    # Separation between Second Price Auctions with Personalized Reserves and the Revenue Optimal Auction

What fraction of the single item n buyers setting's expected optimal revenue MyeRev can the second price auction with reserves achieve? In the special case where the buyers' valuation distributions are all drawn i.i.d. and the distributions satisfy the regularity condition, the second price auction with an anonymous reserve (ASP) is the optimal auction itself. As the setting gets more complex, there are established upper bounds on the fraction of MyeRev that ASP can achieve. On the contrary, no such upper bounds are known for the fraction of MyeRev achievable by the second price auction with eager personalized reserves (ESP). In particular, no separation was earlier known between ESP's revenue and MyeRev even in the most general setting of non-identical product distributions that don't satisfy the regularity condition. In this paper we establish the first separation results for ESP: we show that even in the case of distributions drawn i.i.d., but not necessarily satisfying the regularity condition, the ESP cannot achieve more than a 0.778 fraction of MyeRev in general. Combined with Correa et al.'s result (EC 2017) that ESP can achieve at least a 0.745 fraction of MyeRev, this nearly bridges the gap between upper and lower bounds on ESP's approximation factor.

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## 1 Introduction

Approximating the optimal auction’s expected revenue MyeRev

by simple mechanisms has received a lot of attention in the algorithmic game theory literature, especially in the last decade. The seminal result of

Myerson (1981) describes the optimal auction in a single agent -items setting when buyer valuations are drawn from a product distribution. When the buyer valuation distributions are independent and identical (i.i.d.) and satisfy a technical regularity condition, Myerson’s optimal auction coincides with the second price (Vickrey’s) auction with an anonymous reserve price (ASP). When the setting gets more complex, either because the distributions are not identical or because the distributions don’t satisfy the regularity condition, or both, ASP ceases to be the optimal auction. Bounds on the suboptimality are known in many cases:

• when the distributions are non-identical (independent) and irregular, Alaei et al. (2015) show that ASP cannot get more than a fraction of MyeRev in general, and it is straightforward to show that this is tight;

• when the distributions are non-identical (independent) and regular, Hartline and Roughgarden (2009) show that ASP cannot get more than a fraction of MyeRev in general, Alaei et al. (2015) show that ASP can achieve a fraction of MyeRev in general, and recently Jin et al. (2019) show that ASP cannot get more than a fraction of MyeRev in general;

• when the distributions are identical (independent) and irregular, the pricing problem can be reduced to the prophet inequalities problem on ironed virtual values, for which a approximation via a uniform threshold is known (Samuel-Cahn (1984)). Since the distributions are identical, the uniform threshold in the ironed virtual value space translates into a uniform threshold in the value space, thus showing that ASP can achieve a approximation as well. This approximation factor of ASP is tight, as shown in Hartline (2013), although we will also provide a self-contained proof here as a corollary.

On the contrary, no upper bounds are known on the fraction of MyeRev obtainable by a second price auction with personalized reserve prices. When the reserve prices are personalized, it matters whether the bidders are ranked after eliminating those below the reserve (eager, “ESP”) or before eliminating those below the reserve (lazy, “LSP”). In the eager version, we first eliminate bidders below their respective reserves, and then charge the highest surviving bidder (if any) the maximum of his own reserve price and the second highest surviving bid. In the lazy version, we first decide the potential winner to be the highest bidder, eliminate him if he doesn’t survive his reserve price (in which case we allocate the good to nobody), and then if he survives allocate the good to him at a price of maximum of his reserve price and the second highest bid. Paes Leme et al. (2016) show that for general correlated distributions, the optimal revenue obtainable from ESP and LSP are incomparable (i.e. either could be higher on a specific instance), but are within a factor of two. When the distributions are independent Paes Leme et al. show that the optimal ESP’s revenue dominates the optimal LSP’s revenue. Therefore from the perspective of upper bounding the fraction of MyeRev that is obtainable, any result for ESP implies a result for LSP. ESP’s are also more widely used, and are a more natural auction to run111For instance, when there is at least one buyer that survives their reserve price, ESP always allocates the item, whereas LSP could result in no allocation even then.).

