Two-Player Ruin with Unequal Bet Sizes

Alice and Bob play a sequence of fair-coin games. In each round, Alice wagers $\$1$ and Bob wagers $\$2$. If heads, Alice gains $\$1$ from Bob (and Bob loses $\$1$); if tails, Bob gains $\$2$ from Alice (and Alice loses $\$2$). Alice starts with $\$4$ and Bob starts with $\$6$. The game ends when either player is ruined (reaches $\$0$). Note that the total wealth is not conserved — the net transfer is asymmetric. What is the probability that Alice is ruined?