Study their behaviors. Observe their territorial boundaries. Leave their habitat as you found it. Report any signs of intelligence.

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Friday, March 05, 2021

Fairly Dividing Cake or Chores

There are entire books on the topic of fair cake-cutting.  "Fair cake-cutting is the subject of intense research in mathematics, computer science, economics and political science." As a parent of three self-interested daughters, I'm very interested in this topic!

But I'm not interested in procedures that re-divide shares. A cake should have N contiguous shares, period. Don't ruin the cake.

And I'm not interested in procedures with a moving knife. Nice theory, but players will argue about where and when the knife stopped. (And a moving knife doesn't work on chores, which are somewhat discrete rather than continuous.)

And I'm not interested in envy-free. Players gonna envy. I just want no complaints of favoritism or collusion or election-rigging.

And I don't want a referee. If we can appeal to a fair authority, there is no need for a procedure!

And I want to minimize randomness. Players have their own utility functions, and that's what creates gains from trade.

I've come up with a procedure that I think is better by the above criteria than anything I could find during a hasty scan of the existing scientific literature. The best procedure I could find is this one by Dan Hulme in 2014. But his technique leads to unfairness complaints, as your position in the circle can totally exclude you from the best shares.

The 2nd-best procedure I found is Who Shall Have This. It's inferior to mine because shares are assigned randomly, leaving no room to exploit heterogenous preferences about the shares.

The Goldilocks Veto Procedure

"Player" below means someone with no assigned share yet. Players are assigned a fixed order, e.g. by age, or randomly. It only (and barely) matters that the order is the same during the cutting as during the choosing.

1. While the cake isn't yet cut into N shares, the next player cuts any unassigned share into two shares. (They rest of the rules will make them try to cut out a fair 1/N-size share.)

1A. If it's not obvious which of the two shares is smaller, the players vote on it. If it's a tie, skip rule 2. 

2. Regarding the smaller share, the non-cutting players can abstain, or veto in one of two ways:

2A. A veto of Too Small forces the cutter to take the share. But:

2B. Any vetoes of Too Big gives the share randomly to one of the Too-Big vetoers.

3. The last player picks a share instead of cutting.

4. The remaining players each pick a share, in the same order as they cut.


  • This works for tasks, too: just change Small to Hard, and Big to Easy, and make sure the task pool can be subdivided somewhat finely.
  • Player N is theoretically in the best position, and N-1 the worst, but:
  • If all players envy each other, then all players will try to cut out a fair piece.
  • Any cheater who cuts a Too Small piece gets stuck with it.
  • Any cheater who cuts a Too Big piece loses it to a random other player (and not necessarily an ally).
  • Alliances don't work, because every player has an equal chance at winning any of the shares she thought were Too Big.

I will try this for our family's upcoming Sunday chores. I'm confident my daughters will still be envious of each other and derisive of this procedure, but they won't be able to complain of procedural unfairness or favoritism. (I'll first give them the option of dividing chores using the "Kids' Council" they secretly invented years ago. Long story!)

1 comment:

Unknown said...

Good luck on this. Remember when we lived in Japan, I made a chart for the refrigerator - one person in charge of clearing table, another dishwasher detail, don't think we had a ton of chores. Being in Japan, for the last eight months or so, I had a "maid" one day a week for $8 a day. Mrs. Viscardi and Mrs. Lowery had "maids" every day of the week. And I finally relented to have one for one day a week as I realized this would never happen again. But I was wrong - now have a cleaning lady once a month, but costs a lot more. She and her son are going to wash all the windows sometime soon. We have over 35 windows in this house.