Interdependent Values without Single-Crossing

06/11/2018 ∙ by Alon Eden, et al. ∙ Tel Aviv University University of Washington 0

We consider a setting where an auctioneer sells a single item to n potential agents with interdependent values. That is, each agent has her own private signal, and the valuation of each agent is a known function of all n private signals. This captures settings such as valuations for artwork, oil drilling rights, broadcast rights, and many more. In the interdependent value setting, all previous work has assumed a so-called single-crossing condition. Single-crossing means that the impact of agent i's private signal, s_i, on her own valuation is greater than the impact of s_i on the valuation of any other agent. It is known that without the single-crossing condition an efficient outcome cannot be obtained. We study welfare maximization for interdependent valuations through the lens of approximation. We show that, in general, without the single-crossing condition, one cannot hope to approximate the optimal social welfare any better than the approximation given by assigning the item to a random bidder. Consequently, we introduce a relaxed version of single-crossing, c-single-crossing, parameterized by c≥ 1, which means that the impact of s_i on the valuation of agent i is at least 1/c times the impact of s_i on the valuation of any other agent (c=1 is single-crossing). Using this parameterized notion, we obtain a host of positive results. We propose a prior-free deterministic mechanism that gives an (n-1)c-approximation guarantee to welfare. We then show that a random version of the proposed mechanism gives a prior-free universally truthful 2c-approximation to the optimal welfare for any concave c-single crossing setting (and a 2√(n)c^3/2-approximation in the absence of concavity). We extend this mechanism to a universally truthful mechanism that gives O(c^2)-approximation to the optimal revenue.

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1 Introduction

The most fundamental problem in the theory of auctions is how to sell a single item to potential buyers efficiently; i.e., how to allocate the item to the agent who values it the most. This problem has been fully resolved in 1961 in the case where agents have independent private values (IPV). Indeed, [Vickrey, 1961] introduced the second-price auction, which is dominant-strategy incentive compatible and fully efficient.

While much research has concentrated on the independent private values model, for many high-stake auctions that arise in practice (e.g., auctions for mineral rights [Wilson, 1969]), agents’ valuations are correlated, a special case of which are common values, where agent values are identical [Milgrom, 1979; Wilson, 1969]. In such cases the independent private values model is unsuitable.

To address scenarios of this type, Milgrom and Weber [1982] introduced the model of interdependent values. In their model, every agent has a non-negative private signal , and the valuation of agent

is a function of the entire signal vector; i.e.,

. While the signal is private and unknown by other agents or the auctioneer, the valuation functions are commonly known by all111A variation on this assumption is the asymmetric knowledge scenario, where the valuation functions are common knowledge amongst the agents but not known to the auctioneer. This variant was studied in [Dasgupta and Maskin, 2000; Perry and Reny, 1999]..

Consider the following scenario, which is a variant of the scenario given in [Dasgupta and Maskin, 2000].

Example 1.

Two firms compete for the right to drill for oil on a given tract of land. Firm has a marginal drilling cost of , while firm has a marginal drilling cost of . Suppose oil is sold in the market at a price of . Firm performs a private test and discovers that the expected size of the oil reserve is units. This scenario gives rise to the following valuation functions: , and .

Other examples include: (a) common value auctions, where the value of the item for sale is identical amongst bidders, but bidders may have different information about the item’s value [Wilson, 1969], and (b) auctions with resale, where , where for all . Here, an agent’s value depends on her personal valuation for the item and its resale value, which is reflected by the valuations of others [Klemperer, 1998; Myerson, 1981].

In scenarios with interdependent values, a direct revelation mechanism is one where every agent reports his signal to the mechanism, and the mechanism, knowing the (expected) valuation of each agent as a function of the signal vector, determines the allocation and the payments. The strongest notion of truthfulness relevant in the interdependent value setting is that truth telling is an ex-post Nash equilibrium. Truth-telling is said to be ex-post Nash if, for every bidder , for every possible realization of the other bidders’ signals , and given that other bidders report their signals truthfully, then it is in bidder ’s best interest to report her true signal. Truthfulness in dominant strategies is not viable in interdependent values since the value of one agent depends on the signals of others222To see why a dominant strategy truthful mechanism is hopeless, note that if some agent (Alice) misreports her signal, even if every other agent reports truthfully, the valuation of other agents depends on the signal given by Alice, so valuations computed using the corrupt signal from Alice (corrupt valuations) may vary from the true valuations (based on the non-corrupted signals). Thus, the allocation and prices can be quite different than those computed from the non-corrupted signals, and it may not be a best response of the agents to reveal their true signals. In fact, it is easy to give examples where truth-telling is clearly not a reasonable response..

