It is well known that the sets of the first category in a metric space can be described using the so-called Banach-Mazur game. We will show that if we change the rules of the Banach-Mazur game (by forcing the second player to choose large balls) then we can describe sets which can be covered by countably many closed uniformly porous sets. A characterization of σ-very porous sets and a sufficient condition for σ-porosity are also given in the terminology of games.

We show a general method of construction of non-$\sigma$-porous sets in complete metric spaces. This method enables us to answer several open questions. We prove that each non-$\sigma$-porous Suslin subset of a topologically complete metric space contains a non-$\sigma$-porous closed subset. We show also a sufficient condition, which gives that a certain system of compact sets contains a non-$\sigma$-porous element. Namely, if we denote the space of all compact subsets of a compact metric space $E$ with the Vietoris topology...