Title: A definitive improvement of a game-theoretic bound and the long tightness game,
Abstract: The main goal of the paper is the full proof of a cardinal inequality for a
space with points $G_\delta $, obtained with the help of a long version of the
Menger game. This result, which improves a similar one of Scheepers and Tall,
was already established by the authors under the Continuum Hypothesis. The
paper is completed by few remarks on a long version of the tightness game.