# How do evaluation functions influence the optimal sequence of moves?

Assume you have a game tree and the features $$(f_1, f_2, f_3,\ldots,f_n )$$ that describe the state of the game at any node. Also assume that you are using depth-limited minimax and always expand up to a fixed depth d. Say you have the following four evaluation functions:

$$F_1=w_1f_1+w_2f_2+\cdots+f_n$$.

$$F_2=w_1f_1^2+w_2f_2^2+\cdots+f_n^2$$.

$$F_3=\exp(w_1f_1^2+w_2f_2^2+\cdots+f_n^2)$$.

$$F_4=w_1f_1f_2+w_2f_2f_3+⋯+w_nf_{n-1}f_n$$.

The weights $$(w_1,w_2,\ldots,w_n)$$ are same for all evaluation functions. Then select all the statements that are correct:

1. The optimal sequence of moves for the MAX player would be the different for the function $$F_1$$ and $$F_2$$.

2. The optimal sequence of moves for the MAX player would be the different for the function $$F_2$$ and $$F_3$$

3. The optimal sequence of moves for the MAX player would be the different for the function $$F_1$$ and $$F_3$$

4. The optimal sequence of moves for the MAX player would be necessarily the same for all the functions

5. None of the above are correct

I need help with this problem that I came across. My understanding on this is that unless weights are changed the evaluation function should be the same. that is, in this case statement 2 and 3 are the only ones correct.

Any ideas?

• Why is the weight of the last term of $F_4$ $w_n$? Sep 29 '19 at 4:38
• edited. for first three functions there is no wn in the end. for last one there is. Sep 29 '19 at 5:49
• For example, if $n=4$, what is $F_1$ and $F_4$? Sep 29 '19 at 5:53