effectiveness of quantity support in combinatorial...

Mat-2.108 Sovelletun matematiikan erikoistyö

Effectiveness of quantity support in combinatorial auctions

Valtteri Ervasti 51615N

1 INTRODUCTION............................................................................................................................... 3 2 AUCTION SETTINGS....................................................................................................................... 5

2.1 MATHEMATICAL FOUNDATIONS................................................................................................... 5 2.1.1 Winner determination problem............................................................................................... 6 2.1.2 Decision support problems ..................................................................................................... 6

2.2 AUCTION DESIGN ......................................................................................................................... 8 2.3 SIMULATION DESIGN .................................................................................................................. 11

3 VARIABLE INPUT PARAMETERS ............................................................................................. 13 3.1 PRODUCTION COST FUNCTION SETTINGS: ECONOMIES OF SCOPE ................................................ 13 3.2 PRODUCTION CAPACITIES........................................................................................................... 14 3.3 INITIAL BID GENERATION ........................................................................................................... 15

3.3.1 Method 3............................................................................................................................... 15 3.3.2 Method 4............................................................................................................................... 16 3.3.3 Method 5............................................................................................................................... 16 3.3.4 Setting the initial bid price ................................................................................................... 16

3.4 QUANTITY SUPPORT: AN EXPRESS VERSION................................................................................ 17 3.5 NUMBER OF BIDDERS ................................................................................................................. 18

4 FIXED INPUT PARAMETERS...................................................................................................... 19 4.1 NUMBER OF ITEMS ..................................................................................................................... 19 4.2 VARIANCE BETWEEN THE BIDDERS’ PRODUCTION COST FUNCTIONS .......................................... 19 4.3 EXPECTED PROPORTION OF FIXED COSTS.................................................................................... 19 4.4 OTHER FIXED INPUT PARAMETERS AND SETTINGS ...................................................................... 20

5 EXPERIMENT DESIGN ................................................................................................................. 21 5.1 FACTORIAL ANALYSIS VIEWPOINT.............................................................................................. 21 5.2 THE EFFICIENT ALLOCATION ...................................................................................................... 22 5.3 OUTPUT VARIABLES ................................................................................................................... 22

6 RESULTS .......................................................................................................................................... 24 6.1 EFFECTS ON Q/EFF, Q/SEFF, AND QC/EFF ................................................................................. 24 6.2 EFFECTS ON P2/EFF, P2/SEFF AND P2C/EFF .............................................................................. 25 6.3 EFFECTS ON P2-Q AND P2C/QC................................................................................................. 26 6.4 EFFECTS ON INI-Q ...................................................................................................................... 27 6.5 EFFECTS ON QSUCC ................................................................................................................... 27 6.6 EFFECTS ON EFF/SEFF................................................................................................................ 28 6.7 AUCTION LENGTH ...................................................................................................................... 29 6.8 NUMBER OF WINNING BIDDERS .................................................................................................. 30 6.9 INTERPRETATION OF RESULTS .................................................................................................... 31

7 CONCLUSION ................................................................................................................................. 32 REFERENCES ........................................................................................................................................... 33

2

1 Introduction This study extends the results obtained in my Master’s thesis [1], where the effect of price support and quantity support on the results of a combinatorial auction were studied by means of simulation. An auction is defined (Pekec & Rothkopf 2003) to be combinatorial if:

1) several, distinguishable items are sold at the same time, and 2) indivisible, “all or nothing” bids are accepted for multi-item combinations.

When the bidders’ valuations for the items are expected to be superadditive, a combinatorial auction design is expected to reach a higher level of efficiency than a non-combinatorial one (de Vries and Vohra 2003). It is a consequence of property 2) that winner determination is complex in combinatorial auctions. In the auctions of this study as well as many others, the winners are determined by solving an integer linear programming (ILP) problem, which has been proven (de Vries and Vohra 2003) to be NP-complete. Another consequence of 2) is that in combinatorial auctions, the item combination is another attribute by which bids are evaluated, in addition to the price and other possible attributes. In forward combinatorial auctions, two or more bids cannot be simultaneously accepted if they overlap: in reverse ones, they cannot be simultaneously accepted if they do not add up to total demand. Price support and quantity support are decision support tools for the bidders participating in a combinatorial auction. Their task is to help the bidders place the right bids in order to enter the winning group. Decision support is needed in combinatorial auctions where competition is possible not only with price, but with item combination as well. It is especially needed in sealed bid auctions, where the bidders cannot see each other’s bids, and therefore have no way of knowing which combination should be bid on. Nevertheless, not many studies have addressed decision support in combinatorial auctions. Those that have done so include Adomavicius and Gupta (2005) and Teich et al. (2006) and Leskelä et al., which introduced price and quantity support. Three auction types were defined in [1]. They were different to each other in the decision support system that was used, and similar in every other respect. In each simulation, the three auctions were simulated using the same input data, making pairwise comparisons possible. The same auction and simulation designs are used in this study as well. The results of [1] are extended by studying the effects of some factors that were fixed in [1]. This time, there are:

Two new methods to generate the initial bids. Different economies of scope. An “express” version of quantity support. The highest number of bidders is now 30 instead of 15. The bidders can have equal production capacities.

3

Contrarily, some factors that were variable in [1] are now fixed. The final cost to the auctioneer is considered the most important auction result: in a reverse auction, the cost is being minimized. Some other results, such as auction length and the number of winning bidders, are also studied. The simulations are run with Matlab as was done in [1]. A more efficient optimization engine, the Lindo API, is now being used to solve the winner determination and price/quantity support problems. Using the Lindo API also allows to calculate the so called efficient allocation and cost in each simulation. In this study, Chapter 2 defines the optimization problems related in winner determination and decision support, as well as the auction and simulation designs used in this study. The factors for which different levels are to be tested (also called input parameters), are described in Chapter 3: Chapter 4 describes the input parameters that have fixed values. Chapter 5 presents the experiment design and defines the output variables to represent the results.

4

2 Auction settings

2.1 Mathematical foundations The auctions considered in this study are reverse auctions. This means that the auctioneer is the buyer, and the bidders are sellers. In an auction, there are:

N items, each of which units are demanded by the auctioneer, { Ni ,,2,1 K∈ }

TjNjjj pqqB ;1 L= ij

iD n bidders, m bids.

