This paper provides a sufficient condition for existence and uniqueness of equilibrium, which is in monotone pure strategies, in a broad class of Bayesian games. The argument requires that the incremental interim payoff - the expected payoff difference between any two actions, conditional on a player's realised type - satisfies two conditions. The first is uniform strict single-crossing with respect to own type. The second condition is Lipschitz continuity with respect to opponents' strategies. Our main result shows that, if these two conditions are satisfied, and the bounding parameters satisfy a particular inequality, then the best response correspondence is a contraction, and hence there is a unique equilibrium of the Bayesian game. Furthermore, this equilibrium is in monotone pure strategies. We characterize the uniform monotonicity and Lipschitz continuity conditions in terms of the model primitives. We also consider a number of examples to illustrate how the approach can be used in applications.
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