I started reading "Winning Ways Volume 1" by Conway and I'm confused at the evaluation and notation for the ski-jump positions in Figure 12. on page 9 as shown in the picture.
An example of the notation was given for the 4th board in the last row (with value $2\frac{1}{2}$)
Left's man may jump over Right's, if he wishes. If he does so, the value will be $4-2 = 2$, which is better than the value $3-2 = 1$ he reaches by sliding one place East. If, on the other hand, Right has the move, it will be to a position of value $4-1 = 3$. So the position has value $\{2|3\} = 2\frac{1}{2}$
If we now want to apply this to the figure in question and look at the root of the tree, then: If Left starts, he moves one position eastwards and has $4$ moves left, whereas Right has $5$ positions westwards, so $4-5=-1$ (if left moves more than one, then his move is even smaller). If Right starts and moves one position to the west, then he has $4$ moves left, and Left has $5$ moves, which is $4-5=-1$, so the game should be $\{-1|-1\}$ and not $\{0|1\}$.
If confuses me even more if I try to evaluate other boards, which e.g. arrive at best moves of $\{-1|\frac{1}{2}\}=0$ even though there is no possibility of a jump within the next move. How do the authors calculate the best moves for the ski-jump?
For each board in the first four rows, after the next move we will reach one of the boards in the next row down, so in order to work out the value of the board in the top row you need to know the values of the two boards in the second row. In order to work out these you need to know the values of the boards in the third row, and so on.
The values of the boards in the bottom row are easy to work out because after the next move no jumps will be possible, so the value can be computed as the quotation explains, and then you use these values to work out the values of the boards in the next row up.