  Gambler's ruin estimates can be viewed as harmonic measure estimates for
finite Markov chains which are absorbed (or killed) at boundary points. We
relate such estimates to properties of the underlying chain and its Doob
transform. Precisely, we show that gambler's ruin estimates reduce to a good
understanding of the Perron-Frobenius eigenfunction and eigenvalue whenever the
underlying chain and its Doob transform are Harnack Markov chains. Finite
inner-uniform domains (say, in the square grid $\mathbb Z^n$) provide a large
class of examples where these ideas apply and lead to detailed estimates. In
general, understanding the behavior of the Perron-Frobenius eigenfunction
remains a challenge.
