4.4 Analysis
4.4.1 Unit-wise analysis
In this approach, we use the stand-alone bids (i.e. bids that are placed on the individual units) to analyze the total procurement cost in a VCG outcome. Our goal is to characterize the asymptotic properties of the VCG procurement cost by examining the behavior of the stand-alone bids as the number of bidders increases. We achieve this by constructing an upper bound for the VCG procurement cost only using the second minimum bids placed on each of the individual units, which requires that there are at least two stand-alone bids on each of the units. Moreover our analysis relies on the fact that the bidders’ reported bids are sub-additive. Since sub-additivity of the supplying costs were already assumed in Assumption 4.1, this would be achieved if we assume that bidders place bids on all packages. To ensure this, we first make the following assumption. Later in this section, we will weaken this assumption.
Assumption 4.2. Every bidder places bids on all units and packages.
We now begin our unit-wise analysis by showing that the total procurement cost of a given VCG auction can be bounded by the “second order statistics” of the stand-alone bids. The following lemma is helpful in constructing such a bound.
Lemma 4.1. Suppose Assumptions 4.1 and 4.2 hold. In a given VCG auction, any winning bid is the minimum among the bids placed on that particular package. That is, for any winneri and her winning packagex∗i = a ∈ A, we have:
bia= b(1)a ≡ min
j∈N bja.
The proof of this lemma as well as the proofs of all other lemmas and propositions discussed in this chapter are provided in Section 4.6.2. Basically, the above lemma states that whatever package a bidder wins, the corresponding winning bid should be the minimum bid placed on that particular package. Note that this may not be true in general; the sub-additivity of the costs are essential for this result. For example, consider a two-unit two-bidder CA. Suppose bidder A places stand-alone bids both of which are cheaper than the bids by bidder B. However, if bidder A has significant
synergy in the cost of the bundle so that the costs are strictly super-additive, it could be optimal to assign only one unit to bidder A. In this case, bidder B wins the other unit even if her stand-alone bid on that particular unit is not the minimum.
Using this result, we now characterize an upper bound for the VCG procurement cost only using the second minimum bids placed on each of the individual units. The following proposition establishes the bound.
Proposition 4.1. Suppose Assumptions 4.1 and 4.2 hold. Given a set of realized costs of the bidders, the total procurement cost of the VCG mechanism cannot be larger than the sum of the second minimum stand-alone bids. That is,
PN ≤ X
u∈U
b(2)u . (4.6)
Remark. Proposition 4.1 establishes a useful bound that makes the convergence analysis in a random environment possible – it does not require the knowledge on the final VCG allocation, which is crucial for the analysis. In fact, there is another bound that could be tighter than bound (4.6), which is described as follows:
Similar to bound (4.6), the above bound uses the fact that the payment to a winning bidder cannot exceed the second minimum bid placed on the package that the particular winning bidder wins.
However, to use this bound, one needs to know the VCG allocation x∗ given the placed bids. The difficulty arises by the fact that the winner determination problem of the VCG mechanism (4.1) has no analytical solution, and therefore x∗cannot be characterized a priori using the random bids.
Note that as the number of bidders increase, the second order statistic of each stand-alone bid gets closer and closer to the lower bound of its support, forcing the total procurement cost to the sum of the lower bounds of the stand-alone unit costs. Recall that our objective is to find conditions where the total payoff of bidders vanishes as competition increases. Therefore it would be sufficient if the procurement cost eventually approaches to the lower bound of the total supplying cost. One
possible problem in this unit-wise approach is that the lower bound for the total supplying cost may vary across allocations, but we are bounding the procurement cost only using the allocation that purely consists of the stand-alone costs. That is, if there exists another allocation that achieves a cheaper lower bound for its total cost than the sum of the lower bounds of the stand-alone costs, then inequality (4.6) itself does not imply that the total payoff of bidders will eventually vanish.