Work on interdependent values has typically assumed a prior joint distribution (either correlated or independent) on the signals. One can further categorize this body of work under various informational assumptions (is the distribution of signals commonly known by agents? is it known by the auctioneer?), and solution concepts (Bayes-Nash, ex-post Nash, etc.).

To quote Bergemann and Morris [2005], “The mechanism design literature assumes too much common knowledge of the environment among the players and planner.” Such common knowledge may be unrealistic, and moreover, can lead to miraculous outcomes. For example, Crémer and McLean [1985, 1988] show how to extract full surplus as revenue under appropriate assumptions and solution concepts (in particular, commonly known correlated distributions and Bayes Nash equilibrium). This approach follows the seminal work of Wilson [1985] that posits Wilson’s doctrine333Due to Robert Wilson (1985), and not referring to the Harold Wilson wiretapping doctrine (1966), nor to the Woodrow Wilson extension of the Monroe doctrine (1912).: simple detail-free mechanisms should be preferred in order to alleviate the risks introduced by various assumptions.

In this work we follow this doctrine and propose prior-free mechanisms—mechanisms that do not require any knowledge of the underlying distribution. Moreover, our mechanisms are universally ex-post truthful. Ex-post truthfulness means that bidders maximize utility for any signal profile, so any underlying distribution of bidder values is irrelevant. For universally truthful mechanisms, even if the bidders know in advance the random internal coin tosses made by the mechanism they still have no incentive to bid non-truthfully.

Single-crossing. All previous work on interdependent values assumes some version of a single-crossing (SC) condition on the valuations [Milgrom and Weber, 1982; d’Aspremont and Gérard-Varet, 1982; Maskin, 1992; Ausubel et al., 1999; Dasgupta and Maskin, 2000; Bergemann et al., 2009; Chawla et al., 2014; Che et al., 2015; Li, 2016; Roughgarden and Talgam-Cohen, 2016]. Several definitions of single-crossing have been suggested in the literature. We define the single-crossing (SC) condition as done in Roughgarden and Talgam-Cohen [2016]444Appendix A briefly discusses other definitions of single-crossing, and how the results of our paper can be adapted appropriately.: for every agent , for any set of other players’ signals , and for every agent ,

Thus, as a function of signal (), the valuation of bidder (), increases at least as much as the valuation of any other bidder (, ).

Under the SC condition, a generalization of the VCG mechanism maximizes social welfare [Maskin, 1992; Ausubel et al., 1999; Roughgarden and Talgam-Cohen, 2016; Chawla et al., 2014]; this generalized VCG mechanism is deterministic, ex-post truthful and prior-free. It can be easily verified that the valuations in the oil drilling example described in Example 1 satisfy the SC condition. Indeed, (as a function of ) grows faster than (as a function of ).

However, there are many scenarios where the SC condition does not hold. Consider for example the following scenario of two firms competing for oil drilling rights, given in [Dasgupta and Maskin, 2000].

Example 2.

As before, each firm has a marginal cost for drilling, but this time each firm also has a fixed cost for drilling. Firm has a fixed cost of and a marginal cost of , while firm has a fixed cost of and a marginal cost of . Suppose oil is sold in the market at a price of . Firm performs a private test and discovers that the expected size of the oil reserve is units. This scenario gives rise the following valuation functions: , and . These valuations do not satisfy the SC condition. The increase in firm ’s valuation increases faster than firm ’s valuation as a function of .

As another example, consider the following scenario regarding two retail chains, each interested in renting some location for a shop.

Example 3.

