Auctions in Electronic Commerce - Efficiency versus Computational Tractability - 1

Auctions in Electronic Commerce

- Efficiency versus Computational Tractability -

¹

Peter Gomber

^a

Claudia Schmidt

^a

Christof Weinhardt

^a

aUniversity of Giessen “Business Informatics“

Licher Strasse 70, D-35394 Giessen, Germany phone: (+49) 0641/99-22610, fax: (+49) 0641/99-22619

{peter.gomber, claudia.schmidt, christof.weinhardt}@wirtschaft.uni-giessen.de

1 This work is part of the project ‘Decentralized Planning in Business’ that is supported by DFG under contract We 1436/3-1.

Abstract

In Electronic Commerce, the intra-organizational co- ordination of directly responsible units, e. g. profit- centers or firms within an affiliated group, is of increasing importance. These organizational units can be modeled within a Multi-Agent System (MAS), an interconnection of autonomous information systems. This paper investigates co-ordination mechanisms for MAS in decentralized transportation planning that ensure efficient allocation of scarce resources on the basis of local planning processes. In the domain of transportation, planning problems are characterized by large amounts of data, limitations of time for planning and the intractability of computational problems.

Auctions as market-like co-ordination mechanisms are discussed with respect to the trade-off between theoretical evidence on the quality of the allocation and computational tractability.

Keywords:

Auctions; Transportation Planning; Computational Tractability

Introduction

Increasing (inter)national competition and deregulation lead to co-operation among transportation agencies on various organizational and institutional levels, e. g. strategic alliances, mergers and affiliated groups. Moreover, forwarding agencies tend to divisional structures and profit- center organizations. Consequently, in the domain of transportation planning the problem arises of how to allocate jobs to different organizational units within an association. In particular, co-operating units have to

consider whether such planning problems can be solved adequately with centralized methods or if decentralized planning methods within electronic markets [1, 2] would be more suitable.

In transportation centralized and mostly heuristic OR- methods are usually applied to computer-based planning.

Many data, e. g. the number and load factor of the trucks that are used, the disposable capacities and the idle capacity costs, are necessary for centralized planning and costing. Because of time limitations, the procurement and continual updating of this large volume of data is often not practicable. Even if the immediacy of data can be assured, an optimum solution cannot be found in acceptable time due to the computational complexity of the planning problem [3].

This complexity and short planning periods call for new, decentralized IT-approaches both for the area of vehicle routing and for the allocation of transportation jobs to members of an affiliated group [4]. Decentralized methods will prove to be superior mainly if planning data are extensive, non-deterministic and dynamic. In a decentralized approach all local units plan autonomously, i. e. they pursue their own objectives on the basis of individual valuation methods using local information on the current market situation and their own resources.

Adequate IT-support of decentralized planning processes can be realized on the basis of Multi-Agent Systems (MAS) [5]. It is on their strength that single organizational units, information systems (agents) or persons do not have to gather all relevant information to achieve an overall goal, i.

e. from an abstract point of view, MAS function like real markets. The co-ordination problem in the electronic market lies in assigning transportation jobs to organizational units which are represented by agents. The overall goal is to achieve an efficient allocation, i. e. to determine optimum

utilization (here: cost minimization) of transportation resources within an affiliated group or profit-center organization.

Due to practical applicability the trade-off between theoretical evidence and computational tractability must be taken into account. Therefore, suitable co-ordination and co-operation mechanisms [6] within electronic markets are needed.

Proceeding from these considerations, the paper is organized as follows: In section 2 we consider a scenario that reflects and investigates transportation planning problems and in section 3 a co-ordination principle for MAS is introduced. Section 4 discusses the specifications for and the realization of a price setting that guarantees an efficient allocation. Finally, section 5 concludes the paper by presenting the main results and prospects for further research in this area.

Scenario in Transportation Planning

In a group of affiliated transportation agencies the parent company has to allocate transportation jobs to the consolidated subsidiaries and profit-centers respectively.

The operation department of the parent company receives customer orders and then has to assign the execution of these jobs efficiently.

Jobs can be allocated either one at a time or simultaneously.

This decision is determined by limitations of planning time and the complexity of the problem-solving process [7].

Simultaneous assignment of jobs may improve the quality of the allocation because each organizational unit is able to identify its optimum plan for the whole set of orders, whereas a successive assignment may result in inferior allocation [8]. To illustrate this, Figure 1 shows an example with two jobs, two forwarding agencies, their locations and distances that are assumed to be equivalent to transportation costs.

