This is a puzzle. I dont know if you could describe it as philosophical.

Here’s a fair way to cut a cake into two portions, one for A and one for B. A makes the cut and B chooses. So A has a reason to make the two halves the same size, otherwise B will take the bigger slice.

The question is, can you describe a similarly fair way to cut a cake into three portions, for A, B and C ?. If A cuts and chooses, then A can deliberately cut the cake so that B gets a bigger slice. Someone gets to make a cut, or possibly the first cut. Someone gets to choose the first slice, for him or herself or for someone else, and similarly with the second slice.

You’re not allowed to spin a coin or throw dice, or do any action that involves chance.

Can this be done, without either A, B or C having possible grounds for complaint?

Yes it can.

This is an example of a Fair Division Problem. Division is fair if it is proportional (each of n sharers gets 1/nth) and envy-free (no sharer feels that another gets a bigger share than she did).

I wouldn’t describe it as a specifically philosophical problem, but fair division is crucial, for instance, to (amicable) divorce settlements or territorial divisions/treaties, and so is of interest to lawyers, politicians, and negotiators, as well as to mathematicians/logicians who think up solutions.

Sometimes we are dealing with continuous goods (divisible into arbitrarily small pieces, as with cakes); sometimes with discrete goods (indivisible, as with cats or cars) where we have to come up with a metric for comparison.

You describe a fair 2-person division for continuous goods (one divider, one chooser).

The 3-. 4-, 5,- … n-person procedure is an extension of this in which we have one divider/several choosers (Lone Divider method), or one chooser/several dividers (Lone Chooser method), as well as a more complicated Last-Diminisher method.

Here is the simplest, Lone-Divider method:

A divides cake into three pieces, X, Y and Z.

B and C each states her choice:

Case 1. B chooses X, C chooses Y (or vice-versa).

So, B and C get their chosen slice, A gets Z

Case 2. B and C both choose X (or Y).

So, X (or Y) is merged with say Y (or X) and B/C do a 2-person divide-and-choose on XY.

A gets Z.