Isn’t it easier just to flip a coin and set the order of 1, 2, and 3?
then you can have each person choose the size of their slice. or to promote symmetric shares:
person 1 cuts the pie in 3 person 2 chooses a piece person 3 chooses another piece person 1 receives the rest
this way person one has the incentive to cut it as fair as she possibly could to ensure the overall outcome maximizes her piece (well this assumes that each person wants to maximize the pie share)
of course this is under the premise that all 3 already agree fairness at the minimal includes each person having at least a slice of the pie, and that chance is a fair way of solving this problem.
Now that I’ve think about it more, even if we have the symmetric assumption (each person gets the same share), the pie share is not necessarily 1/n in that the utility of each person is different given a certain amount of pie.
for person 1 is not hungry at all, the pie is worth nothing to her and if she were to get 1⁄3 of the pie, she would not really even enjoy the consumption of it. Thus if person 1 were to get a tiny slice of pie, it could also be consider fair if we look at the symmetry in terms of utility instead of object. Well to achieve this, we can use a bidding system in which people bid for each infinitesimal part of the pie.
Either way, I believe that the argument that “each person gets 1/n of the pie is fair” is not sound because the worth of the pie is different for each person.