An End-to-end Argument in Mechanism Design (Prior-independent Auctions for Budgeted Agents)

This paper considers prior-independent mechanism design, namely identifying a single mechanism that has near optimal performance on every prior distribution. We show that mechanisms with truthtelling equilibria, a.k.a., revelation mechanisms, do not always give optimal prior-independent mechanisms and we define the revelation gap to quantify the non-optimality of revelation mechanisms. This study suggests that it is important to develop a theory for the design of non-revelation mechanisms. Our analysis focuses on single-item auctions for agents with budgets and the objectives of welfare and with a natural regularity assumption on the distribution. Our main results show that the all-pay auction (a non-revelation mechanism) is an optimal prior-independent approximation mechanism. The all-pay auction, however, is not a revelation mechanism. We prove that prior-independent approximation of best revelation mechanism is strictly worse than that of the all-pay auction. Thus, the revelation gap is at least a constant. Finally, we show that the clinching auction (a revelation mechanism) is a prior-independent e ≈ 2.714 approximation. Thus the revelation gap is a constant. Many of our analyses extend to the revenue objective, position environments, and irregular distributions.

Authors

• 6 publications
• 6 publications
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• Simple, Credible, and Approximately-Optimal Auctions

We identify the first static credible mechanism for multi-item additive ...
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• Approximately Optimal Mechanism Design

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• Dashboard Mechanisms for Online Marketplaces

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• Learning in repeated auctions

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• Generalization Analysis for Game-Theoretic Machine Learning

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1 Introduction

The end-to-end principle in distributed systems advocates environment-independent protocols (for the center) that push environment-specific complexity to the applications (the end points) that use the protocol (Saltzer et al., 1984). This principle enabled the Internet protocols designed for the workloads of the 1980s to continue to succeed with workloads of the 2010s. On the other hand, research in mechanism design (which governs the design of protocols for strategic agents and has application both in computer science and economics) almost exclusively adheres to the revelation principle which suggests the design of mechanisms where each agent’s best strategy is to truthfully report her preferences. In revelation mechanisms the agents (the end points) have very a simple “report your true preference” strategies and the mechanism (the center) has the complex task of finding an outcome that both enforces this truthfulness property and also obtains a desirable outcome. Unsurprisingly, optimal revelation mechanisms tend to be complex and dependent on the environment. This paper demonstrates that the end-to-end argument has bite in mechanism design by showing that non-revelation mechanisms are strictly better than revelation mechanisms for a canonical mechanism design problem.

In prior-independent mechanism design, it is assumed that the agents’ preferences are drawn from a distribution that is not known to the designer. The goal of prior-independent mechanism design is to identify mechanisms that are good approximations to the optimal mechanism that is tailored to the distribution of preferences. Specifically, a mechanism is sought to minimize the ratio of its expected performance to the expected performance of the optimal mechanism in worst case over distributions from which the preferences of the agents are drawn. This notion is a standard one that has been applied to revenue maximization (Dhangwatnotai et al., 2010; Roughgarden et al., 2012; Fu et al., 2015; Allouah and Besbes, 2018), multi-dimensional mechanism design (Devanur et al., 2011; Roughgarden et al., 2015), makespan minimization (Chawla et al., 2013), mechanism design for risk-averse agents (Fu et al., 2013), and mechanism design for agents with interdependent values (Chawla et al., 2014a). In none of these scenarios is the optimal prior-independent mechanism known; cf. Fu et al. (2015) and Allouah and Besbes (2018).

The revelation principle suggests that if there is a mechanism with a good equilibrium outcome, there is a mechanism where truthtelling achieves the same outcome in a truthtelling equilibrium. Due to the revelation principle, much of the theory of mechanism design is developed for revelation mechanisms, i.e., ones where truthtelling is an equilibrium. The proof of the revelation principle is simple: A revelation mechanism can simulate the equilibrium strategies in the non-revelation mechanism to obtain the same outcome as a truthtelling equilibrium, i.e., agents report true preferences to the revelation mechanism, it simulates the agent strategies in the non-revelation mechanism, and it outputs the outcome of the simulation. For Bayesian non-revelation mechanisms (where the agents’ preferences are drawn from a prior distribution), the agents’ equilibrium strategies are a function of the prior and thus the corresponding revelation mechanism constructed via the revelation principle is not prior-independent. Thus, the restriction to revelation mechanisms is not generally without loss for prior-independent mechanism design. Non-revelation mechanisms, on the other hand, are widely used in practice and frequently have easily to identify and natural equilibria (e.g., in rank-based auctions, see Chawla and Hartline, 2013). Our proof of a non-trivial revelation gap – that the prior-independent approximation factor of the best non-revelation mechanism is better than that of the best revelation mechanism – gives concrete motivation for a theory of mechanism design without the revelation principle.