Each bidder is required to submit at least one bid, so . The bids entered are of the form , where q is the quantity of item i included in bid j, and

is the price at which the bidder is willing to supply this item combination. Together, all bids in the auction comprise an

nm ≥[ ]

jpmN ×+ )1( bid matrix:

(2-1)

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

m

NmNN

m

pppqqq

qqq

bidmatrix

L

L

MOMM

L

21

21

11211

During the course of an auction, the bidders can submit new bids and remove old ones. To keep track of which bid belongs to which bidder, a separate mn× key matrix is defined:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

nmnn

m

keykeykey

keykeykeykeymatrix

L

MOMM

L

21

11211

, (2-2)

where if bid j belongs to bidder i, and ikeyij = 0=ijkey otherwise1.

As is seen in the next section, the key matrix can also be used to restrict the maximum number of bids that can be accepted from the same bidder.

1 When used to restrict the number of bids, it does not matter what values the nonzero elements of the key matrix assume as long as they are equal within each row. In fact, if they all had value 1 the effect of the corresponding constraints would be exactly the same. But this way the key matrix becomes easier, for the human eye, to interpret.

5

2.1.1 Winner determination problem With information about the bid matrix, the key matrix and the :s, the winners of a reverse, combinatorial auction can be determined by solving the winner determination problem (WDP)

iD

. (2-3)

binaryx

iikeyx

Dqx

Dqxtspx

j

m

jijj

N

m

jNjj

m

jjj

m

jjj

∀≤

≥

≥

≥

∑

∑

∑∑

=

=

==

1

1

11

11

..min

MM

The WDP minimizes the auctioneer’s total cost while satisfying the overall demand as defined by the :s. The constraints based on the key matrix ensure no more than one bid becomes active from each bidder. Reflecting bid indivisibility, each decision variable assumes value 1 if bid j is accepted, and 0 if it is rejected. Bids cannot be partially accepted.

iD

jx

When solved, the WDP produces the binary vector , indicating which bids are to be accepted and which are not. In a notation adopted from [1], the bids for which the corresponding are called active. Conversely, the bids for which are called inactive. In a similar way, the sets of active and inactive bidders are defined. The total cost C to the auctioneer equals the value of the objective function at optimum:

*x

1=jx 0=jx

∑=

=m

jjj pxC

1

*

The WDP can be linearized by leaving the binary constraints of the ’s out of (2-3). The dual of the linearized WDP can then be solved to obtain N dual or shadow prices

. These are later used to solve the quantity support problem.

jx

Ndd ,,1 L

2.1.2 Decision support problems Once the winners of a given auction round have been determined, price and quantity support can be used by any inactive bidder, in order to find a bid that will become active if submitted. The total cost to the auctioneer is required to decrease, so we set

. The size of this decrease is a compromise: as it is effectively inflicted on a single bid, a larger decrease would more likely be unprofitable for the bidder.

CCnew ×= 98,0

6

Conversely, a smaller decrease in total cost would probably cause the auction to last longer. Price support finds a new price P for a bid whose item quantities have already been set: if placed with this new price, the bid would become active. In the corresponding price support problem (PSP), P is maximized while the constraints ensure that the bid becomes active, the auctioneer’s total cost doesn’t exceed 2, and that the overall demand is met.

NQQ ,,1 L

newC

iikeyx

DQqx

DQqx

CPpxtsP

m

jijj

N

m

jNNjj

m

jjj

new

m

jjj

∀≤

≥+

≥

≥+

≤+

∑

∑

∑

∑

=

=

=

=

'

1

'

1

1

'

111

'

1

..max

MM (2-4)

with . jbinaryx

CP

j

new

∀≤≤0

As a solution, the PSP produces the vector { }'1

* ,,; mxxP L . If the bidder now submits the

bid [ , it will become active. ]

TN PQQ *

1 ;L

In quantity support, the item quantities are not fixed. Instead, they are decision variables like P and the ’s. They are required to be non-negative but not to exceed the bidder’s production capacities 3. Other constraints are the same as in the PSP. The objective function to be maximized is now the bidder’s profit, obtained by subtracting from P the cost to produce . The auctioneer does not know the bidder’s production cost function, so it is approximated by the dual prices.

NQQ ,,1 L

jx

Nll ,,1 L

NQQ ,,1 L

2 This constraint ensures the new bid will become active if submitted, since there is no other bid combination whose combined prices would amount to less than C. 3 See Section 3.2 for more information about the production capacities.

7

jbinaryxP

ilQ

Jiikeyx

NiDQqx

CPpxtsQdP

j

ii

m

jijj

i

m

jiijj

new

m

jjj

N

iii

∀≤

∀≤≤

≠∀≤

=≤+

≤+−

∑

∑

∑∑

=

=

==

00

,,1

..max

'

1

'

1

'

11

K

(2-5)

After a solution { }'1

**1

* ,,;,,; mN xxQQP LK to the QSP has been obtained, the bid

will become active if it is submitted. [ TN PQQ ***

1 ;L ]

]

2.2 Auction design Three types of auctions are defined, each with a different type of decision support:

Auction with price support (referred to as a P1 auction) Auction with intelligent price support (P2 auction) Auction with quantity support (Q auction)

Price support is used as the decision support method in both P1 and P2 auctions. The difference is that in the P1 auction it is only used for the bidder’s existing bid, while in the P2 auction it is also used for the corresponding subcombinations, obtained by setting some of the ’s to zero. This is done in an attempt to eliminate possible surplus items from the bid, increasing its chances to become active.

ijq

Intelligent price support should not be thought of as a separate decision support tool. Instead, it attempts to simulate the behaviour of the inactive bidders as they try using price support on different item combinations. For example, suppose a bidder had submitted the bid (2-6) [ T2000030000225300 and it is currently inactive. In a P1 auction, he would use price support for this bid only, to obtain an acceptable price . But in a P2 auction, he would use price support for the bids

20000<P

8

. (2-7)

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

87654321

1503000003003003000000000000000000

5,112022502250225225150003003003000300

PPPPPPPP The collection (2-7) includes the original bid, the original bid with the item quantities halved, and the one- and two-item subcombinations, of which there are six. The collection of bid and price propositions is called the shortlist. In addition to the P2 auctions, a shortlist is also created in the Q auctions: this is described in Section 3.4. In this context, it can be said that the shortlist is also created in the P1 auctions, but it only contains one entry. Since the original bid (2-7) now contains three nonzero item quantities, the shortlist has only If all the item quantities were nonzero in the original bid, the shortlist would contain entries.

