Optimal Deterministic Mechanisms for an Additive Buyer

We study revenue maximization by deterministic mechanisms for the simplest case for which Myerson's characterization does not hold: a single seller selling two items, with independently distributed values, to a single additive buyer. We prove that optimal mechanisms are submodular and hence monotone. Furthermore, we show that in the IID case, optimal mechanisms are symmetric. Our characterizations are surprisingly non-trivial, and we show that they fail to extend in several natural ways, e.g. for correlated distributions or more than two items. In particular, this shows that the optimality of symmetric mechanisms does not follow from the symmetry of the IID distribution.

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1 Introduction

This paper deals with the problem of revenue maximization by a monopolist seller in the simple scenario when a single buyer has an additive valuation over a set of items. Our main focus is on deterministic mechanisms, and we start with the simplest case when there are just two items for sale. In that case, the buyer’s value for the first item is , for the second item is , and for the bundle of both is , where the seller only knows that is sampled from distribution and is sampled from distribution , and that these values are independently sampled. Our seller’s goal is to maximize his expected revenue when the buyer maximizes his quasi-linear utility.

This simplest of scenarios has received much attention lately as it is one of the most fundamental examples where Myerson’s characterization of optimal auctions (for a single item) Myerson (1981) ceases to hold, and indeed optimal auctions for two items may become complex McAfee and McMillan (1988); Alaei et al. (2012); Manelli and Vincent (2007); Hart and Nisan (2012); Hart and Reny (2015); Daskalakis et al. (2015). In particular, different examples are known where the optimal auction sells each of the two items separately, sells both items as a bundle, gives a “discount price” for the bundle Hart and Nisan (2012), is randomized (i.e. uses lotteries) Manelli and Vincent (2007), requires an infinite number of possible randomized outcomes Daskalakis et al. (2015), or is non-monotone (i.e. the revenue it extracts may decrease when the buyer’s valuations increase.) Hart and Reny (2015).

Generally speaking we do not understand the structure of optimal auctions – neither randomized ones nor deterministic ones – even in this simple scenario, and in fact it is known that for the case of items (still with a single additive buyer) determining the optimal randomized auction is -hard Daskalakis et al. (2014) as is determining the optimal deterministic auction Chen et al. (2017).111Conitzer and Sandholm (2004) have proven NP hardness of optimal deterministic auction for a much more general setting. On the other hand, several recent results show that simple auctions (such as selling the items separately or as a single bundle) provide good approximations to the optimal revenue for two items Hart and Nisan (2012), multiple items Li and Yao (2013); Babaioff et al. (2014), as well as further generalizations in scenario such as multiple buyers Yao (2015), or combinatorial valuations Rubinstein and Weinberg (2015), or both Chawla and Miller (2016); Cai and Zhao (2017); Yao (2017) (but not when the item values are correlated Hart and Nisan (2013) — except for special cases Bateni et al. (2015)).

In this paper we prove structural characterizations of the optimal deterministic mechanisms for a monopolist seller that is selling two items to a single222Note that results for a single buyer also hold when there are many buyers but there are no supply constraints (digital goods). additive buyer with item values independently distributed.333A deterministic mechanism for such a problem simply presents prices for each of the items, and for the pair. We also show that these characterizations fail to generalize in any conceivable way: neither to more than two items, nor to items with correlated distributions, nor to randomized optimal mechanisms.



Supermodular menu , where .


Submodular menu , where .

Given a menu, the incentive constraints of the buyer imply a partition of the space of valuations to four regions. The allocation is empty in the South-West region and the revenue is 0. The first item is allocated in the South-East region, the second in the North-West region, and the pair in the North-East region. In each of these regions we mark the revenue for any valuation in that region. We omit the allocation, and payment when there is no sale, in all figures, as they are the same as here. Note that the borders between any two regions are places of indifference, and as ties are broken towards higher payments, each belongs to the neighboring region with the higher payment. The location of the intersection of each of these borders with the axes is also marked (e.g. at the buyer is indifferent between buying nothing and buying the first item.) Note that while our results are not limited to bounded valuations, our figures are drawn as if they are upper bounded, for convenience of illustration. Finally, to avoid clutter, we explicitly draw the axes only for the figure corresponding to the first (supermodular) menu, and eliminate them from all other figures.

