Expressiveness and Robustness of FirstPrice Position Auctions^{†}^{†}thanks: We have benefitted greatly from discussions with Jason Hartline, Robert Kleinberg, and Éva Tardos.
Abstract
Since economic mechanisms are often applied to very different instances of the same problem, it is desirable to identify mechanisms that work well in a wide range of circumstances. We pursue this goal for a position auction setting and specifically seek mechanisms that guarantee good outcomes under both complete and incomplete information. A variant of the generalized firstprice mechanism with multidimensional bids turns out to be the only standard mechanism able to achieve this goal, even when types are onedimensional. The fact that expressiveness beyond the type space is both necessary and sufficient for this kind of robustness provides an interesting counterpoint to previous work on position auctions that has highlighted the benefits of simplicity. From a technical perspective our results are interesting because they establish equilibrium existence for a multidimensional bid space, where standard techniques break down. The structure of the equilibrium bids moreover provides an intuitive explanation for why firstprice payments may be able to support equilibria in a wider range of circumstances than secondprice payments.
1 Introduction
Economic mechanisms are often applied to very different instances of the same problem. It is therefore desirable to find mechanisms that work well in a wide range of circumstances, and specifically do not require any knowledge of agents’ preferences on the part of the designer. This goal has been formulated many times and forms the core of the Wilson doctrine [34] and of the agenda of robust mechanism design [3].
We pursue this goal for a position auction setting with onedimensional types, and specifically seek mechanisms that guarantee good outcomes under both complete and incomplete information. Each of positions is to be assigned to exactly one of agents, and the value of agent for position can be written as , where and . In the complete information case is common knowledge among the agents. In the incomplete information case the components of are independent and identically distributed according to a continuous distribution with bounded support that is common knowledge among the agents, and is known to agent . In both cases, is common knowledge among the agents. A prime example of this setting can be found in the context of sponsored search, where agents correspond to advertisers, positions correspond to slots in which advertisements can be displayed, denotes the fraction of cases where an advertisement in position leads to a conversion, and denotes the value agent has for a conversion.
The goal of the designer is twofold: to provide the best possible service to the agents by allocating positions in a way that maximizes social welfare, i.e., the sum of valuations for the allocated positions; and to maximize revenue subject to this constraint. While the agents agree with the former goal, their interests are diametrically opposed to that of the designer with regard to the latter. From the point of view of the designer, a good mechanism must therefore guarantee existence of an efficient equilibrium and achieve high revenue in every efficient equilibrium. An appropriate revenue benchmark for efficient equilibria is provided by the truthful equilibrium of the wellknown VickreyClarkesGroves (VCG) mechanism [24].
We arrive at the following question:
Does there exist a single mechanism that possesses an efficient equilibrium under both complete and incomplete information, and recovers the truthful VCG revenue in every efficient equilibrium?
To answer this question we consider the three mechanisms commonly used in for position auctions, the VCG mechanism, the generalized firstprice (GFP) mechanism, and the generalized secondprice (GSP) mechanism.^{1}^{1}1Google and Microsoft use the GSP mechanism, Facebook the VCG mechanism. The GFP mechanism was used by Overture, the first company to provide a successful sponsored search service. The variants of these three mechanisms we consider all assign the positions from top to bottom to an agent with maximum bid among those not assigned one of the higher positions.^{2}^{2}2We use this greedy allocation rule rather than one that selects an efficient allocation relative to the bids because it simplifies the analysis and thus enables our main positive result. All negative results also hold for the efficient allocation rule, and the two allocation rules obviously agree in any efficient equilibrium. The VCG mechanism charges each agent the externality it imposes on the other agents, the GFP mechanism charges the agent’s bid on the position it is allocated, and the GSP mechanism charges the nexthighest bid on that position. For each mechanism we moreover distinguish two variants: an expressive variant in which agent submits a bid for each position , and a simplified variant in which agent specifies a single bid and this bid is multiplied by to obtain bids for the different positions. The vector is part of the mechanism, so it is common knowledge among the agents and the designer and may or may not be identical to .