#### Our result.

Our main result (Theorem 2.4) is an upper bound on the maximum fraction of MyeRev that ESP can obtain. We show that in the buyers single item setting, even when the buyer values are drawn i.i.d. (but not necessarily from a regular distribution), the fraction of MyeRev that ESP can extract in general is at most . The i.i.d. irregular setting is the simplest non-trivial setting to consider since, as discussed earlier, the i.i.d. regular setting results in a approximation from even a second price auction with anonymous reserve price.

#### Corollary.

Our construction can also be modified to show that ASP cannot get more than of MyeRev for i.i.d. irregular valuations, establishing that the approximation factor for ASP is tight.

#### Near tightness of ESP result.

A mechanism very related to ESP is the sequential posted price mechanism (SPM) where the seller computes one price per buyer, and then approaches the buyers in the descending order of posted prices until the item is sold. In general single-parameter environments with matroid feasibility constraints (this includes the single item setting we study as a special case), Chawla et al. (2010) showed the the revenue of an ESP  that uses the posted prices of an SPM as its personalized reserve prices is at least as high as the SPM’s revenue. This, combined with the recent result of Correa et al. (2017) showing that SPM can achieve a fraction of MyeRev in the i.i.d. setting with possibly irregular distributions, shows that ESP can also achieve at least a approximation factor in the i.i.d. setting. Thus, our upper bound of on ESP’s approximation factor in the i.i.d. setting is quite close to this lower bound of . In the case of SPM  both the upper and lower bound are known to be : the upper bound of for SPMs was established by Hill and Kertz (1982) about decades prior to the matching lower bound established by Correa et al.

#### Esp vs Spm: the benefit of simultaneity.

The primary advantage of ESP over SPM is that ESP considers all buyers simultaneously while SPM considers buyers sequentially. Thus, as Chawla et al. (2010) show, ESP should only earn better revenue by benefiting from this simultaneity. Our upper bound of on the approximation factor of ESP shows that this benefit of simultaneity is quite small in general, given that SPMs themselves can achieve a approximation. In a sense, our result shows that discrimination is more crucial for revenue maximization than simultaneity. Closing this gap further, or establishing a separation between SPM and ESP remains an interesting open problem.

## 2 Upper bound on ESP approximation factor

We provide an example that establishes the upper bound on ESP’s approximation factor.

###### Example 2.1.

There are buyers, with . Each buyer’s valuation is drawn i.i.d. from a discrete distribution where

 v∼F=⎧⎪⎨⎪⎩n,w.p. 1/n2α/β,w.p. β/n−1/n20,w.p. 1−β/n

Here and are constants to be optimized later to minimize the ratio of MyeRev earned by ESP (note that since they are constants they don’t scale with , and therefore and ; these are things we use in Lemma 2.2 below).

###### Lemma 2.2.

In a buyers setting, when the buyer valuations are drawn i.i.d. from the distribution described in Example 2.1, the expected revenue MyeRev of Myerson’s optimal auction approaches

 1+α−1β(1−e−β).
###### Proof.

We compute the revenue as the expected ironed virtual surplus. To compute the latter, consider the revenue curve , where in this case

denotes the maximum price at which the probability of getting a sale (from a single buyer with valuation drawn from

) is at least . It can be computed that

 R(q)=qF−1(1−q)=⎧⎪⎨⎪⎩qn,0

Note that the revenue curve is not concave, because and , while for sufficiently small . The ironed revenue curve over is formed by joining the points and . The slope of three line segments in the ironed revenue curve are the ironed virtual values corresponding to the three value realizations:

• the slope of the line segment between and is and that is the virtual value (which is equal to the ironed virtual value for this segment) of buyers with value ;

• the slope of the line segment between and is and that is the ironed virtual valuation of buyers with value .