It is not hard to invent pathological scenarios where there is a non-trivial revelation gap. Instead, this paper considers the canonical environment of welfare maximization for agents with budgets and shows such a gap even for distributions on preferences that satisfy a standard regularity property. Moreover, the environment in which we exhibit the revelation gap suggests the end-to-end principle: the agents can easily implement the optimal outcome in the equilibrium of a simple mechanism, while revelation mechanisms that satisfy the constraints must be complex and either prior-dependent or non-optimal.

Main Results.

Our analysis focuses on welfare maximization in a canonical single-item environment with ex ante symmetric budget constrained agents, i.e., each agent’s value is drawn independently and identically from an unknown distribution and the agent cannot make payments that exceed a known and identical budget (cf. Maskin, 2000). Our main treatment of this problem will make a simplifying assumption that the distribution follows a regularity property that implies that the Bayesian optimal mechanism has a nice form (Pai and Vohra, 2014). Our results require a symmetric environment, i.e., an i.i.d. distribution and identical budget. A number of our results extend to the objective of revenue (Laffont and Robert, 1996), to position environments as popularized as a model for ad auctions (Devanur et al., 2013), and beyond regular distributions (Devanur et al., 2013). For clarity the main results are described first for welfare maximization, single-item environments, and regularly distributed agent values.

The main challenge in demonstrating a revelation gap is that it is difficult to identify prior-independent optimal mechanisms, cf. Fu et al. (2015). Though the question has been considered, the prior literature has no examples of optimal prior-independent mechanisms for non-trivial environments. 111Contemporaneously with our results, Allouah and Besbes (2018) showed that the second-price auction is the prior-independent optimal revelation mechanism for revenue when the agents’ values are distributed according to a monotone hazard rate distribution.

Our non-trivial revelation-gap theorem follows from three results. First, the all-pay auction (from the literature, defined below) has a unique equilibrium that is Bayesian optimal and it is prior-independent. Second, we obtain a lower bound on the ability of a prior-independent revelation mechanism to approximate the Bayesian optimal mechanism by identifying the dominant strategy incentive compatible mechanism that is Bayesian optimal for the uniform distribution. The performance of this mechanism is strictly worse than that of the Bayesian optimal mechanism (which is Bayesian incentive compatible); specifically the gap is

. Third, we show that the dominant strategy incentive compatible clinching auction (from the literature, defined below) is an approximation to the Bayesian optimal mechanism. Combining the upper and lower bounds we see a revelation gap between and . 222To better appreciate the magnitude of this lower bound, notice that it is demonstrated for two agents with uniformly distributed values where the optimal expected welfare (even without budgets) is is and the lottery mechanism (which gives the item to a random agent) has expected welfare and is a approximation. The first result follows naturally from the literature; the second and third results are the main technical contributions of the paper.

Three auctions are at the forefront of our study. The all-pay auction solicits bids, assigns the item to the highest agent, and charges all agents their bids. The clinching auction (Ausubel, 2004; Dobzinski et al., 2008; Goel et al., 2015) is an ascending price auction that can be thought of as allocating a unit measure of lottery tickets: a price is offered in each stage, each agent specifies the measure of tickets desired at the given price, each agent is allocated a number of tickets that is equal to the minimum of her demand and the measure of remaining tickets if this agent is only allowed to buy tickets after all other agents have bought as much as they desire first. 333For example, at a price of 0 all agents would want to buy all the tickets, but the agent that arrives last gets no tickets, thus no agents get any tickets at this price; the price increases. The middle-ironed clinching auction – which we identify as the optimal dominant strategy incentive compatible mechanism – behaves like the clinching auction except that values that fall within a middle range are ironed. The allocation that an agent in this middle range receives is the average over he original allocation of for middle range values in the clinching auction. This averaging results in the the budget binding later and more efficient outcomes than in the original clinching auction.

The second step, mentioned above, is to obtain a lower bound on the prior-independent approximation of a revelation mechanism. Our analysis begins with the observation that a prior-independent revelation mechanism must be Bayesian incentive compatible for every distribution. For two agents, this condition is equivalent to being dominant strategy incentive compatible. We ask whether there a gap between the Bayesian optimal dominant strategy and Bayesian incentive compatible mechanism. The comparison between optimal dominant strategy and Bayesian incentive compatible mechanism is standard for multi-dimensional mechanism design problems, e.g., see Gershkov et al. (2013) and Yao (2017)

; we are unaware of previous studies of this phenomenon for single-dimensional agents with non-linear preferences. We answer this question positively by writing the dominant strategy mechanism design problem as a linear program and solving it by identifying a dual solution that proves the optimality of the middle-ironed clinching auction, cf.

Pai and Vohra (2014) and Devanur and Weinberg (2017). The identified gap gives a lower bound on the approximation factor of the optimal prior-independent mechanism.

