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Myerson has a famous theory 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?

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  • $\begingroup$ 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? $\endgroup$
    – Giskard
    Feb 25, 2022 at 12:18
  • $\begingroup$ @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? $\endgroup$ Feb 25, 2022 at 12:58
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    $\begingroup$ @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. $\endgroup$
    – VARulle
    Feb 25, 2022 at 13:35
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    $\begingroup$ @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). $\endgroup$
    – VARulle
    Feb 25, 2022 at 13:38
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    $\begingroup$ @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. $\endgroup$ Feb 26, 2022 at 21:03

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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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