Can't this be solved to some sort of DP way of solving the sub problem?

Do the payout between 0 and 1 as the percentage of the amount.

With a range of 1 to 1 to pay off is obviously one

With a range of 1 to 2 the payout is .5

At three values it becomes more interesting. There are two strategies for the candidate either a binary search for the endpoints.

At four values you still have one level of binary search possible but after that it devolves down to the two value problem.

At five values. If the interviewer thinks the candidate would choose binary search and it becomes too too value problems on each side after removing the middle element.

There's definite problems with this but I wonder if he's already possible pay off matrix

It's possible it could help but it wouldn't be lead to a typical clean DP problem, because you need the full mixed strategy vectors at each level. That requires N real numbers that sum to 1, for each player.

Assuming you've found such a strategy for N, when you go to N+1 you still need to find the (N+1) element vector representing the probability that you select each number as your first guess, and you likewise need to know your opponent's probability vector for adversarially choosing a number. Once you guess at those vectors you can use your recursively built up DP sub-solutions to get the value of the game, but you are still stick with solving the optimization problem of finding those mixed strategy vectors for N+1, and will probably need something like CFRM or a similar technique to find them.