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I have to admit that I find this result unsatisfying. I don't see why mathematicians consider that trimming the remaining pieces cannot trigger envy from earlie

I have to admit that I find this result unsatisfying. I don't see why mathematicians consider that trimming the remaining pieces cannot trigger envy from earlier cake splitters. (They call it dominating in the paper.) Once the remaining slice are modified, it's unclear to me why the original cutter would not envy the new slices. For the three persons scenario and using integers, the first cutter divides the number 3 into 1/1/1. If one of the other participants change the ratio of two slices to 0.5/1.5, it's clear that the original cutter would not be happy to get only 1 in the new 1/0.5/1.5 cuts. The original decision was contingent on the original split. In fact, given this, it seems that being the last to act is the best position as other are locked into the previous choices. I other words, being last allow you to profit from any possible mistakes previous cutters made. This is especially egregious if each have a different way to value things, since once A has chosen, B could modify the remaining pieces so that A preference is now different.

jjnoakes 8y ago
> It's unclear to me why the original cutter would not envy the new slices The original cutter gets second choice of the "trimmed" section's 3-way split, so your 1/1/1 doesn't go to 1/0.5/1.5, it goes to 1.2/1.2/0.6 or something, but (crucially) the original 1.0 guy gets to pick 0.2 more before the 0.5 guy gets to pick the remaining 0.1.
jm_l 8y ago
No participant is incentivized to alter the cake if they agree that it's value is already equally split in 3. The person making the trim is also not incentivized to make it 1.5/0.5, because they don't get first choice of those slices. When dividing cake, if one slice is bigger it may be because there is less frosting on it when frosting is desirable, or it might be because the slicer thought they could trick the others. Then when someone contests this division it's either because their preferences don't match, or they are correctly calling out the trick. In short, the algorithm is meant to deal with differing preferences and incentivized lying.
kgwgk 8y ago
> If one of the other participants change the ratio of two slices to 0.5/1.5 This is not what happens. From the point of view of the first cutter, he got a piece he values at 1 and the others got pieces he values at 1-x and 1. Then they make the repartition of the remaining x. The person who got previously the piece that the first cutter valued at 1-x chooses first, but can’t get more than 1 (as valued by the first cutter) in total. And the first cutter chooses before the other person who got a piece he values at 1, so he cannot end worse off.
- norswap 8y ago
  Thanks I was hung up on why "it's fair for Charlie because he got to choose before Bob" part, but this explains it nicely.

jjnoakes 8y ago

> It's unclear to me why the original cutter would not envy the new slices The original cutter gets second choice of the "trimmed" section's 3-way split, so your 1/1/1 doesn't go to 1/0.5/1.5, it goes to 1.2/1.2/0.6 or something, but (crucially) the original 1.0 guy gets to pick 0.2 more before the 0.5 guy gets to pick the remaining 0.1.

jm_l 8y ago

No participant is incentivized to alter the cake if they agree that it's value is already equally split in 3. The person making the trim is also not incentivized to make it 1.5/0.5, because they don't get first choice of those slices. When dividing cake, if one slice is bigger it may be because there is less frosting on it when frosting is desirable, or it might be because the slicer thought they could trick the others. Then when someone contests this division it's either because their preferences don't match, or they are correctly calling out the trick. In short, the algorithm is meant to deal with differing preferences and incentivized lying.

kgwgk 8y ago

> If one of the other participants change the ratio of two slices to 0.5/1.5 This is not what happens. From the point of view of the first cutter, he got a piece he values at 1 and the others got pieces he values at 1-x and 1. Then they make the repartition of the remaining x. The person who got previously the piece that the first cutter valued at 1-x chooses first, but can’t get more than 1 (as valued by the first cutter) in total. And the first cutter chooses before the other person who got a piece he values at 1, so he cannot end worse off.

norswap 8y ago
Thanks I was hung up on why "it's fair for Charlie because he got to choose before Bob" part, but this explains it nicely.