Each of the two retail chains has conducted a survey to estimate the income level of the population in the area. Let

be the estimate obtained by retail chain (furthermore, assume that ). A good estimate for the income level in the area is the average of and . The valuations of the two retail chains for opening a shop at this location are functions of the income level in the area, since the demand for the goods is a function of the customer income level. Suppose that the first retail chain sells normal goods, whereas the second retail chain sells luxury goods. Let the corresponding valuations be (for normal goods) and (for luxury goods). These valuations do not satisfy the SC condition: as a function of , the rate of increase for is faster than the rate of increase for . (See Figure 1).

Figure 1: Valuations (retail chain 1, normal goods) and (retail chain 2, luxury goods) for opening a shop as a function of the income level. Income level is itself well-estimated as the average of the two signals, and .

It is well known that, without the single-crossing assumption, it is generally impossible to achieve full efficiency [Dasgupta and Maskin, 2000; Jehiel and Moldovanu, 2001]. Thus, the next step is to seek near-optimal guarantees. This is our approach; namely, we study settings with interdependent valuations through the lens of approximately-optimal efficiency.

Unfortunately, without single-crossing, it is not generally possible to give any non-trivial approximation guarantee to the social welfare (see Section 2.5). Thus, the obvious next question is to study the tradeoff between assumptions and the quality of the approximation. The two ends of this spectrum are (a) the single-crossing assumption and full efficiency, and (b) no assumptions on the valuation functions and no efficiency guarantees.

To study this tradeoff, we introduce the following parameterized version of the single-crossing condition: A valuation profile is said to be -single-crossing if for every agent , for any set of other players’ signals , and for every agent ,

This means that as buyer ’s signal changes, the change in any other agent’s value is at most times the change in agent ’s value. For example, the valuations described in Example 2 violate the SC condition, but they are -SC. Similarly, the valuations described in Example 3 are -SC. Can these scenarios, which deviate only slightly from SC, permit good approximation guarantees? We answer this question in the affirmative. In particular, we provide welfare guarantees that depend on the parameter . Our approximation guarantees are strong in the sense that they hold for any signal profile . (This is in contrast to approximation guarantees that are given in expectation over the signal profile).

Moreover, we are also interested in revenue guarantees under relaxed SC valuations. In [Chawla et al., 2014] it was shown that for SC and concave valuations, revenue maximization follows from welfare maximization. Here, concavity essentially means that buyers are more sensitive to a change in some agent’s signal when all of the buyers have lower signals. Like Chawla et al. [2014], we show that, for settings with -single-crossing and concave valuations, [approximate] revenue maximization follows from [approximate] welfare maximization.

1.1 Our Results

Before presenting our results, we present our approximation notion.

Our approximation notion

We say that the value of bidder is -approximated by bidder ’s value at profile , which we denote by , if . We say that an allocation gives a prior-free -approximation to welfare if, for every signal profile , the value of the bidder with the highest value is -approximated by the bidder being allocated to in by the mechanism. A prior-free mechanism gives an -approximation to the welfare without any assumption on prior distributions. The quality of the approximation, for our randomized mechanisms, is in expectation over the internal coin tosses of the mechanism and nothing else555The alternative to prior-free approximation is that the approximation only holds in expectation over some prior distribution; i.e., , where comes from a joint distribution with density ..

[h!] Setting Deterministic Upper Bound Prior-free Randomized Upper Bound Prior-free Deterministic Lower Bound Prior-free Randomized Lower Bound (even w/ Prior) Without Single Crossing -single-crossing arbitrary , c -single-crossing arbitrary , Concavity -single-crossing 2 bidders () -single-crossing 2 Signals ()

  • Choosing a bidder at random gives an approximation.

  • Section 2.5.

  • Section 3.

  • Via the Hypergraph-Coloring algorithm of Section 5.

  • The approximation — Section C.1.

  • Subsection 4.4.

  • For sufficiently large , implied by cell , lower bound example with 2 signals.

  • Implied by cell .

  • The approximation — Section 6. If we only have -concavity, for some , then becomes .

  • Lower bound in Figure 6 has concave valuations (with 2 bidders and 2 signals).

  • See Section 4.1.