The third step, mentioned above, proves that the prior-independent approximation factor of the clinching auction auction is at most and resolves in the affirmative an open question from Devanur et al. (2013). Our proof follows from a novel adaptation of a standard method for approximation results in mechanism design where an auction’s performance is compared to the upper bound given by the ex ante relaxation, in this case, the welfare of the optimal mechanism that sells one item in expectation over the random draws of the agents’ values (i.e., ex ante) rather than for all draws of the agents’ values (i.e., ex post). This method was introduced by Chawla et al. (2007), formalized by Alaei (2011, 2014), generalized by Alaei et al. (2013), and employed in many subsequent analyses.

Extensions.

A number of our results extend beyond regular distributions, single-item environments, and the welfare objective as described above. These extensions all require that the environment be symmetric, specifically, that the agents’ values are independent and identically distributed and their budgets are identical.

For irregular distributions the welfare-optimal auction is not generally the all-pay auction; moreover, it does not generally have a prior-independent implementation. We prove that the all-pay auction is a prior-independent two approximation. Both the regular and irregular prior-independent optimality and approximation results for the one-item all-pay auction extend to the all-pay position auction.

The degradation of the approximation factor by a factor of two for irregular distributions extends to the single-item clinching auction which is an approximation for regular distributions (as described above) and a approximation for irregular distributions.

For the revenue objective (Laffont and Robert, 1996), with appropriate definition of regularity, the -agent single-item all-pay auction is a prior-independent approximation to the revenue optimal auction (cf. Bulow and Klemperer, 1996).

Important Directions.

The most general direction suggested by this work is for a systematic development of non-revelation mechanism design. Unfortunately, it is not generally helpful to do revelation mechanism design and then try to go from the suggested revelation mechanism to a practical and simple non-revelation mechanism. There is a nascent literature on this topic. Papers working to develop a theory of non-revelation mechanism design include Chawla and Hartline (2013), which proves the uniqueness and optimality of equilibria in symmetric rank-based auctions; Chawla et al. (2014b, 2016), which gives data driven methods for optimizing non-revelation mechanisms in symmetric environments; and Hartline and Taggart (2016), which gives a theory for non-revelation sample complexity and the design of approximately optimal non-revelation mechanisms in asymmetric environments.

While the literature has many interesting approximation bounds for prior-independent mechanism design. Rarely have the prior-independent optimal mechanism been identified. Moreover, the prior-independent approximation factors achievable tend to be surprising; for example, Fu et al. (2015) show that the second-price action is not the optimal prior-independent mechanisms for two-agent revenue maximization with agents with regularly distributed values. 444Allouah and Besbes (2018) show that with more restrictive monotone hazard rate distributions, the second-price auction is an optimal prior-independent revelation mechanism. The literature lacks general techniques for answering this question.

We have observed that there is a very simple prior-independent optimal mechanism for welfare maximization in symmetric environments for agents with identical budgets. This mechanism, namely the all-pay auction, achieves its optimal outcome in Bayes-Nash equilibrium. The general question of identifying prior-independent non-revelation mechanisms that optimize a desired objective, like welfare or revenue, needs to be asked with care. Without restrictions to this question, it is asked and answered in the literature on non-parametric implementation theory, see the survey of Jackson (2001). This literature shows that arbitrarily close approximations, called “virtual implementations”, to the Bayesian optimal mechanism can be implemented by an uninformed principal. The mechanisms in this literature tend to be sequential – where agents interact in multiple rounds – and require agents to make reports about their own preferences and crossreports about their beliefs on other agents’ preferences. Our perspective on these results is that they take the model of Bayes-Nash equilibrium too literally and the resulting cross-reporting mechanisms are both fragile and impractical. One approach for ruling out these mechanisms is to restrict attention to mechanism formats that are commonly occurring in practice. Specifically, in the general winner-pays-bid format: agents bid, an allocation rule maps bids to a set of winners, and all winners pay their bids; in the general all-pay format: agents bid, an allocation rule maps bids to a set of winners, and all agents pay their bids. There may be other restricted formats that are also interesting for specific scenarios, e.g., the seller-offer mechanisms that are prevalent as real estate exchange mechanisms (Niazadeh et al., 2014).

Finally, there are still many gaps in our understanding of auctions for identically distributed agents with common budgets. For welfare, the bounds in this paper show that the clinching auction’s approximation factor for the welfare objective is in for regular distributions and for irregular distribution. Sharpening these bounds is an open question. Moreover, we conjecture that the clinching auction is also a prior-independent constant approximation for the revenue objective (with i.i.d. public-budget regular distributions). We also leave open a number of questions with regard to the Bayesian optimal dominant strategy incentive compatible mechanism for agents with budgets. We conjecture that the welfare optimality of the middle-ironed clinching auction extends from uniform distributions to regular distributions. We leave open the question of a similar result for the revenue objective, even for the special case of uniform distributions. There are specific issues that prevent straightforward generalization of our approach of using the dual to certify the optimality of the middle-ironed clinching auction for these questions.

Other Related Work.