• buyers with value have negative ironed virtual value and are irrelevant for revenue purposes.

The complete description of Myerson’s optimal mechanism here is as follows:

• any buyer who bids below is never allocated, and always pays 0;

• if there are no buyers with bid or larger: allocate the item to a uniformly random buyer among all buyers with bids in and charge him ;

• if there is exactly one buyer with bid of or larger, that buyer gets the item with probability , and he pays which is basically the area to the left of the allocation curve of that buyer;

• if there is more than one buyer with bid of n or larger, allocate the item to a uniformly random buyer among all buyers with bids in and charge him .

The expected revenue of Myerson’s optimal mechanism MyeRev equals the expected ironed virtual surplus, which can be computed as follows. With probability , at least one buyer has virtual valuation . With probability , no buyer has virtual valuation but at least one buyer has ironed virtual valuation . Therefore, the expected ironed virtual surplus, as the number of buyers approaches , is

 limn→∞(n(1−(1−1n2)n)+α−1β−1/n((1−1n2)n−(1−βn)n))=1+α−1β(1−e−β),

completing the proof. ∎

###### Lemma 2.3.

In a buyers setting, when the buyer valuations are drawn i.i.d. from the distribution described in Example 2.1, the expected revenue EspRev of ESP as is at most

 1+α−lnα−1β.
###### Proof.

Consider any revenue-maximizing ESP auction. Since the only possible non-zero valuations are and , it does not lose generality to assume that all reserve prices are set to either or . Let denote the fraction of buyers whose reserve is set to , so that the number of buyers with reserves and are and , respectively.

With probability , at least one of the buyers with reserve clears their reserve price, and the ESP auction earns the maximum possible revenue of . Otherwise, all buyers with reserve are eliminated from the auction and we consider the other buyers.

For the other buyers with reserve , at least one of them will clear their reserve with probability , in which case the ESP auction is guaranteed to earn at least . In this case, there is also the possibility of earning the higher revenue of , if at least two buyers report a valuation of . However, the probability of this event is at most times the probability that a given pair of buyers both report , by the union bound. Therefore, the expected revenue collected from this event is at most , and can be ignored as .

Collecting the above terms, for any and , the expected revenue of the corresponding ESP auction is

 n(1−(1−1n2)zn)+αβ(1−1n2)zn(1−(1−βn)(1−z)n) ≤ n(1−(1−(zn1)1n2))+αβ(1)(1−e−β(1−z)) = z+αβ(1−e−β(1−z))+O(1n) (1)

where we used the binomial theorem in the first inequality. The maximum of the concave function over is attained at , and equals , completing the proof. ∎

###### Theorem 2.4.

Let and denote respectively the revenue of the optimal auction (Myerson’s auction) and the second price auction with personalized reserves in a single-item setting with buyers whose values are drawn i.i.d from . Then:

 infn,D\textscEspRev(n,D)\textscMyeRev(n,D)<0.778.
###### Proof.

Consider Example 2.1. By Lemmas 2.2 and 2.3, the fraction of Myerson’s optimal revenue earned by an ESP auction in this example is at most

 β+α−1−lnαβ+(α−1)(1−e−β), (2)

for any values of and . By numerically minimizing over such values of and , it can be checked that when and , the value of (2) is less than 0.778, completing the proof. ∎

### 2.1 Upper bound on ASP approximation factor

Consider the distribution from Example 2.1 with and let . By Lemma 2.2, the optimal revenue of Myerson is which approaches 2 as . On the other hand, ESP must set all of the reserves to either or 1. By the analysis in Lemma 2.3, this is upper-bounded by expression (1) with or , i.e. the revenue of ASP is at most

 max{1−e−n+O(1n),1+O(1n)}

which approaches 1 as . This completes the proof.

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