For this, we further assume that all the feasible allocations have the same lower bounds – this could arise in an environment where the cost synergies tend to be less significant when a firm’s unit costs get more competitive. In Section 4.4.2, where we conduct the allocation-wise analysis, we relax this assumption. The following assumption formalizes this.
Assumption 4.3. There exists a non-negative constant c such that S = c for any partition S ∈ S.
In real-world applications, a bidder may not attempt to win a certain set of packages, by placing no bids on such packages. It may be because she already knows that she is not particularly competitive on those packages and hence very unlikely to win one of those. Another possible reason why she may not place such bids would be that it is too expensive for her to estimate her own supplying costs on such packages. We note that this phenomenon may cause a problem in our analysis. Recall that we rely on the order statistics of the stand-alone bids to bound the total procurement cost, assuming that all the bidders place stand-alone bids on every unit. That is, our analysis may not work if there are bidders that do not bid on individual units but do only on multi-unit packages. As discussed in Chapter 3, however, we observed some notable bidding patterns in the Chilean school meals auction data. First, a firm’s bidding may be concentrated in some subset of units. Especially, bidders who have local cost advantage tend to focus on the units in which they are competitive, placing stand-alone bids on those local units as well as multi-unit bids on packages that contain those local units. Also, firms may not bid on large packages. In our application, for each bidder a maximum package size was imposed by the auctioneer so that a bidder cannot win packages that exceed her size limit. In other applications, it could also be determined by the bidder depending on her service or production capacity.
Our model assumptions on bidders’ bidding behavior is based on these observations. Specif-ically, we assume that each bidder is interested in winning at most d units, and they randomly select d units equally likely, then submit bids on all possible packages that consist of those units only. Therefore, each bidder will place bids on (2d− 1) packages, that include the stand-alone bids on the d units. Under this selection scheme, a particular unit will be selected by a bidder with probability p := d/K (again K is the total number of units in the auction). Before we formalize the assumptions on the bidders’ bidding behavior and provide the main result, let us first examine an illustrating example which establishes the convergence results in a simple setting.
Example 1. In this example, we show that the expected total procurement cost of the VCG mech-anism converges to zero if and only if the average number of interested bidders on each unit np grows to infinity, under the assumptions described so far as well as some additional assumptions – we assume that each of the unit costs follows a uniform distribution, and the lower bounds of the feasible allocations are all c = 0. Now for the analysis, we let B(2:m) be the second order statistic out of m i.i.d. observations from Uni[0, 1]. If each of the n bidders is randomly selecting d units equally likely, then the (random) number of bids placed on a given unit i, denoted by Ni, follows Binomial distribution with parameters (n, p), where p := d/K. Therefore, we have:
E[PN] ≤ X
which can be simplified as:
E[PN] ≤ 2K
p(n + 1) ≤ 2K np.
This provides sufficiency of the convergence, that is: np → ∞ implies E[PN] → 0.
Now we turn our attention to the necessity of the convergence. We note the following observa-tion that we had while analysing the estimated costs in our applicaobserva-tion in Chapter 3. First, the level of synergies tend to depend on the level of the unit costs – when the unit costs are less aggressive, the synergies were more variable and relatively larger on average. Second, when the package costs are very competitive, it was usually the case that the unit costs were also very competitive – and it was less likely that the unit costs are not that competitive but the synergies are significantly large.
To be consistent with these observations, we make further assumption on the relationship between the synergy and the unit costs. Specifically, we assume that the maximum possible synergy level depends on the level of unit costs. Formally, we assume that there exists a constant α ∈ (0, 1]
such that for any package a ∈ A and for any bidder i, we have Pu∈aα(ciu− cu) ≤ cia− ca. We now explain the meaning of this assumption. Notice that by definition of the lower bound, we have ca ≤ ca and by sub-additivity of the costs, we have ca ≤P
u∈acu.That is, these two inequalities define the possible synergy level in this package. However, to capture the pattern of the second observation above, we assume that the smallest possible package cost ca, and hence the maximum possible synergy, also depend on the unit costs Pu∈acu. More specifically, we assume that cais lower bounded by α Pu∈aci+ (1 − α)ca. Then, since Assumption 4.3 implies ca=P
u∈acu, we have Pu∈aα(ciu− cu) ≤ cia− ca. In the special case of this example where we have set c = 0, this condition can be simplified to Pu∈aαciu≤ cia.
