<p>Interesting questions. To go through the list of questions you have...</p>

<ul>
<li><p><strong>Is there a more efficient mechanism? Will it be better to pay the lowest bidder the amount of the second lowest bid?</strong><br>
According to the <a href="https://en.wikipedia.org/wiki/Revenue_equivalence" rel="nofollow noreferrer">revenue equivalence theorem</a>, both types of auctions will result in the same expected revenue to the "cleaner," and will result in the same person doing the cleaning.</p></li>
<li><p><strong>Should the amount of funding of each person depend on his/her bid?</strong><br>
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.</p></li>
<li><p><strong>Should the person cleaning also contribute to the funding?</strong><br>
This doesn't make a difference, as the difference would be constant at a factor of <span class="math-container">$\frac{n}{n-1}$</span> 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 <span class="math-container">$\frac{n}{n-1}$</span>.</p></li>
</ul>

<p>If your constraint is that <em>someone</em> 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.</p>

CC BY-SA 4.0
2020-01-03T04:19:48.850Z
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
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2020-01-03T04:21:56.263Z
See D.W.'s comments [here](https://cstheory.stackexchange.com/questions/46134/allocating-chores-using-auctions), pretty similar.
Giskard
CC BY-SA 4.0
2020-01-03T05:50:48.387Z
@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
CC BY-SA 4.0
2020-01-03T07:48:34.743Z
(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
CC BY-SA 4.0
2020-01-03T08:12:03.550Z
(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
CC BY-SA 4.0
2020-01-03T10:02:24.183Z
(1) I am not claiming the setup violates any condition (I wrote no answer), I am saying that you are making a strong claim without proof! (3) Perhaps I am the one misunderstanding something, but I was under the impression your bid is not an offer of payment here, but a request for payment. (how much you want in exchange for cleaning)  In which case I don't see where this $n/(n-1)$ increase comes from. Could you please make your answer more formal and detailed?
Giskard
CC BY-SA 4.0
2020-01-03T10:13:22.380Z
I'll make (3) more formal. Thanks for the suggestion. For (1), I've always thought that the conditions are more of a formality and that, in real life, first price sealed but auction and second price sealed his auction always yield the same result. Am I wrong?
Art
CC BY-SA 4.0
2020-01-03T11:14:19.190Z
(There are other types of auctions as well.) This is not the usual setup, because players act as bidders and sellers, i.e. in some situations they have to pay, whereas in others they get revenue.
Giskard
CC BY-SA 4.0
2020-01-03T02:24:45.417Z
2020-01-03T04:12:22.347Z
2020-01-03T04:12:22.347Z
Art
<p>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.</p>

<p>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?</p>

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

CC BY-SA 4.0
2020-01-03T03:04:24.137Z
Cross-posted: https://economics.stackexchange.com/q/33478/25335, https://cstheory.stackexchange.com/q/46134/5038.  Please [do not post the same question on multiple sites](https://meta.stackexchange.com/q/64068).
D.W.
CC BY-SA 4.0
2020-01-03T04:18:22.967Z
"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
<game-theory><auctions><mechanism-design>
Allocating chores using auctions
Caspar
# Allocating chores using auctions

(Asked by: username_0 on 2020-01-03)

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.

**Tags:** game-theory, auctions, mechanism-design

> **Comments:**
> **username_1**: Cross-posted: https://economics.stackexchange.com/q/33478/25335, https://cstheory.stackexchange.com/q/46134/5038.  Please [do not post the same question on multiple sites](https://meta.stackexchange.com/q/64068).  
> **username_2**: "Should the person cleaning also contribute to the funding?" You mean should I give myself 10 dollars? This does not seem to change anything.  

---

## 1 answers

### Answer by username_3 on 2020-01-03. Score: 1

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](https://en.wikipedia.org/wiki/Revenue_equivalence), 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.

> **Comments:**
> **username_2**: 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.  
> **username_2**: See username_1's comments [here](https://cstheory.stackexchange.com/questions/46134/allocating-chores-using-auctions), pretty similar.  
> **username_3**: @username_2 (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.  
> **username_2**: (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?  
> **username_3**: (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.  
> **username_2**: (1) I am not claiming the setup violates any condition (I wrote no answer), I am saying that you are making a strong claim without proof! (3) Perhaps I am the one misunderstanding something, but I was under the impression your bid is not an offer of payment here, but a request for payment. (how much you want in exchange for cleaning)  In which case I don't see where this $n/(n-1)$ increase comes from. Could you please make your answer more formal and detailed?  
> **username_3**: I'll make (3) more formal. Thanks for the suggestion. For (1), I've always thought that the conditions are more of a formality and that, in real life, first price sealed but auction and second price sealed his auction always yield the same result. Am I wrong?  
> **username_2**: (There are other types of auctions as well.) This is not the usual setup, because players act as bidders and sellers, i.e. in some situations they have to pay, whereas in others they get revenue.  

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