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We use machine learning to uncover regularities in the initial play of matrix games. We first train a prediction algorithm on data from past experiments. Examining the games where our algorithm predicts correctly, but existing economic models don't, leads us to add a parameter to the best performing model that improves predictive accuracy. We then observe play in a collection of new "algorithmically generated" games, and learn that we can obtain even better predictions with a hybrid model that uses a decision tree to decide game-by-game which of two economic models to use for prediction.