0 1 2 3 4

A B C

transportation job 2 transportation job 1

agency 1 agency 2

Figure 1 - Successive versus simultaneous assignment of jobs

Transportation job 1 (from location B to C) arrives earlier than job 2 (from A to B). If these jobs are allocated one at a time, agency 2 will receive job 1 with costs of 2 and agency 1 job 2 with costs of 3, whereas in the case of a simultaneous assignment agency 1 gets both jobs and has costs of 4. Obviously, the latter leads to reduced overall costs. Therefore, the co-ordination mechanism applied within the MAS has to use simultaneous assignment in

order to exploit synergy effects and to achieve global efficiency.

Matrix Auction

In the following a co-ordination mechanism for an electronic market in transportation planning is introduced. Agents that represent the forwarding agencies of the affiliated group/profit-centers interact in this market. Like the Walrasian auctioneer [9] in traditional markets, this electronic market uses a centralized instance of authority for the co-ordination of agents’ individual intentions. In this context the operation department of the parent company represents the auctioneer (for a detailed investigation of different auction types see e. g. [10, 11]). The mechanism for the allocation of jobs to different organizational units works as follows [12]:

For a simultaneous assignment of jobs, the data relating to each job first have to be transmitted from the auctioneer to all agents by broadcasting or by the use of a black-board.

Each agent then calculates its individual valuation for the jobs. Therefore, a suitable economic concept is needed to measure the local valuation of the transportation jobs.

The concept of the disposition specific contribution margin (in the following: contribution margin) [13, 14]

represents a suitable calculus to meet this need: starting from an optimized state, i. e. an individual transportation plan, each agent inserts a new job into its current plan and reaches a new optimized state. The difference in costs generated by these two states deducted from the sales revenue of the job yields the contribution margin. In general the contribution margin is different for each agent because it depends on the agents’ individual current transportation plan. It may also have negative values. This is of practical relevance if a job is not rejectable, e. g. due to a contractual guarantee given to a customer or to ensure customers’

acceptance. The payoff results from subtracting the auction price from this contribution margin (see Figure 2).

agency A

agency C

agency B transportation

job

sales revenue (= customer price)

marginal costs of the job

contribution margin

payoff

auction price

Figure 2 - Contribution margin and payoff in the transportation scenario

With regard to a simultaneous assignment of n transportation jobs each agent calculates its contribution margin for each potential combination of jobs (2ⁿ-1

combinations altogether). The valuation of a job- combination may differ significantly from the sum of the valuations of each job, e. g. if the combination leads to a circular tour and reduces idle capacity costs.

Agents next transmit their bids to the auctioneer, who determines an efficient allocation by setting up a matrix (see Table 1) with all job-combinations in the columns and the bidding agents in the rows. In the following this coordination mechanism is called matrix auction. The cells of the matrix contain agents’ bids for each job-combination.

The algorithm for this assignment problem has to take into account that the maximum number of assignments in each row equals one. Beyond this, columns/job-combinations that have any job in common must not be selected jointly.

The shaded cells of Table 1 show the optimum allocation for an example with three jobs and four agents/forwarding agencies. The optimum overall contribution margin is 150.

Agent B receives the transportation jobs 1 and 3 and agent C is awarded job 2.

Table 1 - Matrix for the allocation of multiple jobs combina-

tion

{1} {2} {3} {1,2 }

{2,3 }

{1,3 }

{1,2,3 }

bidder

A 10 30 40 65 10 -20 -60

B 5 -10 30 -30 40 80 80

C -10 70 60 40 -20 45 10

D 5 40 35 40 60 -30 50

The capability to determine an efficient allocation depends crucially on whether agents report their contribution margins correctly. This can be achieved by means of an incentive-compatible pricing system, which will be discussed in the next section.

Pricing in the Matrix Auction

The solution of the assignment problem only ensures efficiency, if the bids made by agents coincide with their true valuations of job-combinations. Therefore, a pricing mechanism has to be identified that gives local units a compelling incentive to act in accordance with the parent company’s objectives, i. e. revealing the correct contribution margins is a dominant strategy for each organizational unit/agent. This must hold even for negative valuations because negative local valuations may contribute to global efficiency (see Table 2).

