In the past several years, we have seen a tremendous advance in the field of Bayesian Mechanism Design, based on ideas and concepts rooted in Theoretical Computer Science (TCS). For instance, due to a line of work initiated by Chawla, Hartline, and Kleinberg [CHK07], we now know that posted-price mechanisms are approximately optimal with respect to the optimal Bayesian Incentive Compatible222A mechanism is Bayesian Incentive Compatible (BIC) if it is in every bidder’s interest to tell the truth, assuming that all other bidders’ reported their true values. A mechanism is Dominant Strategy Incentive Compatible (DSIC) if it is in every bidder’s interest to tell the truth no matter what reports the other bidders make.
(BIC) mechanism whenever buyers are unit-demand,333A valuation is unit-demand if . A valuation is additive if . and values are independent444 That is, the random variables
That is, the random variablesare independent (where denotes bidder ’s value for item ). [CHMS10, CMS15, KW12]. Due to a line of work initiated by Hart and Nisan [HN12], we now know that either running Myerson’s auction separately for each item or running the VCG mechanism with a per-bidder entry fee555By this, we mean that the mechanism offers each bidder the option to participate for , which might depend on the other bidders’ bids but not bidder ’s. If they choose to participate, then they play in the VCG auction (and pay any additional prices that VCG charges them). is approximately optimal with respect to the optimal BIC mechanism whenever buyers are additive, and values are independent [LY13, BILW14, Yao15]. Due to a line of work initiated by Cai et al. [CDW12a], we now know that optimal mechanisms are distributions over virtual welfare maximizers, and have computationally efficient algorithms to find them in quite general settings [CDW12b, CDW13a, CDW13b, BGM13, DW15, DDW15]. The main contribution of this work is a unified approach to these three previously disjoint research directions. At a high level, we show how a new interpretation of the Cai-Daskalakis-Weinberg (CDW) framework provides us a duality theory, which then allows us to strengthen the characterization results of Cai et al., as well as interpret the benchmarks used in [CHK07, CHMS10, CMS15, KW12, HN12, CH13, LY13, BILW14] as dual solutions. Surprisingly, we learn that essentially the same dual solution yields all the key benchmarks in these works. We show how to extend this dual solution to multi-bidder settings, and analyze the mechanisms developed in [CHMS10, Yao15] with respect to the resulting benchmarks. In both cases, our analysis yields improved approximation ratios.
1.1 Simple vs. Optimal Auction Design
It is well-known by now that optimal multi-item auctions suffer many properties that are undesirable in practice. For example, with just a single additive buyer and two items, the optimal auction could be randomized [Tha04, Pav11]. Moreover, there exist instances where the buyer’s two values are drawn from a correlated distribution where the optimal revenue achieves infinite revenue while the best deterministic mechanism achieves revenue [BCKW10, HN13]. Even when the two item values are drawn independently, the optimal mechanism might offer uncountably many different randomized options for the buyer to choose from [DDT13]. Additionally, revenue-optimal multi-item auctions behave non-monotonically: there exist distributions and , where stochastically dominates , such that the revenue-optimal auction when a single additive buyer’s values for two items are drawn from achieves strictly larger revenue than the revenue-optimal auction when a single additive buyer’s values are drawn from [HR12]. Finally, it is known that revenue-optimal auctions may not be DSIC [Yao17], and are also #P-hard to find [DDT14].
In light of the aforementioned properties, simple mechanisms are often used in lieu of optimal mechanisms in practice, and an active line of research coined “simple versus optimal” mechanism design [HR09] aims to rigorously understand when simple mechanisms are appropriate in practice. Still, prior work essentially shows that simple mechanisms are never exactly optimal, so the main goal of these works is to understand when simple mechanisms are approximately optimal.666On this front, one should not interpret (say) an -approximation as suggesting that sellers should be happy with of the revenue they could potentially achieve. Rather, these guarantees are meant to be interpreted more qualitatively, and suggest claims like “If simple auction guarantees a small constant-factor approximation in the worst-case, but simple auction does not, maybe it’s safer to use auction in practice.” Some of the most exciting contributions from TCS to Bayesian mechanism design have come from this direction, and include a line of work initiated by Chawla et al. [CHK07] for unit-demand buyers, and Hart and Nisan [HN12] for additive buyers.