Now to show the necessity of the above convergence result in this environment, we will use the fact that for any VCG outcome, the total procurement cost is always larger than the total supplying cost. By Lemma 4.1 and the definition of α, the total supplying cost should be at least αP
i∈UB(1:N
(n,p)
i )
i . Hence, taking expectation and using the fact that Bi’s are i.i.d samples from
, the expected total procurement cost is lower-bounded by:
where the second inequality is by Jensen using the fact that the function g(k) = k+11 is convex.
Therefore, if np does not grow to infinity, E[PN]cannot vanish.
We are now ready to establish the desired asymptotic results, generalizing the above example.
For analytical simplicity, however, we assume further that given n bidders, the number of interested bidders for each unit is deterministic – that is exactly [np]. We call p intensity of the bidders’
interests. Under this assumption together with the assumptions made so far, we show in Theorem 4.1 that the total procurement cost of the VCG mechanism converges to the lower bound c in expectation if and only if the number of bidders (who are interested in winning each unit) grows to infinity. Then the corollary that follows establishes the desired result; the convergence of the VCG payoff profile when competition increases. We first formalize the assumptions we discussed so far.
Assumption 4.4. Given a VCG auction with the set of n bidders N as well as the set of K units U, we assume that:
a) Each bidder places a bid on each of the units with interest intensity p ∈ (0, 1]. That is, each bidder selects a unit with probabilityp and place a stand-alone bid on it.
b) Every package bid placed by a bidder only contains her interested units, which are the units on which she places stand-alone bids.
c) The selection is balanced. That is, it is done in a way that the number of interested bidders for each of the units is exactly[np].
d) There exists a constant α ∈ (0, 1] such that for any package a ∈ A and for any bidder i, we have P
u∈aα ciu− cu ≤ cia− ca.
We now present the main results from our unit-wise analysis.
Theorem 4.1. Consider a fixed set of units U, and suppose Assumptions 4.1, 4.3, and 4.4 hold.
Then the the expected total procurement cost of the VCG mechanism converges to the lower bound
c if and only if the number of bidders n grows to infinity.
The above theorem states that the necessary and sufficient condition for the expected total procurement cost of the VCG mechanism to converge to the lower bound c is that for each of the units there should be infinitely many “interested” bidders. The intuition behind this result is as follows. Recall that the profit that a winning bidder makes in the VCG mechanism is same as her contribution to the minimum possible total cost. As the competition increases and therefore as the number of placed bids increases, a particular winning bidder’s contribution decreases – when the bidding is ample on each of the units, even without the specific winning bidder’s bids, we can find another allocation that achieves the total cost also quite close to that of the optimal (efficient) allocation. Our analysis upper-bounds the total cost of such an alternative allocation using the second order statistics, which approaches to the lower bound c as the number of bids increases.
Note that once the procurement cost approaches to the lower bound of the total cost c, the payoff that the winning bidders make will vanish. That is, even in the complementary environment, increased competition can lead to a competitive VCG outcome. The following corollary provides this result.
Corollary 1. Suppose assumptions 4.1, 4.2, and 4.4 hold. Then the total VCG payoff that is given to the winning bidders converges to zero in expectation if and only if the number of bidders increases to infinity.
Our analysis highlights the impact of competition on the revenue performance of the VCG mechanism. Although the result is intuitive, we believe our analysis provides helpful insights to enhance the practicality of the VCG mechanism. High competition merely measured by the number of bidders may not be enough – in our analysis, ensuring enough competition in each of the unit-wise markets is the key to a competitive revenue performance of the VCG mechanism.