Reny's new exsistence theorem

In the plenary session on the third day of BWGT conference (link), Professor Philip Reny talked about his new research on monotone pure strategy equilibria:
Title: On the Existence of Monotone Pure Strategy Equilibria in Bayesian Games (link to pdf)
Abstract: We generalize 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 semilattices and type spaces to be partially ordered probability spaces. Our proof is based upon contractibility rather than convexity of best reply sets. Several examples illustrate the scope of the result, including new applications to multi-unit auctions with risk-averse bidders.
According to Prof. Reny, while the topic of the paper is related to many fields such as mathematical economics, mechanism design, and auctions, there are two seminal papers that strongly motivated his research. Athey (2001) first establishes the sufficient conditions to guarantee the existence of monotone pure strategy equilibria in Bayesian games with one-dimensional and totally ordered type and action spaces. The key condition is a Spence-Mirlees single-crossing property. McAdams (2003) extends Athey's analysis to multi-dimensional and partially ordered spaces.

Prof. Reny succeeded to derive weaker conditions than McAdams in Bayesian games with multi-dimensional strategy spaces, and also extend to the infinite type and action spaces. The key insight is to use a fixed point theorem derived by Eilenberg and Montgomery (1946) instead of Kakutani's (used by Athey) or Glicksberg's (used by McAdams) ones. The latter two theorems require best reply sets to be convex while the former requires only contractibility, which turns out to be (almost) automatically satisfied in Bayesian games.
His main result says the following:
Theorem: (Under some conditions) If, whenever the other players employ monotone pure strategies, each player's set  of monotone pure-strategy best replies is nonempty and join-closed, then a monotone pure strategy equilibrium exists.
Note that a subset of strategies is join-closed if the pointwise supremum of any pair of strategies in the set is also in the set.

The idea of join-closedness (in the different context, though) recently showed up when I discussed my jointwork on the structure of stable matchings with co-authors. It may have some connection...

Susan Athey (2001), "Single Crossing Properties and the Existence of Pure Strategy Equilibria in Games of Incomplete Information,"  Econometrica, Vol. 69: 861-889.
Samuel Eilenberg and Deane Montgomery (1946), "Fixed Point Theorems for Multi-Valued Transformations," American Journal of Mathematics, Vol. 68: 214-222.
David McAdams (2003), "Isotone Equilibrium in Games of Incomplete Information," Econometrica, Vol. 71:1191-1214.

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