In a setting with heterogeneous items for sale and unit-demand buyers whose values for the items are drawn independently, the state-of-the-art shows that a simple posted-price mechanism777 A posted price mechanism visits each buyer one at a time and posts a price for each item. The buyer can then select any subset of items and pay the corresponding prices. Observe that such a mechanism is DSIC. obtains a constant factor of the optimal BIC revenue (the revenue of the optimal BIC mechanism) [CHK07, CHMS10, CMS15, KW12]. The main idea behind these works is a multi- to single-dimensional reduction. They consider a related setting where each bidder is split into separate copies, one for each item, with bidder ’s copy interested only in item . The value distributions are the same as the original multi-dimensional setting. One key ingredient driving these works is that the optimal revenue in the original setting is upper bounded by a small constant times the optimal revenue in the copies setting.
In a setting with heterogeneous items for sale and additive buyers whose values for the items are drawn independently, the state-of-the-art result shows that for all inputs, either running Myerson’s optimal single-item auction for each item separately or running the VCG auction with a per-bidder entry fee obtains a constant factor of the optimal BIC revenue [HN12, LY13, BILW14, Yao15]. One main idea behind these works is a “core-tail decomposition”, that breaks the revenue down into cases where the buyers have either low (the core) or high (the tail) values.
Although these two approaches appear different at first, we are able to show that they in fact arise from basically the same dual in our duality theory. Essentially, we show that a specific dual solution within our framework gives rise to an upper bound that decomposes into the sum of two terms, one that looks like the the copies benchmark, and one that looks like the core-tail benchmark. In terms of concrete results, this new understanding yields improved approximation ratios on both fronts. For additive buyers, we improve the ratio provided by Yao [Yao15] from to . For unit-demand buyers, we improve the approximation ratio provided by Chawla et al. [CHMS10] from to .
In addition to these concrete results, our work makes the following conceptual contributions as well. First, while the single-buyer core-tail decomposition techniques (first introduced by Li and Yao [LY13]) are now becoming standard [LY13, BILW14, RW15, BDHS15], they do not generalize naturally to multiple buyers. Yao [Yao15] introduced new techniques in his extension to multi-buyers termed “-adjusted revenue” and “-exclusive mechanisms,” which are technically quite involved. Our duality-based proof can be viewed as a natural generalization of the core-tail decomposition to multi-buyer settings. Second, we use basically the same analysis for both additive and unit-demand valuations, meaning that our framework provides a unified approach to tackle both settings. Finally, we wish to point out that the key difference between our proofs and those of [CHMS10, BILW14, Yao15] are our duality-based benchmarks: we are able to immediately get more mileage out of these benchmarks while barely needing to develop new approximation techniques. Indeed, the bulk of the work is in properly decomposing our benchmarks into terms that can be approximated using ideas similar to prior work. All these suggest that our techniques are likely be useful in more general settings (and indeed, they have been: see Section 1.3).
1.2 Optimal Multi-Dimensional Mechanism Design
Another recent contribution of the TCS community is the CDW framework for generic Bayesian mechanism design problems. Here, it is shown that Bayesian mechanism design problems for essentially any objective can be solved with black-box access just to an algorithm that optimizes a perturbed version of that same objective. That is, even though the original mechanism design problem involves incentives, the optimal BIC mechanism can be found via black-box queries to an algorithm (where the input is known/given and there are no incentives), but this algorithm optimizes a perturbed objective instead. One aspect of this line of work is computational: we now have computationally efficient algorithms to find the optimal (or approximately optimal) mechanism in numerous settings of interest (including the aforementioned cases of many additive/unit-demand buyers, but significantly more general as well). Another aspect is structural: we now know, for instance, that in all settings that fit into this framework, the revenue-optimal mechanism is a distribution over virtual welfare optimizers.888Their reduction applies to objectives beyond revenue, such as makespan. The focus of the present paper is on revenue, so we only focus on the projection of their results onto this setting. A mechanism is a virtual welfare optimizer if it pointwise optimizes the virtual welfare (that is, on every input, it selects an outcome that maximizes the virtual welfare). The virtual welfare is given by a virtual valuation/transformation, which is a mapping from valuations to linear combinations of valuations.
