The Auction

Round 1

It is now time to solve the GSP for the first time. The decrement δ by which the total cost to the buyer is required to decrease from round to round is set at 2%. The cost function parameters in Table 23 are inserted in the cost function in the GSP (33).

However, because the cost function is discontinuous (the fixed cost term depends on the combination of items in the bid), the formulation becomes somewhat more complex, and it bears resemblance to the formulation of the QSP with the true cost function (Appendix 2) and of the efficient allocation problem (Appendix 3). A set of auxiliary variables y_i,jkl is defined to construct constraints that guarantee that the correct fixed cost is taken into consideration in the objective function (y_i,jkl = 1 if the items in combination j,k and l all have a non-zero value, y_i,jkl = 0 otherwise). Also, for the same reason we need to create new variables for the item quantities in the incoming bids:

one variable per item per combination it is in. If there are K items in the auction, each item is in 2^K-1 combinations, so in this case each item is in four combinations.

Thus, the cost function ^c~

( )

^Q for bidder i to be inserted into (33) takes the form and that at most one combination is chosen per bidder, the following constraints are added to the GSP for each inactive bidder i∈I ,

0

where M is a constant larger than any conceivable item quantity. In this example, M = 1000 was used. Note that these constraints leave open the possibility that y_i,jkl assumes the value of one, even though one or more of the items in the combination are zero.

For example, when items one and two assume a nonzero value for bidder i, either y_i,12 = 1 or y_i,123 = 1. However, since the fixed cost for a bundle including more items is always larger than for a bundle with less items, the objective function will ensure that the y_i,jkl, which coincides exactly with the combination of nonzero item quantities, assumes the value one.

The bundle of bids suggested by the GSM in the first round is presented in Table 24.

There are always multiple solutions, because the profit or loss in the bids (e_i or s_i) can be divided in an infinite number of ways between the bidders who are offered a bid suggestion. We chose the solution where the estimated profit/loss is divided equally among the bidders. Of course the true profits/losses of the bidders are not equal; we are dividing the profit/loss calculated with the cost function estimates equally among the bidders. Another approach would have been to divide the profit proportionately to the size of the bid. The main effect that the choice of solution has in the auction, is that in some cases it can cause the bidder to accept or reject a bid suggestion. E.g. if our cost estimate is a bit too low, offering a bid suggestion in which the GSP thinks the bidder will just break even, will not be good enough for the bidder and she will reject.

However, if some of the surplus in the auction is allocated to this bidder, it may increase the bid price high enough so that she will accept the bid suggestion. Because we assume we do not know the bidders’ true costs, it is difficult to know which cost

estimates are underestimated, and which way of dividing the surplus would be best.

Thus, we decided to start with dividing the (estimated) profit equally.

Table 24 Bids suggested by the GSM

Bid Item 1 Item 2 Item 3 Price (€)

x_1,new 0 225 0 16,658

x_5,new 75 225 0 21,426

x_7,new ^{300 0 300 43,950}

Bidders 1, 5 and 7 are suggested a bid, and their bids would team up with Bidder 6’s initial bid (225; 150; 300; 49,356 €). The bidders all accept the new suggestions, so the set of provisional winners is now Bidder 1, Bidder 5, Bidder 6 and Bidder 7, and the total cost to the buyer is reduced to 131,390 €.

Next the cost functions are updated. The cost estimates for Bidders 3 and 4, who were inactive but still not offered a new bid, are now lowered by 1%. The cost estimates of the active bidders remain the same. The accepted bids offer no new information on the bidders’ cost functions, because the bid prices are high enough to cover current estimated costs. The reason for the decrease in Bidder 3’s and Bidder 4’s estimates is that it is possible that we have overestimated their cost and that is the reason why they were not offered a bid. Bidders 2, 8, 9 and 10 were active so they could not have received bid suggestions regardless of their cost function estimates, and thus nothing is done to the estimates of their cost functions.

Round 2

The GSP with the updated cost information is solved for the new set of inactive bidders. The solution is presented in Table 25.

Table 25 Bids suggested by the GSM in Round 2

Bid Item 1 Item 2 Item 3 Price (€)

x_4,new ^{300 225 225 53,426}

x_8,new 0 150 225 27,736

This time the GSP suggests bids to Bidder 4 and Bidder 8, which would team up with the initial bids of Bidder 5 (x₅₁ = 1) and Bidder 7 (x₇₁ = 1). The GSP thinks the bids will result in a loss of 474 € for both bidders. However, both bidders accept the suggested bids. The GSP has overestimated their costs, so the true cost of producing the proposed

bundles is below the bid prices. Now we have new information on the costs of Bidders 4 and 8, and their cost estimates are updated. The cost estimates for inactive bidders who did not receive a bid suggestion (Bidders 2, 3, 9 and 10) are lowered by 1%, and the estimates for bidders who previously were active (Bidder 1, 5, 6 and 7), out of which Bidders 5 and 7 are still active, remain untouched. The new cost function parameters are depicted in Table 26.