  • A deterministic approximation is implied by cell .

  • Lower bound in Figure 6 (with 2 bidders and 2 signals).

  • Using the algorithm of Figure 4, Section 4.1.

  • A deterministic approximation is implied by cell .

  • Implied by cell , for sufficiently large .

  • For sufficiently large , see Subsection 4.3.

Table 1: Our Results for Social Welfare maximization. The randomized upper bounds are also universally truthful.

Our results

In Section 2.5, we show that without SC, no deterministic prior-free mechanism can achieve any guarantee for welfare. Moreover, as we show in Section 3, there exists a distribution on signals such that no (randomized) mechanism can do better than allocating the item to a random agent. Such a mechanism would result in no better than fraction of the optimal welfare.

We then study welfare approximation in settings where buyer valuations satisfy -SC. We also consider the impact of the number of possible signals per buyer, i.e., the size of the set of potential signals that a buyer may have. Some of our results depend on this parameter.

In Section 4, we identify two settings that admit a deterministic prior-free -approximation: (a) bidders, any signal space (Theorem 1), and (b) any number of bidders, each with a signal space of size (Theorem 2). This approximation is tight in several senses. First, it is provably impossible to obtain better than a -approximation in these settings. In fact, in the case of signal spaces of size , this impossibility holds even if one considers randomized, truthful-in-expectation mechanisms with a known prior distribution on the buyer signals. We also show that these are the most general settings that admit a -approximation. In particular, for bidders, one with signal space of size and two with signal spaces of size —no -approximation on the social welfare is possible (Proposition 3).

We also give results for general settings with bidders, each of which has one one of possible signals. We start by constructing a family of deterministic prior-free mechanisms that obtain a -approximation (Theorem 3). The mechanism imposes some (arbitrary) ordering over the bidders. In every iteration, the next bidder in is added, and a “tentative allocation” is determined. This includes making appropriate changes to the previous tentative allocation so as to preserve approximation guarantees and monotonicity (truthfulness) with respect to the newly added bidder.

The algorithm is described as computing the full allocation table for all possible signal profiles, which would take non-polynomial time. However, we show that computing the allocation for any profile of signals, on the fly, can be done in polytime ( time—Theorem 4).

We then show that a randomized version of this mechanism has much better welfare guarantees. In particular, if the valuations are concave, then the randomized mechanism obtained by imposing a random ordering over the agents gives a prior-free, universally truthful mechanism that gives a -approximation (Theorem 6). (This guarantee extends to for -concave valuations—a parameterized version of concavity). For general -SC valuations, this mechanism gives a -approximation (Theorem 5).

While the main focus of this paper is social welfare guarantees, our results have implications to revenue optimization. For concave valuations, we establish a black-box reduction from (approximate) welfare to (approximate) revenue, for every implementable allocation rule. To establish this reduction, we use similar ideas to ones used in [Chawla et al., 2014]. In particular, every -single-crossing setting with bidders and concave valuations admits a randomized, universally truthful mechanism that gives an -approximation (and -approximation for -concave valuations) (Theorem 8).

We refer the reader to Table 1, where we give an in depth description of the implications of our results for social welfare. The table shows how the various results relate to one another and gives references to where each upper and lower bound stems from in the organization of this paper.

2 Model and Preliminaries

We consider an auction setting where a single item is sold to agents with interdependent values (Milgrom and Weber [1982]). Each agent receives a single signal which is known only to agent . Let be a signal profile, let denote all signals but , and let denote the profile but where has been replaced with . The set of potential signals for bidder is a discrete signal space . Without loss of generality, assume ; for ease of exposition we may assume that for all .

Each agent also has a publicly known valuation function , which maps every signal profile of the agents to a real (non-negative) value. The valuation functions for all bidders are monotone non-decreasing in every signal for all .

The input to a mechanism is a vector of reported signals . Mechanisms are described by a pair , where is a set of allocation functions satisfying for all possible , and a set of payment functions . An allocation function maps every bid profile

to the probability that agent

gets allocated. A payment rule maps the reported bids to the expected payment from bidder . A bidder’s expected utility is quasilinear, given by where is the true signal profile of the agents.