The structural characterization from previous work roughly ends here: the guaranteed virtual transformations were randomized with no promise any additional properties beyond their existence (and that they could be found in poly-time). Our contribution to this line of work is to improve the existing structural characterization. Specifically, we show that every instance has a strong dual in the form of disjoint flows, one for each agent. The nodes in agent ’s flow correspond to possible types of this agent,999Both the CDW framework and our duality theory only apply directly if there are finitely many possible types for each agent. and non-zero flow from type to captures that the incentive constraint between and binds. We show how a flow induces a virtual transformation, and that the optimal dual gives a virtual valuation function such that:
This virtual valuation function is deterministic and can be found computationally efficiently.
The optimal mechanism has expected revenue its expected virtual welfare, and every BIC mechanism has expected revenue its expected virtual welfare.
The optimal mechanism optimizes virtual welfare pointwise (i.e. on every input, the virtual welfare maximizing outcome is selected).101010This could be randomized
; there is always a deterministic maximizer but in cases where the optimal mechanism is randomized, the virtual transformations are such that there are numerous maximizers, and the optimal mechanism selects one of them from a particular probability distribution.
Here are a few examples of the benefits of such a characterization (which cannot be deduced from [CDW13b]). First, the promised virtual valuation function certifies the optimality of the optimal mechanism: every BIC mechanism has expected revenue its expected virtual welfare, yet the optimal mechanism maximizes virtual welfare pointwise. Second, by looking at the promised virtual valuation function, we can immediately determine which incentive constraints “matter.” Specifically, if the flow corresponding to the promised virtual valuation function sends flow from to , then removing the constraint guaranteeing that prefers to tell the truth rather than report (e.g. through some form of verification) would increase the optimal achievable revenue.111111This claim is only guaranteed to be true in non-degenerate instances with a unique optimal dual - and exactly results from the fact that relaxing tight constraints in non-degenerate LPs improves the optimal solution. Such a characterization should prove a valuable analytical tool for multi-item auctions, akin to Myerson’s virtual values for single-dimensional settings [Mye81].
1.3 Related Work
1.3.1 Duality Frameworks
Recently, strong duality frameworks for a single additive buyer were developed in [DDT13, DDT15, DDT16, GK14, Gia14, GK15]. These frameworks show that the dual problem to revenue optimization for a single additive buyer can be interpreted as an optimal transport/bipartite matching problem. Work of Hartline and Haghpanah also provides an alternative “path-finding” duality framework for a single additive or unit-demand buyer, and has a more similar flavor to ours (as flows can be interpreted as distributions over paths) [HH15]. When they exist, these paths provide a witness that a certain Myerson-type mechanism is optimal, but the paths are not guaranteed to exist in all instances. Also similar is independent work of Carroll, which also makes use of a partial Lagrangian over incentive constraints, again for a single additive buyer [Car16]. In addition to their mathematical beauty, these duality frameworks also serve as tools to prove that mechanisms are optimal. These tools have been successfully applied to provide conditions when pricing only the grand bundle (give the buyer the choice only to buy everything or nothing) [DDT13], posting a uniform item pricing (post the same price on every item) [HH15], or even employing a randomized mechanism [GK15] is optimal when selling to a single additive or unit-demand buyer. However, none of these frameworks currently applies in multi-bidder settings, and to date have been unable to yield any approximate optimality results in the (single bidder) settings where they do apply.
We also wish to argue that our duality is perhaps more transparent than existing theories. For instance, it is easy to interpret dual solutions in our framework as virtual valuation functions, and dual solutions for multiple buyer instances just list a dual for each single buyer. In addition, we are able to re-derive and improve the breakthrough results of [CHK07, CHMS10, CMS15, HN12, LY13, BILW14, Yao15] using essentially the same dual solution. Still, it is not our goal to subsume previous duality theories, and our new theory certainly doesn’t. For instance, previous frameworks are capable of proving that a mechanism is exactly optimal when the input distributions are continuous. Our theory as-is can only handle distributions with finite support exactly.121212Our theory can still handle continuous distributions arbitrarily well. See Section 2. However, we have demonstrated that there is at least one important domain (simple and approximately optimal mechanisms) where our theory seems to be more applicable.
1.3.2 Related Techniques
Techniques similar to ours have appeared in prior works as well. For instance, the idea to use Lagrangian multipliers/LP duality for mechanism design dates back at least to early work of Laffont and Robert for selling a single item to budget-constrained bidders [LR98], is discussed extensively for instance in [Mye97, Voh11], and also used for example in recent works as well [BGM13, Voh12]. It is also apparently informal knowledge among some economists that Myerson’s seminal result [Mye81] can be proved using some form of LP duality, and some versions of these proofs have been published as well (e.g. [MV04]). Still, we include in Section 4 a proof of [Mye81] in our framework to serve as a warm-up (and because some elements of the proof are simplified via our approach).