There is a significant area of research analyzing the performance of simple non-revelation mechanisms in equilibrium, a.k.a., the price of anarchy. Generally these mechanisms are prior-independent and the aim of the literature, e.g. Syrgkanis and Tardos (2013), is to demonstrate that they are approximately efficient. On the other hand, for welfare maximization in many of the studied environments, there is a DSIC revelation mechanism that is (exactly) efficient and, thus, there is no revelation gap. Though this literature focuses on the analysis rather than the design of mechanisms, two conclusions for mechanism design are: (a) that a simple revenue covering property is sufficient (Hartline et al., 2014), necessary (Dütting and Kesselheim, 2015), and potentially optimizable; and (b) that this property (and also a more general smoothness property) is closed under composition, i.e., when multiple independent mechanisms are run simultaneously (Syrgkanis and Tardos, 2013). For a surveys of these and other results see Roughgarden et al. (2017) and Chapter 6 of Hartline (2016).

For agents with budgets, approximation mechanisms have been studied from both a design and analysis perspective for the liquid welfare benchmark proposed by Chawla et al. (2011), Syrgkanis and Tardos (2013), and Dobzinski and Paes Leme (2014). The liquid welfare benchmark is the optimal surplus of a feasible allocation when each agent’s contribution to the surplus is the maximum of her value for her allocation and her budget. These and subsequent papers show that simple mechanisms have welfare that approximate the liquid welfare benchmark. Unfortunately, when evaluated under the formal study of benchmarks for mechanism design developed by Hartline and Roughgarden (2008) and summarized in Chapter 7 of Hartline (2016), the liquid welfare does not satisfy a key property. Specifically, there are i.i.d. distributions where the expected welfare of the Bayesian optimal mechanism is arbitrarily larger than the expected optimal liquid welfare. This bound follows because liquid welfare is at most the sum of the agent budgets which can be arbitrarily close to zero and, in such cases, is unrelated to the welfare possible by a mechanism. 555For example, in a single-item environment with budgets identically equal to zero and agent values identically equal to one (trivially an i.i.d. distribution); the lottery, which allocates the item to a random agent for free, has welfare one while the liquid welfare is zero. Thus, testing mechanisms for their worst-case approximation factor with respect to liquid welfare does not necessarily separate good mechanisms from bad mechanisms.

Organization.

In Section 2 we give the preliminaries of our setup. In Section 3 we analyze the prior-independent approximation factor of the clinching auction for public-budget regular agents. In Section 4 we derive the Bayesian optimal DSIC mechanism for two agents with value uniformly distributed and show it is the middle-ironed clinching auction. In Section 5, we prove that the revelation gap is a constant. In Section 6, we analyze the prior-independent approximation factor of the all-pay auction and the clinching auction for irregular agents. In Section 7, we analyze the approximation ratio of winner-pays-bid mechanisms. In Section 8, we analyze the prior-independent revenue approximation of the all-pay auction for public-budget regular agents. In Appendix A, we give an ascending implementation of the clinching auction with price jumps (a generalization of the middle-ironed clinching auction). In Appendix B, we give a geometric framework for deriving Bayesian optimal mechanisms for agents with budgets and apply this framework to solve for the Bayesian optimal winner-pays-bid mechanism.

2 Preliminaries

Model for auctions with budgets

We consider mechanisms for agents with independent and identically distributed values and identical public budgets. The budget is denoted by . For allocation and payment , an agent with value has utility if is at most the budget and utility otherwise. In other words, the agent cannot under any circumstances pay more than her budget. The agents’ values are drawn independently and identically from distribution with support .

Denote the strategy function of an agent by where is the bid made by the agent when her value is . A bid profile is . A mechanism is given by mapping from bids to allocations and payments which we will denote by and . The outcome of the mechanism and strategy profile on a profile of agent values is denoted by allocation rule and payment rule .

The auction designer typically faces a feasibility constraint that restricts the allocations that can be produced. For example, a single-item auction requires that the number of agents allocated is at most one, i.e.,

. A position environment generalizes a single item auction and is given by a sequence of decreasing probabilities

where without loss of generality the number of positions is equal to the number of agents. The probability that an agent is allocated if assigned to position is . A mechanism then can assign agents to positions (deterministically or randomly) and this process and the position probabilities induce allocation probabilities .

Basic auction theory

A Bayes-Nash equilibrium (BNE) in the mechanism is a profile of agent strategies where each maps a value to a bid that is a best response to the other strategies and the common knowledge that values are drawn i.i.d. from distribution . The mechanism and strategy profile induce for each agent an interim allocation rule . We will consider only symmetric distributions and symmetric auctions. In such auctions, Chawla and Hartline (2013) show that all equilibria are symmetric, thus it is without loss to drop the subscript and refer to the interim allocation rule and payment rule as . The Myerson (1981) characterization of BNE requires that (a) the interim allocation is monotone non-decreasing and (b) the interim payment . Condition (b) is known as the payment identity. A mechanism is Bayesian incentive compatible (BIC) if it induces a BNE where each agents strategy is reporting her value truthfully. A mechanism is interim individual rational (IIR) if the interim allocation is non-negative for all value.

