Three Dollar Table

Player A starts with 1 dollar and player B with 2. Each round the loser pays the winner 1 dollar, and A wins any given round with probability 2/3, independently of the others. They play until one of them is broke.

Tracking A's fortune gives a chain on ${0,1,2,3}$:

from	to 0	to 1	to 2	to 3
0	1	0	0	0
1	1/3	0	2/3	0
2	0	1/3	0	2/3
3	0	0	0	1

Starting from state 1, what is the probability the chain is absorbed at 3, so that A takes everything?

Answer format: round to 3 decimal places.