Figure 1: Submodular vs Supermodular Menus

A general deterministic auction for two items charges prices, and respectively, for each of the two items, as well as a third price for the bundle of both items. There are basically two “forms” of such auctions: the first is the “submodular case” where and the second is the “supermodular case” where .444When the menu is additive - it is both submodular and supermodular. Additive pricing corresponds to selling the item separately. Given a menu, the incentive constraints of the buyer imply a partition of the space of valuations to regions, in each the buyer buys the same bundle. The different shapes of the partition that these two types of auctions induce on the buyer’s two-item value space are depicted in Figure 1. Our first result states that the seller never loses by presenting the buyer with an auction of the submodular type.

Theorem 1.1.

For every two distributions and , and a single additive buyer with two items’ valuations sampled from , the revenue of any deterministic mechanism can be obtained by a deterministic mechanism that is submodular.

This means that the seller will never gain by posting a price for the pair that is higher than the sum of prices for the two items. This is highly desirable property as it implies that in such a mechanism a buyer cannot gain by using “false names” Yokoo et al. (2004) and buying each of the two items separately under a different identity (e.g. when the seller of a digital good faces multiple ex-ante identical buyers, and she posts the same prices to every buyer.) One practical motivation for false-name proof mechanisms is that preventing the usage of false names requires verification of identities, which might be very costly. Another motivation is that for some mechanisms, allowing the buyer to select multiple menu options may result in exponential savings in representation size555For example, item pricing can be represented with linear menu in the case multiple menu entries can be picked, but require exponential menu when the buyer can only pick a single entry from the menu; however such a representation only makes sense for false-name-proof mechanisms. (See also Remark 7.3 for further discussion on “menu-size” vs “additive-menu-size”.)

An additional benefit of such a mechanism is that it satisfies revenue monotonicity - it was shown by Hart and Reny (2015) that submodular deterministic auctions are revenue monotone666For the case of two items it is actually easy to see that any submodular deterministic auction is revenue monotone: increasing values cannot cause the bidder to switch from one item to the other, only from purchasing one item to buying the pair, and the price of the pair is at least the price of any item. (i.e. the revenue that they extract from a player with item values cannot be lower than the revenue that they extract from a player with values with and ). From the theorem and Hart and Reny (2015) we immediately get the next corollary.

Corollary 1.2.

For every two distributions and , and a single additive buyer with two items’ valuations sampled from , the revenue of any deterministic mechanism can be obtained by a deterministic mechanism that is revenue-monotone.

Using Corollary 1.2 we are also able to determine exactly the maximum possible gap between the optimal revenue by deterministic auctions and the optimal revenue from selling the items separately. If we denote the optimal revenue of deterministic auctions by and the optimal revenue from selling the two items separately (each at its optimal Myerson price) by then we get:

Theorem 1.3.

For every two distributions and and a single additive buyer with two items’ valuations sampled from , it holds that , where is the solution of .

This bound is tight as Hart and Nisan (2012) shows that for an “equal revenue distribution777A distribution for which any price in the support has the same revenue. For a formal definition see Section 3. it holds that . It is still open whether the gap between SRev and Rev (the revenue of the optimal randomized mechanism) is larger or not.

We then note several ways in which Theorem 1.1, which holds for deterministic auctions for an additive buyer with two independent items, can not be generalized:

  1. To more than two items: We exhibit an example where the optimal deterministic auction for three independent items fails to be submodular.888For a set of items , a pricing function is submodular if for any it holds that . We leave as an open problem whether optimal deterministic auctions for more than two items must at least be subadditive.999For a set of items , a pricing function is subadditive if for any it holds that .

  2. To correlated item values: We show that the optimal deterministic auction for correlated distributions on the two items may fail to be submodular (or monotone). Nevertheless, the gap between the two is bounded: we show that the maximum possible gap between the revenues of general deterministic versus submodular deterministic101010Our result is actually a little stronger: we show that the gap is at most even if the submodular auction is restricted to either selling each item separately or offering only the grand bundle. auctions in the case of two items with correlated distributions is , and this is essentially tight.

  3. To randomized auctions: while it is not clear what is the exact analog of submodularity for a randomized auction, it is known Hart and Reny (2015) that the optimal randomized auction for two independently distributed items need not be revenue monotone, in contrast to the revenue monotonicity result that we present above for deterministic auction (Corollary 1.2). We also note that while submodular auctions for two items are “false name proof”, optimal randomized auctions need not be so.

