In the previous subsection, we investigated when the buyer can benefit from imple-menting the test auction strategy versus the commonly-employed no-test strategy.
Although we saw that employing test auctions can provide significant benefits, it would be interesting to see how well the test strategy performs compared to an optimal mechanism the buyer may use to source parts. We note that optimal mech-anisms are often difficult to implement but serve as good bounds on the performance of more easily implementable approaches. To this end, we find the optimal mecha-nism for the buyer’s procurement problem in order to investigate our test strategy’s relative performance. The revelation principle is described in Krishna (2002) as fol-lows: “Given a mechanism and an equilibrium for that mechanism, there exists a direct mechanism in which (i) it is an equilibrium for each [supplier] to report his or her value truthfully and (ii) the outcomes are the same as in the given equilibrium of the original mechanism.” We can apply the revelation principle in our setting, where the equilibrium involves the suppliers’ bidding strategy, the buyer’s allocation
and payment rules, and the buyer’s recruitment decision rule. Thus, we restrict our attention to direct mechanisms, as formalized below.
For supplier i, we denote his reported cost as si and his true cost as xi, and we use −i to represent j, ∀j 6= i. The purchase quantity from and payment to supplier i are denoted by Qi(si, s−i) and Pi(si, s−i), respectively. When supplier i reports cost si and all other suppliers report their cost truthfully, supplier i’s expected payment, quantity, and utility are, respectively, pi(si) = EX−i[Pi(si, X−i)], qi(si) = EX−i[Qi(si, X−i)], and ui(si) = pi(si) − xiqi(si). We refer to the vector of incumbents’ costs as xI, and let mT(xI) denote the number of entrants the buyer recruits based on the incumbents’ costs. Similarly, we let xE denote the vector of the recruited entrant’s costs, and let x , [xI, xE].
The objective function states that the buyer minimizes her expected total cost (re-cruitment costs plus payments to suppliers) by choosing the quantity allocation rule, payment rule, and the number of recruited entrants, mT(xI). Equation (2.7) is the standard individual rationality constraint, while equation (2.8) is the incentive compatibility constraint. Equation (2.9) states that the desired Q units has to be procured, while (2.10) addresses the minimum order quantity, y, consistent with
our model described in §2.2. The quantity allocation, payment, and recruitment rules that solve this constrained cost minimization problem will serve as a lower bound on the expected cost the buyer could achieve regardless of sourcing mech-anism — whether it be the test strategy, no-test strategy, or any other sourcing mechanism (e.g., negotiation, first-price sealed-bid auction, Dutch auction, etc.).
Let ψ(xi) = xi + F (xf (xi)
i) denote the virtual cost function, where ψ(·) is increasing because of the regularity assumption.
Proposition II.4. The following constitutes an optimal mechanism: Based on the incumbent suppliers’ reported cost vector xI, the number of entrants recruited by the buyer is
to supply the Q units while the remaining suppliers are not awarded any units and are paid zero.
Under this mechanism, the buyer views the incumbents’ vector of reported costs, xI, and recruits m∗T(xI) entrants based on these reports. After the m∗T(xI) entrants report their costs, the buyer awards the Q-unit contract to the supplier with the lowest report and pays him P∗(x). The first term of (2.12) is the winning supplier’s
true cost, while the second term represents a markup that rewards the supplier for truthfully revealing his cost. If the winning supplier is an entrant supplier, we note that ˜xI = xI and the markup is equal to the difference between the second-lowest report and the winning supplier’s report. Thus, in the case where an entrant is awarded the contract, P∗(x) = x(2:n+m∗
T(xI))· Q and the winning entrant supplier is paid the second-lowest reported cost.
We now address the markup when the winning supplier is an incumbent. The markup paid to a winning incumbent supplier represents the amount by which the supplier could have inflated his reported cost and still won the contract. The im-portant point is that a winning incumbent supplier’s reported cost also influences the buyer’s recruitment decision. Hence, the markup of an incumbent supplier who eventually wins the contract must account for the increased competition (i.e., m∗T(˜xI) − m∗T(xI) additional entrants) associated with an inflated reported cost.
Although the optimal mechanism described in Proposition II.4 results in the min-imum expected cost for the buyer, the buyer may still be unable to use such a mechanism in practice. The described mechanism requires the incumbent suppli-ers to reveal their true cost before the buyer recruits supplisuppli-ers to compete for the units up for bid. Revealing such information prior to the recruitment round will not result in a contract (even for the minimum order quantity) for the incumbents — it will only help the buyer make a more accurate decision on how many entrants to recruit. A buyer may find it difficult to convince a supplier to reveal his cost information just so the buyer can decide how many suppliers to recruit to compete against him. Although the test strategy analyzed in §2.3.1–2.3.3 allows the buyer to gather quotes from the incumbents, the guaranteed award of a minimum of y units to the winning incumbent supplier (given he meets the reserve price, if applicable)
provides an incentive for the incumbents to reveal such information — it is the min-imum quantity required to induce suppliers to bid. There is no such offer under the optimal mechanism. Further, we note that in the test strategy bidders have to beat their competitors’ pricing but do not have to reveal their true cost to win; under this mechanism, the lowest-cost supplier still has to reveal his cost.
Regardless of whether the buyer would be able to implement the optimal mecha-nism in practice, the optimal mechamecha-nism can be used as a theoretical benchmark to quantify the relative performance of the test and no-test strategies; the “optimality gap” between the optimal mechanism and the test and no-test strategies can help the buyer gauge whether the cost and effort of attempting to implement more complex mechanisms than the test and no-test strategies are worth the savings.