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.