Table 26 Cost function parameters after Round 2

Bidder 1 Bidder 2 Bidder 3 Bidder 4 Bidder 5 Bidder 6 Bidder 7 Bidder 8 Bidder 9 Bidder 10 F_i 2222.2 2200.0 2178.0 2200.0 2222.2 2222.2 2222.2 2222.2 2200.0 2200.0 F_ij 3333.3 3300.0 3267.0 3300.0 3333.3 3333.3 3333.3 3333.3 3300.0 3300.0 F₁₂₃ 4444.4 4400.0 4356.0 4400.0 4444.4 4444.4 4444.4 4444.4 4400.0 4400.0

c₁ 66.7 66.0 65.3 64.4 66.7 66.7 65.9 66.7 63.8 66.0

c₂ 61.6 66.0 65.3 66.0 54.5 66.7 66.7 66.7 66.0 66.0

c₃ 66.7 63.5 65.3 66.0 66.7 66.4 66.7 59.1 66.0 66.0

Round 3

First the GSP suggests a bid only for Bidder 10, and that bid (0; 150; 225; 25,160 €) is not acceptable to her. Also according to the estimated cost the bid is not profitable, and therefore we do not receive more information on B10’s cost function. The cost estimates of all the other inactive Bidders (1, 2, 3, 6 and 9) are lowered by 1% and the GSP is solved again. This time the GSP suggests bids for Bidder 1: (150; 225; 0; 26,139

€), Bidder 6: (225; 0; 300; 37,075 €) and Bidder 9: (225; 150; 150; 37,379 €), which would team up with Bidder 5’s initial bid resulting in a total cost to the buyer of 126,187 €. All the three bidders find the suggestions profitable, even though the GSP again thinks the bids would result in losses. The cost estimates are updated for all bidders, except the ones who are active.

Round 4

The first solution of the GSP provides only a suggestion for Bidder 3 (225; 150; 150;

34,855 €), which is not accepted. Thereafter, the GSP is solved seven times, and each time at least one bidder who is suggested a bid declines the suggestion, vetoing the

“group” bid. The accepted bids are added to the bid stream, but the total cost to the buyer does not decline by the required 2% because the complementary bid(s) was not accepted. After each GSP solution the cost functions are updated. Finally, the ninth

iteration produces bid suggestions to Bidder 4: (300; 0; 150; 30,515 €) and Bidder 7:

(300; 150; 300; 50,897 €), which are both profitable for the bidders.

After this round, the GSP does not find bid combinations that would be profitable for all bidders. The auction ends. Going back to the bidders’ cost functions we can conclude that the inactive bidders have such high costs that they could not afford to decrease the total cost to the buyer by the required 2%. The active bidders could have afforded to, but they did not have an incentive to do so, as they were already among the provisional winners. The final bid stream and the winning bids are presented in Table 27.

Table 27 Bid stream and solution of the WDP after Round 4 Bid Item 1 Item 2 Item 3 Price (€) Status

x₁₁ 0 225 150 27,199 1

x₂₁ 150 0 150 22,954 0

x₃₁ 225 225 150 45,990 0

x₄₁ 0 225 0 17,613 0

x₅₁ 0 225 150 25,594 1

x₆₁ 225 150 300 49,356 0

x₇₁ 300 0 0 22,007 0

x₈₁ 0 225 225 33,210 0

x₉₁ 225 150 0 27,831 0

x_10,1 225 225 225 50,076 0

x₁₂ 0 225 0 16,658 0

x₅₂ 75 225 0 21,426 0

x₇₂ 300 0 300 43,950 0

x₄₂ 300 225 225 53,426 0

x₈₂ 0 150 225 27,736 0

x₁₃ 150 225 0 26,139 0

x₆₂ 225 0 300 37,075 0

x₉₂ 225 150 150 37,379 0

x₄₃ 300 150 225 46,781 0

x₄₄ 300 225 225 50,969 0

x₇₃ 300 150 0 31,938 0

x₄₅ 300 0 0 19,666 1

x₇₄ 300 150 300 52,385 0

x₄₆ 300 150 0 30,561 0

x_10,2 225 225 225 46,199 0

x₄₇ 300 0 0 19,778 0

x₄₈ 300 0 0 19,824 0

x₄₉ 300 0 150 30,515 0

x₇₅ 300 150 300 50,897 1

The total cost to the buyer is 123,356 €. In fact, the total cost decreased by more than 2% from Round 3. The previously inactive bid x₄₅ made by Bidder 4 was more advantageous to the buyer now that a good match entered (x₇₅) the auction. The estimated profit for Bidder 4 was larger from the newest bid (x₄₉) and that is why the GSP favored that one (it looks at the auction from the bidders’ perspective), but it is the WDP that determines the set of winners. The mark-up in Bidder 4’s bid x₄₅ is higher than in bid x₄₉. Hence, the bidder may be happier with this outcome.

In document Bidder support in iterative, multiple-unit combinatorial auctions (Page 161-166)