrate the validity of PTA on a pair of games (Blotto, Pokemon). In Blotto, PTA identifies game symmetries, and specifies strategic trade-offs associated with distinct win conditions. These symmetries reveal limitations of PTA unaddressed in previous work. For Pokemon, PTA recovers clusters that naturally correspond to Pokemon types, correctly identifies the designed tradeoff between those types, and discovers a rock-paper-scissor (RPS) cycle in the Pokemon generation type - all absent any specific information except game outcomes.
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