Our second structural result concerns the question of symmetry. Suppose that the two items are symmetric, i.e. sampled IID from the distribution . Does this also imply that there is an optimal deterministic auction that is symmetric (both items have the same price)? For general (not necessarily deterministic) auctions, symmetry may be obtained without loss of generality by averaging over all possible permutations of the items, but this procedure results in a randomized auction even when starting with a deterministic one. It turns out that the answer is still positive, but this is rather delicate.

Theorem 1.4.

For any distribution , and a single additive buyer with two items’ valuations sampled from , the revenue of any deterministic mechanism can be obtained by a deterministic mechanism that is symmetric.

Perhaps surprisingly, we show that this theorem can not be generalized in each of these natural ways:

  1. To correlated symmetric item values

    : We show that there exists a correlated symmetric joint distribution on

    such that the revenue of the best symmetric deterministic auction is strictly less than that of the optimal deterministic auction.

  2. To more than two IID items: We exhibit an example of a distribution where for three items whose values are distributed, independently, according to , the optimal symmetric deterministic auction extracts strictly less revenue than the optimal deterministic auction.

In particular, the above examples show that the symmetry of the joint distribution that is implied by the IID assumption is not sufficient to derive that the optimal pricing is symmetric.

It should be noted that in contrast to the failure of the positive result for deterministic 2-item case to generalize to joint correlated distributions and to more items, for the case of randomized auctions and any number of items, it is trivially true that for any symmetric joint distribution over items, there exists an optimal randomized auction that is symmetric (by going over all item permutations.) This is true when item values are samples IID, and it is also true when the items’ values are correlated in a symmetric way.

1.1 Additional related work

There are countless papers dedicated to the topics we touch on in this work, including tradeoffs between simple and complex mechanisms, structure of optimal mechanisms, false-name proofness, etc. We shall briefly mention a few below. Thirumulanathan et al. (2016)

study revenue maximization for the special case uniforms distributions on intervals of the real line.

Conitzer and Sandholm (2004) prove computational intractability albeit for the much more general setting of correlated valuations. Thanassoulis (2004); Chawla et al. (2007); Briest et al. (2015) study questions related to ours, but for buyers with unit-demand valuations (whereas we focus on additive valuations).

2 Preliminaries

A monopolist seller sets up a mechanism to maximize revenue when facing a single buyer. The buyer has an additive valuation over items, that is, if his value for item is , his value for the set of items is . The values of the items are drawn from a distribution which is known to the seller. We mostly focus on the special cases where and the value for each item is drawn independently from a distribution , and .

We study the revenue that can be achieved by a monopolist seller facing such a buyer. We consider deterministic mechanisms that are truthful and ex-post Individually Rational (IR), which for a single buyer are simply menus that present a price for each bundle, and the buyer picks a bundle which maximizes his quasi-linear utility (the difference between the value for the bundle and the payment), breaking ties in favor of higher payments.

Notation

For any joint distribution of valuations, we use the following notation, mostly due to Hart and Nisan (2012); Babaioff et al. (2014), to denote the optimum revenue for each class of mechanisms:

  • - the supremum revenue among all truthful mechanisms;

  • - the supremum revenue among all truthful deterministic mechanisms;

  • - the supremum revenue obtainable by pricing only the grand bundle;111111Pricing every non-empty bundle at the same price - WLOG the buyer will either buy all the items, or nothing. and

  • - the supremum revenue obtainable by pricing each item separately;121212Every bundle is priced at the sum of prices of the items in the bundle.

  • - the supremum revenue among all truthful deterministic mechanisms with submodular pricing.

When is clear from the context, we simply write etc.

We note that the supremum might not be obtained by any mechanism, even when a single item is for sale to a single buyer, and even if the optimal revenue is bounded from above. To see this, consider the distribution with support over all non-negative real numbers and for which , for such a distribution the revenue with price is which grows to as goes to infinity, yet no price obtains revenue of . On the other hand, for any distribution with a finite support the supremum is clearly obtained. When the supremum is obtained, each of our results that mechanisms with some property (e.g. submodular) can get the same revenue as any mechanisms, implies that there is an optimal mechanism with property .

A deterministic menu for 2 items can be represented by 3 prices: two prices for each of the two items, and another price for the pair. We use triplets as to denote such a menu, with and being the prices for the first and second item, respectively, and being the price for the pair, with all prices being non-negative. Note also that without loss of generality, the seller can restrict her attention to menus with as decreasing the price of an item to the price of the pair never decreases the revenue. We use to denote the expected revenue from menu for the distribution . When the distribution is clear from the context, we use the simpler notation .

