Forcing, games and families of closed sets

We propose a new, game-theoretic, approach to the idealized forcing, in terms of fusion games. This generalizes the classical approach to the Sacks and the Miller forcing. For definable ($\mathbf{\Pi}^1_1$ on $\mathbf{\Sigma}^1_1)$\sigma$-ideals we show that if a$\sigma$-ideal is generated by closed sets, then it is generated by closed sets in all forcing extensions. We also prove an infinite-dimensional version of the Solecki dichotomy for analytic sets. Among examples, we investigate the$\sigma$-ideal$\E$generated by closed null sets and$\sigma$-ideals connected with not piecewise continuous functions.