81123 =+−321125 =+−

The auctions can now be described algorithmically as follows. The graphic outline of an auction is presented in Figure 2.1. Step 0. Initialization. Before the auction actually begins, the set of bidders is defined: this includes defining the bidders’ production capacities and cost functions. After this the bidders place and price their original bids. The initial bid matrix now constitutes n columns. 1. Winner determination. The auctioneer solves the winner determination problem, leaving some bids active and the rest inactive. A set of active bidders is created, as well as a set of inactive bidders. Solving the WDP in its linear form will also produce a set of dual prices for the N items. Proceed to Step 2. Ndd ,,1 K

2. Using price or quantity support. One inactive bidder, with identity j , is chosen at random to use price or quantity support. This is where the auctions differ from each other:

In a P1 auction, the PSP is solved for bidder j’s existing bid. In a P2 auction, the PSP is solved for bidder j’s existing bid and the

corresponding subcombinations as defined above. In a Q auction, the QSP is solved several times as defined in Section 3.4.

9

As a result, a shortlist of bid propositions is then presented to bidder j. The bidder then compares bid prices in the shortlist to the costs of producing the item quantities involved4:

lp

NlNjljjNl qcqcFqf +++= ...)( 11,12K (2-8) and finds the most profitable bid, that is, the bid for which )(qfp ll − reaches its maximum value. If 0)( >− qfp jl , this bid is added at the end of the bid matrix, and another column is added to the key matrix, with bidder identity number at the correct row. (In a P1 auction, no new bid is added in the bid matrix: the original bid price is altered instead.) If bidder j has previously placed a bid whose item quantities are identical to the new one but the price is higher, the older bid is deleted from the bid matrix. Also the corresponding column is deleted from the key matrix in this case. The first round of the auction has ended. Return to Step 1. If no bids on the shortlist are profitable, the bidder doesn’t choose any of them and is removed from the set of inactive bidders. If the set has now become empty, proceed to Step 3. If not, the turn to use quantity support is then passed on to another bidder in the inactive set. Repeat Step 2. Step 3. End auction. Coming here means all inactive bidders have used price/quantity support and rejected its propositions for new bid, while the bid matrix hasn’t changed. The auction has ended, and currently active bidders are the final winners. The winners, and the item combinations included in their bids, are called the winning allocation.

Figure 2.1. The outline of an auction.

4 For information about the production cost functions, see Section 3.1.

10

To summarize, an auction round can end in one of two ways:

1. A bidder accepts a proposition of price or quantity support and places it as a new bid (or in a P1 auction, updates the price).

2. The set of inactive bidders becomes empty. The former case means the auction moves into the next round, where the WDP is solved again to obtain a new set of winners and a new, decreased total cost to the auctioneer. The latter case means the auction has ended. In this auction design, the round length is variable, as is the number of rounds. Finally, it should be noted that the main interest now is to study the differences in results between the P2 and Q auctions. However, the P1 auctions are included in each simulation.

2.3 Simulation design A simulation is defined as an independent set of three auctions that is initialized by a set of input parameter and produces a set of output variable values. Each simulation contains three phases:

Initialize the simulation. Equals Step 0 described above. o Define the set of bidders, production cost functions and capacities o Calculate the efficient allocation o Submit the initial bids

Run the auctions Record the results

Because the initial bid matrix is the same in all three auctions, also the initial winning bidders (first results of Step 1) are the same. The auctions begin to divert when a randomly selected bidder uses price/quantity support in Step 2. The same input parameter values are used in all three auctions, making pairwise comparisons possible within a simulation. An outline of a simulation is presented in Figure 2.2.

11

Figure 2.2. The outline of a simulation.

Several parameters must be assigned values before a simulation can take place: these are called input parameters. The intention is now to run many simulations with different input parameter values, to see what effect their variation has on the simulation results. Those input parameters that are assigned multiple values are defined in Chapter 3. The rest of the input parameters are assigned fixed values, which remain unchanged throughout the simulations. These are defined in Chapter 4. The output variables, used to represent the auction results, are defined in Chapter 5. For more details on the mathematical foundations, the auctions and the simulations, see Chapters 3 and 4 of [1].

12

3 Variable input parameters This chapter defines the input parameters that are assigned multiple values.

3.1 Production cost function settings: economies of scope Each bidder j has his production cost function defined as NjNjjjjIj qcqcFqf +++= ...)( 11, , (3-1) where

= variable cost to bidder j to produce one unit of item i. ijc = fixed cost to bidder j to produce nonzero quantities of the combination

. jIF ,

{ }L,, 21 iiI = Therefore, each bidder has a well defined production cost for each item combination

. As stated in Section 2.2, the bidders evaluate the propositions of price and quantity support by looking at the difference of the proposed bid price P and the production cost

NQQ ,,1 L

)(qf j . If 0)( >− qfP j , the proposition is profitable. Economies of scope are defined to exist if the cost of producing a combination of items is lower than the sum of the costs of producing each item separately. Formally defined, if

is the fixed cost involved in producing the item combination I, economies of scope exist whenever

jIF ,

is true for all jJIjJjI FFF ,,, ∪>+ JI ≠ . Two levels of economies of scope are defined and tested in this study. In practice, this means assigning different values to the ’s. When I is a combination of the two items i1 and i2, the expected value of is

jIF ,

jIF ,

( ))(5,1)( ,1, jijI FEFE ×= with Scope 1, and

( ))(4,1)( ,1, jijI FEFE ×= with Scope 2. With larger combinations, each entering item adds ( )jIFE ,5,0 × to the expected fixed cost when Scope 1 is used. With Scope 2, ( )jIFE ,4,0 × is added instead. The numerical values of the ’s are presented in Table 3.1. jIF ,

13

Table 3.1. Fixed cost parameters and their distributions under different economies of scope.

Economies of scope: normal Economies of scope: large

F j,I Minimum Expected Maximum F j,I Minimum Expected Maximum1 item 16000 18000 20000 1 item 16000 18000 20000

2 items 24000 27000 30000 2 items 22400 25200 280003 items 32000 36000 40000 3 items 28800 32400 360004 items 40000 45000 50000 4 items 35200 39600 440005 items 48000 54000 60000 5 items 41600 46800 52000

Unlike the ’s, the variable cost parameters are fixed. They are drawn from the uniform distribution [ ], with

jIF , ijc67,6633,53 60)( =ijcE .

3.2 Production capacities As was done in [1], the production capacities of each bidder j and item i are denoted with

’s: ijl = the largest quantity of item i that can be produced by bidder j. ijl The total demand for each item is defined to be 600 units. With inequal production capacities, the :s are drawn from the discrete distribution:

iD

ijl

= 300 with 50% probability. ijl = 225 with 25% probability. ijl = 150 with 25% probability. ijl

This distribution is intended to model the division of the bidders into small, medium, and large players in the industry. The ’s drawn do not affect each other in any way, so a bidder may well be a “small” player in one item and “large” in another. The maximum value = 300, equaling 50% of total demand, also simulates the auctioneer not wishing to be too dependent on one supplier.

ijl

ijl

With equal production capacities, we set for all ’s: ijl

= 300 ijl When the production capacities are equal, all bidders can bid on every combination, and the competitiveness of the auction is expected to increase.

