On the Existence of Monotone Pure Strategy Equilibria in Bayesian Games

We extend and strengthen both Athey’'s (2001) and McAdams’' (2003) results on the existence of monotone pure strategy equilibria in Bayesian games. We allow action spaces to be compact locally-complete metrizable semilatttices and can handle both a weaker form of quasisupermodularity than is employed by McAdams and a weaker single-crossing property than is required by both Athey and McAdams. Our proof which is based upon contractibility rather than convexity of best reply sets —demonstrates that the only role of single-crossing is to help ensure the existence of

monotone best replies. Finally, we do not require the Milgrom-Weber (1985) absolute continuity condition on the joint distribution of types.

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