Revenue maximization is one of the fundamental problems in auction theory. The well-celebrated result of Myerson  characterized the revenue-maximizing mechanism when there is only one item for sale. Specifically, in the single buyer case, the optimal solution is to post a take-it-or-leave-it price. Since Myerson’s work, the optimal mechanism design problem has been studied extensively in computer science literature and much progress has been made [12, 13, 14, 15, 2, 26]. The problem of finding the optimal auction turned out to be so much more complex than the single-item case. Unlike the Myerson’s single-item auction, the optimum can use randomized allocations and price bundles of items already for two items and a single buyer. It is also known that the gap between the revenue of the optimal randomized and optimal deterministic mechanism can be arbitrarily large [11, 38], the optimal mechanism may require a menu with infinitely many options [42, 27], and the revenue of the optimal auction may decrease when the buyer’s valuation distributions move upwards (in the stochastic dominance sense).
In light of these negative results for optimal auction design, many recent papers focused on the design of simple mechanisms that are approximately optimal. One such notable line of work initiated by Hart and Nisan  concerns a basic and natural setting of monopoly problem for the buyer with item values drawn independently from given distributions and whose valuation for the sets of items is additive111A buyer has additive valuations if his value for a set of items is equal to the sum of his values for the items in the set. (linear). A remarkable result by Babaioff et al.  showed that the better mechanism of either selling items separately, or selling the grand bundle extracts at least -fraction of the optimal revenue. It was also observed [38, 4, 45] that the independence assumption on the items is essentially necessary and without it no simple (any deterministic) mechanism cannot be approximately optimal.
Auction design with budget constraints is an even harder problem. Because buyer’s utility is no longer quasi-linear, many standard concepts do not carry over222E.g., the classic VCG mechanism may not be implementable and social efficiency may not be achievable in the budgeted-setting .. For example, even for one buyer and one item, the optimal mechanism may require randomization when the budget is public , and may need an exponential-size menu when the budget is private . Despite many efforts [40, 23, 35, 10, 1, 24, 31, 9, 8, 18, 21, 7, 34, 29, 28, 30, 46], the theory of optimal auction design with budgets is still far behind the theory without budgets.
In this paper, we investigate the effectiveness of simple mechanisms in the presence of budgets. Our work is motivated by the following questions:
How powerful are simple mechanisms in the presence of budgets? In particular, is there a simple mechanism that is approximately optimal for a budget-constrained buyer with independent valuations?
To this end we consider one of the most basic and natural settings of extensively studied monopoly problem for an additive buyer. In this setting, a monopolistic seller sells items to a single buyer. The buyer has additive valuations drawn independently for each item from an arbitrary (non-identical) distribution. We study two different budget settings: the public budget case where the buyer has a fixed budget known to the seller, and the private budget case where the buyer’s budget is drawn from a distribution. The seller wishes to maximize her revenue by designing an auction subject to individual rationality, incentive compatibility, and budget constraints. We consider the Bayesian setting where the buyer knows his budget and his values for each item, but the seller only knows the prior distributions.
1.1 Our Results and Techniques
Our first result is that simple mechanisms remain approximately optimal when the buyer has a public budget.
For an additive buyer with a known public budget and independent valuations, the better of selling each item separately and selling the grand bundle extracts a constant fraction of the optimal revenue.
Theorem 1.1 is among the few positive results in budget-constrained settings that hold for arbitrary distributions. Before our work, it is not clear that any mechanism extracting a constant fraction of the optimal revenue can be computed in polynomial time.
In Sections 3 and 4, we present two different approaches to prove Theorem 1.1. Both approaches truncate the valuation distribution according to the budget (in different ways) and then relate the revenues of the optimal/simple mechanisms on the truncated distribution to the revenues on the original valuations. The first approach uses the main result of  in a black-box way, and the second approach adapts the duality-based framework developed in .
It is worth pointing out that many of our structural lemmas hold for correlated valuations as well. Using these lemmas, we can generalize Theorem 1.1 with minimum effort to allow the buyer to have weakly correlated valuations. We call a distribution weakly correlated if it is the result of conditioning an independent distribution on the sum of being at most : (See Definition 1 for the formal definition).
Let be a weakly correlated distribution. For an additive buyer with a public budget and valuations drawn from , the better of selling separately and selling the grand bundle extracts a constant fraction of the optimal revenue.