14

3.3 Initial bid generation After the production cost functions and capacities have been defined, the next step is to place the initial bids. It consists of two phases: setting the item quantities, and setting the price. In [1], three methods to generate the initial item quantities were defined. These were:

Method 1: the item quantities were drawn from the uniform distribution [ ]ijl,0 . Method 2: the item quantities were drawn from normal distribution with

parameters: o Expected value )1( ijij advl −× o Standard deviation ijl×25,0 .

Method 3: the item quantities were discrete: either 0=ijq or , depending on the variable production costs.

ijij lq =

In Method 1, the production cost didn’t affect the item quantities of the bid at all. In Method 2, the variable costs were considered so that if a bidder’s variable cost to produce item i was lower than average, he was more likely to bid for a large quantity of this item. In Method 3, the bidder would always bid for a large quantity if he had an advantage in variable costs. The initial bids were found to significantly affect the outcome of an auction, with Method 3 generally amounting to the best results. In this study, it will be compared to two other methods to generate initial bids. The next three sections describe the bid generation methods tested in this study.

3.3.1 Method 3 With N items in the auction, each bidder j has N variable cost parameters , whose values have been drawn from the uniform distribution

jic[ ]ii ba . Cost advantage indices

are then defined by 3ijadv

ii

iijij ab

aFadv

−

−=3 . (3-2)

The item quantities are then set at zero if the bidder has higher than average , and at maximum capacity if he has lower than average :

ijq jic

jic

if . 0=ijq 5,0≥ijadv if . ijij lq = 5,0<ijadv

15

3.3.2 Method 4 Method 4 considers fixed cost parameters rather than variable ones. Other than that, it is similar to method 3. With N = 5 items in the auction, each bidder j has fixed cost parameters , one for each item combination I. They are drawn from the uniform distribution

. As a result, each bidder will have 31 cost advantage indices , defined by

31125 =− IjF

[ II BA ] 4Ijadv

II

IIjIj ab

aFadv

−

−=4 . (3-3)

For each bidder, the minimum of is found subject to I. Then we set for all

.

4Ijadv ijij lq =

4minIi∈

3.3.3 Method 5 The idea of method 5 is to find the combination each bidder is most efficient producing the maximum amount of, when both fixed and variable costs are considered. For each item combination I, there is the bidder’s true production cost

, and the expected production cost that all bidders share:

. Based on these, we now define 31 cost advantage indices

for each bidder:

∑∈

+=Ii

ijijIjIjTRUE lcFf ,

∑∈

+=Ii

ijijIjIjEXP lcEFEf )()(,

IjEXP

IjTRUEIj F

Fadv

,

,5 = . (3-4)

As in the previous method, the minimum of is found subject to I, and we set

for all .

5Ijadv

ijij lq = 5minIi∈

3.3.4 Setting the initial bid price

16

Once the item quantities of a bid have been set, the bidder has to decide the price at which he offers to supply this item combination. As stated earlier, the bidder’s cost to produce the combination equals

Njj qq ,,1 L

jp

jNjNjjNjj qcqcFqf +++= ...)( 1112, K . The initial price is then set 20% higher than the production cost: jp )(2,1 qfp jj ×= . This creates an initial profit margin of 16,67%.

3.4 Quantity support: an express version In Section 2.2, the quantity support problem (QSP) was defined as

jbinaryxP

iLQ

Jiikeyx

NiDQqx

CPpxtsQdP

j

ii

m

jijj

i

m

jiijj

new

m

jjj

N

iii

∀≤

∀≤≤

≠∀≤

=≥+

≤+−

∑

∑

∑∑

=

=

==

00

,,1

..max

'

1

'

1

'

11

K

(2-5).

To approximate the bidder’s production cost function, the dual prices are used in the objective function. As this is only an approximation, it can happen that the optimal solution to the QSP is not the most profitable one to the bidder. In the worst case, the QSP can produce a solution that is unprofitable to the bidder, while a profitable one would have been found using a different objective function.

id

To account for the inaccuracy of the approximation, alternative solutions of the QSP can be generated by adding extra constraints of the types 1=jx and 0=iQ one or more at a time. The collection of these solutions is called the shortlist. It is now studied how the effectiveness of quantity support is affected by the length of the short list. We define the full version of the shortlist to contain:

The original solution obtained with no additional constraints. Solutions obtained with all the 1=jx constraints, one at a time. Solutions obtained with the 0=jQ constraints for all -item

combinations. 1,,2,1 −nK

17

The express version of the shortlist does not apply the 1=jx constraints, and therefore contains:

The original solution obtained with no additional constraints. Solutions obtained with the 0=jQ constraints for all -item

combinations. 1,,2,1 −nK

These two versions of the shortlist are now compared, particularly to see if the transition from the full version to the express version significantly worsens the auction results. In the simulations, the original solution, obtained with no additional constraints, was placed at the end of the shortlist. This is done to make it easier to monitor the performance of quantity support.

3.5 Number of bidders The number of bidders was a variable input parameter in [1]: its assigned values were 10 and 15. Changes in the number of bidders were found to affect many output variables: for example, the difference in the final cost of a P2 auction and a Q auction decreased when there were more bidders. It would be interesting to see if this trend continues when the number of bidders is further increased. This is straightforward and it doesn’t affect the winner determination or the quantity support problems, although it does make the calculations more time-consuming. In this study, the simulations are run with 15 and 30 bidders. The variable input parameters are summarized in Table 3.2. Table 3.2. Variable input parameters and their values.

VARIABLE INPUTS

Short list Full ExpressProduction capacities Equal DifferentInitial bid generation B3 B4 B5Economies of scope Normal LargeNumber of bidders 15 30

18

4 Fixed input parameters In the previous chapter, five input parameters were assigned multiple values in order to study the effect of their variation on the auction results. To keep the size of the experiment reasonable, the rest of the input parameters are assigned fixed values. The first three sections of this chapter specify those input parameters that were variable in [1], but are fixed now. The fourth section briefly describes the rest of the input parameters. For a thorough description of the input parameters and auction rules, see Chapter 4 of [1].

4.1 Number of items The simulations in [1] were run with 3 and 5 items in the auction. While it would be interesting to further increase the number of items, this could only be done with some simpler form of production cost function. This is because a separate fixed cost parameter

is defined for each bidder, and each item combination I. The number of combinations increases exponentially with the number of bidders: with 5 items, there are

such combinations, with 7, there are , and with 9, there are . In its present form, the efficient allocation (see Section 5.2.) would be very

difficult to calculate with more items. Therefore in this study, the number of items is set

jIF ,

31125 =− 127127 =−511129 =−

at 5.

