Optimal Auctions vs. Anonymous Pricing: Beyond Linear Utility

05/10/2019
by   Yiding Feng, et al.
Northwestern University
0

The revenue optimal mechanism for selling a single item to agents with independent but non-identically distributed values is complex for agents with linear utility (Myerson,1981) and has no closed-form characterization for agents with non-linear utility (cf. Alaei et al., 2012). Nonetheless, for linear utility agents satisfying a natural regularity property, Alaei et al. (2018) showed that simply posting an anonymous price is an e-approximation. We give a parameterization of the regularity property that extends to agents with non-linear utility and show that the approximation bound of anonymous pricing for regular agents approximately extends to agents that satisfy this approximate regularity property. We apply this approximation framework to prove that anonymous pricing is a constant approximation to the revenue optimal single-item auction for agents with public-budget utility, private-budget utility, and (a special case of) risk-averse utility.

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1 Introduction

In Bayesian mechanism design, a central question “simple versus optimal” focuses on how well simple mechanisms can approximate the revenue of the optimal mechanism in complex environments. The main motivation comes from the fact that the optimal mechanisms for asymmetric agents are usually both difficult to derive and implement. For agents with linear utility, Myerson (1981) characterized the optimal mechanism; it is sophisticated and involves discrimination. For more general models (e.g., budgeted agents, risk-averse agents), either no closed form characterization is known in the literature, or the optimal mechanisms take the model too literally and are both fragile and impractical. On the other hand, some simple mechanisms (e.g., posting an anonymous price, second-price auction with anonymous reserve) are prevalent broadly.

Alaei et al. (2018) study the simplest mechanism for asymmetric agents with linear utility, namely, an anonymous pricing where an anonymous price is posted for selling a single good, and the first agent (in an arbitrary order) who values the good at higher price will buy the good. They upper bound the optimal revenue by considering the ex ante relaxation which sell at most one item in expectation over randomness of all agents’ values, i.e., the ex post feasibility constraint of selling at most one item is relaxed to that of selling at most one item ex ante. They derive a tight -approximation bound for the anonymous pricing to the ex ante relaxation for independent but non-identical agents with linear utility and regular valuation distributions (see below). Their result implies that, up to an factor, discrimination and simultaneity are unimportant for optimizing revenue in single-item auctions. A natural question motivating our work is

Does the approximate optimality of anonymous pricing generalize to agents with non-linear utility under a suitable generalization of the regularity assumption?

Regularity is a common assumption in mechanism design that simplifies the derivation of the optimal mechanism (Myerson, 1981) and enables approximation mechanisms for agents with linear utility (e.g., Hartline and Roughgarden, 2009). Fixing any class of mechanisms and a single agent, the revenue curve is a mapping from a constraint

on the ex ante probability of sale, over randomness in the agent’s type and the mechanism, to the revenue of the optimal mechanism with the ex ante constraint. Specifically, the

price-posting revenue curve is generated by fixing mechanism class to all price-posting mechanisms; and the ex ante revenue curve is by fixing mechanism class to all possible mechanisms. The regularity for linear utility is defined as the equivalence of the price-posting revenue curve and the ex ante revenue curve (cf. Bulow and Roberts, 1989). These two revenue curves are sufficient to pin down the revenue from anonymous pricing, and ex ante relaxation, respectively.

Following the perspective of Alaei et al. (2013), the methods of our paper can be viewed as reductions in the following two senses. First, we approximately reduce the analysis of revenue of anonymous pricings for non-linear-utility agents to the analysis of revenue of anonymous pricings for linear-utility agents. Thus, relative to anonymous pricings, non-linear agent models can be considered well approximated by linear agent models. Second, our analysis reduces the multi-agent question of approximation by an anonymous price to a collection of single-agent approximation questions. These single-agent approximation questions ask whether or not the price-posting revenue curve is a good approximation to the ex ante revenue curve. Relative to Alaei et al. (2013), the single-agent problem to which we reduce gives simpler mechanisms.

Main Results

We introduce a generalization of regularity that characterizes the gap between the price-posting revenue curve and the ex ante revenue curve. Based on this generalization, we give a reduction framework to approximately reduce the analysis of the approximation bound for anonymous pricing for agents with non-linear utility to that of agents with linear utility. As the instantiations of the framework, we analyze the approximation bound for the anonymous pricing for asymmetric agents with public-budget utility, private-budget utility, and (a special case of) risk-averse utility, respectively.

Public-budget Utility:

The first classical non-linear utility model we consider is agents with public but non-identical budgets. For asymmetric agents with arbitrary distributions, the optimal mechanism can be solved by a polynomial-time solvable linear program over interim allocation rules

(cf. Alaei et al., 2012; Che et al., 2013), but no closed-form characterization is known. With the characterization of the optimal mechanism under ex ante constraints in Alaei et al. (2013), and a generalization of an argument from Abrams (2006), our framework yields -approximation and -approximation bounds, assuming the valuation distributions are regular and regular with decreasing density, respectively (Theorem 4.1).

