Iterative Combinatorial Auctions

In combinatorial auctions, we iteratively try to match a set of packages of items with bidders in a way that the revenue for the auctioneer is maximized. Let S be the set of items being auctioned. Then at each round t, we need to solve the following winning determination problem (WDP):

max X j∈St cjxj s.t. X j∈St a_ijx_j ≤ 1, ∀i ∈ S (4.16) xj ∈ {0, 1}, ∀j ∈ St,

where St_{is the set of considered packages in round t, c}

j is the bid amount of package j,

aij =      1, if item i is in package j; 0, otherwise. , and xj =     

1, if package j is a winning package; 0, otherwise.

.

In the above set packing formulation, bidders who have submitted bids on the winning packages are awarded the corresponding items in those packages. Constraints ensure that each item is awarded at most once. Depending on our assumptions, we may also formulate WDP as a set partitioning problem max X j∈St cjxj s.t. X j∈St a_ijx_j = 1 ∀i ∈ S (4.17) xj ∈ {0, 1} ∀j ∈ St

so that all items are ensured to be awarded in round t. Although this formulation is used under the assumption of zero disposal value, bidders are also allowed to win more than one package in each round.

At each round, auctioneer solves the WDP and provides bidders with feedback information. Receiving this feedback after each round, the non-winning bidders can

• increase their bids for the packages already in the auction, or

• create bids for new packages.

according to the auction rules Therefore, in the next round, we need to solve a modified WDP with updated objective coefficients, new columns or both.

In our computational experiments, we use SYMPHONY’s SPP+CUTS package which gen- erates at each node star cliques, odd holes and odd anti-holes cuts (Eso [1999]) to solve set par- titioning or packing problem. These cuts remain valid in our case for the branching tree from round-to-round since they are generated from the constraint matrix and we only change the objec- tive coefficients or add new columns. We do lift the star cliques inequalities greedily. That is when new columns are added, we check whether the inequalities can be lifted with the new variables representing added columns.

We test warm-starting for both set packing and set partitioning cases for auction instances generated by Combinatorial Auction Test Suite (CATS - Brown et al. [2007]) including number of bidders varying between 5 and 10 and number of items varying between 3 and 18. Note that the smallest instance at any round might have at most 5 ∗ (23 − 1) columns whereas the largest

instance might have at most 10 ∗ (218− 1) columns.

In each case, we try resetting the warm-start tree either at each round (Figure 4.16, 4.18) or dynamically (Figures 4.17, 4.19). In the latter case, though many other rules specific to the structure of the auction problem can be used, we only reset the warm-start at the end of current round if the total # of columns added since the last reset is over p∗(# of bidders) or the solution time for the current round is over q∗(the solution time) of the WDP solved in the very next round of the last reset. For the following tests, the best outcome we have obtained was for p = 5 and

q = 1.25 and as a result, we have observed the reset counts for Figure 4.17:

WS-20% → 8.8% (59/672) WS-30% → 8.7% (58/668) WS-50% → 13.2% (36/272) WS-100% → 9.2% (25/272),

and for Figure 4.19:

WS-20% → 9.3% (27/290) WS-30% → 8.9% (44/497) WS-50% → 11.4% (33/290)

0 20 40 60 80 100 Relative time in ws instances Reset WS - RCan - pt1 rel-WS rel-noWS ws-tl-100% ws-tl-50% ws-tl-30% ws-tl-20%

Figure 4.16: WDP-Set Packing Formulation / Reset WS each round / WS consists of nodes below the r% level of the tree, r ∈ {20, 30, 50, 100}

WS-100% → 8.9% (26/289),

where WS-r denotes the corresponding case where only the nodes below level r% of the branch- ing tree used as a warm-start. Each column consists of the normalized solution times for the corresponding instance. In other words, if we let for a specific instance λW S be the total solution time with warm-starting, λN oW Sbe the total solution time without warm-starting, then rel-WS = 100 ∗ λ_{W S}/(λ_{W S}+ λ_{N oW S} and rel-noWS = 100 - rel-WS.

In all experiments, warm-starting for both strategies is dominating solving the instances from scratch. For set packing formulation, warm-starting at each round with a small tree size seems to be the best option. For set partitioning formulation, although both strategies are close to each other, the best option seems to warm start dynamically with a large tree size. One reason for this result is that the feasible solution space is restricted with equality constraints and the number of partitions of the feasible region in a larger tree helps to identify them quicker.

0 20 40 60 80 100 Relative time in ws instances Reset WS - Dyn - RCan - pt1

rel-WS rel-noWS

ws-tl-100% ws-tl-50%

ws-tl-30% ws-tl-20%

Figure 4.17: WDP-Set Packing Formulation / Reset WS dynamically / WS consists of nodes below the r% level of the tree, r ∈ {20, 30, 50, 100}

0 20 40 60 80 100 Relative time in ws instances Reset WS - BCan - pt1 rel-WS rel-noWS ws-tl-100% ws-tl-50% ws-tl-30% ws-tl-20%

Figure 4.18: WDP-Set Partitioning Formulation / Reset WS each round / WS consists of nodes below the r% level of the tree, r ∈ {20, 30, 50, 100}

0 20 40 60 80 100 Relative time in ws instances Reset WS - Dyn - BCan - pt1

rel-WS rel-noWS

ws-tl-100% ws-tl-50%

ws-tl-30% ws-tl-20%

Figure 4.19: WDP-Set Partitioning Formulation / Reset WS dynamically / WS consists of nodes below the r% level of the tree, r ∈ {20, 30, 50, 100}