Revenue Monotonicity

A menu is revenue monotone if the revenue obtained by the menu does not decrease when the value of every item weakly increases. That is, a revenue monotone menu satisfies the condition that the revenue from a player with item values cannot be lower than the revenue extracted from a player with values with and .

3 Deterministic Pricing for Two Independent Items

We prove that for two independent items and an additive buyer, a deterministic seller does not lose revenue by restricting herself to mechanisms that are submodular. We do so by showing that the revenue of any supermodular menu is dominated by the revenue of one out of two additive menus that are derived from the supermodular menu.

Theorem 3.1 (Theorem 1.1 restated).

For every two distributions and , and a single additive buyer with two items’ valuations sampled from , the revenue of any deterministic mechanism can be obtained by a deterministic mechanism that is submodular. Thus, ,

Proof.

Given a strictly supermodular menu (selling item , item , and the grand bundle for prices , respectively), assume wlog that . By strict supermodularity, . We argue that at least one of the following additive menus achieves at least as much revenue: selling the items separately for prices , respectively (menu ), and selling separately for prices (menu ()).131313It may not be immediately intuitive why charging

for the first item is a good idea. However, we expect to have probability mass at

because this is where, when facing the original supermodular menu, the buyer switches from buying item to the grand bundle.

The first step of the proof is to consider the partition of the valuation space induced by the incentive compatibility constraints of the three menus, and to compute the payments in each region (see Figure 2 for details). We now split into cases depending on the ratio between and . (Notice that this step only makes sense for independent valuations.) We show that when



Payments to seller with strictly supermodular menu , satisfying .



Payments to seller with the additive menu .



Payments to seller with the additive menu .


In each region, we compute the difference in revenues of two menus. Above : vs. ( dominates ).
Below : vs. ( and are the same).
Figure 2: Submodular menus are optimal
(1)

the menu obtains at least as much revenue as menu , while when this is violated, the menu obtains at least as much revenue as menu .

Detailed explanations

Below we explain why in each case the respective submodular menu is at least as good as supermodular menu . Refer also to Figure 2.

First, we argue that when Inequality (1) holds, menu obtains at least as much revenue as menu . This is so as both menus obtain the same revenue for any valuations with , and for any , due to independence, the contribution to the expected gain of the event is at least as large as the contributed expected loss of the event (note that for such that , the menu obtains at least as much revenue as the menu , so it is enough to look at and use independence).

Finally, we argue that when Inequality (1) fails to hold, menu obtains at least as much revenue as menu . This is so as menu obtains at least as much revenue as menu for any valuations with , and for any , due to independence, the contribution to the expected gain of the event is at least as large as the contributed expected loss of the event . ∎

Hart and Reny (Hart and Reny, 2015, Corollary 9) have shown that submodular menus are revenue-monotone, i.e. when facing a higher (stochastically-dominating) valuation distribution, their revenue can never decrease. The following corollary thus follows immediately from Theorem 3.1.

Corollary 3.2 (Corollary 1.2 restated).

For every two distributions and , and a single additive buyer with two items’ valuations sampled from , the revenue of any deterministic mechanism can be obtained by a deterministic mechanism that is revenue-monotone.

3.1 SRev vs DRev

From Corollary 3.2 we are able to determine exactly the maximum possible gap between the optimal deterministic auction and the optimal revenue from selling the two items separately.

Theorem 3.3 (Theorem 1.3 restated).

Let be the solution of the equation . For every two distributions and it holds that . That is, any revenue-optimal deterministic mechanism for selling two items with independent distributions obtains at most -times more revenue than selling each item separately. Furthermore, this is tight even for IID item distributions.

The tightness of our result follows from a result of Hart and Nisan (2012), that shows that for any “equal revenue distribution” , it holds that . We next formally define equal revenue distributions - distributions such that any price that generates positive revenue, generates the same revenue.

Definition 3.4 (Equal Revenue Distribution).

For , let the -equal revenue distribution, denoted by , be the single-item valuation distribution whose cdf satisfies:

We next show that the revenue-optimal deterministic mechanism for selling two items whose valuations are drawn from independent (but possibly different) equal revenue distributions always sells both items together (i.e. for two independent equal revenue distributions).

Lemma 3.5.

For any it holds that

Proof.