4.2 Variance between the bidders’ production cost functions The variable and fixed cost parameters are drawn from the uniform distribution [ ]ba, . The variance is the length of this interval, as a percentage of the lowest possible cost:

%100×−a

ab . It represents the highest percentage by which one bidder’s cost to produce

some item combination can be higher than another’s. In [1], variance was tested at 25% and 100%. It was found that with 100% variance, the final cost of the auction tended to be higher, and that quantity support’s success rate tended to be lower. It seems unnecessary to test higher levels of variance. In this study, the variance is set at 25%.

4.3 Expected proportion of fixed costs This input parameter is defined as “the proportion of fixed costs out of expected total production costs, when units of item i are produced by one bidder”. In [1], the values of 10% and 50% were tested for this input parameter. This time, there are no

iD×5,0

19

resources to test its effect, as other parameters have been chosen for closer study. The expected proportion of fixed costs is set at 50%.

4.4 Other fixed input parameters and settings The remaining input parameters are fixed at the same values as they were in [1]. Most of these have already been mentioned in the previous sections, but a summary is presented here for clarity.

Total demand for each item is set at 600. iD The maximum number of units of each item that can be allocated to one bidder is

set at . This is taken into account in the production capacities , which never exceed 300.

3005,0 =× iD ijl

At most one bid is to become active from any one bidder. The price decrement on each round is set at 2%. In other words, each auction

round decreases the total cost by 2%. The profit margin included in the initial bids is set at 16,67%. On each auction round, one inactive bidder at a time is selected to use price or

quantity support. This selection is made at random. When evaluating the bid propositions of price and quantity support, the bidders

are prepared to accept prices that include, at least, a price equal to the production cost. Therefore, the minimum profit margin is 0%.

The bidders evaluate the bid propositions in terms of absolute profit rather than relative.

The number of auction rounds is unlimited. The auction ends when no bidder has accepted the propositions of price or quantity support.

All fixed input parameters are described in Table 4.1.

Table 4.1. Fixed input parameters.

FIXED INPUTS & AUCTION SETTINGS

Number of items 5Cost variation 25 %Proportion fixed cost / total 50 %Total demand of each item 600Max. allocation given to one bidder 300Max. accepted bids from one bidder 1Price decrement 2 %Initial profit margin 16,67 %Minimum profit margin 0 %Selection of PS and QS users RandomProfit calculation AbsoluteShort list generation, int. price support See 2.2Number of rounds Unlimited

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5 Experiment design With the input parameters defined in Chapters 3 and 4, it is possible to initiate and run an individual simulation. The purpose of this study is to run many simulations with different input parameter values, to see what effect their variation has on the results. In this chapter, Section 5.1. defines how the simulations are arranged. The output variables that represent the simulation results are defined in Section 5.3.

5.1 Factorial analysis viewpoint In the context of factorial analysis, the variable input parameters defined in Chapter 3 are referred to as factors. The effect of a factor is defined to be the change in response5 produced by a change in the level of the factor (Montgomery 1991). In Chapter 3, a total of five factors were defined to be tested. Letters A to E are now introduced to represent the factors as described by Table 5.1. Factor C (Initial bids) was previously assigned three levels, while the other four were assigned two levels. All factor level combinations are tested, equaling a factorial design. Each set of factor levels is called a block. To gain statistical significance for the results, each block contains 50 simulations as they did in [1]. The total number of simulations then becomes

.

48324 =×

24005048 =× Table 5.1. The factors and their levels.

FACTOR LEVELS

A: Quantity support Full ExpressB: Production capacities Equal DifferentC: Initial bid generation B3 B4 B5D: Economies of scope Normal LargeE: Number of bidders 15 30

Of the factors presented in Table 5.1, the type of quantity support and the number of bidders define rules that can be directly used to initialize and run an auction. The remaining three factors define distributions from which the actual input data are drawn for each simulation. Therefore, any two simulations will have somewhat different input data even if their factor levels are the same. The output data is divided into groups sorted by different factor levels. Means and medians are then calculated for each group, and the median test is used to determine if the differences between groups are statistically significant. The median test is non-parametric, meaning it does not assume the data to be normally distributed, or the variances in different subgroups to be equal. The data in [1] was found in general not to 5 Here, the response is referred to as output.

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meet such assumptions: the same is expected here. For more information about the median test, see Conover (1991). Before defining the output variables, a brief outlook into the efficient allocations is in order.

5.2 The efficient allocation When the production capacities and cost functions have been defined for all the bidders, it is possible to determine the set of bidders who can produce the total item quantities

at the lowest cost. This can be done by solving a specific linear optimization problem: a detailed description can be found in Section 3.4. of [1]. This set of bidders, and the item quantities each of them is supposed to produce, is called the efficient allocation. The sum of the production costs is called the efficient cost. Given the particular set of bidders, the efficient cost is the lowest possible total cost that can theoretically be reached in an auction.

NDDD ,,, 21 K

In [1], the efficient allocation was not calculated because of insufficient computing capacity. Instead, it was approximated by a “semi-efficient” allocation where the items were allocated to the bidders one by one, ignoring the fixed cost parameters of multi-item combinations. There was no information of how close the semi-efficient cost was to the real efficient cost, and its main purpose was to serve as a benchmark with which the auction results could be compared across the whole range of factor levels. In this study, the true efficient cost is calculated in every simulation.

5.3 Output variables The following abbreviations are used when denoting the simulation results:

Ini = Initial auction cost P1 = Final cost of a P1 auction P2 = Final cost of a P2 auction P2C = The sum of the production costs for the winning allocation of a P2 auction Q = Final cost of a Q auction QC = The sum of the production costs for the winning allocation of a Q auction SEff = Production cost of the semi-efficient (heuristic) allocation Eff = Production cost of the true efficient allocation

It should be noted that the final cost variables P1, P2 and Q are not directly comparable across all factor levels. For example, the different levels of factor D (Scope) assign different values to the fixed cost parameters, affecting the expected production costs of all item combinations: with Scope 2, the :s are generally lower than with Scope 1. Therefore, also P2 and Q are expected to be lower.