A dominant strategy equilibrium (DSE) in the mechanism is a profile of agent strategies where each maps a value to a bid that is a best response regardless of what other agents are doing. The characterization of DSE follows from the BNE characterization: (a) the allocation is monotone non-decreasing in and (b) the payment . A mechanism is dominant strategy incentive compatible (DSIC) if it induces a DSE where each agents strategy is reporting her value truthfully. A mechanism is ex-post individual rational (ex-post IR) if the allocation is non-negative for all value profile.

Optimal auctions with budgets

Laffont and Robert (1996) and Maskin (2000) for the revenue and welfare objectives, respectively, characterize the Bayesian optimal mechanisms for agents with public budgets. With the following regularity assumptions on the distribution, defined distinctly for revenue and welfare, the optimal mechanism has a nice form.

Definition 2.1.

A single-dimensional public budget agent is public-budget regular for welfare

is concave; she is public-budget regular for revenue if additionally is non-decreasing.

The results of Laffont and Robert and Maskin can be reinterpreted, à la Alaei et al. (2013), as solving a single-agent interim optimization problem that is given by an interim constraint . An interim allocation is interim feasible under the interim constraint if for all values , the probability of allocating item to an agent with value greater than with allocation rule is at most that with allocation rule , i.e. . In many cases solution to these interim optimization problems will take the form of the original constraint with ironed interval and reserve. Ironing on arbitrary interval corresponds to the distribution weighted averaging as follow, for all . Reserve at value corresponds to rejecting all value below as follows, for all . An important allocation constraint is that given by the highest-bid-wins allocation rule. The highest-bid-wins allocation rule for agents and with values from cumulative distribution function is , e.g., for two agents with uniform values it is .

Theorem 2.1 (Laffont and Robert, 1996; Maskin, 2000; Alaei et al., 2013).

For public-budget regular i.i.d. agents and interim allocation constraint , the welfare-optimal single-agent mechanism allocates as by except that values in are ironed for some and the revenue-optimal single-agent mechanism additionally reserve prices values in for some ; payments are given deterministically by the payment identity.

For single-item environments, one possible implementation of Theorem 2.1 is the all-pay auction. The all-pay auction has a unique Bayes-Nash equilibrium which is identical to outcome described in Theorem 2.1 for the allocation constraint given by the highest-bid-wins allocation rule.

Definition 2.2 (all-pay auction).

The all-pay auction is a mechanism where allocates item to the agent with highest bid with tie broken at random and charges each agent their bid, i.e. .

For public-budget regular i.i.d. agents, the all-pay auction is welfare optimal.

3 Welfare Approximation of the Clinching Auction

In this section, we study a prior-independent revelation mechanism called the clinching auction in single-item environments. Dobzinski et al. (2008) gave the following formulation of the clinching auction and characterized properties of its outcome. See Figure (b)b.

Definition 3.1 (clinching auction).

The clinching auction maintains an allocation and price-clock starting from zero. The price-clock ascends continuously and the allocation and budget are adjusted as follows.

1. Agents whose values are less than price-clock are removed and their allocation is frozen.

2. The demand of any remaining agent is the remaining budget divided by the price-clock.

3. Each remaining agent clinches (and adds to their current allocation) an amount that corresponds to the largest fraction of their demand that can be satisfied when all other remaining agents are first given as much of their demand as possible.

4. The budget and allocation are updated to reflect the amount clinched in the previous step.

Proposition 3.1 (Dobzinski et al., 2008).

For public-budget agents, the clinching auction always allocates all items, is ex-post IR, and is DSIC.

Lemma 3.2 (a special case of Dobzinski et al., 2008).

In single-item environment, for public-budget agents with budget and value profile , and let be the largest integer such that

 v(k)≥B⋅k where v(k) is the k-th highest value in value profile v.

Then, the agents with highest values win with same probability greater or equal to and the agent with the -th highest value wins with the remaining probability.

We use the following approach to show that the clinching auction is an -approximation for public-budget regular agents. We relax the feasibility constraint to an ex ante constraint and show that the optimal mechanism that sells to each agent with ex ante probability simply posts a price (of exactly assuming that the budget binds) for a chance to win the item (Lemma 3.3, below). This simple form of mechanism is closely approximated by the clinching auction which sells lotteries of probability (full details given subsequently). A key property is that in this clinching auction with lotteries, the budget does not bind with constant probability. The probability that the budget does not bind in the clinching auction with lotteries allows a lower bound on the allocation probability in the clinching auction which allows its welfare to be compared to the ex ante relaxation.

Consider the welfare-optimal auction. Since agents are symmetric, each agent will win with ex ante probability exactly in the welfare-optimal auction where is the number of agents. We replace the feasibility constraint that ex post allocation cannot allocate more than one item (i.e. for all ) with a ex ante constraint that each agent cannot be allocated more than in expectation (i.e. ). Ex ante optimal mechanisms for agents with public budgets were proposed and studied by Alaei et al. (2013).

