A week later, you have been asked to develop a solver for a single$-$player non$-$deterministic game

called chumpkins$-$with$-$dice. This time, you have a trick up your sleeve, and your heuristic

eval$($x$)$ now returns the exact value of a state under optimal play. However, now the gremlin

has changed the search algorithm $($but hasn$$t told you$).$ Every time the search algorithm runs

it also rolls a dice and chooses with equal probability to either $1)$ add $3$ to every evaluation

value $2)$ take the square of every evaluation value $3)$ leave the evaluation values

unchanged. The search algorithm always outputs a policy that uses just one of these rules for

play on a given game tree. You test the algorithm empirically by running it repeatedly on many

game trees $($of your choosing!$)$ and averaging values of each policy that the gremlin$$s algorithm

returns but with your eval$($x$)$ function. You compare the average values of the gremlin$$s policies

to the true values computed by your evaluation function at each root state. When do you get

policies that play optimally and why?