Facebook and the VCG Auction

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this means that bi= vi − x v b_iⁱ₁

(

)

x⁻ − ⁺ for all i = 2, ..., S); (iv) the bid of the top bidder is not uniquely pinned down, the only restriction being that its value exceeds that of the next bid.

To illustrate this competitive equilibrium, as well as other points in the subsequent discussion, we will repeatedly refer to the following example (the example is taken from Decarolis, Goldmanis and Penta, 2017):

Example 1. Consider an auction with four slots and five bidders, with the following valuations: v1 = 5, v2 = 4, v3 = 3, v4 = 2 and v5 = 1. The CTRs for the four positions are the following: x1 = 20, x2 = 10, x3 = 5, x4 = 2. In this case, the competitive equilibrium benchmark in the GSP auction is as follows:

b5 = 1, b4 = 1.6, b3 = 2.3 and b2 = 3.15. The highest bid b1 > b2 is not uniquely determined, but it does not affect the revenues because it doesn’t affect the payment of the highest bidder (it only determines the fact that he gets the highest slot). In this example, the total revenues are 96, and the resulting allocation is clearly efficient.

In the discussion above we intentionally disregarded a feature that was prominently pushed through by Google when it launched its GSP model: quality scores. The main insight is that some advertisers might value appearing on keywords that are a poor match for their products with the logic of creating a potential “lead” (i.e., building a name recognition that might generate future sales) at a very low price (a click on their link will be unlikely). This, however, would hurt the search engine both in the short run, through the low click- through-rate, and in the long run, as consumers using the search engine might find particularly annoying to be exposed to advertisements unrelated to their queries. To solve these problems, Google’s version of the GSP ranks advertisers not only by their bids but by the product of their bids and a quality score. The latter is a function of past click behavior and, like the algorithm for Google’s organic search results, assigns more weight to advertisers with a greater likelihood of being clicked. The mechanics of the auction with quality scores is nearly identical to what we illustrated above, but with a more involved notation. For that extension we therefore defer to our more technical study, Decarolis, Goldmanis and Penta (2017).

auctions was complete. These display ad auctions are different from those of the search engines we discussed so far. That is because these auctions are not generated by keywords and because they raise specific challenges to integrate ads within Facebook’s organic content. But these technicalities aside, they boil down to the same kind of economic problem we have been discussing all along:

a multi-unit auction problem.

Before John Hegeman, an economics MA graduate from Stanford, took the role of Facebook’s chief economist, the (multi-unit) VCG had had a limited impact outside of academia. Perhaps for this reason, or for the somewhat byzantine VCG payment rule, the industry’s initial reaction to Facebook’s innovation was one of surprise (cf. Wired, 2015). But Facebook and its VCG auction are now essential parts of this industry: in the second quarter of 2015, Facebook pulled in $4.04 billion and, together with Twitter, it has become one of the largest players in display ad auctions. According to Varian and Harris (2014), around 2012 also Google considered a transition to the VCG auction for its search auctions, but ultimately decided to switch to VCG exclusively for its contextual ads sales, because of the perceived risks associated with communicating to bidders the complex VCG payment rule.

The VCG is a classic and widely studied auction in the academic literature that involves a fairly complex payment scheme. As we will explain in Section III.3.1., it is designed to price the externalities that each bidder forces on others in the efficient allocation. On the other hand, as we will also discuss in Section III.3.1., the VCG has the advantage that bidding truthfully is a dominant strategy, just as in the baseline second-price auction. The resulting allocation therefore is efficient. The GSP auction in contrast has very simple rules (the k-highest bidder obtains the k-highest slot at a price-per-click equal to the (k + 1)-highest bid), but it gives rise to more complex strategic interactions.

The relative merits of the two auctions therefore appear unclear, at least at first glance.

However, consider once more our earlier auction problem from Example 1:

Example 2. There are four slots and five bidders, with the following valuations: v1 = 5, v2 = 4, v3 = 3, v4 =2 and v5 = 1. The CTRs for the four positions are the following: x1 = 20, x2 = 10, x3 = 5, x4 = 2. But this time suppose that the seller uses a VCG auction, rather than the GSP. As we will discuss shortly, bidding truthful is a dominant strategy in the VCG. In this equilibrium, everybody bids bi = vi, and hence the resulting allocation is the same as the GSP auction. Moreover, applying the formula for the VCG payments, it is easy to check that the total revenues are exactly the same which would be obtained in the benchmark competitive equilibrium of GSP auction: 96.

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Hence, based on this example, it seems that the GSP auction is both simpler and ensures the same revenues and allocation as the VCG: while the increased complexity of the VCG ensures that bidding truthfully is a dominant strategy, it does not seem to yield higher revenues in this setting, nor a better allocation.

Economic theorists have shown that this outcome-equivalence result between the VCG auction and the benchmark competitive equilibrium of the GSP auction holds in general (See Edelmann, Ostrovsky and Schwarz, 2007).

Combined with the simplicity of the GSP rules, this result has provided a rationale for the GSP’s striking success and, until recently, its almost universal diffusion.

