An infinite game with topological consequences

We introduce a two player infinite game for which a winning strategy is preserved by Σ-products and implies the existence of a winning strategy in Gruenhage's W-space game. A space exhibiting a winning strategy is collectionwise normal and countably paracompact. Consequently, a Σ-product of spaces is collectionwise normal if each space possesses a winning strategy. This is a generalization of a well-known result on Σ-products of metrizable spaces. Finally we show certain uniform box products have a winning strategy.