Lemma 3.3 (Alaei et al., 2013).

For public-budget regular i.i.d. agents with budget , the ex ante welfare-optimal mechanism is either:

1. Budget binds: Post the price for allocation probability with set to satisfy . Values select the lottery.

2. Allocation probability binds: Post price for allocation probability one.

We build the connection between the clinching auction and the ex ante optimal mechanism by considering the an additional auction: the clinching auction with lotteries which allocates lotteries with winning probability per lottery, using the clinching auction framework under the same public budget. Lemma 3.4 below shows that by selecting an appropriate , the probability that an agent with value wins in the clinching auction with lotteries is at least an fraction of the probability that the agent (with value ) wins in the ex ante relaxation. See Figure 2.

Lemma 3.4.

For public budget i.i.d. agents, at value defined in Lemma 3.3, there exists , such that the interim allocation of the clinching auction with lotteries is an -approximation of the interim allocation of the ex ante optimal mechanism , i.e.,

Proof.

Denote the notation as the -th order statistic among

i.i.d. random variables from distribution

.

We denote the posted pricing in Lemma 3.3 as PP. Let where is the interim allocation at the highest value . By the construction of PP, and (equality holds when the budget binds in PP). Let be the smallest integer which is greater or equal to . Consider the clinching auction which allocates lotteries with winning probability per lottery, using the clinching auction framework under public budget .

First, fix an arbitrary agent and fix her value to be , we consider the following event : in , this agent with value is one of the highest valued agents and the budget does not bind. Recall that when the budget does not bind, the highest agents in each receive lotteries (with allocation probability ) and pay the value of the -st highest agent divided by (i.e. ). The budget bids in if and only if and we can lower bound the lower bound the probability of the event as follows,

 \bf Pr[E] =\bf Pr[(v(k:n−1)/k≤B)∧(v(k:n−1)≤v†)] =\bf Pr[(v(k:n−1)/k≤v†/k0)∧(v(k:n−1)≤v†)] =\bf Pr[v(k:n−1)≤v†] =k−1∑i=0(n−1i)(k0n)i(n−k0n)n−1−i. Above, the third line is derived from the second line using the definition of k≥k0. Denote by xE and x¯E the allocation rule x conditioned on the events E and ¯E, respectively. The interim allocation for Clinchk at value v† can be lower bounded as follows. xClinchk(v†) =xEClinchk(v†)⋅\bf Pr% [E]+x¯EClinchk(v†)⋅\bf Pr[¯E] ≥xEClinchk(v†)⋅% \bf Pr[E] =k0k⋅xPP(h)⋅\bf Pr[E] ≥1e⋅xPP(h).

The final inequality follows because the term achieves the minimum at when and goes to infinity. ∎

We now prove our main theorem about the approximation ratio for the clinching auction.

Theorem 3.5.

For public-budget regular i.i.d. agents, the clinching auction is an -approximation to the welfare-optimal mechanism.

Proof.

By Lemma 3.3 the interim allocation rule of the ex ante optimal mechanism is a step function that steps at value . By Lemma 3.4, at value , the allocation rule of the clinching auction with lotteries is an -approximation to that of the ex ante optimal mechanism. The allocation rule of the clinching auction with lotteries is monotone, so its allocation rule is an -approximation to that of the ex ante optimal mechanism at every value. Consequently, the expected welfare of the clinching auction with lotteries is at least an -approximation to that of the ex ante relaxation. See Figure 2.

Finally, Lemma 3.2 implies that for every ex post value profile, the welfare of the clinching auction is at least that of the clinching auction with lotteries. ∎

For public-budget regular i.i.d. agents, the all-pay auction is optimal while the clinching auction is not, since the budget binds for more value profiles in the clinching auction than in the all-pay auction. Based on this, we give a lower bound of the approximation ratio for the clinching auction and leave the actual approximation ratio as an open problem.

Lemma 3.6.

There exists the instance of public-budget regular agents where the clinching auction is a -approximation of the welfare-optimal mechanism.

Proof.

Consider a simple setting: there are 2 public-budget regular agents with value drawn uniformly from and the budget .

By Theorem 2.2, the all-pay auction is welfare-optimal for public-budget regular agents. The interim allocation rule of it is if and otherwise. The expected welfare of all-pay auction is .

The interim allocation rule of the clinching auction is if and otherwise. The expected welfare of the clinching auction is .

By setting , it optimizes the ratio as . ∎

4 Bayesian Optimal DSIC Mechanism

In Theorem 2.2, the all-pay auction is welfare-optimal under public-budget regular distribution. Hence, applying the revelation principle to the all-pay auction, it produces a Bayesian optimal revelation mechanism. This mechanism is BIC but not DSIC. In this section, we characterize the optimal DSIC mechanism for two agents with uniformly distributed values. We obtain a lower bound on its approximation ratio with the BIC optimal mechanism.

