Need to find expectation of the prize in the game

Question

Player plays game, where he starts with $i$ tokens, and there are also $n$ tokens "at all".
Game lasts $t$ rounds at most.
At every round player wins one more token with probability $\frac{1}{3}$, or he loses one (with probability $\frac{2}{3}$).
There are 3 cases when the game ends:
1)The player is out of his tokens (which were $i$ by the start)
2)Player got all the n tokens
3)There were $t$ rounds played
When game ends, player gets his prize with size $prize = 2^{result}$, where $result$ is amount of tokens player got by the end of the game.
Need to find expectation ($E[prize]$) of the prize.
Please, could you give me any ideas of solving the problem? I have tried to use binomial distribution to solve that problem, but it doesn't fit really good as it seems to me.
By the way, this problem was met in my "algorithm and data structures" course (which is over already), so may be the problem has some decision related to one?
Thank you a lot!

Answer 1

Note that $2^k$ with $k$ the number of tokens is a martingale. If the player has $k_n$ tokens before a round, the expectation of $2^{k_{n+1}}$ with $k_{n+1}$ the number of tokens after the round is

$$ \frac132^{k_n+1}+\frac232^{k_n-1}=2^{k_n}\;. $$

Thus the expected prize is $2^i$.