Private-budget Utility: Mechanism design for agents with private-budget utility is challenging because the agent types are multi-dimensional. A lot of work has focused on characterizing the optimal mechanism and approximation mechanisms in this setting (discussed subsequently). Our framework shows that with independent value and budget distributions, regular value distributions, and with some assumptions on the budget distribution that anonymous pricing is a constant approximation to the optimal revenue (Theorem 5.4, Theorem 5.12), e.g., for monotone hazard rate budget distributions anonymous pricing is a approximation.

Risk-averse utility: It is standard to model risk-averse utility as a concave function that maps agents’ wealth to a utility. This introduces a non-linearity into the incentive constraints of the agents which in most cases makes mechanism design analytically intractable. Most results for agents with risk-averse utility consider the comparative performance of the first- and second-price auctions, cf., Riley and Samuelson (1981), Holt Jr (1980), Maskin and Riley (1984). Matthews (1983) and Maskin and Riley (1984), however, characterize optimal mechanisms for symmetric agents for constant absolute risk aversion and more general risk-averse model. In this paper, we restrict attention to very specific form of risk aversion studied in Fu et al. (2013), which is called capacitated utility. We derive a constant approximation bound (Theorem 6.3) for anonymous pricing for asymmetric agents, under an assumption on the capacities.

public budget independent private budget risk averse
regular, decreasing density regular value regular, budget MHR value regular, budget exceeds expectation w.p. at least regular, support , capacity at least
approx ratio
Table 1: Approximation bounds for anonymous pricing with asymmetric agents

All the approximation bounds and the corresponding assumptions can be found in Table 1. In each section, we also provide examples showing that without the assumptions we make, the constant approximation for anonymous pricing cannot be guaranteed.

Related Work

A prominent line of research has studied anonymous pricing in single-item environment for agents with linear utility. Hartline and Roughgarden (2009) show that second-price auction with an anonymous reserve is a -approximation to the optimal revenue. Alaei et al. (2018) improve this result by showing that anonymous pricing is a tight -approximation to the optimal ex ante relaxation. Jin et al. (2019b) prove that the tight ratio between anonymous pricing and the optimal (discriminatory) sequential post pricing is , and Jin et al. (2019a) show that the same tight ratio holds between anonymous pricing and the optimal mechanism.

One of the main contributions in our paper is to show that anonymous pricing is a constant approximation for asymmetric agents with private budgets. There are several papers in the literature that consider similar anonymous-pricing problems but with no unit-demand constraint on agents. Abrams (2006) shows that market clearing (the anonymous pricing where demand meets supply) gives a two approximation to the revenue of the optimal mechanism for selling multiple units to a set of asymmetric agents with public values and public budgets. Richter (2016) show that under the assumption that the valuation distribution is regular with decreasing density, a price-posting mechanism is optimal for selling a divisible good to a continuum of agents with private budgets. The lack of a unit-demand constraint on agents is crucial for their analyses and its introduction poses significant challenges for ours.

In the single item environment, for a single agent with private budget constraint, when her valuation distribution satisfies declining marginal revenues, 111The decreasing marginal revenue assumption considers revenue as a function of price and requires its derivative be decreasing. Che and Gale (2000) characterize the optimal mechanism by a differential equation. Devanur and Weinberg (2017) characterize the optimal mechanism for a single agent with an arbitrary distribution by a linear program and use an algorithmic approach to construct the solution. For multi-agents settings, Laffont and Robert (1996) show that the all-pay auction is optimal for symmetric agents in the public budget setting when the valuation distribution is regular and decreasing density. Pai and Vohra (2014) generalize the characterization of the optimal mechanism for symmetric agents in the private budget setting, with budgets distributed uniformly.

For more general feasibility constraints and private budgets, Chawla et al. (2011) show that when the feasible allocations form a matroid and the valuation distribution is monotone hazard rate (MHR), a simple lottery mechanism is a constant approximation to the optimal pointwise individually rational mechanism. 222 The pointwise individually rational mechanism requires that the payment of the agent is at most her value of the allocation after the realization of the randomness of the mechanism. Note that, via an example in Chawla et al. (2011), there can be a linear gap in revenue between the optimal pointwise individually rational mechanism and the optimal interim individually rational mechanism. Under the more classical interim individual rationality constraint, Alaei et al. (2012) reduce the multi-agent problem to the single-agent interim optimization problem, and solve it via a convex program. Alaei et al. (2013) approximately reduce the multi-agent problem to the single-agent ex ante optimization problem, and show, for example, that in the special case of the single item environment, (discriminatorally and) sequentially posting the single agent ex ante optimal mechanisms gives an -approximation. In comparison with these reduction results, the advantage of our approach is that while the ex ante optimal mechanisms are used in our bound, the approximately optimal mechanisms we identify are based only on single-agent posted pricing and are, thus, much simpler. Moreover, our resulting simple mechanisms can be optimized over directly.

2 Preliminaries

We consider the single item environment from auction theory. A seller has a single indivisible good. Agents have private types drawn independently, but not necessarily identically from some distributions. We are interested in optimizing the revenue, namely the sum of payments made by agents, of a mechanism for selling the good.

Auction Instance

An auction instance is defined as . Here is the set of agents and is the number of agents. , and are the type space, distributions and utility functions for each agent. In this paper, agents are non-identical, and the outcome for an agent is the distribution over the pair , where allocation and payment . The utility function of each player is a mapping from her private type and the outcome to her von Neumann-Morgenstern utility for the outcome.