jIF ,

22

In a similar way, the efficient cost is also expected to vary with different factor levels: for example, Eff is expected to assume lower values with Scope 2 than with Scope 1. As a consequence, P2/Eff and Q/Eff are expected to be comparable across all factor levels, with 1 being the lowest value they can possibly assume. In other words, Eff serves as a benchmark that makes the final costs comparable at all factor levels. SEff is calculated so it can be compared with Eff, and also to make the results of this study easier to compare to those of [1], where only SEff was calculated. The production costs P2C and QC are interesting because they can be directly compared to SEff and Eff which also represent production costs. It is true that . )( QinincludedprofitQQC −= If a value of 1 is observed for QC/Eff in a simulation, it means the efficient allocation has been found in the Q auction. This would not be not be noticed just by looking at Q/Eff, because even when the efficient allocation is reached, there is usually still some profit margin included in the total cost Q. In such a case the variable Q/Eff would assume a value higher than 1. In other words, the variable QC/Eff contains information about how close the winning allocation is to the efficient allocation, and the variable Q/Eff contains information about how close the final cost is to the efficient cost. Both variables are interesting in their own right. In the next chapter, mean and median results are presented for the following output variables:

Q/Eff, Q/SEff and QC/Eff represent the final cost of the Q auctions. P2/Eff, P2/SEff and P2C/Eff represent the final cost of the P2 auctions.

2

2P

QP − and CPQCCP

22 − represent the pairwise difference between the final costs

of the P2 and the Q auctions of the same simulation.

Ini

QIni − represents the cost decrease achieved during the course of a Q auction.

Qsucc represents the success rate of quantity support. Eff/SEff represents the accuracy of SEff in approximating Eff. Auction length and the number of winning bidders are also observed.

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6 Results This chapter presents the mean and median results sorted by the different levels of the five factors. The results for all 48 factor level combinations are available in Appendix 1.

6.1 Effects on Q/Eff, Q/SEff, and QC/Eff The mean and median results for the variables Q/Eff, Q/SEff and QC/Eff are shown in Table 6.1. The results are sorted by different factor levels: also the standard deviations are shown. Table 6.1. Factorwise mean and median results of Q/Eff, Q/SEff, and QC/Eff. Q/Eff

Mean Median St.Dev.All 1,062 1,048 0,048

A: Shortlist Express 1,066 1,049 0,053Full 1,058 1,046 0,042

B: ProdCap Equal 1,025 1,022 0,014Inequal 1,099 1,092 0,041

C: Bid 3 1,065 1,051 0,0514 1,061 1,046 0,0485 1,060 1,047 0,045

D: Scope 1 1,062 1,048 0,0482 1,062 1,049 0,048

E: Bidders 15 1,070 1,055 0,05430 1,054 1,041 0,039

Q/SEffMean Median St.Dev.

All 0,892 0,881 0,064



C: Bid 3 0,905 0,898 0,0674 0,885 0,875 0,0635 0,884 0,874 0,059

D: Scope 1 0,897 0,886 0,0642 0,886 0,876 0,064

E: Bidders 15 0,910 0,898 0,06930 0,873 0,867 0,052

QC/Eff




C: Bid 3 1,027 1,017 0,0284 1,025 1,016 0,0265 1,027 1,015 0,029

D: Scope 1 1,026 1,016 0,0282 1,026 1,016 0,028

E: Bidders 15 1,028 1,017 0,03030 1,025 1,015 0,025

Judging by the observations obtained for Q/Eff, it can be seen that the final cost of a Q auction was around 5 to 6 per cent higher than the theoretical minimum Eff. This seemed to be very stable across different factor levels, with the exception of factor B (Production

24

capacities). When all the bidders had equal production capacities , the final cost Q was little more than 2% higher than Eff. With inequal production capacities, this difference was around 9% to 10%. Also factor E (Bidders) had a minor, but statistically significant effect, with Q/Eff decreasing when there were more bidders. The other factors were labeled by the median test as statistically insignificant.

ijl

The variables Q/SEff and QC/Eff behaved in a similar way to Q/Eff. The factor effects were qualitatively similar, but larger for Q/SEff and smaller for QC/Eff. The results of QC/Eff show that the winning allocation of a Q auction was usually quite close to the efficient one, with the production cost, on average, only 2,6% higher than the efficient cost.

6.2 Effects on P2/Eff, P2/SEff and P2C/Eff The mean and median results for the variables P2/Eff, P2/SEff and P2C/Eff are shown in Table 6.2. Table 6.2. Factorwise mean and median results of P2/Eff, P2/SEff, and P2C/Eff. P2/Eff



C: Bid 3 1,168 1,143 0,0984 1,164 1,142 0,0915 1,261 1,257 0,117

D: Scope 1 1,196 1,178 0,1112 1,200 1,177 0,113

E: Bidders 15 1,239 1,231 0,12030 1,157 1,133 0,085

P2/SEffMean Median St.Dev.

All 1,006 0,987 0,115


C: Bid 3 0,993 0,975 0,1074 0,972 0,956 0,1005 1,053 1,046 0,120

D: Scope 1 1,011 0,993 0,1122 1,001 0,982 0,117

E: Bidders 15 1,054 1,043 0,12030 0,958 0,941 0,086

P2C/Eff



C: Bid 3 1,065 1,051 0,0484 1,067 1,058 0,0425 1,109 1,096 0,062

D: Scope 1 1,079 1,066 0,0552 1,082 1,069 0,056

E: Bidders 15 1,096 1,082 0,06430 1,065 1,059 0,039

Overall, the final cost of a P2 auction was about 18% to 20% higher than Eff, with factors B and E having visible effects. The behavior of P2/Eff was somewhat similar to that of Q/Eff, but with higher standard deviations, and factors B and E having more effect. The

25

big qualitative difference was the effect of factor C (Bids). While the initial bids had no significant effect on Q/Eff, the values of P2/Eff were clearly increased when the initial bids had been generated with Method 5. The median test labeled factors B, C and E as statistically significant. Table 6.2. demonstrates that the results obtained for P2/SEff and P2C/Eff qualitatively resembled those of P2/Eff. Factor effects and the standard deviations were somewhat smaller for P2C/Eff.

6.3 Effects on P2-Q and P2C/QC

The variable 2

2P

QP − (shortened to P2-Q) exists to represent the decrease in final cost

when quantity support was used instead of intelligent price support. Additionally, the

variable CPQCCP

22 − (shortened to P2C-QC) was observed to study the decrease the

corresponding decrease in production cost. The results are shown in Table 6.3. Table 6.3. Factorwise mean and median results of P2-Q and P2C-QC. P2-Q

Mean Median St.Dev.All 10,81 % 9,61 % 6,45 %

A: Shortlist Express 10,59 % 9,61 % 6,48 %Full 11,02 % 9,61 % 6,42 %

B: ProdCap Equal 8,84 % 7,76 % 5,29 %Inequal 12,77 % 13,19 % 6,90 %

C: Bid 3 8,40 % 7,76 % 5,59 %4 8,52 % 7,76 % 5,00 %5 15,50 % 14,92 % 5,98 %

D: Scope 1 10,69 % 9,61 % 6,46 %2 10,93 % 9,61 % 6,44 %

E: Bidders 15 13,06 % 13,19 % 6,75 %30 8,56 % 7,76 % 5,25 %

P2C-QCMean Median St.Dev.