Fix a menu and assume WLOG that . To prove the claim we show that any revenue that can be obtained by a deterministic menu, can also be obtain by a menu in which the buyer never strictly prefers to buy a single item (over buying the pair or getting nothing). A sufficient condition for such a menu is that . Consider any deterministic menu such that , that is, it sells item for a cheaper price than the price charged for the bundle. We will show that increasing the price of item to can only increase the revenue (the same arguments will clearly hold for the second item as well). That is, the menu generates at least as much revenue as the menu .

For the allocation and payments for both menus and are the same, as it is never the case that the first item is bought alone, see Figure 3 for an illustration. We next argue about the case that . For any such case, it holds that for any fixed the revenue with menu is (since item 2 is never sold and is distributed according to the equal revenue distribution ). On the other hand, for , for any such the buyer either gets nothing or buys the pair, and the pair is sold in the event that which happen with probability at least as large as the probability that . As selling item at price alone generates revenue , the menu which sells with at least as high probability, generates at least as high revenue.


Payments with menu .


Payments with menu . Item is never sold alone. For values the allocation and payments for both menus are the same.
Figure 3: DREV equals BREV for ER distributions

We use the next Lemma from Hart and Nisan (2012) in the proof of Theorem 3.3 below.

Lemma 3.6 ((Hart and Nisan, 2012, Lemma 13)).

For every two distributions and ,

Furthermore, for some distributions this holds as equality (the result is tight).

We are now ready to prove Theorem 3.3.

Proof of Theorem 3.3.

For let be the optimal revenue from selling item that is sampled from the distribution . We note that any item value distribution for which the optimal revenue is , is stochastically dominated by the distribution . This implies, by Corollary 3.2, that . We use this as the first step in our proof:

(DRev is monotone by Cor. 3.2)
(Lemma 3.5)
(Lemma 3.6)
(Def. of SRev, ER distributions)

4 Deterministic Pricing for Two IID Items

Assume that the two items are ex-ante symmetric, will a deterministic seller lose revenue by restricting her menu to be symmetric ()? For independent items, symmetric items means that they are sampled IID. We show that the optimum revenue by deterministic mechanisms for selling two IID items can be achieved by symmetric deterministic mechanisms. We later show that the symmetry of the joint distribution of two IID items is not sufficient to prove this result - for correlated symmetric distribution the result does not hold. We demonstrate this by presenting an example in which an asymmetric deterministic mechanism generates higher revenue than the best symmetric deterministic mechanism (Example 6.3).

We prove that for two IID items restricting to symmetric menus does not decrease the revenue. The proof is similar in spirit to the proof of Theorem 3.1 but is much more involved and delicate.

Theorem 4.1 (Theorem 1.4 restated).

For any value distribution , and a single additive buyer with two items’ valuations sampled from , the revenue of any deterministic mechanism can be obtained by a deterministic mechanism that is symmetric.

Proof.

Given an asymmetric deterministic menu (that prices the two items differently), we construct a symmetric menu that obtains at least as much revenue.

Let denote the prices that the asymmetric menu charges for item , item , and the bundle, respectively. Wlog, we can assume that . Furthermore, by Theorem 3.1, we can assume wlog that . It follows immediately that also , but we do not know whether . Our proof proceeds by analyzing separately the two possible cases.

The case where

The case when is the easy one. We show below that in this case, the symmetric menus and (i.e. charge both items , respectively ) perform on average exactly as well as the original asymmetric menu . Thus at least one of those symmetric menus has to yield at least as much revenue as the asymmetric menu .

To prove that menus and yield as much revenue, on average, as the asymmetric menu, we partition the valuation space across the line . For , we argue that a buyer facing menus or never buys item by itself (without item ). For menu , since and , if then and thus whenever item gives non-negative utility, the pair is preferred to buying item by itself. For menu , and implies , thus buying item by itself is preferred to buying item by itself. Similarly, for , a buyer facing menus or never buys item by itself, so they yield the same revenue. For menu , since and , if then and thus , so whenever item gives non-negative utility, the pair is preferred to buying item by itself. For menu , implies , thus buying item by itself is preferred to buying item by itself.

By symmetry, menus and yield the same revenue; is identical to on , and identical to on . Therefore,

(2)

where denotes the expected revenue from menu for the given distribution. See also Figure 4. Notice that this argument also works for correlated symmetric distributions.

Incentive constraints induced by menus , (dotted), , and .

