Any mathematical claim about the fairness properties of Thue-Morse must be qualified with technical assumptions about the underlying game, e.g., how value would be distributed across the sequence.
Further, discussing any fairness properties, it is important to distinguish between the following:
- Fairer: reduces the advantage disparity.
- Fairest: reduces the advantage disparity as much as possible.
- Perfectly Fair: reduces the advantage disparity to zero.
Remarks:
- Each condition is a stronger statement that the previous.
- Fairer and fairest do not necessarily imply that the disparity has been reduced to zero.
- Fairer and fairest are relative statements. Fairer requires comparison to at least one other alternative (e.g., simple alternation). A reasonable claim of “fairest” should compare to a reasonably large and comprehensive set of alternatives.
- As a quirk of how we use the language, “perfectly fair” is often just shortened to “fair”, which makes the phrase appear linguistically weaker than “fairer” or “fairest”, while it is actually the strongest claim.
One might be able to argue that Thue-Morse is fairer or even fairest (among some group of alternatives), but that is not enough to show that it is perfectly fair.
One can easily construct examples where Thue-Morse is not perfectly fair, even if it can be shown to be fairest among several alternatives. For example, dividing an odd number of equally valued items.
5, 4, 3, 2, 2
The perfectly fair (and hence fairest) turn sequence is ABABB. Thue-Morse and the greedy scheme that you described produce sub-optimal results.
Finding a fair way to divide a finite set of items is also known as a partition problem, which is NP-complete. Although this problem has efficient practical algorithms (due to the existence of a pseudo-poly-time dynamic programming solutions), it’s NP-hardness suggests that simple greedy algorithms (or a fixed sequence like Thue-Morse) are unlikely to be optimal in general.