Suppose that a parent brings home from a trip $2n$ gifts of roughly equal value for his/her two children. The children get to choose one at a time which gifts they want. What is the fairest way to do this?

For instance, if $n=1$ then clearly one child chooses first (determined by a coin flip) and the other child chooses second. If we denote the children by 0 and 1, then this method is described by the choice sequence 01 (assuming, as I do from now on, that 0 choose first). Now suppose $n=2$. The choice sequence 0101 is clearly biased toward 0, since 0 has the first choice at the beginning and after both have chosen one gift. The fairest sequence by any reasonable criterion is 0110.

What about general $n$? If $n=2^k$, an argument can be made that the fairest sequence is the first $n$ terms of the Thue-Morse sequence (https://mathworld.wolfram.com/Thue-MorseSequence.html).

Another argument can be made that the fairest sequence $a_1,\dots, a_n$ is one that maximizes the value of $k$ for which the polynomial $(1-2a_1)x^{n-1} + (1-2a_2)x^{n-2}+\cdots+(1-2a_n)$ and its first $k$ derivatives vanish at $x=1$. (The Thue-Morse sequence does not have this property, though I cannot recall where I once saw this.)

Has this problem received any attention? What is a reference for the problem of maximizing $k$?