First Phase: Setting Up the Auctions

In any simulation study, some initial values have to be assumed. In this study we needed to choose the number of bidders, number of items and the quantities demanded of each item. The bidders needed to be assigned cost functions so that it became possible to evaluate the profitability of the shortlist items offered by the QSM.

Also, a set of initial bids from the bidders had to be generated, because the QSM cannot be used without some bids already in place in the bid stream.

8.1.1.1 The Cost Function

We wanted the cost function to be as simple as possible, but also we wanted it to portray both economies of scale and scope. The use of combinatorial auctions is justified in a situation in which there are synergies between the items. In reverse auctions, synergies between items can be understood as synergies in the production process of the items, i.e. economies of scope. Thus, it would not make sense to use a cost function that would not allow for synergies in production.

The simplest form for a cost function exhibiting economies of scale is a linear function with a fixed cost element:

C(q_k) = F_k + c_kq_k (14)

where F_k is the fixed cost, and c_k the per-unit cost of producing item k alone.

For the two-item case the cost function would take the form of

C(q₁, q₂) = F₁₂ + c₁q₁ + c₂q₂ (15)

The existence of economies of scope in this framework implies simply that the inequality of Equation (16) between the fixed cost parameters holds:

F₁₂ < F₁ + F₂ (16)

The above presented multi-product cost function (Equation (15)) is very simplistic. It is theoretically very restrictive, as it implies constant marginal costs and monotonically decreasing average costs. The function is discontinuous at points when the level of one or more outputs is zero, which makes it difficult to use in optimization problems. The function is easy and convenient to use only in situations with relatively few products. It can easily be seen that the number of different fixed cost elements increases rapidly as the number of products increases. In the case of two products, there are only three parameters F₁, F₂, and F₁₂. With three products there are seven parameters: F₁, F₂, F₃, F₁₂, F₁₃, F₂₃,, and F₁₂₃, where F_ij indicates the fixed cost of producing goods i and j.

When the number of products is increased to five there are already 31 fixed cost parameters, with six products there are 63, and with seven products 127 parameters.

However, the simplicity of the function makes it intuitive and it is very flexible as it can represent economies of scope of different magnitudes between different items – and even diseconomies of scope between so items, if necessary. Thereby it is very appealing in a theoretical framework such as ours.

Another reason for our choice of cost function was that there are not very good alternatives available. Cobb-Douglas and CES (constant elasticity of scale) forms can be used for multi-product cost functions, but they would have to be linearized before they could be used in linear or integer programming problems. A commonly used form for the cost function is the translog cost function (see Equation (17)), which is a function of the output quantities (y_i) and input prices (p_j).

∑∑

elasticity on the cost structure. Due to its flexibility, the translog cost function has been popular in empirical studies which aim at estimating real world cost functions for some firm or industry (see e.g. Murray and White, 1983, and Cho, 2003).

The translog cost function, however, is too complex for the purposes of this study. In a simulation study it would be good to minimize the number of parameters to choose.

Thus, the inclusion of the input prices in the cost function is an unnecessary complication. We chose to use the simple form of the cost function presented in Equation (15). It is intuitive, easy to use, and sufficient for the purposes of this study, where we only want to find out whether the QSM works under some circumstances.

8.1.1.2 Parameters

In order to reduce the sensitivity of the results to the initial values, we decided to vary some of them. The number of bidders was fixed at 10, but the number of items to be auctioned was either 3 or 5. The quantity demanded was 1000 for each item. The variable cost parameters were drawn from the same uniform distributions in each design. In the three-item auctions the variable cost parameters were drawn from the range [30, 50] for c₁, [40, 60] for c₂, and within [60, 70] for c₃. For the five-item auction the variable costs fort items 1, 2 and 3 were drawn from the same range as in the three-item auction, and for the additional three-items from the range [15, 45] for c₄ and [20, 55] for c₅. The distributions for the variable cost parameters (and fixed cost parameters) were the same for all bidders, so I have omitted the index indicating the bidder from the notation.

The uniform distribution from which the fixed cost parameters (F_ijk) were drawn had two possible levels: “high” and “low”. The ranges for the fixed cost parameters were chosen so that it was very likely that economies of scope would exist. This meant that the lower bound of F₁₂ was less than or equal to the sum of the lower bounds of F₁ and

F₂. A similar logic was applied to the upper bounds. It was also kept in mind that total fixed cost should not decrease from the addition of a new product. Thus the lower bound of F₁₂ was set higher than the upper bounds of F₁ and F₂.

When the costs were low, the lower bounds ranged from 700 (for F₃) to 2300 (F₁₂₃) and the upper bounds from 1000 (F₃) to 3200 (F₁₂₃) in the three-item auctions. Again, in the five-item auctions the parameter ranges were the same for the first three items. The lower bounds ranged from 700 (for F₃) to 5500 (F₁₂₃₄₅) and the upper bounds from 1000 (F₃) to 7500 (F₁₂₃₄₅). The exact ranges for all fixed cost parameters can be seen in Appendix 1. When the costs were high, the lower bounds for the fixed costs ranged from 5000 (F₃) to 42000 (F₁₂₃₄₅) and upper bounds from 7000 (F₃) to 50000 (F₁₂₃₄₅). The new ranges can also be seen in Appendix 1. The higher fixed cost values were used to test the effect of more pronounced economies of scope on the results. With the lower level of fixed costs, the proportion of fixed costs in the total cost was only about 10%.

After the increase the proportion of fixed costs was almost 30%.

The initial bids in the bid stream were created so that each bidder was assumed to have placed one bid. Thus, the number of bids in the initial bid stream equaled the number of bidders. The quantities in the initial bids were drawn randomly from a uniform distribution [100, 500], and rounded to the nearest 50. Some of the bid quantities, however, were chosen to be zero in order to create some sparsity in the bid matrix. It is realistic to assume that all bidders would not place bids for all products but a subset of them. The level of sparsity was 20%. The constraint q_new,k ≤ 500 was added to the quantity support problem to simulate the capacity constraints of the sellers. The capacity constraint was set to simulate the fact that no bidder alone would be capable of producing the whole demand. The bid price in the initial bid was the cost for the bidder of producing that specific bundle, to which an initial mark-up of 30% was added.

The simulation study consisted of four different experiment settings displayed in Table 4. Five replications of each setting were conducted.

Table 4 Design of the first simulation study

Experiment Items Bids Fixed Cost

I 3 10 Low (~10%)

II 5 10 Low (~10%)

III 3 10 High (~30%)

IV 5 10 High (~30%)

In document Bidder support in iterative, multiple-unit combinatorial auctions (Page 111-115)