2.1 Solution Concepts

We focus on the design of incentive-compatible and individually rational mechanisms. In the interdependent setting, we cannot hope for truth-telling to be a dominant strategy: one agent’s misreport could cause the auctioneer to overcharge a different agent. The strongest incentive-compatibility (IC) concept in this setting is thus that truth-telling is an ex-post Nash Equilibrium, or that it is in every agent ’s best interest to report his true signal given that all other agents reported the true signal profile :

Similarly, individual rationality (IR) cannot possibly hold if signals are corrupted, so the appropriate notion with respect to individual rationality is that of ex-post IR, i.e.,

Thus, in this paper, incentive-compatibility (IC) refers to truth-telling being an ex-post Nash and individual rationality (IR) refers to ex-post individually rational. We use the term truthful mechanism for mechanisms that are both; all the mechanisms we present are truthful.

We emphasize that the information state is the following: (a) agents know their own valuation function , and their own private signal ; (b) the auctioneer knows the valuation functions of the agents participating; and (c) signals are private and arbitrary. Except for bidder , no other bidder, nor the auctioneer, knows anything about .

An allocation is said to be implementable if there exist payment functions such that the mechanism is truthful.

We give both deterministic and randomized truthful mechanisms. Our randomized mechanisms are a distribution over a family of deterministic mechanisms, all of which are truthful (i.e., they are universally truthful [Nisan and Ronen, 1999; Dobzinski and Dughmi, 2009]).

All the mechanisms we present are prior-free; i.e., there is no assumption of an underlying prior over the agents’ signals; neither in the design of the mechanism, nor in the truthfulness notion, nor in the approximation guarantees. Every assertion holds for every signal profile .

We note that weaker solution concepts appear in the literature; i.e., truthful-in-expectation (vs. universally truthful), Bayesian truthful (vs. ex-post truthful), and interim IR (vs. ex-post IR). The reader is referred to [Roughgarden and Talgam-Cohen, 2016] for formal definitions. All of our positive results hold for our solution concept, which is analogous to dominant-strategy IC in the private value setting. Many of our impossibility results hold even with respect to weaker solution concepts.

2.2 Monotone Allocations

The following definition is key in characterizing which mechanisms are truthful.

Definition 1 (Monotonicity).

An allocation function is said to be monotone if for every , is monotone non-decreasing in the signal .

Similar to Myerson’s characterization for the independent private value setting, Roughgarden and Talgam-Cohen [2016] characterized the class of truthful mechanisms, as follows.

Proposition 1.

Monotonicity is a necessary and sufficient condition for allocation functions to be implementable, i.e., there exist payment functions such that the mechanism is truthful. Moreover, an analogue of Myerson’s payment identity holds, so the payment is uniquely determined by the allocation function.

It follows that constructing a truthful mechanism is equivalent to constructing a monotone allocation function.

An allocation function is called deterministic if for all and all . For a deterministic mechanism, we use the notation

Given a deterministic monotone allocation function and signals for agents , , the critical signal for is as follows: if there exists some such that then set , otherwise there is no critical signal for .

For deterministic truthful mechanisms, the payment identity of Roughgarden and Talgam-Cohen [2016] implies the following.

Proposition 2.

Let agent be the allocated winner at bid profile in a deterministic truthful mechanism. Then his payment is his value at the critical bid, i.e., .

2.3 Single-Crossing

A single-crossing condition captures the idea that bidder ’s signal has a greater effect on bidder ’s value than on any other bidder’s value. Formally:

Definition 2 (Single-Crossing).

A valuation profile is said to satisfy the single-crossing condition if for every agent , for any set of other players’ signals , and for every agent ,

In the context of discrete signal spaces, for , define .

Whenever single-crossing holds, full efficiency can be achieved: once an agent has the highest value, by the single-crossing condition, his value continues to be the highest of all bidders as his signal increases. Therefore, allocating to the bidder with the highest value defines a monotone allocation rule, and therefore, according to Proposition 1, it is implementable. The payment of that agent is then just his value at his critical signal. Note also that this mechanism is deterministic and prior-free. This is precisely the generalized VCG mechanism used in [Maskin, 1992].

