A few days ago +John Baez wrote about famous Thue-Morse sequence, which looks like to be in limit, dual fair share sequence of geometric sequence (a_i) for small multiplier p. ( It would be great if someone can find the link to his post and put it in comments. I don’t know why, but I cannot find it looking at his stream, and Google+ search is not good, unfortunately…)

In the article below, Fair Share Sequence for Fibonacci sequence is mentioned.

Such sequence should exist for every sequence (a_i), but of course for many sequences partial differences for particular Number may be far from being fair 🙂 As mentioned below, if sequence (a_i) has with very large value at some point, and the rest is sequence, even summed, do not exceed this one element, there is large difference in partial values between two players. However fair share sequence exist, only the best way of division is not very fair…

Another possibility is to take into account families of sequences, like geometric sequences, arithmetic, or other, depending on various parameters. It seems that at least for geometric or arithmetic sequences, there’s limiting dual fair share sequence which is the best method of division of values for certain limiting cases ( like for geometric sequence with small multiplier).

Certainly there are families of sequences for which there is no limiting dual fair share sequence, and even for the same family, some limiting cases may do not show convergence to any particular sequence of fair share.

All the above may be generalized of course to many players, and probably even continuous functions….

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