In Section 5, we examine the private budget setting. The budget is no longer fixed but is drawn from a distribution . The seller only knows the prior distribution but not the value of . We first show that if the valuations can be correlated with the budget, the problem is at least as hard as budget-free mechanism design with correlated valuations, where simple mechanisms are known to be ineffective. In light of this negative result, we focus on the setting where the budget distribution is independent of the valuations . In this setting, we show that simple mechanisms are approximately optimal when the budget distribution satisfies the monotone hazard rate (MHR) condition.
When the budget distribution is MHR, the better mechanism of pricing items separately and selling a grand bundle achieves a constant fraction of the optimal revenue.
We will show that it is sufficient to pretend the buyer has a public budget . The proof of Theorem 1.2 uses the MHR condition, as well as the fact that for a public budget , the (budget-constrained) optimal revenue is nondecreasing in , but optimal revenue divided by is nonincreasing in .
1.2 Related Work
The most closely related to ours are the following two lines of work.
In a line of work initiated by Hart and Nisan [39, 41, 4],  first showed that for an additive buyer with independent valuations, either selling separately or selling the grand bundle extracts a constant fraction of the optimal revenue. This was later extended to multiple buyers , as well as buyers with more general valuations (e.g., sub-additive , valuations with a common-value component , and valuations with complements ). Others have studied the trade-off between the complexity and approximation ratio of an auction, along with the design of small-menu mechanisms in various settings [38, 48, 32, 25, 3].
Auctions for Budget-Constrained Buyers.
There has been a lot of work studying the impact of budget constraints on mechanism design. Most of the earlier work required additional assumptions on the valuations distributions, like regularity or monotone hazard rate ([40, 23, 9, 44]). We mention a few results that work for arbitrary distributions. For public budgets,  designed approximately optimal mechanisms for several single-parameter settings and multi-parameter settings with unit-demand buyers. For private budgets,  characterized the structure of the optimal mechanism for one item and one buyer.  gave a constant-factor approximation for additive bidders whose private budgets can be correlated with their values. However, they require the buyers’ valuation distribution to be given explicitly, which is of exponential size in our setting. There are also approximation and hardness results in the prior-free setting [10, 1, 29], as well as designing Pareto optimal auctions [31, 34].
Other Related Work.
Our work concerns revenue maximization for additive buyer. Another natural and basic scenario extensively studied in the literature concerns buyers with unit-demand preferences [19, 20, 22]. Our work studies monopoly problem for additive budgeted buyer in the standard Bayesian approach. In this framework, the prior distribution is known to the seller and typically is assumed to be independent. Parallel to this framework, the (budgeted) additive monopoly problem has been studied in a new robust optimization framework [17, 36]. Another group of papers on budget feasible mechanism design [7, 24, 47, 46] studies different reverse auction settings and are concerned with value maximization.
2.1 Optimal Mechanism Design
We study the design of optimal auctions with one buyer, one seller, and heterogeneous items labeled by . There is exactly one copy of each item, and the items are indivisible. The buyer has additive valuation ( for any set ) and a publicly known budget 333 In this paper, we mostly focus on the public budget case. So we define notations and discuss backgrounds assuming the buyer has a public budget..
We use to denote the buyer’s valuations, where is the buyer’s value for item . We consider the Bayesian setting of the problem, in which the buyer’s values are drawn from a discrete444Like previous work on simple and approximately optimal mechanisms, our results extend to continuous types as well (see, e.g.,  for a more detailed discussion). distribution . Let be the set of all possible valuation profiles in . We use for any
to denote the probability mass function of: . Let . We say the valuation distribution is independent across items if it can be expressed as .
We assume the buyer is risk-neutral and has quasi-linear utility when the payment does not exceed his budget.
Let and denote the allocation and payment rules of a mechanism respectively.
That is, when the buyer reports type , the probability that he will receive item is , and his expected payment is (over the randomness of the mechanism).
Thus, if the buyer has type , his (expected) value for reporting type is exactly , 555We use to denote the inner product of two vectors
to denote the inner product of two vectorsand . and his (expected) utility for reporting type is
By the revelation principle, it is sufficient to consider mechanisms that are incentive compatible (i.e., “truthful”). A mechanism is (interim) incentive-compatible (IC) if the buyer is incentivized to tell the truth (over the randomness of mechanism), and (interim) individually rational (IR) if the buyer’s expected utility is non-negative whenever he reports truthfully. We use for the option of not participating in the auction (), and let . Then, the IC and IR constraints can be unified as follows:
To summarize, when the seller faces a single buyer with budget and valuation drawn from , the optimal mechanism
is the optimal solution to the following (exponential-size) linear program (LP):
A mechanism is called ex-post IC, ex-post IR, or ex-post budget-preserving respectively, if the corresponding constraints hold for all possible outcomes, without averaging over the randomness in the mechanism. We will show the better of pricing each item separately and pricing the grand bundle, which is a deterministic ex-post mechanism, can extract a constant fraction of the revenue of any interim mechanism.
