Fair Division: the Continuous Case the Cut-and-Choose Method

Before we proceed, we need a definition and a few assumptions

Original object (cake) Ava Bert Carlos 1 or 100% Ava must value the whole cake as 1 or 100% 1 or 100% Bert must value the whole cake as 1 or 100% 1 or 100% Carlos must value the whole cake as 1 or 100%

STEP 1: “A” cuts into equal parts (for herself), “B” & “C” place values (for themselves) Ava Bert Carlos ½ = .5 ½ = .5 These are the values Bert could place on the pieces…he does NOT have to see them as equal .4 .6 These are the values Carlos could place on the pieces…he does NOT have to see them as equal .8 .2

STEP 2: “B” chooses the larger piece (to him) & “A” gets other piece Ava Bert Carlos Ava keeps the piece Bert did not choose; to her, both were exactly 1/2 1/2 1/2 Ava gets this piece Bert chooses this piece Ava gets this piece Bert chooses this piece Bert chooses the biggest piece from his perspective .4 .6 Ava’s Bert’s These are the values Carlos still places on the pieces…he does NOT have to see them as equal .8 .2

STEP 3: “A” & “B” cut their respective piece into equal parts (for themselves) Ava Bert Carlos Ava’s piece started as 1/2 of the original, so… 1/2 ÷ 3 = 1/6 1/6 1/6 1/6 Bert’s piece started as .6 of the original for him, so… .6 ÷ 3 = .2 .2 .2 .2 These are the values Carlos could place on the pieces…he does NOT have to see them as equal .5 .1 .2 .05 .05 .1 These must sum to .8, the total value Carlos placed on this piece These must sum to .2, the total value Carlos placed on this piece

Ava Bert Carlos STEP 4: “C” chooses 1/6 1/6 1/6 .2 .2 .2 .05 .1 .5 .1 Ava keeps the 2 pieces Carlos did not choose. Ava’s piece started as 1/2 of the original, so 1/2 ÷ 3 = 1/6 1/6 1/6 1/6 Bert keeps the 2 pieces Carlos did not choose Bert’s piece started as .6 of the original, so .6 ÷ 3 = .2 .2 .2 .2 Ava’s Bert’s Carlos chooses 1 piece from Ava and 1 piece from Bert. Carlos chooses the two pieces with the largest value .05 .1 .5 .1 .2 .05 These must sum to .8 These must sum to .2

Final Division – Everybody gets at least their fair share Ava Bert Carlos 1/6 + 1/6 = 2/6 = 1/3 >= 1/3 1/6 1/6 1/6 .2 + .2 = .4 = 40% >= 1/3 .2 .2 .2 .5 + .1 = .6 = 60% >= 1/3 .1 .5