Table 2 - Example of an efficient allocation including negative valuations

combina- tion

{1} {2} {1,2}

bidder

A 90 40 60

B 30 -20 50

C 20 -10 65

The GVA Pricing Mechanism

Varian [15] describes a mechanism - the so-called Generalized Vickrey Auction2 (GVA) - that can be applied to the pricing problem in the matrix auction. In this context the GVA works as follows:

Suppose there are i = 1,..,I bidders and n jobs, i. e. j = 1,..,2ⁿ-1 job-combinations. bkj represents the bid of a bidder k for job-combination j, vkj his true valuation concerning job- combination j.

x

ij

* are the binary variables of the optimum assignment (with

x

ij

* = 1 if bidder i receives job- combination j and

x

ij

* = 0 otherwise) and

x

_ij^*¬k are the variables of the optimum assignment with the row of bidder k skipped. The price for a bidder k in the efficient allocation is computed by deducting the sum of the bids of all other bidders in

x

ij

* from the sum of the bids in

x

_ij^*¬k, i. e. pk =

b x

_{ij j} ^k

j n

i i k

I

i

*¬

=

−

≠=

∑

∑

1 2 1

1

-

b x

_{ij ij}

j n

i i k

I

*

=

−

=≠

∑

∑

1 2 1 1

. (1)

In the example of Table 1 with an efficient allocation of 150 the price of bidder C results from deducting the sum of the bids of all other bidders in the efficient allocation (here: 80) from the sum of the bids in the optimum assignment (here:

120) with bidder C skipped (see Table 3).

2In the VICKREY AUCTION [11] the highest bidder is awarded the

object at the price of the second-highest bid. For the VICKREY AUCTION it can be shown that it is a dominant strategy to bid one’s true valuation. This results from the Vickrey principle: a bid does not determine the purchase price, but it determines its rank among all bids. A bidder who makes a bid below his valuation may lose the award. A bid exceeding his individual valuation embodies the risk of a loss if another bid ranks between the price he has offered and his true valuation of the job.

Table 3 - Optimum assignment with bidder C skipped combina-

tion

{1} {2} {3} {1,2 }

{2,3 }

{1,3 }

{1,2,3 }

bidder

A 10 30 40 65 10 -20 -60

B 5 -10 30 -30 40 80 80

D 5 40 35 40 60 -30 50

Consequently, C has to pay pC= 120 - 80 = 40 for job {2}.

Bidder B has to pay pB = 125 - 70 = 55 for the job- combination {1, 3}.

The payoff gk of a bidder k results from the difference between his individual valuation and the price he has to pay (see also Figure 2), i. e.

gk=

v i

kj

( )

- (

b x

ii

ij j k j

n

i i k

I

i

*

( )

¬

=

−

=≠

∑

∑

1 2 1 1

-

b x

iii

ij ij j n

i i k

I

*

( )

=

−

=≠

∑

∑

1 2 1 1

). (2)

Each bidder aims to maximize his payoff gk. The sum of the bids in the optimum solution with the row of bidder k skipped (ii) cannot be influenced by bidder k, i. e. bidder k’s objective is the maximization of vkj +

b x

_{ij ij}

j n

i i k

I

*

=

−

=≠

∑

∑

1 2 1 1

. The

parent company maximizes bkj +

b x

_{ij ij}

j n

i i k

I

*

=

−

=≠

∑

∑

1 2 1 1

. Obviously,

the objective functions of each agency and the parent company are equivalent and bidder k makes a bid bkj = vkj. Therefore, each agency reports its correct valuation and receives a payoff which corresponds to its contribution to efficiency.

In the example (Table 1) bidder C receives a payoff gC = 70 - 40 = 30 which corresponds to the difference between the optimum allocation and the optimum allocation without the participation of C (150 - 120 = 30). The same holds for B (gB = 80 - 55 = 150 - 125 = 25).

The results discussed above also apply to negative valuations. In the example of Table 2 bidder A has to pay pA= 65 - (-10) = 75 and his payoff gA = 90 - 75 = 15 is equivalent to his contribution to efficiency. C’s price is pC= 70 - 90 = - 20. He receives a payment of 20 and his payoff is gC = (-10) - (-20) = 10.

To sum up, the pricing mechanism of the GVA ensures efficiency by establishing compatibility of local and global

goals and provides an incentive to contribute negative valuations as well.

