# Unable to understand V* at infinite time horizon using Bellman equation for solving MDP

I've been following the Berkeley cs188's assignment (I'm not taking the course). Currently, they don't show the solution in the gradescope unless I get it correct.

My reasoning was

$$V^*(a)$$ = 10 fixed, because the optimal action is to terminate and receive the reward 10.

$$V^*(b) = 10 \times 0.2 = 2$$ using Bellman optimality eqn $$V^*(s) = R(s)+ \gamma \ \rm{max}_{a} \sum_{s'} P(s'|s,a) V^*(s')$$, where the optimal actionfrom b is to left.

Similarly, I get $$V^*(c) = 10 \times (0.2)^2 = 0.4$$

For the state $$d$$, it is optimal to move to the right and exit at $$e$$ to receive 1, therefore $$V^*(d) = 1 \times 0.2 = 0.2$$.

And $$V^*(e) = 1$$ fixed.

However, there autograder says it's incorrect and doesn't show explanation. Can anyone explain what the right approach or answer is?