There are two natural ways to generalize the concept of Positional Game: one way is the (1) discrepancy version, where Maker wants (say) 90% of some hyperedge instead of 100%. Another way is the (2) biased version like the (1 : 2) play, where underdog Maker claims 1 point per move and Breaker claims 2 points per move.
Chapter VI is devoted to the discussion of these generalizations.
Neither generalization is a perfect success, but there is a big difference. The discrepancy version generalizes rather smoothly; the biased version, on the other hand, leads to some unexpected tormenting(!) technical difficulties.
The main issue here is to formulate and prove the Biased Meta-Conjecture. The biased case is work in progress; what we currently know is a bunch of (very interesting!) sporadic results, but the general case remains wide open.
We don't see any a priori reason why the biased case should be more difficult than the fair (1:1) case. No one understands why the general biased case is still unsolved.
The Biased Meta-Conjecture is the most exciting research project that the book can offer. We challenge the reader to participate in the final solution.
The biased Maker–Breaker and Avoider–Forcer games remain mostly unsolved, but we are surprisingly successful with the biased (1:s) Chooser–Picker game where Chooser is the underdog (in each turn Picker picks (s + 1) new points, Chooser chooses one of them, and the rest goes back to Picker).