Our contribution
It turns out that most candidate mechanisms are disqualified by prior work. The expressive VCG and GSP mechanisms have an efficient complete information equilibrium with revenue zero for all possible valuations of the agents [29].^{3}^{3}3This result requires that the agents can bid arbitrary nonnegative numbers. It can be circumvented by forcing the agents to submit nonincreasing bids. But then there are still efficient equilibria with revenue arbitrarily smaller than the truthful VCG revenue [14]. For the simplified variants of these mechanisms the situation is somewhat better, and this has in fact been used as an argument in favor of simplification [29]. However, revenue in an efficient completeinformation equilibrium may still be arbitrarily small compared to the truthful VCG revenue [14], and the simplified GSP mechanism may not have an equilibrium at all when information is incomplete [22]. The simplified GFP mechanism, on the other hand, has a unique equilibrium under incomplete information [22] but may not have an equilibrium under complete information [16].
This only leaves the expressive GFP mechanism, and we show that it indeed possesses the desired robustness property: an efficient equilibrium under both complete and incomplete information, and the truthful VCG revenue in every efficient equilibrium. While good outcomes under either complete or incomplete information can be obtained with a simplified mechanism, expressiveness thus turns out to be both necessary and sufficient for robustness. This provides an interesting counterpoint to previous work on position auctions that has highlighted the benefits of simplicity [29, 14]. An additional advantage of the expressive GFP mechanism is that it is independent of . It can therefore be used in settings where the designer is uncertain about the exact value of , and our results extend to such settings.
Our analysis of the complete information case is similar to the classic analysis of Bernheim and Whinston [5] that links equilibria of firstprice auctions to the core, and to more recent approaches that also make this connection [11, 23]. The common feature is the use of what Milgrom [28] has called targetprofit strategies. Specifically, we show that having each agent bid its value for position minus its truthful VCG utility , or zero if this is negative, yields an efficient equilibrium. Notable differences concern our use of a greedy rather than efficient allocation rule, and the fact that we show revenue in every efficient equilibrium to be at least the truthful VCG revenue. Unlike prior work we also explicitly handle ties in choosing an allocation.
As types are onedimensional, our incomplete information analysis can use Myerson’s classic characterization result [30] to identify equilibrium candidates. The standard technique to verify that a particular candidate is an equilibrium involves integrating the derivative of an agent’s utility, as a function of both valuation and bid, along a path between two bids. This technique breaks down in our setting, where the bid space has higher dimension than the valuation space and the utility function may not be defined everywhere on the path. We overcome these difficulties by performing an induction from the last position to the first, and showing that the conjectured equilibrium bid on position is optimal for agent given that the other agents bid according to the conjectured equilibrium, and agent bids according to the conjectured equilibrium on positions to . We believe that similar techniques can be used to show equilibrium existence in more general settings, including settings with multidimensional types.
Each step of the induction considers only one dimension of the bid space and can use the standard technique, but identifying the equilibrium bids and deriving the utility function is a nontrivial task. Myerson’s theorem only provides a necessary condition for bids that lead to an efficient equilibrium, namely that payments in expectation equal the truthful VCG payments. Since the bid space is multidimensional, many different bids satisfy this condition. In the eventual equilibrium, the bid of agent on position equals its expected truthful VCG payment conditioned on being allocated position . These bids again have a natural interpretation in terms of targetprofit strategies and also provide an intuitive explanation for why firstprice payment rules may be able to support equilibria in a wider range than secondprice payment rules: the expected truthful VCG payment of an agent subject to allocation of a given position depends on the agent’s valuation and on the distribution from which the valuations of the other agents are drawn, which is exactly the information available to the agent.
Related work
The design of more expressive mechanisms for specific applications is an important topic of contemporary mechanism design [e.g., 1, 19, 10, 15, 13, 20, 21]. In addition, it has been argued more abstractly that the expressiveness of a mechanism is positively correlated with the quality of the outcomes it is able to support. Benisch et al. [2] showed that for combinatorial auctions, the maximum social welfare over all outcomes of a mechanism strictly increases with expressiveness, for a particular measure of expressiveness based on notions from computational learning theory. Implicit in this result is the intuition that more expressiveness is generally desirable, as it allows a mechanism to achieve a more efficient outcome in more instances of the problem.
