$A$ and $B$ play a game. $A$ has $n_{A}\geq 0$ dollars and player B has $n_{B}\geq 0$ dollars. A fair coin is tossed. If it is heads, $B$ gives a dollar to $A$. If tails, $A$ gives a dollar to $B$. The game stops when one of the players loses all of his/her money. What is the average number of steps until the end of the game?
(Hint: Let $m_{j}$ be the expected number of steps required, when player $A$ has $j$ dollars and try to set up a recursive equation for $m_{j}$. Find out if the recursive equation is solvable.)
Thank you for your help.