<p>The second-price auction without a reserve price is in this case the revenue maximizing auction with symmetric bidders, independent private values and increasing virtual values. The answer is in Myersons paper. He rewrites the problem of the seller such that she should maximize the weighted sum of the virtual values. As those can be negative, a reserve price improves the outcome. However, if the seller needs to sell the item, she still maximizes the weighted sum of virtual values. This can be done pointwise. Thus, yielding that the item should be allocated to the bidder with the highest virtual value. With the assumptions above, a second price auction achieves this. If symmetry does not hold, there is an appropriate dominant strategy auction that deals with asymmetry. If increasing virtual values are violated, ironing still does the trick.</p>

CC BY-SA 4.0
2022-04-28T14:09:36.680Z
2022-04-28T14:09:36.680Z
GameTheorist1982
<p>Myerson has a <a href="https://www.jstor.org/stable/3689266" rel="nofollow noreferrer">famous theory</a> that can be used to design truthful auctions maximizing the revenue of the seller. The simplest case is when a seller sells a single item to buyers whose values are independent identically-distributed random variables. In this case, Myerson's auction is equivalent to a second-price auction with a reserve price, where the reserve price is determined by the distribution (for example, if all buyers' valuations are distributed uniformly between 0 and 100, then the revenue-maximizing auction is a second price auction with reserve price 50).</p>
<p>Implementing this optimal auction may require to destroy the item, in case all buyers bid below the reserve price. But discarding items might be impossible or illegal. For example, according to John Locke, the &quot;natural law&quot; allows a person to appropriate parts of nature for his personal use, but does not allow to destroy items.</p>
<p>This raises the following question: what is a revenue-maximizing auction in a situation in which it is not allowed to discard items?</p>

CC BY-SA 4.0
2022-02-25T12:18:00.510Z
Are you familiar with the revenue equivalence theorem? If yes: what are the exact conditions that stop us from doing a first or second-price auction without a reserve price?
Giskard
CC BY-SA 4.0
2022-02-25T12:58:20.757Z
@Giskard sure, we can do a second-price auction without a reserve price, but the expected revenue will be lower than with the optimal reserve price. My question is: what is the truthful auction that maximizes the revenue, under the constraint that the item must be sold? Is it indeed the second-price auction without reserve prices?
Erel Segal-Halevi
CC BY-SA 4.0
2022-02-25T13:12:12.227Z
Again I ask, "are you familiar with the revenue equivalence theorem?"
Giskard
CC BY-SA 4.0
2022-02-25T13:19:12.367Z
Also, "in a situation" is kind of vague, you probably need to provide more details; e.g.: are the bidders risk-neutral, is their number common knowledge?
Giskard
CC BY-SA 4.0
2022-02-25T13:35:24.623Z
@Giskard: Under standard IPV assumptions, the (first- or) second-price auction without a reserve price is not revenue-maximizing. Being familiar with the RET doesn't help here.
VARulle
CC BY-SA 4.0
2022-02-25T13:38:09.917Z
@ErelSegal-Halevi: Why should it sometimes be necessary to destroy the item? As I understand the model, the item is sometimes just not sold (and the seller keeps it).
VARulle
CC BY-SA 4.0
2022-02-25T20:05:08.697Z
@VARulle I was under the impression that reserve prices can increase the auction revenue because the efficient allocation assumption of the RET is violated, but it is true that I am not too knowledgeable about auctions. Also, it occurs to me that you can still violate efficient allocation without destroying/not selling the item; e.g., random allocation if the reserve price is not met by any bids.
Giskard
CC BY-SA 4.0
2022-02-26T21:03:21.853Z
@Giskard the RET says that, if the allocations of two auctions are identical, then their revenues are identical too. But with reserve prices, the allocations are not identical to an auction without reserve prices.
Erel Segal-Halevi
<game-theory><auctions><mechanism-design>
Revenue-maximizing auction with no free disposal
Erel Segal-Halevi
# Revenue-maximizing auction with no free disposal

(Asked by: username_0 on 2022-02-25)

Myerson has a [famous theory](https://www.jstor.org/stable/3689266) that can be used to design truthful auctions maximizing the revenue of the seller. The simplest case is when a seller sells a single item to buyers whose values are independent identically-distributed random variables. In this case, Myerson's auction is equivalent to a second-price auction with a reserve price, where the reserve price is determined by the distribution (for example, if all buyers' valuations are distributed uniformly between 0 and 100, then the revenue-maximizing auction is a second price auction with reserve price 50).

Implementing this optimal auction may require to destroy the item, in case all buyers bid below the reserve price. But discarding items might be impossible or illegal. For example, according to John Locke, the "natural law" allows a person to appropriate parts of nature for his personal use, but does not allow to destroy items.

This raises the following question: what is a revenue-maximizing auction in a situation in which it is not allowed to discard items?

**Tags:** game-theory, auctions, mechanism-design
> **Comments:**
> **username_1**: Are you familiar with the revenue equivalence theorem? If yes: what are the exact conditions that stop us from doing a first or second-price auction without a reserve price?  
> **username_0**: @Giskard sure, we can do a second-price auction without a reserve price, but the expected revenue will be lower than with the optimal reserve price. My question is: what is the truthful auction that maximizes the revenue, under the constraint that the item must be sold? Is it indeed the second-price auction without reserve prices?  
> **username_1**: Again I ask, "are you familiar with the revenue equivalence theorem?"  
> **username_1**: Also, "in a situation" is kind of vague, you probably need to provide more details; e.g.: are the bidders risk-neutral, is their number common knowledge?  
> **username_2**: @Giskard: Under standard IPV assumptions, the (first- or) second-price auction without a reserve price is not revenue-maximizing. Being familiar with the RET doesn't help here.  
> **username_2**: @ErelSegal-Halevi: Why should it sometimes be necessary to destroy the item? As I understand the model, the item is sometimes just not sold (and the seller keeps it).  
> **username_1**: @VARulle I was under the impression that reserve prices can increase the auction revenue because the efficient allocation assumption of the RET is violated, but it is true that I am not too knowledgeable about auctions. Also, it occurs to me that you can still violate efficient allocation without destroying/not selling the item; e.g., random allocation if the reserve price is not met by any bids.  
> **username_0**: @Giskard the RET says that, if the allocations of two auctions are identical, then their revenues are identical too. But with reserve prices, the allocations are not identical to an auction without reserve prices.  

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## 1 answers

### Answer by username_3 on 2022-04-28. Score: 1 | Accepted answer

The second-price auction without a reserve price is in this case the revenue maximizing auction with symmetric bidders, independent private values and increasing virtual values. The answer is in Myersons paper. He rewrites the problem of the seller such that she should maximize the weighted sum of the virtual values. As those can be negative, a reserve price improves the outcome. However, if the seller needs to sell the item, she still maximizes the weighted sum of virtual values. This can be done pointwise. Thus, yielding that the item should be allocated to the bidder with the highest virtual value. With the assumptions above, a second price auction achieves this. If symmetry does not hold, there is an appropriate dominant strategy auction that deals with asymmetry. If increasing virtual values are violated, ironing still does the trick.


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