2.2 Simple Mechanisms
For a buyer with valuation distribution , we frequently use the following notations in our analysis:
: the revenue of the optimal truthful mechanism.
: the maximum revenue obtainable by pricing each item separately.
: the maximum revenue obtainable by pricing the grand bundle.
: the revenue of the optimal truthful mechanism, when the buyer has a budget .
: the maximum revenue that can be extracted by pricing each item separately, when the buyer has a public budget .
: the maximum revenue that can be extracted by pricing the grand bundle, when the buyer has a public budget .
We know that is obtained by running Myerson’s optimal auction separately for each item, and is obtained by running Myerson’s auction viewing the grand bundle as one item. Similarly, is a single-parameter problem as well, with the minor change that the posted price is at most .
The case of is more complicated. For example, when a budgeted buyer of type participates in an auction with posted price for each item , he will maximize his utility by solving a Knapsack problem. There exists a poly-time computable mechanism that extracts a constant fraction of (e.g., ). We focus on the structural result that the better of and is a constant approximation of . A better approximation for is an interesting open problem that is beyond the scope of this paper.
2.3 Weakly Correlated Distributions
We call a distribution like weakly correlated if the only condition causing the correlation is a cap on its sum.
For an -dimensional independent distribution and a threshold , we remove the probability mass on any with and renormalize. Let denote the resulting distribution. Formally,
Weakly correlated distributions arise naturally in our analysis. We will show that if the buyer’s valuations are weakly correlated, then the better of selling separately and selling the grand bundle is approximately optimal, and this holds with or without a (public) budget constraint.
2.4 First-Order Stochastic Dominance
Stochastic dominance is a partial order between random variables. A random variablewith (weakly) first-order stochastically dominates another random variable with if and only if
This notion of stochastic dominance can be extended to multi-dimensional distributions. In this paper, we use the notion of coordinate-wise dominance.
Given two -dimensional distributions and , we say coordinate-wise stochastic dominates ( or ) if there exists a randomized mapping such that when , and coordinate-wise for all with probability .
This notion helps us express the monotonicity of optimal revenues in some cases. For example, we can show that when . The mapping allows us to couple the draws and , so that for a set of fixed prices, if the buyer buys an item under , he will also buy it under .
3 Public Budget
In this section, we focus on the public budget case and prove our main result (Theorem 1.1). The buyer has a fixed budget and valuations drawn from an independent distribution .
Theorem 1.1. .
It follows that the better of and is at least . 666We do not optimize the constants in our proofs. In Section 4, we will give an alternative proof of Theorem 1.1 that shows , thus improving this constant from 32 to 11.
Overview of Our Approach.
Instead of taking the Lagrangian dual of LP (1) to derive an upper bound on the optimal objective value , we adopt a more combinatorial approach. Intuitively, we come up with a charging argument that splits and charges each part to either or .
First, we partition the buyer types into two sets: high-value types where and low-value types where . Note that we can already charge the revenue of high-value types to : If we sell the grand bundle at price , all high-value types will exhaust their budgets.
We now examine the low-value types. Let denote the valuation distribution conditioned on the buyer having a low-value type. Observe that is independent because it is defined using -norm, and we can remove the budget to upper bound its revenue. For a budget-free additive buyer with independent valuations, we can apply the main result of , which states that either selling separately or grand bundling works for : .
Next, we will relate to . We can assume the sum of is usually much smaller than . Similar to standard tail bounds, if the sum is often small and the random variables are independent and bounded (each is at most ), then must have an exponentially decaying tail. Therefore, we can add back the budget, because the sum , which upper bounds the buyer’s payment, is rarely very large.
Finally, we will show that and . The BRev statement is easy to verify, but the SRev statement is more tricky. The monotonicity of in the budget-free case (see Section 2.4) no longer holds when there is a budget. Fortunately, we can pay a factor of two and circumvent this non-monotonicity due to budget constraints.
We will now make our intuitions formal and present three key lemmas. Throughout the paper, we will always use as defined below.
Fix an -dimensional distribution . Let be the independent distribution where every coordinate of is capped at . That is, , and is given by .
Assume . Then, and .
3.1 Proof of Lemma 1
We will prove the following lemma, which is a generalization of Lemma 1.
Fix and . For any distribution with and for any , we have .
Intuitively, Lemma 4 upper bounds the optimal revenue by splitting the buyer types into two sets: when , we upper bound the seller’s revenue by the budget ; when , we run the optimal mechanism for .
Proof (of Lemma 4)
Let and , and
denote the support and probability density function ofand respectively. Let be the optimal mechanism that obtains . Recall that and are the allocation and payment rules, and is the optimal solution to LP (1) for and .
We split the optimal revenue into two parts:
Since , the first term is at most , because we can sell the grand bundle at price .
The second term is at most , because is a feasible solution to the LP for . In other words, satisfies the IC and IR constraints for . The revenue of is at least the revenue of on :
Combining the upper bounds, we get .
3.2 Proof of Lemma 2
Lemma 2 states that when the sum of is often small, the budget does not matter too much for . Intuitively, because each coordinate of is independent and upper bounded by , a concentration inequality implies that the sum has an exponentially decaying tail. Therefore, the budget constraint is less critical because it is very unlikely that the buyer’s value for the grand bundle is much larger than the budget.
We formalize this intuition by proving the following lemma, which is similar to standard tail bounds. The main difference is that, instead of knowing the mean of is small, we only know that is small.
If is independent and for all , then
In particular, if , then for all integer ,
Proof (of Lemma 2)
Let and . We know that from the assumption .
3.3 Proof of Lemma 3
Lemma 3 states that and are both (up to constant factors) monotone in . We prove a more general version of the lemma that does not require to be independent. Recall that means is coordinate-wise stochastically dominated by .
Fix and . For any distribution , and .
Intuitively, we would like to prove that for any . While this is true in the budget-free case (See Section 2.4), it is actually false in the presence of a budget. We give a counterexample in Appendix 0.B. Fortunately, we can prove . The intuition is that we can cap the price of each item at , then the buyer either spend at least , or he will purchase everything he likes.
Proof (of Lemma 6)
First consider BRev. Because and ,
For SRev, let be the optimal mechanism that achieves by pricing each item separately. We construct a mechanism to mimic except the prices are capped at . Consider applying to a buyer with valuation drawn from and a budget . As , we can couple the realizations and such that (coordinate-wise). For every pair:
If gets a revenue of at least on . This is at least -fraction of the revenue gets on , because the latter is at most .
If gets a revenue less than on , then the buyer has enough budget left to buy any item. Therefore, the buyer can buy everything he wants. Because , the revenue of on is at least that of on .
Thus, can get at least -fraction of the revenue that gets on , which implies . ∎
4 Public Budget and Weakly Correlated Valuations
In this section, we present an alternative approach to prove our main result (Theorem 1.1). Recall that the buyer has a public budget and valuations drawn from an independent distribution .
Theorem 1.1. .
Overview of Our Approach.
We will truncate the input distribution in a different way: instead of truncating in -norm (as in Section 3), we will truncate in -norm. This truncation produces a correlated distribution . The upshot of truncating in -norm is that we always have , so can ignore the budget. In addition, as in Section 3, we can relate the optimal revenue to the revenue of (Lemma 4), and we can relate the revenue of simple mechanisms on back to revenue of simple mechanisms on (Lemma 6).
We still need to argue that simple mechanisms work well for . This is the main challenge in this approach. Because is correlated, we cannot apply the result of  in a black-box way. Instead, we need to modify the analysis of previous work [41, 4, 16] and build on the key ideas like “core-tail” decomposition. More specifically, we generalize the duality-based framework developed in  to handle the specific type of correlation has.
Weakly Correlated Valuations.
It is worth mentioning that our structural lemmas (Lemmas 4 and 6) do not require the input distribution to be independent. This is why our techniques can be applied to more general settings. For example, in this section, we will generalize Theorem 1.1 with minimum effort to handle weakly correlated valuations (see Definition 1 for the formal definition).
Fix . Let for an independent distribution . We have .
If , then the seller can price the grand bundle at and the buyer always buys it. In this case, the revenue is and . Thus, we focus on the more interesting case where . 777Throughout the paper, when we consider the conditional distribution , we will always have , so that the event we condition on happens with non-zero probability.
Let for . We will reuse Lemmas 4 and 6 from Section 3. We can reuse both lemmas because they do not require or to be independent, does not modify the small-sum part of , and (which we will prove as Lemma 15 in Appendix 0.D).
We now prove Corollary 1. Intuitively, Corollary 1 holds because simple mechanisms work well for weakly correlated valuations, and the the weakly-correlated notion is closed under further capping the sum.
Let be the input distribution. If , then we can remove the budget constraint and apply Lemma 7 directly. If , then we can cap at to obtain a weakly correlated distribution . One can verify that Lemmas 4 and 6 still hold for and , and Lemma 7 holds for . The only difference is that we need to show for . We will prove this (Lemma 14) in Appendix 0.D. ∎
5 Private Budget
In this section, we consider the case where the budget is no longer fixed but instead drawn from a distribution . One natural model is that the buyer’s budget is first drawn from , and then depending on the value of , the buyer’s valuations are drawn independently for each item.
We show that in this case, the problem is at least as hard as finding (approximately) optimal mechanisms for correlated valuations in the budget-free setting. Consider an instance in which all possible budgets are larger than so they are irrelevant. However, the budget can still be used as a signal (or a correlation device) to produce correlated valuations. It is known that for correlated distributions, the better of selling separately and bundling together , or even the best partition-based mechanism , does not offer a constant approximation.
This negative result motivates us to study the private budget setting when the budget distribution is independent of the valuation distributions .
5.1 Monotone-Hazard-Rate Budgets
We focus on the case where the budget is independent of valuations, and it is drawn from a continuous888If the distribution is a discrete MHR distribution, similar results still hold. For discrete distributions we have instead of . monotone-hazard-rate (MHR) distribution. Let and
be the probability density function and cumulative distribution function of. The MHR condition says is non-decreasing in .
Let be the expectation of an MHR distribution . Let be the optimal mechanism for a buyer with a public budget . Then in expectation, extracts at least -fraction of the expected optimal revenue when the buyer has a private budget drawn from .
Let denote the expected revenue of when the buyer has a public budget and valuations drawn from . Let denote the expected revenue of when the buyer’s budget is drawn from .
The second last step uses when , because provides a menu of allocation/payment pairs for the buyer to choose from; A buyer with budget can afford any option on the menu so he will choose the same option as if he had budget . The last inequality comes from the fact that for any MHR distribution , (see, e.g., ).
Let denote the optimal revenue we can extract when the buyer has private budgets drawn from .
The first line is because the seller can only do better if she knows the buyer’s budget . The third line is because . The second line uses the fact that when and when .
We have when because a buyer with budget can afford all options from the menu that achieves . When , consider the menu that achieves and cap all prices at . A buyer with budget either chooses the same option as if he had budget , or chooses a different option whose price must be , and therefore .
By definition . Therefore, . ∎
Theorem 1.2. When the budget distribution is MHR, the better of pricing items separately and bundling them together achieves a constant fraction of the optimal revenue.
6 Conclusion and Future Directions
In this paper, we investigated the effectiveness of simple mechanisms in the presence of budgets, and showed that for an additive buyer with independent valuations and a public budget, either selling separately or selling the grand bundle gives a constant approximation to optimal revenue.
The area of designing simple and approximately optimal auctions with budget constraints is still largely unexplored. Our work leaves many natural follow-up questions. We only considered selling to a single buyer. An immediate open question is whether our results can be extended to multiple bidders. A generalization to multiple bidders is known in the budget-free case [49, 16].
Question 1. Is there a simple mechanism that is approximately optimal for multiple additive buyers, when each buyer has the same public budget ?
For private budgets where the budget is independent of the valuations, we showed that if the budget distribution satisfies monotone hazard rate, then we can extract a constant fraction of the revenue. The general case with arbitrary budget distributions appears to be nontrivial and is an interesting avenue for future work.
Question 2. Is there a simple mechanism that is approximately optimal for an additive buyer with private budgets, when the budget distribution is independent of the valuations?
Yu Cheng is supported by NSF grants CCF-1527084, CCF-1535972, CCF-1637397, CCF-1704656, IIS-1447554, and NSF CAREER Award CCF-1750140. Kamesh Munagala is supported by NSF grants CCF-1408784, CCF-1637397, and IIS-1447554; and by an Adobe Data Science Research Award. Kangning Wang is supported by NSF grants CCF-1408784 and CCF-1637397.
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