Comparison of the Optimal Mechanism with Test and No-Test

We can now compare the buyer’s expected cost under the optimal mechanism to her costs when employing the test and no-test strategies. We calculated the test and no-test strategies’ performance relative to the optimal mechanism in a wide range of situations by generating 81 problem instances. We chose a low, medium, and high value for the recruitment cost, number of incumbents, width of the suppliers’ cost distribution, and minimum order quantity that needs to be offered to suppliers in the test auction to induce them to participate, where the medium value corresponded to the value from the example in §2.3.3. Specifically, the parameters studied included k(m) ∈ {12500m, 17500m, 22500m}, n ∈ {2, 5, 8}, U[a, b] ∈ {U[2.5,17.5], U[5,15], U[7.5,12.5]}, and y ∈ {500, 2500, 4500}.

We calculated the optimal mechanism’s expected cost as follows: For the given problem parameters, costs are randomly generated according to the distribution F for the incumbent suppliers. Given these costs, the optimal number of entrants

are recruited and their costs are revealed; finally, the buyer pays the amount given by (2.12). Because this calculation depends on the randomly-generated costs of the incumbents, we run this simulation for 1,000,000 replications for that problem instance and calculate the average. We call this average the optimal mechanism’s expected cost for the given problem instance.

For these 81 scenarios, the optimal mechanism provides a relatively substantial average net savings of 8.4 percent over the minimum expected cost of the no-test and test (without reserve price) strategies, while it only provides an average net savings of 1.9 percent over the minimum of the no-test and test with reserve price strategies. In these calculations, we compare the optimal mechanism to the mini-mum of the no-test and test strategies because the buyer will decide a priori which strategy to use based on the problem instance’s parameters. Further, in these 81 scenarios the test with reserve price strategy provides an average net savings of 10.2 percent over the no-test strategy. Thus, when the buyer can use a reserve price, the intuitive and easily-implementable test strategy provides the buyer with significant savings and near-optimal results. The implications of this observation are important to reiterate: The buyer can use an easily-implementable strategy with an auction mechanism very familiar to practitioners that will provide average net savings of over 10 percent compared to the commonly-used no-test strategy, and furthermore this easily-implementable strategy only performs a mere 2 percent worse than the optimal mechanism. For these reasons, we believe that the test with reserve price strategy is a powerful sourcing mechanism for buyers.

Finally, we are interested in further exploring when the test with reserve price strategy performs well and when it performs poorly compared to the optimal mecha-nism. To this end, we vary the problem parameters beyond the aforementioned 81

in-stances. Once again, we use the example from §2.3.3 as a base case, where the default parameter values were F ∼ U[$5, $15], Q = 50,000, y = 2,500, k(m) = $17,500m, and n = 5.

First, the (constant) marginal cost of recruiting an entrant was varied from $2,000 to $160,000: Figure 2.5 shows that the net savings of the optimal mechanism over the minimum of the no-test and test with reserve price strategies is increasing and then decreasing in the recruitment cost. For extremely low recruitment costs, the buyer will use the no-test strategy and recruit a large number of entrants prior to holding an auction; likewise, under the optimal mechanism the buyer will (most likely) recruit many entrants after viewing the incumbents’ costs. However, as the recruitment cost increases the value of viewing the incumbents’ costs increases until a certain point, after which the value decreases because the buyer will be less inclined to recruit entrants due to the recruitment cost. We note that the slight “noise” in the graph (and subsequent graphs) is due to the numerical calculation via simulation of the optimal mechanism’s expected cost, as described above.

Second, the number of incumbent suppliers was varied from 2 to 30. Figure 2.6 shows that the net savings of the optimal mechanism increases and then decreases in the number of incumbents. Using similar intuition from the recruitment cost case, for very small and large n, the buyer’s recruitment decision hinges less on the incumbents’ cost information; the buyer will be more inclined to recruit a lot (in the case of very small n) or few (in the case of large n) entrants regardless of the cost information, so the value of the optimal mechanism with respect to the test and no-test strategies decreases.

The width of the suppliers’ cost distribution (originally U[$5,$15]) was varied while holding the midpoint of the cost distribution constant at $10 – the widest cost

Figure 2.5: Buyer’s net savings as a function of the constant marginal recruitment cost.

0 5 10 15 20 25 30

0 0.5 1 1.5 2 2.5 3

Number of Incumbent Suppliers

Net Savings of Optimal Mechanism (%)

Figure 2.6: Buyer’s net savings as a function of the number of incumbent suppliers.

distribution studied was U[$0,$20] and the narrowest was U[$9.4,$10.6]. Figure 2.7 shows that the optimal mechanism’s net savings grow at a decreasing rate as the width of the suppliers’ cost distribution increases; as the variability of the incum-bents’ costs increase, the value of being able to view the incumincum-bents’ cost information without having to allocate any quantity to incumbents increases due to the increased uncertainty in cost.

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0.5 1 1.5