The idea to use “paths” of incentive compatibility constraints to upper bound revenue in single-bidder problems dates back at least to work of Rochet and Choné studying revenue optimization in general multi-item settings [RC98], and Armstrong [Arm96, Arm99], which studies the special case of a single bidder and two items. More recently, Cai et al. use this approach to prove hardness of approximation for a single bidder with submodular valuations for multiple items [CDW13b], Hartline and Haghpanah provide sufficient conditions for especially simple mechanisms to be optimal for a single unit-demand or additive bidder [HH15], and Carroll proves that selling separately is max-min optimal for a single additive buyer when only the marginals are known but not the (possibly correlated) joint value distribution [Car16]. Indeed, many of these works also observe that the term “virtual welfare” is appropriate to describe the resulting upper bounds. Still, these works focus exclusively on providing conditions for certain mechanisms to be exactly optimal, and therefore impose some technical conditions on the settings where they apply. In comparison, our work pushes the boundaries by accommodating both approximation (in the sense that our framework can prove that simple mechanisms are approximately optimal and not just that optimal mechanisms are optimal) and unrestricted settings (in the sense that our framework isn’t restricted to a single buyer, or additive/unit-demand valuations).
Finally, we note that some of the benchmarks used in later sections can be derived without appealing to duality [CMS15]. Therefore, a duality theory is not “necessary” in order to obtain our benchmarks. Still, prior to our work it was unknown that these benchmarks were at all useful outside of the unit-demand settings for which they were developed. Additionally, both the primal and dual understanding of these benchmarks is valuable for extending the state-of-the-art, discussed in more detail below. Prior work has also obtained approximately optimal auctions via some sort of “benchmark decomposition” in the unrelated setting of digital goods [CGL15].
1.3.3 Approximation in Multi-Dimensional Mechanism Design
Finally, we provide a brief overview of recent work providing simple and approximately optimal mechanisms in multi-item settings. Seminal work of Chawla, Hartline, and Kleinberg proves that a posted-price mechanism gets a 3-approximation to the optimal deterministic mechanism for a single unit-demand buyer with independently drawn item values [CHK07]. Chawla et al. improve the ratio to 2, and prove a bound of 6.75 against the optimal deterministic, DSIC mechanism for multiple buyers [CHMS10]. Chawla, Malec, and Sivan show that the bound degrades by at most a factor of 5 when comparing to the optimal randomized, BIC mechanism [CMS15]. Kleinberg and Weinberg improve the bound of 6.75 to 6 [KW12]. Roughgarden, Talgam-Cohen and Yan provide a prior-independent “supply-limiting” mechanism in this setting, and also prove a Bulow-Klemperer [BK96] result: the VCG mechanism with additional bidders yields more expected revenue than the optimal deterministic, DSIC mechanism (with the original number of bidders), when bidder valuations are unit-demand, i.i.d., and values for items are regular and independent (possibly asymmetric) [RTCY12]. All of these results get mileage from the “” benchmark initiated in [CHK07].
More recent influential work of Hart and Nisan proves that selling each item separately at its Myerson reserve131313The Myerson reserve of a one-dimensional distribution refers to the revenue-optimal price to set if one seller is selling only this item to a single buyer. gets an approximation to the optimal mechanism for a single additive buyer and independent (possibly asymmetric) items [HN12]. Li and Yao improve this to , which is tight [LY13]. Babaioff et al. prove that the better of selling separately and bundling together gets a 6-approximation [BILW14]. Bateni et al. extend this to a model of limited correlation [BDHS15]. Rubinstein and Weinberg extend this to a single buyer with “subadditive valuations over independent items” [RW15]. Yao shows that the better of selling each item separately using Myerson’s auction and running VCG with a per-bidder entry fee gets a -approximation when there are many additive buyers and all values for all items are independent [Yao15]. Goldner and Karlin show how to use these results to obtain approximately optimal prior-independent mechanisms for many additive buyers [GK16]. More recently, Chawla and Miller show that a posted-price mechanism with per-bidder entry fee gets a constant-factor approximation for many bidders with “additive valuations subject to matroid constraints” [CM16]. All of these results get mileage from the “core-tail decomposition” initiated in [LY13].
