Generating random graphs in biased Maker-Breaker games

We present a general approach connecting biased Maker-Breaker games and problems about local resilience in random graphs. We utilize this approach to prove new results and also to derive some known results about biased Maker-Breaker games. In particular, we show that for b=o(n), Maker can build a pancyclic graph (that is, a graph that contains cycles of every possible length) while playing a (1:b) game on E(K_n). As another application, we show that for b=Θ(n/lnn), playing a (1:b) game on E(K_n), Maker can build a graph which contains copies of all spanning trees having maximum degree Δ=O(1) with a bare path of linear length (a bare path in a tree T is a path with all interior vertices of degree exactly two in T).