Unfortunately, according to Proposition 1, monotonicity of the allocation rule is also necessary. Hence, without single-crossing, it is impossible to have a truthful mechanism that maximizes welfare.

2.4 Approximation

While full efficiency is unattainable without single-crossing, one might hope for approximate efficiency. We say that the value of bidder is -approximated by bidder ’s value at profile , which we denote by , if . We say that an allocation gives a prior-free -approximation to welfare if

A prior-free mechanism gives an -approximation to the welfare without any assumption on prior distributions. The quality of the approximation (welfare, revenue), for our randomized mechanisms, is in expectation over the internal coin tosses of the mechanism and nothing else666The alternative to prior-free approximation is that the approximation only holds in expectation over some prior distribution; i.e., , where comes from a joint distribution with density ..

The generalized VCG mechanism (that gives optimal efficiency under single-crossing) is a prior-free mechanism.

2.5 Impossibility Results for Settings Without Single-Crossing

Here we show that no truthful, prior-free, and deterministic mechanism can obtain any bounded approximation ratio when the valuations do not satisfy single-crossing.

Example: [Impossibility for deterministic prior-free mechanisms] Consider a scenario with two bidders (bidder and bidder ), where and , and the following valuation functions:

It is easy to see that does not satisfy single-crossing since when increases, does not increase but increases by , making go from being times greater than to being times smaller than it.

We claim that, for these valuations, no truthful, deterministic, and prior-free mechanism has an approximation ratio better than . To see this, consider the signal profile . To get a better than -approximation for this profile, bidder must win the item. Truthfulness requires the allocation to be monotone in each bidder’s signal, hence bidder 1 must also win at report , which results in an allocation that is a factor of off from the optimal allocation. Since is arbitrary, the approximation ratio is arbitrarily bad.

In Section 3 we show that without single-crossing, even when we consider randomized mechanisms, no truthful randomized mechanism can achieve a better approximation than the simple mechanism that allocates the item to a random bidder, disregarding the reported signals altogether. This impossibility holds even if signals come from a known joint distribution.

2.6 Approximate Single-Crossing and Its Implications

The impossibility results motivate the following relaxed notion of single-crossing.

Definition 3 (-Single-Crossing).

A valuation profile is said to satisfy the -single-crossing condition if for every agent , for any set of other players’ signals , and for every agent ,

We now explore some useful properties of -single-crossing, depicted in Figure 3.

Lemma 1.

For any profile and , if and the valuations satisfy -single-crossing, then for any such that , .

Proof.

By assumption, . By -single-crossing, . In addition, , so . Then adding the first two inequalities and using this fact gives the desired result: . ∎

Corollary 1.

For valuations that satisfy -single-crossing, the following are implied by Lemma 1:

  • The contrapositive: For any , if , then at any profile for , . See Figure 3.

  • For any , if , then at any profile for , .

Figure 2: Left: An illustration of Lemma 1. Right: An illustration of the contrapositive, mentioned in Corollary 1.
Figure 3: An illustration of a propagation conflict.

Remark.

We have defined our notion of single-crossing as a relaxation of the single-crossing definition in [Roughgarden and Talgam-Cohen, 2016]. If a set of valuation functions obey the [Roughgarden and Talgam-Cohen, 2016] definition then they also obey the various other definitions of single-crossing used in [Milgrom and Weber, 1982; d’Aspremont and Gérard-Varet, 1982; Maskin, 1992; Ausubel et al., 1999; Dasgupta and Maskin, 2000; Bergemann et al., 2009; Chawla et al., 2014; Che et al., 2015; Li, 2016], but not vice versa. A discussion regarding alternative definitions of single-crossing and the applicability of our results under these definitions appears in Appendix A.

2.7 Concave valuations

Definition 4 (Concave Valuations).

Valuations are said to be concave if for all bidders and for any and such that for all , it holds that

That is, when anyone aside from ’s signals are lower, then every bidder is more sensitive to the change in ’s signal as when those bidders (aside from ) have higher signals.