The classic analysis of position auctions is due to Varian [33] and Edelman et al. [16]. Followup work has emphasized the benefits of simplicity in this context. Milgrom [29] and Dütting et al. [14] considered a complete information setting and showed that simplification can eliminate zerorevenue equilibria without introducing new, and potentially undesirable, equilibria. The authors also pointed out certain advantages of the GSP mechanism over the VCG mechanism in this regard. Gomes and Sweeney [22] and Chawla and Hartline [8] showed that under complete information the GSP mechanism may fail to have an efficient equilibrium, whereas the GFP mechanism always possesses a unique equilibrium, which is efficient and yields the truthful VCG revenue. Paes Leme and Tardos [31], Lucier and Paes Leme [25], Caragiannis et al. [7], and Syrgkanis and Tardos [32] showed that the GSP mechanism has a small constant price of anarchy under both complete and incomplete information.^{4}^{4}4The price of anarchy compares the minimum social welfare in any equilibrium to the maximum social welfare of any outcome. That greedy algorithms can achieve a small price of anarchy, potentially smaller than that of an efficiently computable outcome, was previously highlighted by Gairing [18] in the context of covering games. Lucier et al. [26] established lower bounds on the revenue of the GSP mechanism: for complete information it can be arbitrarily small compared to the truthful VCG revenue, for incomplete information it always is a constant fraction of the latter.
Our work also has connections to the literature on nonparametric BayesNash implementation, robust full implementation, and priorfree approximation. Nonparametric BayesNash implementation shows that an uninformed designer can implement essentially the same outcomes in equilibrium as an informed designer [e.g., 27]. Robust full implementation seeks to obtain mechanisms that implement a desired outcome in every equilibrium and for any prior the agents may have [e.g., 4]. Priorfree approximation seeks to approximate a desired outcome for any prior [e.g., 12].
To the best of our knowledge, the study of mechanisms for position auctions that admit efficient equilibria and yield high revenue in every efficient equilibrium under both complete and incomplete information, and the use of additional expressiveness to achieve this goal, are both novel.
2 Preliminaries
We study a setting with a set of positions ordered by quality and a set of agents with unit demand and onedimensional valuations for the positions. More formally, write for the set of dimensional vectors whose entries are positive and nonincreasing. For , let be the onedimensional subspace of spanned by . Agent ’s valuation can then be represented by a vector in this subspace, such that is the agent’s value for position . Our goal is to assign the positions to agents in order to maximize total value. We refer to an assignment of agents to positions that achieve this as efficient. Because the base of the subspace is the same for all agents, this can be achieved by allocating positions in decreasing order of . We assume that is common knowledge among the agents.
We compare two kinds of auctions. An expressive auction solicits a vector of bids from each agent , where is interpreted as agent ’s bid on slot . A simplified auction^{5}^{5}5We refer to these mechanism as simplified as they can be viewed as resulting from the expressive mechanism by restricting the dimensional bid space to a dimensional subspace. is parameterized by vector and solicits a singledimensional bid from each agent . The singledimensional bid is extended to a dimensional bid by multiplying it with . Agent ’s bid on slot is thus
More specifically, we consider simplified and expressive variants of the generalized firstprice (GFP), generalized secondprice (GSP), and the VickreyClarkeGroves (VCG) auctions. We denote these auctions by , , and and by GFP, GSP, and VCG. We assume that all three mechanism assign the items greedily. That is, starting from the first position and proceeding to the last position, they assign the current position to the agent with the highest bid that has not yet received a position. We focus on greedy winner determination algorithms because it simplifies the equilibrium analysis, and also because it is consistent with the current practice in sponsored search. The payment rules are defined identically for the simplified and expressive variants of each auction. In the generalized firstprice auctions, the payment of agent assigned position is equal to the bid value associated with position . In the generalized secondprice auctions, the payment of agent for position is equal to the nextlower bid for that position. In the VCG auctions, agent assigned position is charged an amount equal to the total loss in value of all other agents, according to their bids, caused by assigning position to agent .
We make the usual assumption of quasilinear utilities, such that the utility of agent with value , in a given auction and for a given bid profile , is equal to its valuation for the position it is assigned minus its payment for that position. To be able to reason about the strategic behavior of agents we need to specify what the agents know about each others’ valuations. In the complete information setting values are common knowledge among the agents. A vector of bids is a Nash equilibrium of a given mechanism if no agent has an incentive to change its bid assuming that the other agents don’t change their bids, i.e., if for every and every ,
In incomplete information environments, values are drawn independently from a distribution supported on for some finite . Distribution is assumed to be common knowledge among the agents. In this setting, a vector of bidding functions is a BayesNash equilibrium of a given auction if no agent has an incentive to change its bidding function assuming that the other agents don’t change their bidding functions and values of the other agents are drawn from , i.e., if for every , every , and every bidding function ,
Because our environment is onedimensional we can appeal to Myerson’s characterization of the expected payments in a BayesNash equilibrium.