The present paper unifies these two lines of work by showing that the benchmark and the core-tail decomposition both arise from essentially the same dual in our duality theory. These benchmarks provide a necessary starting point for the above results, but proving guarantees against these benchmarks of course still requires significant work. We believe that our duality theory now provides the necessary starting point to extend these results to much more general settings, as evidenced by the follow-up works discussed below.
1.3.4 Subsequent Work
Since the presentation of an earlier version of this work at STOC 2016, numerous follow-up works have successfully made use of our framework to design (approximately) optimal auctions in much more general settings. For example, Cai and Zhao show that the better of a posted-price mechanism and an anonymous posted-price mechanism with per-bidder entry fee gets a constant-factor approximation for many bidders with “XOS valuations over independent items” [CZ17]. This extends the previous state-of-the-art [CM16] from Gross Substitutes to XOS valuations. Eden et al. show that the better of selling separately and bundling together gets an -approximation for a single bidder with “complementarity- valuations over independent items” [EFF17b]. The same authors also prove a Bulow-Klemperer result: the VCG mechanism with additional bidders yields more expected revenue than the optimal randomized, BIC mechanism (with the original number of bidders), when bidder valuations are “additive subject to downward closed constraints,” i.i.d., and values for items are regular and independent (possibly asymmetric) [EFF17a]. Brustle et al. design a simple mechanism that achieves of the optimal gains from trade in certain two-sided markets, such as bilateral trading and double auctions [BCWZ17]. Devanur and Weinberg provide an alternative proof of Fiat et al.’s solution to the “FedEx Problem,” and extend it to design the optimal auction for a single buyer with a private budget [FGKK16, DW17]. Finally, Liu and Psomas provide a Bulow-Klemperer result for dynamic auctions [LP16], and Fu et al. design approximately optimal BIC mechanisms for correlated bidders [FLLT17].
Organization. We provide preliminaries and notation below. In Section 3, we present our duality theory for revenue maximization in the special case of additive/unit-demand bidders. In Section 4, we present a duality proof of Myerson’s seminal result, and in Section 5 we present a canonical dual solution that proves useful in different settings. As a warm-up, we show in Section 6 how to analyze this dual solution when there is just a single buyer. In Section 7, we provide the multi-bidder analysis, which is more technical. In Section 8, we conclude with a formal statement of our duality theory in general settings.
Optimal Auction Design. For the bulk of the paper, we will study the following setting (in Section 8, we will show that our duality theory holds much more generally). The buyers (we will use the terms buyer and bidder interchangeably) are either all unit-demand or all additive, with buyer having value for item . Recall that a valuation is unit-demand if and a valuation is additive if . We use to denote buyer ’s values for all the goods and to denote every buyer except ’s values for all the goods. is the set of all possible values of buyer for item , , and . All values for all items are drawn independently. We denote by the distribution of , , , and , and (or , etc.) the densities of these finite-support distributions (that is, ). We define to be a set system over that describes all feasible allocations.141414When bidders are additive, only allows allocating each item at most once. When bidders are unit-demand, contains all matchings between the bidders and the items.
A mechanism takes as input a reported type from each bidder and selects (possibly randomly) an outcome in , and payments to charge the bidders. A mechanism is Bayesian Incentive Compatible (BIC) if it is in each buyers interest to report their true type, assuming that the other buyers do so as well, and Bayesian Individually Rational (BIR) if each buyer gets non-negative utility for reporting their true type (assuming that the other bidders do so as well). The revenue of an auction is simply the expected sum of payments made when bidders drawn from report their true values. The optimal auction optimizes expected revenue over all BIC and BIR mechanisms. For a given value distribution , we denote by the expected revenue achieved by this auction, and it will be clear from context whether buyers are additive or unit-demand. For a specific BIC mechanism , we will also use to denote the expected revenue achieved by when bidders with valuations drawn from report truthfully.
