# Allocating chores using auctions

I live in a student dorm with 16 people and we share a kitchen. Every week someone has to clean the kitchen. Right now we just take turns in a way determined by the numbers of our respective rooms. This is obviously not very efficient. I was thinking about the possibility of using auctions to determine who should do the cleaning. One possibility could be a sealed bid type thing where everyone states how much money they want in order to do the cleaning. Then the person who wants least does the cleaning and the other people fund this by contributing an equal amount.

Is there a more efficient mechanism? Some points I am considering: Should the amount of funding of each person depend on his/hers bid? Should the person cleaning also contribute to the funding? Will it be better to pay the lowest bidder the amount of the second lowest bid?

I am well aware that this kind of thing might be socially/morally objectionable. It is more of a thought experiment.

• – D.W. Jan 3 '20 at 3:04
• "Should the person cleaning also contribute to the funding?" You mean should I give myself 10 dollars? This does not seem to change anything. – Giskard Jan 3 '20 at 4:18

Interesting questions. To go through the list of questions you have...

• Is there a more efficient mechanism? Will it be better to pay the lowest bidder the amount of the second lowest bid?
According to the revenue equivalence theorem, both types of auctions will result in the same expected revenue to the "cleaner," and will result in the same person doing the cleaning.

• Should the amount of funding of each person depend on his/her bid?
This is the same question of choosing first-price or second-price auction. Indeed, any type of auction will depend on the bidders' bids to an extent.

• Should the person cleaning also contribute to the funding?
This doesn't make a difference, as the difference would be constant at a factor of $$\frac{n}{n-1}$$ depending on which "rule" you apply. That is, if you switch the rule to saying that the person cleaning woulnd't have to contribute to the funding, then everyone could just increase their bids by a factor of $$\frac{n}{n-1}$$.

If your constraint is that someone in your dorm will have to do it, then I'd say this (along with simply keep raising the "price" until someone says, "I'll do it") would be the most efficient. If this is not a constraint, though, an outsider (cleaning company, random guy outside of your dorm) might be willing to do it for less.

• 1. Is it trivial the conditions of the revenue equivalence theorem hold in this situation? 2. What you write under the second question is more of a comment, not an answer, though what "should" means in this question is unclear. 3. For the third question see my comment under the Question. 4. What is the concept of "efficiency" that you use? Seeing that you write "most efficient" it does not seem to be the standard efficiency concept from auction theory. – Giskard Jan 3 '20 at 4:19
• See D.W.'s comments here, pretty similar. – Giskard Jan 3 '20 at 4:21
• @Giskard (1) I think so. (2) I'll admit that the question is not very clear, but I was simply stating that no matter what type of auction we choose, the amount of funding will depend on everyone's bid. (3) We're saying the same thing. If I'm willing to clean the kitchen for \$16, then I'll bid \$16 if I don't have to pay, and 17.067 if I have to pay. (4) Efficiency for me simply means that the person who gets to clean is the one who's willing to do it for the lowest cost. Having an outsider would be weakly more efficient, as there's a chance that that person would be willing to do it for less. – Art Jan 3 '20 at 5:50
• (1) Why? Seems like this would be the actual question. (3) I am afraid we are not saying the same thing. Why is what you are saying true? Does it hold in the case of asymmetric distribution of valuations? – Giskard Jan 3 '20 at 7:48
• (1) What conditions do you think the setup violate? (3) We're saying the same thing that it doesn't change the outcome. Payment would still be the same, but the bidding would be increased by a factor of 16/15. – Art Jan 3 '20 at 8:12