We first introduce the middle-ironed clinching auction (for two agents).

Definition 4.1.

The two-agent middle-ironed clinching auction is parameterized by and and its outcome is highest-bid-wins on values less that , a fair lottery on values in , and the clinching auction on values exceeding ; a precise formulation for two-agents is given in Figure (a)a and a general formulation is given in Appendix A.

For two-agents case, the middle-ironed clinching auction allocates the item efficiently except for value profiles in (both agents with values in ) or (both agents with values in ). For the value profile in , it randomly allocates the item to one of the agent with probability with payment . For the value profile in , it allocates the item such that the budget binds for the agent with higher value and the allocation rule depends on the lower value only. Figure (b)b depicts the allocation rule of the clinching auction for comparison. The middle-ironed clinching auction can be implemented with an ascending price via a generalization of the clinching auction that allows for price jumps which we develop in Appendix A (this generalization is non-trivial).

We will show that by selecting the proper thresholds and , the middle-ironed clinching auction is the Bayesian optimal DSIC mechanism for two agents with uniformly distributed values. An intuition behind the optimality of the middle-ironed clinching auction is as follows: Dobzinski et al. (2008) show that for two public budget agents, the clinching auction is the only Pareto optimal (i.e. there is no outcome which is weakly better for all agents and strictly better for one agent) and DSIC auction. Moreover, after the price increases past the point where the budget binds, a differential equation governs the allocation of any DSIC mechanism. Our goal is to optimize expected welfare rather than satisfy Pareto optimality. Sacrificing welfare for lower-valued agents by ironing can delay the budget from binding and enable greater welfare from higher-valued agents. From our proof of optimality, it is sufficient to only iron one region in the middle of value space.

Theorem 4.1.

For two public-budget agents with budget and value uniformly drawn from , Bayesian optimal DSIC mechanism is the middle-ironed clinching auction with some thresholds and .

The approach of the proof is to write down our problem as a linear program (primal), assume the middle-ironed clinching auction to be the solution, and then construct the dual program with a dual solution which witnesses the optimality of the primal solution by complementary slackness. This approach is reminiscent of that of Pai and Vohra (2014) and Devanur and Weinberg (2017); however, our multi-agent DSIC constrained program presents novel challenges and for this reason we only solve the problem of two agents and uniform distributions.

We first solve a discrete version of the problem. Then, we solve the continuous version as the limit from the discrete version. Consider the value distribution with finite value space with equal probability each. We begin by writing down the optimization program for welfare maximization among all possible DSIC mechanism.

 sup(x,p)∑v1,v2∈[h](v1⋅x1(v1,v2)+v2⋅x2(v1,v2))⋅1h⋅1hs.t. (x,p) are DSIC, ex-post % IR,and feasible(x,p) is budget balanced

By the characterization of dominant strategy equilibrium, we simplify this optimization program into a linear program as follows,

 max(x,p)≥0∑v1,v2∈[h]v1⋅x(v1,v2)s.t. h⋅x(h,v2)−∑ht=1x(t,v2)≤B% for all v2∈[h][Budget Constraint]x(v1,v2)+x(v2,v1)≤1for all v1,v2∈[h][% Feasibility Constraint]x(v1,v2)≤x(v1+1,v2)for all v1∈[h−1],v2∈[h][Monotonicity Constraint]

where we assume for all since it is an agent-symmetric linear program. 666 Note the program in turns of is asymmetric.

Additionally, we relax the monotonicity constraint by replacing it with which is common for Bayesian mechanism design with public budget agents.

 x(v1,v2)≤x(h,v2)for all v1∈[h−1],v2∈[h][Relaxed Monotonicity Constraint]

Then we write down the corresponding dual program. Let denote the dual variables for budget constraints; denote the dual variables for feasibility constraints (for simplicity, we use both and to denote the same dual variable); and denote the dual variables for monotonicity constraints. The dual program is,

 min(Λ,\mathboldβ,\mathboldμ)≥0∑v2∈[h]B⋅Λ(v2)+12∑v1,v2∈[h]β(v1,v2)s.t. −Λ(v2)+β(v1,v2)+μ(v1,v2)≥v1for all v1∈[h−1],v2∈[h][x(v1,v2)](h−1)Λ(v2)+β(h,v2)−∑h−1t=1μ(t,v2)≥h% for all v2∈[h][x(h,v2)]

We give a short overview of the plan to solve the program. For each possible thresholds chosen in the middle-ironed clinching auction, we first construct a solution in dual which satisfies the complementary slackness with this middle-ironed clinching auction as a solution in primal. These induced dual solutions may be infeasible. Next, we will show that there exists a pair of thresholds which induces a feasible dual solution. This feasible dual solution witnesses the optimality of the middle-ironed clinching auction.

