Absorption time and absorption probabilities for a family of multidimensional gambler models

For a family of multidimensional gambler models we provide formulas for the winning probabilities (in terms parameters system) and distribution game duration eigenvalues underlying one-dimensional games). These were known case - initially proofs purely analytical, later probabilistic construction has been given. Concerning duration, in many cases our approach yields sample-path constructions. We heavily exploit intertwining between (not necessary) stochastic matrices (for results), notion Siegmund duality winning/ruin probabilities), Kronecker products.