Type structures and utility functions

The general framework of the paper will be instantiated in four specific models.

  1. Linear utility: A private type is a private value

    of the good. We denote the cumulative distribution function and the density function for the valuation distribution by

    and respectively. Given allocation and payment , an agent’s utility is . Our development of the approximation bound for anonymous pricing is based on a reduction from general utility function environments to linear utility environments.

  2. Public-budget utility: A private type is a private value . The utility function also encodes a public budget , which is not necessarily identical across agents. Given allocation and payment , an agent has utility if the payment is at most the budget and utility otherwise.

  3. Private-budget utility: A private type is a pair that consists of a private value and a private budget . We denote the cumulative distribution function and the density function for the budget distribution by and respectively. Given allocation and payment , her utility is if the payment is at most the budget and is otherwise.

  4. Risk-averse utility: A private type is a private value of the good. The utility function is a concave function mapping from the wealth of an agent to a utility. Specifically, we restrict attention to a very specific form of risk aversion studied in Fu et al. (2013), which is both computationally and analytically tractable: utility functions that are linear up to a given capacity and then flat. Given allocation and payment , an agent has utility . We refer to this utility function as capacitated utility. The capacity is encoded in the utility function and is not necessarily identical across agents.

The assumption that the supports of the distributions are bounded intervals in the models above is for technical simplicity and is without loss of generality.

Ex ante revenue curves and price-posting revenue curves

We introduce the quantile space, ex ante revenue curves and price-posting revenue curves.

Definition 2.1.

The quantile of a single-dimensional agent with value drawn from distribution is the measure with respect to of stronger values, i.e., ; the inverse demand function maps an agent’s quantile to her value, i.e., .

Definition 2.2.

Given ex ante constraint , the single-agent ex ante revenue-maximization problem is to find the optimal mechanism with ex ante allocation probability (i.e. the expected allocation over the draws of the agent’s type) exactly . The optimal ex ante revenue, as a function of , is denoted by the ex ante revenue curve .

Since the revenue (i.e., expected payment) is a linear objective, and the space of feasible mechanisms is convex, the ex ante revenue curve is concave.

Fact 2.1.

The ex ante revenue curve is concave.

Definition 2.3.

A per-unit price for selling an indivisible good probabilistically is a mapping from a lottery with winning probability to a payment . Per-unit pricing for a single agent is a mechanism where a per-unit price is posted, and the agent can select a lottery with an arbitrary winning probability and pay . The item remains unsold with probability .

Definition 2.4.

Given ex ante constraint , the single-agent ex ante price-posting problem is to find the per-unit pricing with ex ante allocation probability exactly . The optimal ex ante price-posting revenue, as a function of , is denoted by the price-posting revenue curve . The market clearing price for the ex ante constraint is .

For an agent with linear utility, the price-posting revenue curve at any quantile  is achieved by posting per-unit price , i.e., ; and Bulow and Roberts (1989) give a geometric connection between the ex ante revenue curve and price-posting revenue curve.

Lemma 2.2 (Bulow and Roberts, 1989).

For an agent with linear utility, the ex ante revenue curve is equal to the concave hull of the price-posting revenue curve .

For agents with linear utility, an agent’s valuation distribution is regular if the price-posting revenue curve is concave; equivalently, if the price posting and ex ante revenue curves are identical, i.e., . The price that maximizes the revenue from a single agent is called monopoly reserve . Lemma 2.2 implies that the ex ante optimal mechanisms for a single agent with linear utility, is either a price posting or the randomization over two price postings. For general utility agents, the ex ante revenue curve can be much larger than the price-posting revenue curve almost everywhere (Example A.2).

It is straightforward to map per-unit price to quantile with the price-posting revenue curve, i.e., . In this paper, it will be useful to imagine that an ex ante revenue curve is generated by per-unit pricing, even when it is not, and mathematically define the mapping from effective price to quantile in the same way, i.e., . Equivalently, this definition imagines a regular linear-utility agent with the same revenue curve and for which is the largest quantile that accepts price .

Definition 2.5.

Effective price posting to an ex ante revenue curve is a mechanism where a per-unit price is posted to an agent with a price-posting revenue curve equivalent to .

Different mechanisms of interest in this paper

Through this paper, we focus on optimal mechanisms in the following three classes.

  1. Auctions: An auction is any mechanism that maps types to allocations and payments subject to incentive and feasibility constraints. Under the linear utility assumption, the optimal auction was characterized by Myerson (1981) and this characterization, though complex, is the foundation of modern auction theory. For more general utility models with asymmetric agents (e.g. public-budget utility, private-budget utility, risk-averse utility), there is no closed form characterization in the literature.

  2. Ex ante relaxations: The ex ante relaxation considers the problem of selling at most one item in expectation over draws of all agents’ types, i.e., the ex post feasibility constraint of selling at most one item is relaxed to that of selling at most one item ex ante. Fixing the ex ante probability of serving each agent, the ex ante optimal mechanism solves the single-agent ex ante revenue-maximization problem for all agents individually. The ex ante optimal mechanism for asymmetric agents usually discriminates and may, ex post, simultaneously serve multiple agents. This relaxation was identified as a quantity of interest in Chawla et al. (2007) and its study was refined by Alaei (2011) and Yan (2011). The revenue of the ex ante optimal mechanism gives an upper bound on the revenue of the (point-wise feasible) optimal auction, see e.g., Alaei (2011). We denote the optimal revenue achieved by the ex ante relaxation for a specific collection of ex ante revenue curves by .