The PPC Pricing Mechanism

With regard to practical applicability, the GVA suffers from its computational complexity, especially in the framework of planning time limitations. Up to I + 1 NP-complete assignment problems have to be solved in order to determine an efficient allocation and the prices for the bidders.

Therefore, we consider an alternative pricing mechanism for the matrix auction that reduces the number of assignment problems to one. This mechanism works as follows: the bidders in the optimum assignment receive a job- combination at a price that exclusively refers to the bids for this combination/column. It is based upon the Vickrey principle: each bidder has to pay the price of the bid next in rank with regard to this combination. In the following this price setting is called Pricing Per Column (PPC).

In the example of Table 1 bidder C has to pay 40 for job {2}

and B has to pay 45 for the job-combination {1, 3}. C (B) receives a payoff gC= 70 - 40 = 30 (gB= 80 - 45 = 35) which corresponds to the difference between his valuation and the price he has to pay.

In addition, this mechanism can handle negative contribution margins. In the example of Table 2 bidder A has to pay pA = 30 for job {1}, his payoff is gA= 90 - 30 = 60. C’s price for job {2} is pC= -20 . He receives a payment of 20 and his payoff is gC= (-10) - (-20) = 10.

In contrast to the GVA, in the PPC it is not a dominant strategy in the game-theoretic sense to contribute correct valuations, i. e. there is no stringency for an efficient allocation: a bidder k who expects to have the highest valuation for a job-combination may have an incentive to assure his participation in an efficient allocation by bidding a high price. But he risks suffering either a loss if another bid exceeds his valuation for this job-combination or a loss in the sense of opportunity costs if he would have been better off in the globally efficient allocation. Only in the case of complete information on other bids can these risks be avoided. Assuming that complete information might be available, however, counteracts the motives for the application of decentralized planning (see section 1).

The PPC ensures that each bidder who is awarded a job- combination has a non-negative payoff and therefore a compelling incentive to participate in the bidding process regardless of positive or negative contribution margins.

Concerning computational tractability it is on the strength of the PPC-mechanism that, apart from the efficient allocation, no additional assignment problems have to be solved. Especially in the case of a great number of jobs that are allocated simultaneously and time limitations, the PPC is superior to the GVA concerning practical applicability.

Concluding Remarks

In agent-based commerce co-ordination mechanisms especially auctions are often applied without providing detailed arguments for their selection, although it is obvious that the co-ordination principles are crucial for the quality of solutions generated within the electronic market [16]. It is on the strength of the mechanisms discussed in this paper that - in comparison to existing approaches in transportation planning on the basis of intelligent agents [see e.g. 4, 17] - the participating agents do not benefit from lying and multiple jobs can be assigned efficiently.

Proceeding from a scenario that presents transportation planning problems in affiliated groups of forwarding agencies and profit-center organizations respectively, this paper introduces a co-ordination principle within MAS. The main objective is the efficient allocation of multiple transportation jobs within a limited time for planning. Hence, the paper focuses on alternative price settings considering both the required degree of goal accomplishment and practical applicability.

The results of considering alternative mechanisms in the matrix auction are the following: On the one hand the GVA ensures a compelling incentive for the revelation of the true valuations and therefore an efficient allocation can be determined. In the PPC truthfully revealing one’s willingness to pay is not a dominant strategy in the game- theoretic sense, but it is ensured that each participant will receive a non-negative payoff even in the case of negative valuation. On the other hand the PPC obviously is superior to the GVA due to computational tractability.

The relative superiority of these alternative price settings depends on the particular planning determinants:

• the computing capacities of the parent company’s operation department (the auctioneer) and of each forwarding agency,

• the time available for planning and especially

• the size of the problem, i. e. the number of participating forwarding agencies and the number of jobs that have to be allocated simultaneously.

Currently, both mechanisms are implemented in ADAMCO [18], a multi-agent architecture that is designed for decentralized operational planning processes and that provides different market-like coordination mechanisms. The next step in the project ‘Decentralized Planning in Business’

is an empirical investigation and comparison of the GVA- and the PPC-mechanism in co-operation with the PTV GmbH, Karlsruhe, a german consultancy in the area of transportation planning. The future research focuses on the transfer of these concepts to other classes of planning problems in transportation, e. g. the allocation of decomposable jobs or identical jobs/parts of a job, and on planning problems in other domains of Electronic

Commerce, e. g. inter-organizational production planning or team-oriented collaboration based on intranets or extranets.

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