Theorem 1 (Myerson [30]).
Consider a position auction, and assume that agents use bidding functions such that agent with valuation is assigned position with probability . Then the bidding functions are a BayesNash equilibrium of the auction only if, for every agent ,

the expected allocation is nondecreasing in

the expected payment is
where .
Since an efficient allocation satisfies monotonicity, we have the following corollary.
Corollary 1.
In an efficient BayesNash equilibrium of any position auction, the expected payment of every agent is equal, for every value , to the expected payment of the agent in the truthful equilibrium of the expressive VCG auction.
3 Complete Information Analysis
We begin our analysis by reviewing the properties of the expressive GFP mechanism in settings with complete information. We show that expressive GFP always has a Nash equilibrium, that all its Nash equilibria are efficient, and that payments in every Nash equilibrium are at least the truthful VCG payments. The proof is given in Appendix A.
Proposition 1.
Assume that valuations are taken from . Then,

the expressive GFP mechanism has an efficient Nash equilibrium with payments equal to the truthful VCG payments,

every Nash equilibrium of the expressive GFP mechanism is efficient, and

the payments in every Nash equilibrium of the expressive GFP mechanism are at least the truthful VCG payments.
4 Incomplete Information Analysis
Next we consider environments with incomplete information and show our main result, that the expressive GFP mechanism always has an efficient equilibrium that yields the truthful VCG revenue.
Theorem 2.
Assume that valuations are drawn independently from a continuous distribution on with bounded support. Then the expressive GFP mechanism has an efficient BayesNash equilibrium with the same payments as the truthful equilibrium of the VCG auction.
We prove this result by constructing a bidding function and showing by induction that an agent maximizes its utility by bidding according to assuming that all other agents bid according to as well. To this end, we define in Section 4.1 a function for each position that maps a valuation to the expected truthful VCG payment an agent with valuation would face if it was allocated position . The equilibrium bidding function will then be given by . We will say that an agent with valuation bids truthfully on position (according to ) if he bids , and that he bids truthfully if he bids truthfully on all positions. The property we show by induction is that independently of the bids on positions and assuming truthful bids on positions , it is optimal to bid truthfully on position . For this we apply the usual technique and integrate the derivative of the utility function from the truthful bid on position to a conjectured beneficial deviation on position to derive a contradiction.
Denote by the expected utility of an agent with valuation who bids on position while all other agents bid truthfully. The proof of Theorem 2 uses the following lemmata, which we prove in Sections 4.2 and 4.3.
Lemma 1.
Fix a particular agent. Assume that all other agents bid truthfully and that the agent bids truthfully on positions . Then the derivative in the bid on position of the agent’s expected utility vanishes at the truthful bid, i.e.,
Lemma 2.
Fix a particular agent. Assume that all other agents bid truthfully and that the agent bids truthfully on positions . Then, the derivative in the valuation of the derivative in the bid on position of the agent’s expected utility is nonnegative, i.e.,
Proof of Theorem 2.
Fix a particular agent and assume that all other agents bid truthfully. Suppose that we have established the claim for positions , and that we want to establish it for position . The claim trivially holds for , so we know from the induction hypothesis that
To show that the claim holds for position , assume for contradiction that there exists such that
First assume that . Then,
where the inequality and the last equality respectively hold by Lemma 2 and Lemma 1. This is a contradiction. It is important to note here that when all other agents bid according to , it is without loss of generality to consider only bids where is in the support of , because any other bid will be dominated by a bid of this type.
If we can proceed analogously to show that the deviation is not beneficial. ∎
4.1 Truthful VCG Payments and Allocation Probabilities
We begin by formally defining the positionspecific bidding functions and computing their derivative in the valuation. We then derive a recursive formulation of the allocation probabilities, which will be used in the proofs of Lemma 1 and Lemma 2. Bid equals the truthful VCG payment for position given valuation and conditioned on allocation of position . This quantity is equal to the sum over the differences multiplied by the expected value of the highest valuation among all agents conditioned on being the highest valuation and assuming that valuations are drawn independently from distribution . Formulaically,
Using that and defining we have that
Using that we obtain
(1) 
Denote by the probability that an agent is assigned position against opposing agents if he reports a valuation vector . Then can be written recursively as
(2)  
The intuition behind this formulation is that the agent is assigned position if of the opposing agents have valuations smaller than and the agent is not assigned one of the positions against the remaining agents. An important observation at this point is that does not depend on for .
