Customizing Cost Functions Approximations

The formulation of the GSP – just like the formulation of the QSP – requires an approximation of the bidders’ cost functions. Thus, the second building block of the GSM is the procedure through which the cost functions are approximated. In the QSM, the cost function approximation was a linear cost function using dual prices as per-unit cost parameters. The QSM used the same cost function estimate for all bidders, and thereby the bid suggestions of QSM were anonymous (i.e. it gave the same suggested bids regardless of the bidder). However, it does not make sense for the GSM to be anonymous. If each bidder did not have a unique approximation of the cost function, the optimization algorithm would assign the bid suggestions randomly to some bidders, (as there would be numerable alternative optima) and it would pool the quantities into one large bid or few large bids (as allowed by the capacity constraints).

We hold on to the assumption that bidders do not want to disclose any cost information, and thus the only information we have on the bidders’ costs is information from the bidders’ bids in the bid stream. We use the bid information to customize the cost function approximations for each bidder.

To start off, the form of the cost function to be used in the approximation was chosen.

We decided to try out the same functional form as was used in the two simulation studies in Chapter 8. This way we could compare the results of the GSM auctions to the QSM simulations. To approximate the bidders’ cost functions we designed an inverse optimization problem (in the spirit of Beil and Wein, 2003; see also Zionts and Wallenius, 1976), which utilizes the information we get in the form of bids in the bid stream. We assume that bidders do not place bids in which costs exceed the price.

Thus, the task of the inverse optimization problem is to find a set of cost function parameters, which are consistent with a bidder’s bidding behavior (see (42)). Also, we made some assumptions that allowed us to pose some constraints on the cost parameters. First, we assumed that by including an additional item into the bundle should not decrease the total cost, i.e. F₁₂ ≥ F₁ and F₁₂ ≥ F₂, (see constraints (45)).

Secondly, we assumed that there would be economies of scope between the items, i.e.

F₁₂ ≤ F₁ + F₂, (see constraints (46)). Also, in order to constrain the feasible set a little more, we assumed that upper and lower limits for all the cost parameters can be derived for any particular industry (constraints (43) and (44)).

The constraints of the Cost Estimation Problem (CEP) are:

[ ]

where ε is a small positive scalar, Γ is the set of all possible item combinations, and L, L’ are subsets of Γ, Par[L] refers to all possible partitions of L, and L_t is an element of Par[L]. A partition of set L is the group of disjoint sets, which together form L.

These constraints (42) – (46), however, still leave a vast range of feasible options to choose from. Therefore, the choice of the objective function to a large extent

determines the values for the cost parameters. Thus, the question becomes how to choose the objective function.

The use of different objective functions will lead the CEP to choose different points in the feasible set. Without any further information on the bidders’ cost functions besides the constraints that make up the feasible set, there is no way of knowing, which point would be better than some other point. In other words, it is impossible to say which objective function would provide the best – or even good – approximations of the bidders’ cost functions. Thus, we chose simply to maximize the sum of the cost function parameters as the objective of the CEP, that is,

max

1 ij K

k ik L

iL c p

F

∑

∑

= Γ

⊆

+ ₍₄₇₎

and designed the following iterative scheme to approximate the bidders’ cost function parameters and to narrow down the feasible set as the auction progresses.

First, lower and upper bound estimates for the cost function parameters are set. At the beginning of the auction the bounds coincide with “industry estimates”, or in our experiments, the ranges of the distributions. The objective function will drive all the parameters to their upper bounds in the absence of any bid information to provide contradicting evidence. This may naturally be an over estimation of the cost functions, but it will not prohibit the GSP from finding bid suggestions, since losses are allowed in the formulation. Thus, it can still suggest the bids to the bidders even though the cost function approximations suggest that the bids would result in losses, and it is possible that the bidders will actually find them profitable. If the bidders accept what appeared to be unprofitable bids, it has the added benefit that now we get contradicting information and can update the estimates for the cost function parameters. The updated estimates are then set as the new upper bounds for the parameters, and the auction continues. If we started from a lower bound estimate for the cost parameters, and the GSP suggests bids in which all the estimated profits are positive, the acceptance of the bids is expected and would not give us any new information on the cost function parameters (the old estimates are still consistent with the new evidence).

Naturally, in this case, if the bidders declined the bid suggestions we would get new

information. However, it is desirable that the bidders accept the bid suggestions because that way the auction progresses. In order to maximize the information obtained from new bids and to speed up the auction process it makes more sense to start from the upper bound estimates.

It is worth noting that the GSP will not offer bid suggestions to all inactive bidders unless there is room for everybody in the set of provisional winners, which is more unlikely the more there are participants in the auction. Thus, some bidders are not suggested a bid by the GSP, and without any bid information the cost function estimate would not be lowered, which decreases the probability that the bidder would be offered a bid suggestion the next time. Some bidders could get stuck in this loop, and never be suggested anything. We could consider adding a constraint requiring that the bidder requesting support would be guaranteed a bid suggestion (not to upset the bidder), but so far we have not added any such constraints. Instead, recognizing that the estimates are above the true parameters, we decided to decrease the estimates by 1% for each bidder who is not suggested anything to improve their chances to receive a suggestion in the next round. We chose the decrement to be 1% in order to make only small adjustments in the estimates. Increasing the decrement could reduce the number of iterations needed to get the GSM to suggest a bid for the bidder, but a smaller decrement allows us to get closer to the true estimates. If, in the next round, the bidder receives a bid suggestion and accepts it, the upper bounds are replaced by the 1% lower estimates. If it turns out that the new estimate was too optimistic (the GSM offers a bid it thinks is profitable, but the bidder declines), we solve the cost function estimation problem again with the rejected bid added to the constraints, and receive an updated estimate of the parameters.