The next subsection provides a more in-depth look at the VCG auction, and its relation with the GSP and the baseline second-price auction. An attempt to explaining why the VCG might be actually preferable to the GSP is provided in Section IV, in which we discuss further recent trends in the market, which operate along with the changes in the auction formats and affected their performance.

3.1. Economic Analysis of the VCG Auction

We begin by explaining why bidding truthfully is a dominant strategy in the VCG auction. To this end, we recall the rules of this auction:

■ In the (VCG) Auction, the k-th highest bidder gets the k-th slot, and pays a price equal to a weighted sum of all lower bids, where weight of the l-th highest bid (for l > k) is equal to (xl−1 − xl), where we set xk = 0 for all k > S.

First note that, similar to both the GSP and the baseline (single-unit) second price auction, each bidder’s own bid does not affect directly the price he pays for the slot he obtains (besides determining which slot he gets). If i places the k-highest bid, he obtains the k-th slot, and pays a price which only depends on the lower bids (each weighted by the term (xl−1 − xl) for all l > k). It is thus clear that, unlike the GFP auction, bidders in the VCG wouldn’t have a strict incentive to lower their bids, holding the allocation constant. In fact, when there is a single-object on sale (S = 1), then the VCG coincides with the baseline second- price auction, just like the GSP does.

To see that it would never be optimal to bid more than one’s own valuation, note that (similar to the baseline second-price auction), bidding bi > vi would

only affect the outcome in the event that some of the other bids were above vi. But, in that case, the gain due to the increased CTR would be more than offset by the higher price: suppose that, by bidding truthfully, agent i obtained position k, whereas by bidding bi > vi he climbed up one position, to slot k−1.

Then, this means that there exists exactly one opponent, say j, whose bid bj is such that vi < bj < bi. Now, bidder i’s increase in utility due to climbing one position up from k to k − 1 is equal to (xk−1 − xk) . vi. But the increase in price is equal to (xk−1 − xk) . bj, since now bidder j has fallen below bidder i, increasing his payment. But note that, by assumption, bj > vi in this case, and hence the increase in payment is larger than the increase in payoff due to the higher slot.

Increasing one’s bid above one’s own valuation in order to climb one position up therefore would never be optimal. A similar argument applies to the case in which bidding bi > vi allows bidder i to climb more than one position up.

In all these cases, increasing one’s bid above one’s own valuation either has no effect on the ultimate allocation, or it lowers the overall payoff, since it induces an increase in payment higher than the increase in utility due to obtaining a better slot. A symmetric argument also shows that lowering one’s bid below one’s own valuation never increases the payoff: it either has no effect on the resulting allocation and payoffs, or it induces a lower slot in a suboptimal way, in that climbing up to the original slot would induce an increase in utility which is larger than the increase in payment it is associated with.

In conclusion, exactly like in the baseline second-price auction, bidding truthfully is an optimal strategy in the VCG regardless of what others do. Recall that this was not the case in the GSP auction, in which in fact bidding truthfully was not an equilibrium (see Example 1). In this sense, the VCG truly is the correct way of generalizing the properties of the baseline second-price auction to the case in which multiple objects are on sale. Despite the seemingly closer connection between the GSP and the baseline second-price auction, the GSP has very different properties from it. Those properties are instead inherited by the more complicated VCG auction: bidding truthfully is dominant, and it induces an efficient allocation.

This is not by chance. In fact, academic economists designed the VCG auction and its generalizations precisely to achieve these goals. These ideas have been applied for instance to ensure socially efficient outcomes not only in auctions, but also in environmental economics, or for solving the problem of optimal provision of public goods. The key idea behind the VCG payments, and the reason why they induce efficient allocations, is that they provide a sophisticated way of pricing the externalities which may otherwise induce

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inefficiencies, very much like the Pigouvian taxes used to reduce firms’ polluting emissions.

To see this, note that if everybody bids truthfully in the VCG auction, then each bidder i obtains the i-th slot, and pays a price equal to a weighted sum of the valuations of all agents j > i, where each vj is weighted by the term (xj−1 − xj).

Formally, the payment for the i-th position is equal to ⁿ ₁( j i j) j

j i x− x v

= +

− ⋅

∑

words, bidder i pays for the i-th slot the total value of the externality that he imposes on others. To see that this is actually the case, it is useful to pause for a moment and consider what is i’s externality on others: if bidder i and his bid were removed from the system, then the bidders with valuation higher than i (that is, those indexed with j < i) would still obtain the same slots. However, if i and his bid were removed from the auction, then all bidders below him (the j’s such that j > i) would each climb up one position. Hence, each j would move from CTR xj to CTR xj−1. The expected gain in utility for such j is thus (xj−1 − xj) . vj. Hence, the total externality that i’s presence forces on others is that it prevents all bidders with lower valuation to each climb up on position in the ranking of slots, which displaces a utility of (xj−1 − xj) . vj for each j > i. The total externality of agent i in slot i therefore is precisely ( )

1 n

j i j j

j i x− x v

= + − ⋅

∑

the VCG payment for the i-th slot if everybody bids truthfully.

IV. RECENT DEVELOPMENTS: NEW PLAYERS AND AGENCY