From the relevant wikipedia page: "envy-free divisions of connected intervals to 3 or more people cannot be found by any finite protocol." - I can agree that yo

From the relevant wikipedia page: "envy-free divisions of connected intervals to 3 or more people cannot be found by any finite protocol." - I can agree that your algorithm is quadratic, and therefore, that your algorithm is not envy-free. Your suggested algorithm seems to be unfair because the first party to choose has a clear advantage over the parties who choose later and did not cut: depending on the order of the parties there is also likely to be collusion. The immediately easiest (and politically realistic) way to game your system is to split yourself into multiple parties and then cut your allies/sock puppets a massive slice.