Learning or estimating game models from data typically entails inducing separate models for each setting, even if the games are parametrically related. In empirical mechanism design, for example, this approach requires learning a new game model for each candidate setting of the mechanism parameter. Recent work has shown the data efficiency benefits of learning a single parameterized model for families of related games. In Bayesian games -- a typical model for mechanism design -- payoffs depend on both the actions and types of the players. We show how to exploit this structure by learning an interim game-family model that conditions on a single player's type. We compare this to the baseline approach of directly learning the ex ante payoff function, which gives payoffs in expectation of all player types. By marginalizing over player type, the interim model can also provide ex ante payoff predictions, as necessary for Bayes-Nash equilibrium approximation. We also leverage the interim model to compute new beneficial piecewise best-response strategies, without any additional sample data. We validate our method through a case study of a dynamic sponsored search auction. For both payoff accuracy and Nash-approximation error, the interim model matches the ex ante model on the trained range, and outperforms ex ante in extrapolation. Our case study demonstrates that Bayesian game-family models can support comprehensive mechanism design, and that through interim-stage modeling we can enhance expressivity and reliability.

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