Alice is taking CS$234$ and has just learned about the Q$-$values. She is trying to explore a large nitehorizon MDP with $\backslash$gamma $=1.$ The transitions are deterministic and QH$+1($s$,$ a$)=0$ for all s$,$ a$.$ To

help her with her MDP you tell her the optimal policy $\backslash$pi

$$

$($s$,$ t$),$ dened in every state s and timestep

t$,$ that Alice should follow to maximize her reward. Denote with Q$$

t

$($s$,$ a$)$ the Q$-$values of the optimal

policy upon taking action a in state s at timestep t$.$

A$)$ First Step Error

In the rst timestep t $=1$ Alice is in state s$1$ and chooses action a$,$ which is suboptimal. If she then

follows the optimal policy from t $=2$ until the end of the episode, what is the value of this policy

compared to the optimal one? Express your result only using Q$$

$1$

$($s$1,).$