Three-Player Elimination Game

Three players $A$, $B$, $C$ hold tokens: $A$ has $2$, $B$ has $1$, $C$ has $1$ (total $4$). Each round, two players are chosen uniformly at random (from the $\binom{3}{2} = 3$ pairs) and they play a fair game: each of the two has probability $\tfrac{1}{2}$ of winning one token from the other. A player with $0$ tokens is eliminated. Play continues until one player holds all $4$ tokens.

What is the probability that player $A$ (who starts with $2$ tokens) ends up with all $4$?