  3. Anonymous pricings: An anonymous pricing mechanism posts an anonymous per-unit price and the agents arrive in an arbitrary order. Each agent can select a lottery with an arbitrary winning probability and pay . The lottery executes immediately before the next agent arrives. Once an agent wins the item, the mechanism halts. We denote the revenue achieved by posting anonymous per-unit price for a specific collection of price-posting revenue curves by . We denote the revenue from the optimal anonymous price by .

    Consider an arbitrary agent . Suppose all agents who come earlier than her do not win the item. The probability that she wins the item is , where the randomness comes from her own type and the lottery which she selects. Thus, .

Linear utility -approximation bound

Alaei et al. (2018) show that anonymous pricings give a tight -approximation to ex ante relaxations for agents with regular valuation distributions and linear utility. For regular linear agents, the price-posting revenue curves are concave and equal to the ex ante revenue curves. Moreover, for any non-negative concave function on domain , there is a regular value distribution with this function as its revenue curve. Thus, the result of Alaei et al. (2018) can be restated as follows.

Theorem 2.3 (Alaei et al., 2018).

Any set of non-negative concave functions on domain satisfies for which numerically evaluates to ; i.e., anonymous pricing is a approximation to the ex ante relaxation for linear utility agents with values drawn from regular distributions with these revenue curves.

3 A General Reduction Framework

In this section, we give two approaches that approximately reduce the approximation of anonymous pricing for non-linear agents to the approximation of anonymous pricing for linear agents. The first approach applies when a “closeness” condition between the price-posting revenue curve and the ex ante revenue curve holds. The approximation guarantee degrades with this closeness condition. The second approach applies under a much weaker closeness condition but additionally requires that the price-posting revenue curve is concave. The high-level steps of these reductions are depicted in Figure 1.

Thm. 3.4: -approx,

if is -close for ex ante

optimization to .

Thm. 3.1: -approx,

if is -close for

price posting to .

Thm. 2.3: -approx,

if is concave.

Thm. 2.3: -approx.
Figure 1: The reduction framework. The upper path uses the approximate robustness of anonymous pricing. A better bound for the upper path is obtained from Theorem 3.2 but omitted from the figure. The lower path uses the approximate robustness of the ex ante relaxation and requires that the price-posting revenue curves are concave.

These reductions are based on the facts that the ex ante revenue is given mathematically by the ex ante revenue curves , the anonymous pricing revenue is given by the price-posting revenue curves , and the approximation of anonymous pricing to the ex ante relaxation for linear agents implies, as stated in Theorem 2.3, an approximation bound between anonymous pricing and the ex ante relaxation for any set of concave revenue curves.

3.1 Approximate Robustness of Anonymous Pricing

We develop the upper path of Figure 1 for reducing non-linear anonymous pricing problem to the linear anonymous pricing problem. This upper path is based on identifying a closeness property on revenue curves that approximately preserves anonymous pricing revenue. This closeness property, illustrated in the left hand side of Figure 2, can be viewed as an approximate notion of regularity.

Definition 3.1.

The agent’s price-posting revenue curve is -close for price posting to her ex ante revenue curve , if for all . Such an agent is -close for price posting.

Theorem 3.1.

Given agents with ex ante revenue curves and price-posting revenue curves , if each agent , is -close for price posting, anonymous pricing on the price posting revenue curve is an -approximation to anonymous pricing on the ex ante revenue curve, i.e., .

Proof.

Let be the anonymous price on ex ante revenue curve, and let the per-unit price . For each agent , let and . If the quantile , since is -close for price posting to , . Therefore, , which implies .

If , and if , then . If , -closeness for price posting implies that , namely, , which contradicts to the assumption that .

Therefore, the revenue of posting price on the price posting revenue curve is



Figure 2: The left hand side illustrates an example that the price-posting revenue curve is -close for price posting to ex ante revenue curve . The definition requires that is at least on , i.e., that the black thin line is above the black dotted line. The right hand side illustrates an example that the price-posting revenue curve is -close for ex ante optimization to ex ante revenue curve . Depicted with the gray dashed line, is defined as and -closeness requires that , i.e., that the gray dashed line is above the black dotted line.

The bound in Theorem 3.1 can be improved if we have extra information about the closeness between the price posting revenue curve and the ex ante revenue curve. Specifically, let denote a bound on the ratio between the single agent optimal mechanism and the single agent optimal price posting, i.e., . It is easy to see that if and are -close then . Often, a better bound on than can be obtained and, when it can, Theorem 3.2 improves the bound of Theorem 3.1.

Theorem 3.2.

Given agents with ex ante revenue curves and price-posting revenue curves , if each agent is -close for price posting and the optimal price posting revenue for each agent is an -approximation to the optimal single agent revenue, then anonymous pricing on the price posting revenue curve is a -approximation to anonymous pricing on the ex ante revenue curve, i.e., .

Proof.