Figure 4: Symmetric menus are optimal (Case )

The case where

We consider three alternative symmetric menus: , , and .141414Informally, we expect to have probability mass at (for ) because this is where the buyer switches from item to the bundle in the asymmetric menu. We show that at least one of them yields at least as much revenue as the asymmetric menu. Specifically, we consider the following two cases:

  • When

    (3)

    we show that

    thus one of the menus or has revenue as least as high as the menu .

  • When

    (4)

    we show that

    thus one of the menus or has revenue as least as high as the menu .

Both proofs proceed as follows. First, we consider the partition of the valuation space induced by the incentive constraints of the respective four mechanisms (we consider a total of five menus, but we only need four for each case). Then, for each region of the space, we sum the payments to the seller in the symmetric menus, and subtract the payments in the asymmetric menus. At this point, we still have some regions with negative and positive contributions. We now use the fact that the two items are symmetrically distributed to observe that the revenue contribution of one region of valuations is exactly equal to the contribution of the symmetric region (when reflecting over the line), and we rearrange the revenue contributions without changing the total revenue using this fact. Finally, we use the inequality on the single dimensional distributions (Inequality (4) or (3)) to show that the positive contributions outweigh the negative contributions. We note that this last argument is only true for the IID case and it fails for correlated distribution – and indeed the claim is false for such distributions, see Example 6.3. See Figures 5 and 6 for more details.


Payments to seller with menu . Note that the menu is submodular as , so this plot as a submodular menu is correct (see Figure 1 for the difference in plots of submodular and supermodular menus).


Payments to seller with menu .


Payments to seller with menu .


Sum of payments from menus and (dotted).151515See also the right plot in Figure 1.
Figure 5: Symmetric menus are optimal (Case ): the menus

Menus , (dotted), , and .

On the left: the number inside each region represents the sum of contributions from and , minus the contributions from and .

On the right: we rearranged the payments, moving from the right-center region to the top-center region. By symmetry, the probability mass on those two regions is the same, so the total revenue does not change. Whenever (3) holds, we have that the total contribution in each horizontal line (fixed ) is nonnegative.

Menus , (dotted), , and .

On the left: The number inside each region represents the sum of contributions from and , minus the contributions from and .

On the right: We rearranged the payments, moving from the left-center region to the center-bottom region. By symmetry, the probability mass on those two regions is the same, so the total revenue does not change. Notice that whenever (4) holds, we have that the total contribution in each horizontal line (fixed ) is nonnegative.

Figure 6: Symmetric menus are optimal (Case ): averaging arguments

5 More than Two Items

In this section we show that our results for two items do not extend even to three items, even when the values of the three items are sampled IID from a distribution with a finite support. While for two independent items the revenue of a deterministic menu can be obtained with a submodular menu, for three IID items this is not true. Additionally, for three items symmetric menus are losing revenue compared to arbitrary deterministic menus.

Theorem 5.1.

There exist a distribution with finite support such that for three items sampled IID from (), the following holds161616Note that since the support is finite, the optimal mechanism in each case is indeed obtained.: The expected revenue of the optimal deterministic mechanism is strictly larger than both the expected revenue of the optimal symmetric deterministic mechanism and of the optimal submodular deterministic mechanism.

The theorem directly follows from the next example which further shows that imposing both symmetry and submodularity decreases the revenue even more than imposing only one of them.

Example 5.2 ().

Let be following discrete distribution that samples values uniformly from the multi-set , that is: , and . It holds that:

  • The expected revenue of the optimal deterministic mechanism is .

  • The expected revenue of the optimal symmetric deterministic mechanism is .

  • The expected revenue of the optimal submodular deterministic mechanism is .

  • The expected revenue of the optimal symmetric and submodular deterministic mechanism is .

First, observe that it is enough to only consider integer prices, bounded above by appropriate prices (e.g., not more than 6 for a single item, not more than 12 for 2 items, etc.). Using this fact, by exhaustive search we found the optimal mechanisms in each class. These optimal mechanisms, with prices , are as follows:

  • is an optimal deterministic mechanism;

  • is an optimal symmetric deterministic mechanism;

  • is an optimal submodular deterministic mechanism; and

  • is an optimal symmetric and submodular deterministic mechanism. ∎

This example does not rule out, however, that subadditive deterministic mechanisms can get as much revenue as any other deterministic mechanism. Resolving whether this is indeed the case is the main open problem we propose in this work:

Open Question 5.3.