All 4,83 % 4,27 % 4,01 %



C: Bid 3 3,44 % 3,04 % 3,50 %4 3,81 % 3,48 % 3,08 %5 7,24 % 6,64 % 4,22 %

D: Scope 1 4,68 % 4,18 % 3,95 %2 4,98 % 4,38 % 4,06 %

E: Bidders 15 5,95 % 5,21 % 4,43 %30 3,71 % 3,53 % 3,17 %

It can be seen that overall, the decrease P2-Q was about 10%, but less (about 8%) with more bidders or equal production capacities, and more (about 15%) when the initial bids were generated with Method 5. Judging by the results obtained for P2/Eff in Section 6.2, the effect of factor C on P2-Q can be accounted to P2 being higher with C (Bids) = 5. When, instead, the production costs included in the winning allocations of the P2 and Q auctions were compared, the difference was not as large but still positive. Qualitatively, the effects of the factors on P2C-QC were similar to those on P2-Q. If efficiency is defined as the ability to produce the items at the lowest cost, it can now be stated that the Q auctions allocated the items more efficiently.

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6.4 Effects on Ini-Q

The variable Ini

QIni − (shortened to Ini-Q) represents the price decrease that occurred

during a Q auction. The mean and median results are shown in Table 6.4. Table 6.4. Factorwise mean and median results of Ini-Q.

Mean Median St.Dev.All 18,56 % 18,29 % 3,81 %



C: Bid 3 17,77 % 16,66 % 3,74 %4 17,45 % 16,78 % 3,12 %5 20,45 % 19,93 % 3,80 %

D: Scope 1 18,50 % 18,29 % 3,79 %2 18,62 % 18,29 % 3,83 %

E: Bidders 15 19,61 % 18,51 % 4,42 %30 17,50 % 17,78 % 2,69 %

The values of Ini-Q were fairly stable, with the means and medians varying between 16% and 21%. Larger decreases in total cost were observed when there were more bidders (factor E) and when the initial bids were generated with Method 5 (factor C). This latter case is suspected to have occurred because Ini was generally higher with C = Bid 5 than with other bid generation methods.

6.5 Effects on QSucc The variable Qsucc measures the success rate of quantity support. It is defined by:

( )( )usedQSN

succeededQSNQsucc = .

As defined in Section 3.4, the last entry of the shortlist corresponds to the solution obtained for the QSP without any additional constraints. Quantity support is therefore defined to have succeeded when the bidder using it identifies the last entry of the shortlist as the most profitable one. When this happens, the bidder’s production cost function jNjNjjNjj qcqcFqf +++= ...)( 1112, K has successfully been approximated by the objective function

27

∑=

−N

jiiQdP

1

of the quantity support problem. The mean and median results of Qsucc are shown in Table 6.5. Table 6.5. Factorwise mean and median results of Qsucc.




C: Bid 3 0,748 0,769 0,2464 0,750 0,772 0,2505 0,775 0,829 0,229

D: Scope 1 0,757 0,797 0,2452 0,758 0,778 0,239

E: Bidders 15 0,740 0,755 0,25330 0,776 0,827 0,229

Large variation was typically observed for Qsucc, as demonstrated by the high standard deviations. The overall mean was 0,783. Very high success rates occurred when all the bidders had equal production capacities: with unequal production capacities, the results were closer to those obtained in [1], where the production capacities were also unequal. Factor A (Shortlist) had statistical significance, as did E (Bidders).

6.6 Effects on Eff/SEff The relation between the true (Eff) and the approximative (SEff) efficient cost is very interesting, but it was not investigated in [1], as Eff was not calculated. In this study, both Eff and SEff were calculated in each simulation, and the results for Eff/SEff are shown in Table 6.6. Factors A and C are not shown, as

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Table 6.6. Factorwise mean and median results of Eff/SEff.

Eff/SEffMean Median St.Dev.

All 0,839 0,837 0,032


D: Scope 1 0,844 0,843 0,0322 0,833 0,831 0,032

E: Bidders 15 0,850 0,849 0,03330 0,828 0,827 0,028

Table 6.6 shows that the means and medians of Eff/SEff were very stable between 0,82 and 0,86. The standard deviations were also low. Individual values observed for Eff/SEff were also quite stable, usually variating between 0,75 and 0,9. All three factors were labeled statistically significant by the median test, but the effects were very small. Based on these results, it can be stated that under the cost function settings of this study, SEff was about 20% higher than Eff.

6.7 Auction length Auction length can be measured by the number of rounds. But since each round consisted of a variable number of bidders using price or quantity support, the round length was variable. Another, perhaps more accurate measure of auction length would be the number of times price or quantity support was used during the entire auction. These are referred to as iterations. Nevertheless, the number of rounds is interesting in its own right, as it corresponds to the decrease in total cost (because each round decreased the total price by 2%). It should also be noted that the number of iterations is not comparable across all factor levels: when the number of bidders increases, so does the number of iterations. On the other hand, the number of rounds can be compared across all factor levels. Table 6.7 shows the means, medians and standard deviations of the numbers of iterations and rounds. Overall results are shown, as well as those obtained with different levels of B (Production capacities) and E (Bidders), and with C (Bids) = 5. It can be observed that changing from equal to unequal production capacities increased the number of iterations in the Q auctions, but decreased them in the P1 and P2 auctions. To a lesser extent, the same is true of the number of rounds.