Reduced Forms. The reduced form of an auction stores for all bidders , items , and types , the probability that bidder will receive item when reporting to the mechanism (over the randomness in the mechanism and randomness in other bidders’ reported types, assuming they come from ) as . It is easy to see that if a buyer is additive, or unit-demand and receives only one item at a time, that their expected value for reporting type to the mechanism is just (where we treat and
as vectors, anddenotes a vector dot-product). We say that a reduced form is feasible if there exists some feasible mechanism (that ex-post selects an outcome in with probability ) that matches the probabilities promised by the reduced form. If is defined to be the set of all feasible reduced forms, it is easy to see (and shown in [CDW12a], for instance) that is closed and convex.
We will also use to refer to the expected payment made by bidder when reporting to the mechanism (over the randomness in the mechanism and randomness in other bidders’ reported types, assuming they come from ).
Simple Mechanisms. Even though the benchmark we target is the optimal randomized BIC mechanism, the simple mechanisms we design will all be deterministic and satisfy DSIC. For a single buyer, the two mechanisms we consider are selling separately and selling together. Selling separately posts a price on each item and lets the buyer purchase whatever subset of items she pleases. We denote by the revenue of the optimal such pricing. Selling together posts a single price on the grand bundle, and lets the buyer purchase the entire bundle for or nothing. We denote by the revenue of the optimal such pricing. For multiple buyers the generalization of selling together is the VCG mechanism with an entry fee, which offers to each bidder the opportunity to pay an entry fee and participate in the VCG mechanism (paying any additional fees charged by the VCG mechanism). If they choose not to pay the entry fee, they pay nothing and receive nothing. We denote the revenue of the mechanism that charges the optimal entry fees to the buyers as , and the revenue of the VCG mechanism with no entry fees. The generalization of selling separately is a little different, and described immediately below.
Single-Dimensional Copies. A benchmark that shows up in our decompositions relates the multi-dimensional instances we care about to a single-dimensional setting, and originated in work of Chawla et. al. [CHK07]. For any multi-dimensional instance we can imagine splitting bidder into different copies, with bidder ’s copy interested only in receiving item and nothing else. So in this new instance there are single-dimensional bidders, and copy ’s value for winning is (which is still drawn from ). The set system from the original setting now specifies which copies can simultaneously win. We denote by the revenue of Myerson’s optimal auction [Mye81] in the copies setting induced by .151515Note that when buyers are additive that is exactly the revenue of selling items separately using Myerson’s optimal auction in the original setting.
Continuous versus Finite-Support Distributions. Our approach explicitly assumes that the input distributions have finite support. This is a standard assumption when computation is involved. However, most existing works in the simple vs. optimal paradigm hold even for continuous distributions (including [CHK07, CHMS10, CMS15, HN12, LY13, BILW14, Yao15, RW15, BDHS15]). Fortunately, it is known that every can be discretized into such that , and has finite support. So all of our results can be made arbitrarily close to exact for continuous distributions. We conclude this section by proving this formally, making use of the following theorem proved in [RW15], which draws from prior works [HL10, HKM11, BH11, DW12]. Note that the theorem below holds for distributions over arbitrary valuation functions , and not just additve/unit-demand.
[RW15, DW12] Let be any BIC mechanism for values drawn from distribution , and for all , let and be any two distributions, with coupled samples and such that for all . If , then for any , there exists a BIC mechanism such that , where denotes the expected welfare of the VCG allocation when buyer ’s type is drawn according to the random variable .
To see how this implies that our duality is arbitrarily close to exact for continuous distributions, let be the distribution that first samples from , then outputs such that .161616That is, if the value of for the grand bundle satisfies , then . Otherwise, for all . It is easy to see that as , : for every mechanism and every , there exists an such that a fraction of ’s revenue when buyers’ types are drawn from comes from buyers with 171717Similarly, if achieves infinite revenue, then for every , there exists an such that the revenue of when buyers’ types are drawn from coming from buyers with is at least . So our approach will still show that whenever is infinite, the revenue of the approximately optimal mechanisms we use is unbounded.. For the chosen , is still a BIC mechanism when buyers’ types are drawn from , and its revenue under is at least fraction of its revenue under . So we can get arbitrarily close while only considering distributions that are bounded.
Now for any bounded distribution , define to first sample from , then output such that . Similarly define to first sample from , then output such that . Then it’s clear that and can be coupled so that for all , and that taking either of the two consecutive differences results in a such that for all . So for any desired , applying Theorem 1 with as the optimal mechanism for , we get a mechanism for with revenue at least . Similarly, applying Theorem 1 with as the optimal mechanism for , we get a mechanism for with revenue at least . Together, these claims imply that . Finally, we just observe that (the revenue of the optimal mechanism increases by exactly going from to ), as every buyer values every outcome at exactly more in versus . So as , both approach . Note that both and have finite support, so our theory will directly design that achieve constant-factor approximations for .
