The Quantity Support Problem

If the bidder could express her costs in a functional form, and if she would be willing to disclose the cost function to the bid taker (or, if it could be arranged, to a neutral third party), it would be fairly straightforward for the bid taker to solve for the bid that maximizes the bidder’s profit. The quantity support problem for bidder m (QSP_m) would reduce to a standard mixed integer programming problem. The objective (5) of the problem is to maximize the profit of bidder m by solving for the new price p_m,new, the vector Q_m,new of item quantities q_m,new,k and the values for the bid status variables x_ij. In order for the new bid to become active (provisional winner), the current total cost to the buyer C* is required to decrease by a predetermined decrement δ as a result of the new bid (6), and the demand for each item d_k must be fulfilled (7). The item quantities

q_m,new,k should not exceed the bidder’s corresponding capacities a_mk (10)¹⁴. It is also

assumed that at most one bid per bidder can be active at a time (8), (9) to simplify the bidding language. The formulation of the QSP_m is thus:

14 We have included only these simple, per item capacity constrains in our formulation. Allowing the bidders to announce capacity constraints for combinations of items would increase the number of constraints in the formulation, reducing its readability. If desired, such more complex capacity constraints can be incorporated in the design.

(QSP_m) which bids are among the winners, and p_ij indicates the price and q_ijk the quantity of item k in bidder i’s j^th bid.

Note that even though we call our tool the “quantity support mechanism”, it also suggests a price to attach to the quantities. The quantity support problem in (QSP_m) is presented for a price-only auction, but if we use the “pricing out” –method as in Teich et al. (2001, 2006), also multiple attributes could be included in the bids. Also, if the bids in the bid stream were disclosed to all bidders (an open-cry auction), each bidder could formulate her own quantity support problem similar to (QSP_m) replacing ~ (.)

cm

with her own cost function (or an approximation of it).

However, it is not realistic that the QSP_m as such could be applied into practice. It is possible that the bidders are not able to express their costs in a functional form.

However, we do assume that the bidders are still capable of comparing the profitability of different bids. Also, it is unlikely that they would be willing to disclose their cost functions to the auction owner or even a neutral third party, even if they could specify the functions. Therefore we need to find a way to approximate the bidders’ cost functions, and preferably without having to ask for information from the bidders.

In this study we used a linear approximation of the bidders’ cost using the dual prices of the demand constraints of the WDP as the variable cost parameters. The dual prices

can be interpreted as market prices for the items (see discussion on combinatorial

where µ_k is the dual price of the k^th quantity constraint in the linear relaxation of the WDP, and thereby the dual price for item k.

According to economic theory, firms have two kinds of costs: variable and fixed. We have considered only the variable costs in our linear approximation. A fixed cost term could easily be added, but as it is a constant, it would not affect the solution of the maximization problem. The lack of the fixed cost element affects the value of the objective function, but in this problem the approximated profit indicated by the objective function is not interesting – only the allocation and bundle price are.

The resulting linear cost function has several limitations that need to be taken into account. First of all, it cannot portray economies of scale. Thus, in case the bidders experience economies of scale, the bidders’ costs for large bundles are systematically overestimated. This would mean that there would be a bias in the QSP towards smaller bids. However, since the objective is to maximize absolute profit (not relative), we expect the very small bids to be eliminated anyway. Secondly, a linear cost function is incapable of portraying economies of scope (= subadditive cost function). Because of that the QSP has no incentive to try to bundle many items into the same bid. It is the existence of economies of scope, which was the reason to organize a combinatorial auction in the first place, hence we should assume the bidders’ true costs to exhibit economies of scope. Therefore, one purpose of this study is to determine, whether a linear approximation of the cost functions is good enough.

Thirdly, notice that the linear cost function estimate (11) is the same for all bidders, hence it does not account for individual differences in the bidders’ cost functions.

Thereby the bid solutions of QSP are anonymous (i.e. the QSP gives the same suggested bids regardless of the bidder). The only differences in the suggestions can arise from the requirement that maximum one bid per bidder can be active at a time.

Thus, the bid suggestion for bidder m cannot be such that it would team up with bidder m’s previous bids, whereas for any other bidder these bids can be teamed-up with. We also recognize that there are differences between the bidders’ cost structures, so the “anonymous” suggestion offered by the QSP may not be the best for all bidders.

Due to these limitations, it is possible that the solution of the QSP is not an acceptable bid suggestion for any of the bidders. Therefore a shortlist of alternative solutions is generated, and the shortlist alternatives are presented to the bidders together with the solution from the QSP, and the bidder can choose the bid which is the most profitable one for her. This can be done in two ways. First, we can go through the neighboring pivots of the original quantity support problem. Pivoting in the integer case is interpreted as solving the QSP over and over again, but each time forcing one status variable x_ij with a zero value in the original QSP solution to assume the value “one”.

Hence, the number of pivots depends on the number of bids in the bid stream (as there is one status variable for each bid), and the number of bids in the optimal combination (those which assume the value “one” in the original QSP solution). The second alternative is to solve the QSP over and over again, but this time setting different Q_i’s to zero to generate new combinations. If all the possible combinations are searched through, there are 2^K-2 combinations to go through, so in larger auctions it would not be feasible. Oftentimes, though, the different pivots, as well as the different combinations produce the same bid suggestion, so the number of non-identical items on the shortlist hardly ever reaches the theoretical maximum. The QSP together with the shortlist forms the core of the bidder support tool we call the QSM.