Remark[-concave valuations]: We also consider a parameterized version of concavity. Valuations are said to be -concave if the inequality above is replaced by

That is, every bidder is at least more sensitive to a change in ’s signal when bidders’ signals are lower.

2.8 Monotonicity and propagation

We prove that several mechanisms are truthful and prior-free approximations to social welfare. In each proof, we have two components: monotonicity and approximation. To demonstrate that the approximation is prior-free, the approximation component is established for every profile . To demonstrate monotonicity in randomized mechanisms, we need to prove that monotonicity holds for any allocation that might be realized after the random choices of the mechanism.

In the construction of monotone allocations, we use the term an allocation propagation. Suppose we assign . Then, to ensure monotonicity of ’s allocation as his signal increases, we propagate the allocation to to all profiles . That is, the propagation operation is the assignment of for all whenever .

A propagation conflict, as depicted in Figure 3, refers to the event where we allocate in a way such that multiple propagations cause multiple conflicting winners at some signal profile. Let and . If we assign , this must propagate such that . If we also assign , this allocation must propagate such that . Then at , we have a propagation conflict because both and must be the winner to satisfy monotonicity, which is not possible.

Having propagation in our algorithm ensures that the allocation is monotone, so instead of verifying monotonicity of the allocation, we only need to verify that no propagation conflicts occur.

3 Impossibility result for randomized mechanisms without single-crossing

Consider the case where every bidder ’s signal space is , and each agent has a valuation ; that is, the bidder has a value if and only if every other agent has signal . For every bidder , for some small ,

The optimal expected welfare is whenever at least bidders have a signal. This happens with probability . Therefore,

(1)

Consider any truthful mechanism at profile . At this profile, the mechanism gets bidder ’s value in welfare with probability that he is allocated, , and otherwise gets zero since no other bidder has non-zero value. By monotonicity, for every , we have that , and by feasibility, . Under any other profile (where at least two signals are ), all agents have zero value, so welfare is zero. The expected welfare of any truthful mechanism is thus bounded by

Welfare (2)

Combining (1) with (2), we get that the approximation ratio of any monotone mechanism is which can be made arbitrarily close to ; this is the same as the welfare attained by just allocating to a random bidder.

These last examples motivate the following notion of approximate single-crossing.

4 -Approximations for Settings with 2 Bidders or 2 Signals

In this section we give deterministic truthful mechanisms for two special cases.

4.1 A -Approximation Mechanism for Agents

A very simple idea gives a -approximation for two bidders, independent of the sizes of their signal spaces. Start at the signal profile that is the origin and allocate to the bidder who has the largest value at this signal profile, say bidder . Propagate the allocation as ’s signal increases. Then move to the profile and repeat this: allocate to the bidder who has the highest value here, propagate as that bidder’s signal increases, and then move to the profile where the losing bidder’s signal has increased by one. This algorithm is illustrated in Figure 4.

Two-bidder coloring:

  • Let and .

  • While or :

    • Let and let .

    • For all , set .

    • Let .

Figure 4: An illustration of the two-bidder coloring algorithm. (a) At , bidder 1 has the highest value. Allocate to 1 and propagate this to all profiles for . (b) Move to , where bidder 2 has the highest value. Allocate to 2 and propagate this allocation to all profiles where . (c) Move to where is highest. Allocate to and propagate.
Theorem 1.

When we have two bidders whose valuations satisfy -single-crossing, the allocation function given from the two-bidder coloring defines a truthful, deterministic, prior-free mechanism that guarantees a -approximation to social welfare.

Proof.

(Monotonicity.) By construction, whenever we set , we set for every . Similarly, if , then for every .

(Approximation.) Suppose for any profile that, without loss of generality, the item is allocated to bidder 1. The algorithm sets either because , in which case, the highest valued bidder is the winner, or because it was true that for some . By Lemma 1, then . (A symmetric argument proves the approximation for .) ∎

As we will show by the example in Figure 6, the bound of is tight. Observe that these valuations are also concave.

4.2 A -Approximation to Welfare for Settings with 2 Signals

In this section, we consider the case where the size of the signal space for each bidder is at most : a bidder’s signal is either low () or high (). We will denote by the set of high-signal-bidders at , that is, .