3 Our Duality Theory
In this section we provide our duality framework, specialized to unit-demand/additive bidders. We begin by writing the linear program (LP) for revenue maximization (Figure 1). For ease of notation, assume that there is a special type to represent the option of not participating in the auction. That means and . Now a Bayesian IR (BIR) constraint is simply another BIC constraint: for any type , bidder will not want to lie to type . We let . To proceed, we will introduce a variable for each of the BIC constraints, and take the partial Lagrangian of LP 1 by Lagrangifying all BIC constraints. The theory of Lagrangian multipliers tells us that the solution to LP 1 is equivalent to the primal variables solving the partially Lagrangified dual (Figure 2).
Let be the partial Lagrangian defined as follows:
3.1 Useful Properties of the Dual Problem
In this section, we make some observations about the dual problem to get some traction on what duals might induce useful upper bounds.
Definition 3 (Useful Dual).
A feasible dual solution is useful if .
Lemma 4 (Useful Dual).
A dual solution is useful if and only if for each bidder , forms a valid flow, i.e., iff the following satisfies flow conservation (flow in = flow out) at all nodes except the source and the sink:
Nodes: A super source and a super sink , along with a node for every type .
Flow from to of weight , for all .
Flow from to of weight for all , and (including the sink ).
Let us think of using expression (2). Clearly, if there exists any and such that
then since is unconstrained (note that we do not include the constraints in Figure 1, so these variables are indeed unconstrained) and has a non-zero multiplier in the objective, . Therefore, in order for to be useful, we must have
for all and . This is exactly saying what we described in the Lemma statement is a flow. The other direction is simple, whenever forms a flow, only depends on . Since is bounded, the maximization problem has a finite value. ∎
Definition 5 (Virtual Value Function).
For each , we define a corresponding virtual value function , such that for every bidder , every type , Note that for all , is a vector-valued function, so we use to refer to the component of , and refer to this as bidder ’s virtual value for item .
Theorem 6 (Virtual Welfare Revenue).
Let be any useful dual solution and be any BIC mechanism. The revenue of is less than or equal to the virtual welfare of w.r.t. the virtual value function corresponding to . That is:
Equality holds if and only if for all such that , the BIC constraint for bidder between and binds in (that is, bidder with type is indifferent between reporting and ). Furthermore, let be the optimal dual variables and be the revenue-optimal BIC mechanism, then the expected virtual welfare with respect to (induced by ) under equals the expected revenue of , and
When is useful, we can simplify by removing all terms associated with (because all such terms have a multiplier of zero, by Lemma 4), and replace the terms with . After the simplification, we have , which equals , exactly the virtual welfare of . Now, we only need to prove that is greater than the revenue of . Let us think of using Expression (1). Since is a BIC mechanism, for any and , . Also, all the dual variables are nonnegative. Therefore, it is clear that is at least as large as the revenue of . Moreover, if the BIC constraint for bidder between and binds in for all such that , then we in fact have , so the revenue of is equal to its expected virtual welfare under (because the Lagrangian terms added to the revenue are all zero).
When is the optimal dual solution, by strong LP duality applied to the LP of Figure 1, we know equals the revenue of . But we also know that is at least as large as the revenue of , so necessarily maximizes the virtual welfare over all , with respect to the virtual transformation corresponding to . ∎
To summarize: we have shown that every flow induces a finite upper bound on how much revenue a BIC mechanism can possibly achieve. We have also observed that this upper bound can be interpreted as the maximum virtual welfare obtainable with respect to a virtual valuation function that is decided by the flow. In the next two sections, we will instantiate this theory by designing specific flows, and obtain benchmarks that upper bound the optimal revenue.