4.2 Proof of Lemma 1
To prove the lemma, we write the expected utility that agent can achieve with a report given value as a sum of the contributions of position . We then group these contributions into those of positions , those of positions and , and those of positions , and argue for each group that their derivative in vanishes at .
For the contribution of positions this is rather straightforward, as neither the allocation probability , nor the utility subject to allocation, depends on . Hence the derivative in is zero everywhere, and in particular at .
To prove the claim for , we first apply the recursive formulation of the allocation probabilities to compute the derivatives in of and . We then observe that the derivative of vanishes at if and only if a certain differential equation involving the bids and is satisfied. Finally, we use the formulas for the truthful VCG payments conditioned on allocation and their derivatives to show that this differential equation is satisfied.
Lemma 3.
Fix a particular agent. Assume that all other agents bid truthfully and that the agent bids truthfully on positions . Then,
Proof.
Now consider the contribution of position . By applying (2) to ,
By pulling out of the sum and applying (2) to it, we obtain
and thus
We conclude that the derivative in of the contribution from positions and vanishes at if and only if
Using to simplify and rearranging leads to the following differential equation:
We first observe that the parts of and cancel and . This is the case because for the part of is equal to
It remains to show that times the part of minus the part of is equal to the derivative in of at . Formulaically, the former can be expressed as
(3)  
We prove the identity by showing that for all and , the terms in (3) are identical to the corresponding terms in (1).
Next we consider the contribution from positions .
Lemma 4.
Fix a particular agent. Assume that all other agents bid truthfully and that the agent bids truthfully on positions . Then,
Note that for position the contribution only depends on through the allocation probability . It therefore suffices to show that the derivative in of vanishes at . We establish this claim by means of two auxiliary lemmata, which again exploit the recursive formulation of the allocation probabilities. The proofs are given in Appendices B and C.
Lemma 5.
Fix a particular agent. Assume that all other agents bid truthfully and that the agent bids truthfully on positions . Then, for all ,
Lemma 6.
Fix a particular agent. Assume that all other agents bid truthfully and that the agent bids truthfully on positions . Then, for all and such that ,
4.3 Proof of Lemma 2
We now turn to Lemma 2, and begin by recalling the results for the onedimensional case. In this case the expected utility for report given value is equal to
which for truthful report simplifies to
(4) 
We will use this formula below to express the expected utility from positions for which both agent and the other agents report their valuations truthfully.
To compute the derivative in of the expected utility we first observe that the contribution is independent from for , and thus
For the contribution from position ,
For the contributions from positions we use (4) to obtain
Taking the derivative in yields
The final step now is to argue that this expression is nonnegative. That the beta fraction won increases in the report on position holding everything else fixed follows by an expost argument. If the agent was allocated a position then changing his reported valuation for position has no effect and he will still be allocated position . If the agent was allocated position , he will still be allocated this position for higher . If the agent was allocated a position or no position at all, then by increasing the reported valuation for position he will either be allocated the same position as before or position , which means that the beta fraction won will increase weakly.
5 Conclusion and Future Work
In this paper we analyzed position auctions through the lens of robustness. We asked whether there exists a single mechanism that works well under complete and incomplete information settings. Specifically, we were looking to identify a mechanism that achieves the truthful VCG revenue in every efficient equilibrium. By recalling results from prior work we were able to exclude both simplified and expressive variants of the VCG and the GSP mechanism as well as simplified variants of the GFP mechanism. We then showed that an expressive GFP mechanism indeed achieves the desired property.
Our work has a clear message: If the goal is robustness against uncertainty about the information agents have about one another, then expressiveness beyond the type space is both necessary and sufficient. It also provides a nice counterpoint to recent work on position auctions which has highlighted the benefits of simplicity.
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Appendix A Proof of Proposition 1
Proof of Part 1
Assume that the agents are ordered by decreasing value, i.e., that . Then in an efficient assignment agent is assigned position , for . Denote by the truthful VCG utility for agent and denote by the truthful VCG payment for position . Then for and for We claim that the bid profile with
for and is an equilibrium of GFP that is efficient and yields the truthful VCG payments.
With this bid profile, an efficient allocation assigns position to agent at price