Let be the anonymous price on ex ante revenue curve. For a threshold to be tuned subsequently, we say an agent is a significant contributor to the anonymous pricing revenue if is at least . We analyze separately the case that there is a significant contributor and the case that all agents are insignificant contributors.

When Significant Contributor Exists

Suppose there exists an agent such that is at least . Consider the optimal price-posting for this single agent . Its revenue is a -approximation to the ex ante relaxation, i.e.,

(1)

where the first inequality is due to our -approximation assumption for a single agent, and the second inequality is due to the fact that agent is a significant contributor.

When Significant Contributor Does Not Exist

Suppose there is no significant contributor, i.e., for all . By a similar argument to Theorem 3.1, we can lower bound for every agent as follows,

where the latter case uses the facts that and .

Therefore, for parameter , the the revenue of posting price is

(2)

Notice that the approximation bounds of equations (1) and (2) have in the denominator and numerator, respectively. Thus, setting to equalize these bounds (when possible) will optimize approximation bound. Specifically, if , setting optimizes the approximation bound as ; and if , setting optimizes the approximation bound as . ∎

Theorem 3.1 and Theorem 3.2 reduce the approximation of anonymous pricing for non-linear agents to that of regular linear agents. For example, if , Theorem 3.1 implies . By creftypecap 2.1, the ex ante revenue curves are concave and thus, by Theorem 2.3, the approximation ratio is at most .

Corollary 3.3.

Given agents with ex ante revenue curves and price-posting revenue curves , if each agent is -close for price posting, then the worst case approximation factor of anonymous pricing to the ex ante relaxation is , i.e., . If additionally, the optimal price posting revenue for each agent is an -approximation to the optimal single agent revenue, then the worst case approximation factor of anonymous pricing to the ex ante relaxation is , i.e., .

The reduction framework in Corollary 3.3 is tight with the given assumptions. See Appendix C for a lower bound instance.

3.2 Approximate Robustness of the Ex Ante Relaxation

We develop the lower path of Figure 1 for reducing the non-linear anonymous pricing problem to the linear anonymous pricing problem. This lower path is based on identifying a closeness property on revenue curves that approximately preserves the optimal revenue of the ex ante relaxation. The revenue of ex ante relaxation is approximately preserved under weaker conditions than the revenue of anonymous pricing. On the other hand, to make use of this approximate robustness to derive an anonymous pricing approximation, our framework additionally requires the assumption that non-linear agents’ price-posting revenue curves are concave.

Definition 3.2.

The agent’s price-posting revenue curve is -close for ex ante optimization to her ex ante revenue curve , if for all , there exists a quantile such that . Such an agent is -close for ex ante optimization.

The above definition is illustrated in the right hand side of Figure 2. Note that -close for price posting implies -close for ex ante optimization.

Theorem 3.4.

Given agents with ex ante revenue curves and price-posting revenue curves , if each agent is -close for ex ante optimization, then the optimal ex ante relaxation on revenue curves is a -approximation to the optimal ex ante relaxation on revenue curves , i.e., .

Proof.

Let be the optimal ex ante relaxation for ex ante revenue curves , and let  be the quantile assumed to exist by -closeness such that and for each . We have

When the price posting revenue curves are concave, Theorem 3.4 can be combined with Theorem 2.3 applied to to obtain the following corollary.

Corollary 3.5.

Given agents with ex ante revenue curves and price-posting revenue curves , if each agent has concave price-posting revenue curve and is -close for ex ante optimization, then the worst case approximation factor of anonymous pricing to the ex ante relaxation is , i.e., .

3.3 Heterogeneous Agent Utility Models

Our closeness definitions are monotonic, formalized in the subsequent lemma. With this observation, our framework can be applied to environments with heterogeneous utility functions. For example, suppose some of the agents have private budget constraints and some of the agents are risk averse. If each agent is -close for price posting, then anonymous pricing for these agents is a -approximation to the optimal ex ante relaxation.

Lemma 3.6.

For any , , and , -close for price posting implies -close for price posting, and -closeness for ex ante optimization implies -close for ex ante optimization.

4 Public-budget Utility

In this section, we consider the case where agents have public-budget utility. We make the following two standard assumptions on the valuation distributions. In the linear utility model, Alaei et al. (2018) show that the regularity on valuation distribution is necessary for the constant approximation bound of the anonymous pricing, so we make the same assumption under the public-budget utility. We also consider an additional assumption that the density function is decreasing on its support. These two assumptions together are sometimes called public-budget regularity (cf. Pai and Vohra, 2014). The following theorem and corollary summarize the main results of this section.

Theorem 4.1.

An agent with public-budget utility and regular valuation distribution is -close for price posting, and with regular and decreasing density valuation distribution the agent is -close for price posting.

Corollary 4.2.

For a single item environment with agents with public-budget utility and regular valuation distributions, the worst case approximation factor of anonymous pricing to the ex ante relaxation is at most ; with regular and decreasing density valuation distributions, the worst case approximation factor of anonymous pricing to the ex ante relaxation is at most .

To prove Theorem 4.1, we start with a characterization of the ex ante optimal mechanism from Alaei et al. (2013).

Lemma 4.3 (Alaei et al., 2013).

For a single agent with public-budget utility and regular and decreasing density valuation distribution, the ex ante optimal mechanism is a per-unit pricing, i.e., .

Notice that the outcome of the optimal per-unit pricing of Lemma 4.3 is clear. If the agent’s value is below the per-unit price, then she does not buy. If her value is above the per-unit price then she buys until either the budget constraint binds or the unit-demand constraint binds. Since the ex ante optimal mechanism is a per-unit pricing, the revenue curves are equal and the agent is -close for price posting. The second part of Theorem 4.1 is completed, and the following lemma implies the first part.

Lemma 4.4.

For a single agent with public-budget utility and regular valuation distribution, the optimal price-posting revenue is a -approximation to the ex ante optimal revenue, i.e., .

Proof.

Fix any ex ante constraint . Let be the market clearing price with ex ante allocation exactly . If is smaller than the budget (i.e. the budget constraint does not bind), since the valuation distribution is regular, posting is optimal for ex ante constraint.

Consider the remaining case where the market clearing price is larger than the budget. View the ex ante mechanism as a menu with quantity-price options (the taxation principle).

  • Consider the options in the ex ante optimal mechanism with per-unit price smaller than . The total demand of these options is at most and thus, at per-unit prices below , the total revenue from these options is at most .

  • Consider the options in the ex ante optimal mechanism with per-unit price larger than . The types that buy these options all have value at least . The payment from each of these types is at most their budget. Under the market clearing price all of these types pay exactly their budget. Thus, the total revenue from market clearing is at least the revenue from these options.

In conclusion the ex ante optimal revenue is upper bounded by twice the price posting revenue. ∎

The second part of the proof of Lemma 4.4 is similar to an analysis by Abrams (2006). In contrast, her setting lacks the unit-demand constraint, which simplifies her argument since an agent will pay her full budget even when the market clearing price is smaller than her budget. In contrast, our analysis uses regularity when the unit-demand constraint binds.

5 Independent Private-budget Utility

In this section, we consider the case where agents have private-budget utility. To obtain a constant approximation bound for anonymous pricing, we assume that values and budgets are independently distributed, the valuation distribution is regular and one of two possible assumptions on the budget distribution: (a) the probability that the budget exceeds its expectation is at least a constant ; and (b) the budget distribution has monotone hazard rate (MHR), i.e, is monotonically non-decreasing with respect to . Since for a MHR distribution, a sample exceeds its expectation with probability at least (cf. Barlow and Marshall, 1965), assumption (b) is special case of assumption (a) with . We give two examples in LABEL:apx:privatenegative_example that demonstrate the necessity of these assumptions to guarantee the constant approximation of anonymous pricing.

5.1 Unconstrained 2-approximation

Before giving results about the closeness of ex ante revenue curves and price-posting revenue curves, we first show that the optimal pricing is a 2-approximation to the optimal mechanism (with no ex ante constraint). With this result we will be able to invoke Theorem 3.2 with . Though it will not result in any improvements in our main results, this approximation bound does not make any assumptions on the budget distribution.

Theorem 5.1.

For selling a single item to a single private-budget agent with independent value and budget distributions and regular value distribution, the optimal per-unit pricing is a 2-approximation to the optimal mechanism.

We give the proof of this theorem below. It follows from a characterization of the optimal revenue for public-budget agents from Laffont and Robert (1996) and a generalization of a geometric argument from Dhangwatnotai et al. (2010) that analyzes the revenue from a random price from the value distribution, both stated below. We employ this random-price argument later in Lemma 5.14 as well.

Lemma 5.2 (Laffont and Robert, 1996).

For a single item and single public-budget agent with regular valuation distribution, the optimal mechanism posts a price equal to the smaller of the agent’s budget and the monopoly reserve.

































Figure 3: In the geometric proof of Lemma 5.3, for any fixed budget the optimal revenue is the area of the light gray striped rectangle and the revenue from posting random price is the area of the dark gray region. By geometry, the latter is at least half of the former. The black curve is the price-posting revenue curve with no budget constraint . The figure on the left depicts the large-budget case (i.e., ), and the figure on the right depicts the small-budget case (i.e., ).

The following lemma generalizes a lemma and proof approach by Dhangwatnotai et al. (2010) to public-budget agents.

Lemma 5.3.

For a single item and a single public-budget agent with regular valuation distribution, posting a random per-unit price drawn from the valuation distribution is a -approximation to the optimal revenue.

Proof.

Denote the agent’s fixed budget by and let be a random per-unit price drawn from valuation distribution . For the depiction of Figure 3, we claim that the dark grey shaded area is the expected revenue of posting the random price , and the area of the light grey striped rectangle is the optimal revenue. Concavity of the price-posting revenue curve with no budget constraint (by regularity of the value distribution) implies that a triangle with half the area of the light gray rectangle is contained within the dark gray region and, thus, the random price is a 2-approximation. The remainder of this proof shows that the geometry of the regions described above is correct.

Let be the price-posting revenue curve with no budget constraint. When , the revenue of posting price is and, when , the revenue of posting price is . Let . LABEL:lem:public_budgetoptimal_under_regular implies that the optimal revenue for budget is by posting the minimum between the monopoly reserve and the budget , i.e., it is the area of the light gray striped rectangle. On the other hand, the revenue of posting is , which is exactly the dark grey shaded area. ∎

Proof of Theorem 5.1.

Consider the revenue from types with each fixed budget level separately. At each budget level, LABEL:lem:single_agent_approxpublic guarantees that the revenue from posting a random per-price is a 2-approximation to the optimal revenue from these types. Since value and budget are independent, the random per-unit prices offered at all budget levels are identically distributed, thus simultaneously offering the same random per-unit price across all budget levels guarantees a 2-approximation to optimal revenue for the private-budget agent. Furthermore, the optimal deterministic per-unit price obtains revenue that is no worse than that of the randomized price. ∎

5.2 Budgets Exceeding Expectation with Constant Probability

We first analyze the approximation bound for anonymous pricing mechanism under the assumption that the budget exceeds its expectation with constant probability at least .

Theorem 5.4.

A single private-budget agent is -close for price posting if value and budget are independently distributed, the valuation distribution is regular, and the budget exceeds its expectation with probability at least .

With Theorem 5.1 and Theorem 5.4, Corollary 3.3 implies that under the same assumptions in Theorem 5.4, the approximation factor of anonymous pricing is .

Corollary 5.5.

For a single item environment with private-budget agents, anonymous pricing is a -approximation to the ex ante relaxation if values and budgets are independently distributed, the valuation distributions are regular, and the budgets exceed their expectations with probability at least .

We take the following high-level approach which is similar to the analysis of Abrams (2006). Fix any ex ante constraint less than . Consider the per-unit price induces ex ante allocation probability exactly ; henceforth, the market clearing price. For any allocation and payment obtained by an agent in the ex ante optimal mechanism, refer to the agent’s per-unit price as . We decompose the ex ante optimal mechanism into two mechanisms where the per-unit price in the first mechanism for each type is at most the market clearing price, and the per-unit price in the second mechanism for each type is at least the market clearing price. The revenue of the former mechanism is smaller than the revenue of posting the market clearing price since it sells less of the item and with a lower per-unit price and, under our assumptions, the revenue of the latter can be bounded within a constant factor of the revenue of posting the market clearing price. The latter analysis considers separately the case of high and low market clearing prices, and in the former derives bounds via the geometry of revenue curves and in the latter by the assumption on budgets.

In order to decompose the ex ante optimal mechanism, we first provide a definition and a characterization of all incentive compatible mechanisms for a single agent with private-budget utility, and her behavior in the mechanisms.

Definition 5.1.

An allocation-payment function is a mapping from the allocation purchased by an agent to the payment she is charged.

Lemma 5.6.

For a single private-budget agent, any incentive compatible mechanism, and all types with any fixed budget; the mechanism provides a convex and non-decreasing allocation-payment function, and subject to this allocation-payment function, each type will purchase as much as she wants until the budget constraint binds, the unit-demand constraint binds, or the value binds (i.e., her marginal utility becomes zero).

The formal proof of Lemma 5.6 is given in Appendix B. At a high level, in private budget settings, by relaxing the incentive compatibility constraint for both values and budgets to the incentive compatibility constraint only for values, the payment identity at every fixed budget level is sufficient to induce an allocation-payment function for all types with that fixed budget.

Fix an arbitrary ex ante constraint and denoting be the ex ante optimal mechanism. Consider the decomposition of into two mechanisms and . These decomposed mechanisms will be incentive compatible for values, but not necessarily for budgets; a setting we refer to as the random-public-budget utility model. We require that (a) the per-unit prices in for all types are at most the market clearing price; (b) the allocation for all types in is at most the allocation in ; (c) the per-unit prices in for all types are larger than the market clearing price; and (d) the revenue from (for a private-budgeted agent) is upper bounded by the revenue from and (for a random-public-budgeted agent with same distribution), i.e., .

The construction is as follows: For each budget , let be the allocation-payment function for types with budget in mechanism , and be the largest allocation such that the marginal price is below the market clearing price , i.e., . Define the allocation-payment functions and for and respectively below,



Figure 4: Depicted are allocation-payment function decomposition. The black lines in both figures are the allocation-payment function in ex ante optimal mechanism ; the gray dashed lines are the allocation-payment function and in and , respectively.

In this construction, since the original allocation-payment functions in are convex and non-decreasing for each budget level , the constructed allocation-payment functions and are also convex and non-decreasing. Hence, for each type, the payment in is upper bounded by the sum of the payments in and , and the requirements above are satisfied.

Lemma 5.7.

For a single agent with random-public-budget utility, independently distributed value and budget, and any ex ante constraint ; the revenue of is at most the revenue from posting the market clearing price, i.e., .

Proof.

The ex ante allocation of is at most the ex ante allocation of , i.e., . Combining with the fact that the per-unit prices in for all types are weakly lower than the market clearing price, its revenue is at most the revenue of posting the market clearing price. ∎

Now, we introduce two lemmas Lemma 5.8 and Lemma 5.9 to upper bound the revenue of ; the former for the case that the market clearing price is large, and the latter for the case that it is small. At a high level, when the market clearing price is large, we utilize the geometry of revenue curves to bound the revenue; and when the market clearing price is small, we use the assumption on the budget distribution to bound the revenue.

Lemma 5.8 (Large Market Clearing Prices).