4 Canonical Flow for a Single Item
In this section, we provide a canonical flow for single-item settings, and show that it implies the main result from Myerson’s seminal work [Mye81]. Essentially, Myerson proposes a specific virtual valuation function and shows that for this virtual valuation function, the expected revenue of any BIC mechanism is always upper bounded by its expected virtual welfare. Moreover, he describes an ironing procedure to guarantee that this virtual valuation is monotone, and proves that the revenue-optimal mechanism simply awards the item to the bidder with the highest virtual value. Myerson’s proof is quite elegant, and we are not claiming that our proof below is simpler.181818If one’s goal is simply to understand Myerson’s result and nothing more, the original proof and ours are comparable in simplicity. Many alternative comparably simple proofs exist as well, some of which are not much different than ours (e.g. [MV04]). The purpose of the proof below is:
Serve as a warm-up for the reader to get comfortable with flows and virtual valuations.
Separate out parts of the proof that can be directly applied to more general settings (e.g. Theorem 6).
Provide a specific flow that will be used in later sections to provide benchmarks in multi-item settings.
In this section, we will have and drop the item subscript . We begin with a definition of Myerson’s (ironed) virtual valuation function, adapted to the discrete setting.
Definition 7 (Single-dimensional Virtual Value).
For a single-dimensional discrete discribution , if , let , where . If , let .191919Actually we could define arbitrarily and everything that follows will still hold. If the distribution is clear from context, we will just write .
The ironing procedure described below essentially finds any non-monotonicities in and “irons” them out. Note that Steps 4 and 5 maintain that ironed virtual values are consistent within any ironed interval.
Definition 8 (Ironing).
Let be an equivalence relation on the support of , and be the ironed virtual valuation function defined in the following way. We say that an interval is ironed if for all .
Initialize , the highest un-ironed type.
For any , define the average virtual value .
Let maximize the average virtual value. That is, (break ties in favor of the maximum such ).
Update for all .
Update for all .
Update , the highest un-ironed type.
Return to Step 2.
These definitions in the discrete case might be slightly different than what readers are used to in the continuous case. We provide some observations proving that this is “the right” definition for the discrete case briefly in Section 4.1. Our proof continues in Section 4.2.
4.1 Discrete Myersonian Virtual Values
In the continuous setting, Myerson’s virtual valuation is defined as , where and are the CDF and PDF of . We first show that for any continuous distribution, discretizing it into multiples of and taking virtual valuations as in Definition 7, we recover Myerson’s virtual valuation in the limt as .
Let be any continuous distribution, and be the discretization of with point-masses at all multiples of . That is, for all . Then for all ,
For fixed , consider the set of . Then clearly for all outside this set, . For any in this set, we have . It’s also clear that as , we have , and (the latter is simply the definition of probability density). ∎
A second valuable property of Myersonian virtual values is that they capture the “marginal revenue.” That is, if the seller was selling to a single bidder at price just above (i.e. above) , and decreased the price to just below (i.e. below) , the revenue would go up by exactly . We confirm that discrete virtual values as per Definition 7 satisfy this property as well.
For any single-dimensional discrete distribution , we have , where . In other words, captures the marginal change in revenue as we go from setting price to price .
This follows immediately from the definition of . But to be thorough:
|(using that )|
|(definition of )|
We need one more definition specific to discrete type spaces before we can get back to the proof. A little more specifically: Myerson’s payment identity, which shows that allocation rules uniquely determine payments for any BIC mechanism over continuous type spaces, does not apply for all BIC mechanisms when types are discrete. Fortunately, the payment identity still holds for any mechanism that might possibly maximize revenue, but we need to be formal about this.
A BIC mechanism has proper payments if it is not possible to increase payments while keeping the allocation rule the same without violating BIC. Formally, a BIC mechanism has proper payments if for all and all subsets , and all , increasing by for all while keeping the same does not result in a BIC mechanism. Note that all revenue-optimal mechanisms have proper payments.
Let be monotone non-decreasing for all . Then there exist such that is BIC and has proper payments.
For ease of notation in the proof, label the types in so that ( for ). Then for a given , define:
We first claim that all are indifferent between telling the truth and reporting . This is clear, as by definition of we have , which implies:
Now, consider any set and any , and imagine raising the payments of all types by . If we have for any , then increasing all payments in by will cause to prefer reporting instead of telling the truth. So must contain all of , and in particular . But if we increase the payment of by , we violate individual rationality, as we defined . So no such , can exists, and has proper payments.
Finally, we just need to show that is BIC. Notice that by definition of , for any , we have . The lower bound corresponds to the case that all the change from to occurs going from to (i.e. ), and the upper bound corresponds to where all the change occurs going form to (i.e. ).
The lower bound directly implies that prefers telling the truth to reporting as: