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