For a single agent with random-public-budget utility, independently distributed value and budget, and a regular valuation distribution:

  1. for any ex ante constraint and market clearing price which is at least the monopoly reserve , i.e., , the revenue of posting the market clearing price is at least the revenue of , i.e., ;

  2. (otherwise) for any ex ante constraint and market clearing price for some , the revenue of posting the market clearing price is an -approximation to the revenue of , i.e., .

Note that the third case, where the market clearing price is small, will be handled subsequently by Lemma 5.9.

Proof of Lemma 5.8.

In both and the mechanism that posts the market clearing price, the types with value lower than the market clearing price will purchase nothing, so we only consider the types with value at least in this proof. Each budget level is considered separately. If a -approximation is shown separately, then -approximation holds in combination.

For types with budget , by posting the market clearing price , those types always pay their budgets , which is at least the revenue from those types in .

For types with budget , by posting the market clearing price , those types always pay . Since the budget constraints do not bind for these types, it is helpful to consider the price-posting revenue curve without budget, which we denote by . The regularity of the valuation distribution guarantees that is concave. Now we consider two cases: (i) the market clearing price is at least the monopoly reserve ; (ii) the market clearing price is less than the monopoly reserve but at least . In both cases, we use the concavity of .

For case (i), the concavity of implies that higher prices above extracts lower revenue than . Since the per-unit prices in for all types are at least , the concavity of guarantees that the expected revenue of posting for types with budget larger than is at least the expected revenue for those types in .

For case (ii), the concavity of implies that lower prices below extracts lower revenue. Thus, posting extracts higher revenue than posting . Again, the concavity of guarantees that the revenue of posting is an -approximation to the optimal revenue (i.e., posting the monopoly reserve ) generated from these types, even with the relaxation of ignoring their budget constraint.

Combining these bounds above, if the market clearing price is at least the monopoly reserve , then ; and otherwise if the market clearing price is at least , then . ∎

Lemma 5.9 (Small Market Clearing Prices).

For a single agent with random-public-budget utility, independently distributed value and budget, and a budget distribution such that the budget exceeds its expectation with probability at least , for any ex ante constraint with market clearing price , the revenue of posting the market clearing price is a -approximation to the revenue of , i.e.,

Before the proof of Lemma 5.9, we introduce an intermediate lemma used in our argument.

Lemma 5.10.

Fix any convex and non-decreasing allocation-payment function, and any value , for any budgets and such that , subject to this allocation-payment function, the payment from the type with value and budget is a -approximation to the payment from the type with value and budget .

Proof.

If the type pays her budget (i.e., the budget constraint binds), her payment is a -approximation to the payment from the type , since the type pays at most .

If the type pays less than her budget (i.e., the unit-demand constraint binds, or the value binds), her allocation is equal to the allocation from the type . Hence, their payments are the same. ∎

Proof of Lemma 5.9.

Similar to the proof of Lemma 5.8, we only consider the types with value at least . Let be the expected budget. We claim that all types with budget pay their budget in posting the market clearing price. Otherwise, the unit-demand constraint binds for all types such that , ; and the assumption that implies the ex ante allocation for them is , which exceeds the ex ante constraint , a contradiction.

Let be the expected revenue of providing the allocation-payment function in to the types with budget ; and let be the expected revenue of posting the market clearing price to the types with budget .

The following three facts allow comparison of to : (a) Posting the market clearing price makes the budget constraints bind for the types with budget at most , so for all . (b) Lemma 5.10 implies that for all . (c) Since the revenue of posting the market clearing price to an agent with budget is at most the revenue to an agent with budget ; with the assumption that budgets exceed the expectation with probability at least , it implies that

We upper bound the revenue of as follows,

where the first inequality is due to facts (a) and (b); in the second inequality, the first term is due to , the revenue is monotone increasing in , and by definition , and the second term is due to fact (a); and the last inequality is due to fact (c). ∎

We combine the three lemmas into a proof for Theorem 5.4.

Proof of Theorem 5.4.

Set the parameter . Lemma 5.7, Lemma 5.8 and Lemma 5.9 implies that for all ex ante constraints , the price posting revenue is a -approximation to the ex ante revenue. ∎

5.3 MHR Budget Distributions

Assuming that the value and the budget are independently distributed, the valuation distribution is regular, and the budget distribution is MHR, the price-posting revenue curve is concave. This result is formally stated in Lemma 5.11, whose proof is deferred to Appendix B. Due to the concavity of the price-posting revenue curve, we show a better approximation bound for anonymous pricing using the reduction in Theorem 3.4.

Lemma 5.11.

A single private-budget agent has a concave price-posting revenue curve if her value and budget are independently distributed, the valuation distribution is regular, and the budget distribution is MHR.

Theorem 5.12.

A private-budget agent is -close for ex ante optimization if her value and budget are independently distributed, the valuation distribution is regular, and the budget distribution is MHR.

Corollary 5.13.

For a single item environment with private-budget agents, anonymous pricing is a -approximation to the ex ante relaxation if values and budgets are independently distributed, the valuation distributions are regular, and the budget distributions are MHR.

When the market clearing price is larger than the monopoly reserve, Lemma 5.7 and Lemma 5.8 guarantees that posting the market clearing price is a 2-approximation to