For items with independent valuations, is it true that the revenue of any deterministic mechanism can be obtained by a deterministic mechanism that is subadditive?

Additionally, we have shown that for two items, revenue-monotone deterministic mechanisms can get as much revenue as any other mechanisms. We leave open the question if this is true for more than two items.

Open Question 5.4.

For items with independent valuations, is it true that the revenue of any deterministic mechanism can be obtained by a deterministic mechanism that is revenue monotone?

6 Correlated Valuations

We next aim to understand to what extent the results for deterministic auction for selling two items with values sampled independently, extend to two item with values sampled from a correlated joint distribution. We have seen that for two independent items restricting to submodular menu does not result in any revenue loss for the seller. We next show that for correlated items, the situation is very different - moving from submodular menu to an unrestricted one can increase the revenue of the seller, and that increase can be as large as almost 50%. We also show that this is tight and the gain is never larger than 50%.

6.1 Price of Submodularity

The following example shows that for two items with values sampled from a correlated distribution, the optimal deterministic auction can gain almost 50% more revenue than the optimal deterministic submodular auction — and this is true even for a symmetric correlated distribution.

Example 6.1 ().

Fix a small . Consider two items with valuations drawn from the following distribution:

There is a high-revenue deterministic auction that charges for each item or for the grand bundle. Its revenue is . In contrast we show below that any submodular auction has revenue at most .

We consider three different cases:

  • If the auction prices each of the two items at a price larger than , the buyer does not buy anything for any valuation in her support except . The seller’s revenue is at most the welfare from this event, .

  • Suppose the auction charges at most for one of the items, and more than for the other item. Then the grand bundle never sells for price more than — otherwise even with valuration the buyer prefers buying the cheaper item over the grand bundle. Furthermore, the buyer never buys the more expensive item. Therefore the seller’s revenue is bounded from above by from selling the cheaper item and from selling the grand bundle.

  • Finally, any submodular auction that charges at most for both items (and hence, by submodularity, at most for the grand bundle) can make at most revenue from the high valuation.

The following lemma shows that the previous example is tight.

Lemma 6.2.

For any joint value distribution over two items it holds that

Proof.

Throughout the proof we use to denote the expected revenue from menu over the distribution . Fix any deterministic menu . Assume that the menu is not submodular, that is, . Let denote the events that, given prices , , and , the buyer buys the empty set, only item , etc. Now, the revenue from the menu is given by:

Let . To complete the proof we will show that we can bound by half the revenue obtained by selling only the grand bundle at price (with the menu ) plus the revenue obtained by selling the two items separately at prices and (with the additive menu ). This implies that times the better revenue of the two menus, is at least the revenue of the menu . I.e. we show that

which is clearly sufficient to complete the proof as the menus and are both submodular.

For any valuation for which the buyer buys the grand bundle (i.e. event occurs) with menu , it holds that he prefers the bundle over any item , thus . So by summing these two up, in any such case it holds that . Thus with menu , for any such valuation the buyer prefers buying the grand bundle over the empty set. Therefore,

Additionally, for the mechanism that offers each item separately for price (and the bundle for ) it holds that for buyer’s valuations in event , the buyer buys both items since for any items as it holds that (by supermodularity of the menu ), thus

Combining the above two inequalities:

6.2 Symmetric vs Asymmetric Menus

Consider a joint distribution that is symmetric. Will a deterministic seller lose revenue by restricting her menu to be symmetric when items are correlated? We next show that for correlated distributions over two items, the fact that the joint distribution is symmetric is not sufficient to ensure that there is an optimal deterministic menu that is symmetric. Moreover, there is a constant factor gap between the achievable revenues.

Example 6.3 (Asymmetric menus for symmetric distributions).

Fix a small and consider the following correlated distribution over two-item valuations:

There is a deterministic asymmetric menu with prices and it obtains revenue of .

In contrast, we claim that any symmetric menu yields revenue at most . We consider three different cases, depending on the price of any of the items by itself (notice that by the symmetry restriction they are priced the same).

  • Case : the buyer buys nothing unless she has valuation . Thus the expected revenue is at most .

  • Case : the buyer never buys a single item when his valuation is . If a buyer with valuation does not buy the bundle, the revenue is bounded by . If a buyer with valuation does buy the bundle, it must be the case that and so . If the revenue is bounded from above by . Otherwise, players with types other than do not buy the bundle, and the revenue is bounded by