Having a random person cut slices and then randomly allocating slices is not a bad idea, but has a problem too. That procedure is unfair to those who value reliably-sized slices.
Say A and B use this procedure. A and B both hate each other and want to make the pizza selection turn out bad for the other person. But while A is just normally hungry, B is at death’s door from starvation and needs at least 1/3 of a pizza to live. This procedure gives a 1/4 possibility that A can starve B to death, by letting A cut a tiny piece and a huge piece, and randomly assigning A the huge piece. Whereas no matter what B does, even if he is lucky enough to cut and receive a huge piece, A is not going to die. So A has the advantage. But a truly fair procedure would assure both A and B at least 1/2 a pizza, a property that would assure that B would live.