For monotonicity in an allocation, if we allocate to bidder at , then for all where , we must propagate the allocation, allocating to bidder at these profiles as well. The 2-signal case is special because (1) an allocation can only propagate to at most one profile, from to = 1, and (2) if and we allocate to , then no propagation is necessary, so bidders with high signals are in a sense special. We can capitalize on these properties to achieve a -approximation, independent of the number of bidders .

The mechanism is simple: consider profiles in order of the number of high signals (equivalently, in increasing hamming distance from the origin). For any profile in which the allocation has not already been determined, if the high-signal bidder with the largest value -approximates the bidder with the largest value , then allocate to . Otherwise, allocate to and propagate.

High-if-possible:

  • For all profiles increasing in , if is undefined:

    • Let and .

    • If , set .

    • Otherwise, set and propagate to .

Theorem 2.

When we have bidders where the size of the signal space is at most 2 and the valuations satisfy -single-crossing, the allocation from high-if-possible gives a monotone, deterministic, and prior-free -approximation to social welfare.

Figure 5: An example with 2 bidders and 2 signals for each, where no deterministic mechanism can improve upon a approximation.
Figure 6: An argument for why the algorithm results in no conflicts via the common ancestor of any two nodes that might conflict.
Proof.

(Approximation.) At every profile , the allocated bidder -approximates the highest valued bidder. If the allocation at is to bidder and is determined by propagation, by the allocation rule, it must be that at that was the highest valued bidder, hence for all bidders . By Lemma 1, since only increases from to , then , and thus the maximum value is -approximated by ’s value. Otherwise, by definition of the allocation rule, we allocate to a bidder who -approximates the maximum valued bidder at .

(Monotonicity.) Our propagation step ensures monotonicity; we just need to verify that we do not cause any propagation conflicts. The argument as to why not follows, and is illustrated in in Figure 6. Suppose that at we allocate to with and propagate ’s allocation to . To have a propagation conflict at due to propagation of both and some other winner , we observe that a propagation of can only come from allocation at a profile where to a cell where : there could only be a conflict at if . By definition of the allocation rule, we know that , or we would have allocated to at . Hence, by Corollary 1, , and thus . By Lemma 1, this implies that . Hence, there exists a high-signal bidder whose value -approximates ’s value, so the algorithm would never allocate to at this profile. Thus, there are no propagation conflicts. ∎

We next show that the -approximation is tight. In the example depicted in Figure 6, the valuations satisfy -single-crossing, and any allocation that achieves better than a -approximation must allocate to bidder 1 at profile and to bidder 2 at profile . However, for monotonicity, we need to propagate these allocations, which would cause a propagation conflict at ; hence no monotone allocation that achieves better than a -approximation is possible. In fact, in Subsection 4.3, we show that the is tight even if we consider random and truthful-in-expectation mechanisms that are given the prior distribution over signals.

Further, the approximations for this -signal case and the -bidder case of Section 4.1 are tight in another sense, as proven in Subsection 4.4.

4.3 Randomized lower bound with -single-crossing

In Section 4.2, we presented a deterministic, prior-free, and universally truthful mechanism guaranteeing a -approximation for the case where each bidder has two signals. In this section, we show this is essentially tight, even if one considers randomized, truthful in expectation mechanism that are given the prior. This essentially shows that the algorithm in Section 4.2 is optimal for that case.

Consider the case where there are bidders, and for , agent ’s value is defined as follows:

It is easy to see the valuations are -single-crossing since whenever ’s signal increases from to , his value increases by , while every other bidder’s value increases by at most . Consider the following (known) correlated distribution over signal profile. The probability that is an arbitrarily small , and the probability for any vector profile where for exactly one coordinate and for every other coordinate is . It is immediate that

Consider any truthful mechanism at profile . At this profile, the mechanism gets bidder ’s value in welfare with probability that he is allocated, , and otherwise gets . By monotonicity, for every , we have that , and by feasibility, . We have that

Welfare

Therefore, , which as tends to and as is arbitrarily small, tends to , hence we get a lower bound that tends to as tends to .

4.4 No -approximation exists for more than