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Table 6.7. Auction length measured in rounds and iterations. AUCTION LENGTH

Overall Equal production capacities Inequal prod. capacitiesMean Median St. Dev, Mean Median St. Dev, Mean Median St. Dev,

Iter(P1 ) 40,41 35 25,30 46,56 43 23,75 34,26 26 25,31Iter(P2 ) 42,75 38 24,84 45,99 44 23,10 39,50 32 26,08Iter(Q ) 47,27 43 18,96 41,04 39 13,95 53,50 49 21,14

N(P2 ) 5,42 6 2,54 6,23 7 2,13 4,61 4 2,66N(Q ) 11,19 11 2,37 10,86 11 1,58 11,53 11 2,91

15 bidders 30 bidders Bid 5Mean Median St. Dev, Mean Median St. Dev, Mean Median St. Dev,

Iter(P1 ) 22,00 20 12,03 58,82 59 21,38 33,85 26 24,17Iter(P2 ) 24,95 24 11,74 60,55 58 21,51 35,84 29 23,88Iter(Q ) 34,56 33 9,60 59,98 57 17,43 48,59 44 18,90

N(P2 ) 4,81 5 2,63 6,03 7 2,29 3,92 4 2,28N(Q ) 11,86 11 2,77 10,52 10 1,62 12,36 12 2,42

6.8 Number of winning bidders Quantity support was found to decrease the number of winning bidders. As described in Table 6.8, the mean number of winning bidders in a Q auction was 2,56, as opposed to about 4,1 in the P1 and P2 auctions. The number of winners also varied less in the Q auctions. The most effective factor was B (Production capacities): when all bidders had equal production capacities, there were less winners in all three auctions than when the production capacities were unequal. But even in this case, the Q auctions nearly always finished with less winning bidders than the P1 and P2 auctions. It can also be noted that when the initial bids were generated with Method 5, the number of winners increased strikingly in the P1 and P2 auctions, from around 3,6 to around 5. Meanwhile, the Q auctions were largely unaffected. Table 6.8. The number of winning bidders NUMBER OF WINNING BIDDERS

Overall Equal production capacities Inequal prod. capacitiesMean Median St.Dev, Mean Median St.Dev, Mean Median St.Dev,

P1 4,10 4 1,25 3,41 3 0,87 4,80 5 1,18P2 4,13 4 1,21 3,45 3 0,84 4,82 5 1,14Q 2,56 2 0,62 2,06 2 0,24 3,07 3 0,44

Bid 3 Bid 4 Bid 5Mean Median St.Dev, Mean Median St.Dev, Mean Median St.Dev,

P1 3,65 4 0,94 3,60 3 0,89 5,07 5 1,26P2 3,73 4 0,92 3,62 3 0,86 5,04 5 1,26Q 2,58 3 0,61 2,58 3 0,62 2,54 2 0,62

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6.9 Interpretation of results The length of the shortlist (factor A) did not greatly affect the results. For the variables Q/Eff, Q/SEff and QC/Eff, denoting the final cost of a Q auction, similar results were observed with both levels of factor A. The success rate of quantity support was slightly increased with the express version of the shortlist. It can therefore be said that the full version of the shortlist was not necessary to minimize the final cost. Instead, the simulations where the full version was used usually took 25%-50% more time, due to the increased amount of calculations. The production capacities (factor B) were found to strongly affect the results. The effective difference between intelligent price support and quantity support, denoted by the variables P2-Q and P2C-QC, decreased when the bidders had equal production capacities. But even then, the Q auction tended to reach a lower final cost: in fact, the median of QC/Eff in such cases was as low as 1,005, suggesting the efficient allocation was nearly always found. With equal production capacities, the efficient allocation usually consisted of two bidders, each producing 300 units of each item. The success rate of quantity support was also strongly affected The initial bids (factor C) had little effect on the outcome of the Q auctions. But in the P2 auctions, method 5 was clearly the one producing the worst results, increasing the final cost and the number of winning bidders. Meanwhile, methods 3 and 4 were virtually indistinguishable from each other. Economies of scope (factor D) had very little effect on any of the results. One reason could be that the difference between the two levels was rather small: with Scope, the fixed cost to produce two items was 1,5 times the cost to produce one item, and in Scope 2 the ratio was 1,4. A higher ratio could have been tested for Scope 1: a lower one for Scope 2 seems unrealistic. Nevertheless, there was an incentive to study the effect of economies of scope, as combinatorial auctions are only expected to have an advantage over non-combinatorial ones when economies of scope exist. The number of bidders (factor E) had an effect on the final costs of both the Q and the P2 auctions: the final cost was observed to decrease when there were more bidders. The decrease was sharper in the P2 auctions. Similar results were obtained in [1], suggesting that quantity support increases competition in an auction, otherwise achieved by allowing more bidders to enter. But even with 30 bidders, the Q auctions produced better results than the P2 ones, as denoted by the positive mean and median of the variable P2-Q.

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7 Conclusion In my Master’s Thesis [1], evidence was obtained that quantity support improved the results of a combinatorial auction in many ways. This was done by simulating combinatorial auctions with and without quantity support, using the same input data. Five input parameters were considered particularly important, and were assigned multiple values in order to study what effect their variation has on the auction results: these parameters were referred to as factors. As the effects of many potentially interesting factors could not be studied in [1], this study has attempted to extend its results. The effects of three entirely new factors were studied now, and two old ones were assigned new values. To allow for a direct comparison, the auction and simulation designs of this study were identical to those of [1]. The new optimization engine enabled the so called efficient allocation to be calculated, as well as auctions to be simulated with up to 30 bidders. The results were quite similar to those of [1]. The auctions with quantity support were found to finish with lower total cost to the auctioneer, and to allocate the items more efficiently. Some of the new factors were found to affect the results, but in all cases quantity support was preferable to price support. Quantity support also had a stabilizing effect on the auction results, with less variation within each block of certain factor levels. The approximated efficient cost, introduced in [1] when the true one could not be calculated, was found to be about 20% higher than the true efficient cost.

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References [1] Ervasti, Valtteri, 2006. The Effects of Quantity Support on the Results of a Combinatorial Auction. Master’s Thesis, Helsinki University of Technology, Institute of Strategy and International Business. [2] Adomavicius, G., Gupta, A., 2005. Toward Comprehensive Real-Time Bidder Support in Iterative Combinatorial Auctions. Information Systems Research 16(2), 169-185. June 2005. [3] Conover, W.J., Practical Nonparametric Statistics. John Wiley & Sons, Inc. New York, 1999. [4] De Vries, S., Vohra, R.V., 2003. Combinatorial Auctions: A Survey. INFORMS Journal on Computing 15(3), 284-309. [5] Leskelä, R.-L., Teich, J., Wallenius, H., Wallenius, J. Decision Support for Multi-Unit Combinatorial Bundle Auctions. Forthcoming in Decision Support Systems. [6] Montgomery, D. C. Design and Analysis of Experiments. John Wiley & Sons, Inc. New York, 1991. [7] Pekec, A., Rothkopf, M., 2003. Combinatorial Auction Design. Management Science 49(11), 1485-1503. [8] Teich, J., Wallenius, H., Wallenius, J., Zaitsev, A., 2006. A Multi-Attribute e-Auction Mechanism for Procurement: Theoretical Foundations. European Journal of Operational Research 175(1), p. 90.

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