Alternating die game with unequal winning faces

Two players take turns rolling one fair die, with player 1 going first. Player 1 wins immediately on rolling a $1$ or $2$; player 2 wins immediately on rolling any of $3,4,5,6$. Turns alternate until somebody wins. If $p_1$ and $p_2$ are the two players' winning probabilities, what is the larger of the two?

$\tfrac47$. Player 1 wins at once with probability $\tfrac13$; otherwise both players must miss (probability $\tfrac23\cdot\tfrac13=\tfrac29$) and the game restarts. So $p_1=\tfrac13+\tfrac29p_1$, giving $p_1=\tfrac37$ and $p_2=\tfrac47$.