We will partition the dual variables into following five areas (, , , and ) as in Figure 4; and construct the dual solution for them separately. We denote as the discrete derivative of the dual variable , i.e. .

in :

Since the budget constraints do not bind, by complementary slackness,

 Λ(v)=0 for all v∈L.
in :

Let be a value profile in area such that . By complementary slackness on , if ; otherwise (i.e. . We let

 β(v,v′)=v and μ(v,v′)=0.\lx@notefootnoteAnintuitionhereis:$\mathboldμ$arethedualvariablesfortherelaxedmonotonicityconstraintandcanbethoughtasindicatorsofironing.Thoughthemonotonicityconstraintbinds,thisisnotbecauseofironingbutbindingallocation(i.e.$x(⋅)≤1$).Therefore,weset$\mathboldμ$aszero.

Since the relaxed monotonicity constraint does not bind at , i.e. , the corresponding dual variable is

 μ(v′,v)=0.
in :

Let be a value profile in area such that . Since both agents win with non-zero probability, by complementary slackness on and , the corresponding dual constraints bind. Since the relaxed monotonicity constraint does not bind at , the monotonicity dual variable is

 μ(v′,v)=0.

The binding dual constraint of is . Hence,

 β(v,v′)=β(v′,v)=v′+Λ(v).\lx@notefootnoteRecallthat$β(v,v′)$and$β(v′,v)$denotethesamedualvariable.

The binding dual constraint of is . Note the relaxed monotonicity constraint is tight for . Hence,

 μ(v,v′)=v−v′+Λ(v′)−Λ(v).

Here we write as terms of . In the next paragraph, we will solve for .

in :

Let . Consider the binding dual constraint of , . Notice that by complementary slackness, for all . Plugging and as terms of into the these dual constraints of , we can solve for as

 λ(v)=h−vv for all v∈H and Λ(h)=0.\lx@notefootnoteRecallthat$\mathboldλ$isthediscretederivativeofdualvariables$Λ$,so$Λ(v)=h−1∑t=vλ(v)$.
in and in :

Let be a value profile in area such that . Since the relaxed monotonicity constraints do not bind for either or , the corresponding dual variables are

 μ(v,v′)=μ(v′,v)=0.

The binding dual constraints of implies . On the other hand, the binding dual constraints of implies . Recall that and denote the same variable, hence,

 λ(v)=−1 for all v∈M∖{v‡−1},\lx@notefootnoteComplementaryslacknessdoesnotpindown$λ(v‡−1)$.Weleaveitasavariableandidentifyitlaterwhenwechoosethethresholds$v†$,$v‡$toensurethatthedualsolutionisfeasible.
 β(v,v′)=Λ(v‡)+λ(v‡−1)+v+v′−v‡.
in and :

Let be a value profile in area such that . With the similar argument for area ,

 μ(v′,v)=0 and μ(v,v′)=v−v′+Λ(v′)−Λ(v),
 β(v,v′)=v′+Λ(v) if v

Plugging as terms of into the binding dual constraint of ,

 β(h,v′)=(h−1)(v‡−v′)+1+(v‡−1)λ(v‡−1).

With the analysis above, we construct the following dual solution which satisfies complementary slackness with the middle-ironed clinching auction as a solution in primal,

 Λ(v2)=⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩0if v2v2 (1)
Lemma 4.2.

For the middle-ironed clinching auction with arbitrary thresholds and , the dual solution (1) satisfies the complementary slackness.

Proof.

The complementary slackness is directly implied by the construction above. ∎

Though the this dual solution satisfies the complementary slackness, it may be infeasible. Therefore, we argue that there exists some thresholds , and under which the dual solution is feasible.

Lemma 4.3.

There exists and such that the constructed dual solution (1) is feasible.

Proof.

We define function to simplify the argument. Notice that in the dual solution (1) if .

Due to complementary slackness, all dual constraints corresponding to some bind, so they are satisfied automatically. Hence, to ensure the constructed dual solution is feasible, there remain four groups of constraints which need to be satisfied. For each group of constraints, there is a “pivotal” constraint such that if it is satisfied, all constraints in that group is satisfied. We list these four groups of constraints and “pivotal” constraint for each group below,

All dual constraints of where and :

The pivotal constraint is the dual constraint of , which can be simplified as

 λ(v‡−1)≤Z(v‡).
All dual constraints of where and :

The pivotal constraint is the dual constraint of , which can be simplified as

 −1≤Z(v‡).
All dual constraints of where :

The pivotal constraint is the dual constraint of , which can be simplified as

 λ(v‡−1)≤hv‡−1−1.
:

The pivotal constraint is , which can be simplified as

 λ(v‡−1)≥Z(v‡)−1.

We now show how to relate , and to satisfy the four inequalities identified above.

Notice that when and , the interval becomes empty. In that case, the first and second groups of constraints disappear. The combination of these four inequalities is equivalent to

1. and ; or

2. .

Without loss of generality, we assume that Condition (i) does not hold and then argue that Condition (ii) holds in this case.

The construction of implies the following two facts,