EFFICIENCY AND INCENTIVES IN MAS-COORDINATION
Peter Gomber Claudia Schmidt Christof Weinhardt {peter.gomber, claudia.schmidt,
christof.weinhardt}@wirtschaft.uni-giessen.de University of Gießen
BWL-Wirtschaftsinformatik Licher Straße 70 D-35394 Gießen
Germany
ABSTRACT
This paper focuses on market-like coordination mechanisms in Multi-Agent Systems (MAS) and business planning has been chosen as the application area. Several fundamental criteria are derived in order to evaluate market- like coordination mechanisms. The central criterion is the efficient allocation of jobs to agents. Assuming a relationship between classes of operational planning problems and certain coordination mechanisms, planning problems are first classified on the basis of their relevant attributes. Secondly, adequate coordination mechanisms for each of these classes are introduced on the basis of auction theory. All these mechanisms prove to have a common basis: the Vickrey auction.
1. INTRODUCTION
„Coordination, the process by which an agent reasons about its local actions and the (anticipated) actions of others to try and ensure the community acts in a coherent manner, is perhaps the key problem of the discipline of Distributed Artificial Intelligence (DAI)“ (Jennings 1996, p. 187). Nevertheless, in many DAI-projects coordination mechanisms for MAS are applied without providing detailed arguments for their selection, although it is obvious that the coordination principles are crucial for the quality of solutions generated by MAS.
Coordination mechanisms for MAS should be capable of achieving an overall goal while implicitly taking into account different local intentions. Their strength should lie in the fact that it is not necessary for single organisational units,
information systems/agents or persons to collect all the local information that is relevant for the overall goal. If we consider coordination as an assignment of orders/jobs to organisational units represented by agents, the overall goal is to achieve an efficient allocation, i.e. no other assignment leads to a higher degree of goal accomplishment. Hence, we investigate coordination mechanisms in the light of their ability to provide such a solution.
In DAI and especially in modern economic theory hierarchical principles of coordination are often discussed in comparison to market-like mechanisms (see e.g. Malone/Yate/Benjamin 1987). In general, hierarchical mechanisms that require a centralisation of data and competence are not suitable for MAS. While market-like coordination mechanisms implicitly apply local information to determine an allocation (Bond/Gasser 1988, p. 15), they raise the problem of agents’ strategic behaviour. In this context, strategic behaviour means that
• it is profitable to reveal prices/costs that do not equal one’s individual valuation of orders/jobs and therefore may prevent an efficient allocation and
• organisational units/agents can increase their profit by gathering information about the valuations of the others (counterspeculation).
Consequently, we must identify market-like coordination mechanisms that provide a compelling incentive to reveal one’s true valuation, i.e. that prevent strategic behaviour, and assure the required quality of the solution.
Within the scope of our project `Decentralised planning in business´ 1 we analyse planning situations in business administration, where several organisational units, e.g. profit centers, jointly try to find solutions to planning problems in various business domains. In this sense planning is concerned with the allocation of the limited resources of the participating organisational units, i.e. the assignment of orders/jobs that have to be executed by them. More abstractly, these planning situations can be considered as markets where problem-solving competence and capacities are supplied by the agents whose intention is to obtain profits by contributing to the problem-solving process.
Assuming a relationship between classes of planning problems and certain coordination mechanisms, different classes of planning problems have first to be identified. Secondly, they have to be assigned to suitable market-like mechanisms within the MAS in order to coordinate efficiently the competence and capacities of all participating agents with their local intentions.
In this context , we need a suitable economic concept to measure the local valuation of the order(s)/job(s). The concept of disposition specific contribution margin (in the following: contribution margin) (Riebel 1988) represents an adequate calculus to meet this requirement: starting from an optimised state each agent plans a new order/job for its current program based on its local information and reaches a new optimised state. The difference in costs generated by these
1 This work is part of the program ‘Distributed Information Systems in Business’ supported by the DFG under contract We 1436/3-1.
two states deducted from the sales revenue of the order/job yields the contribution margin. Generally, this contribution margin is different for each agent, depending on its individual situation, and it may also have negative values. This is of importance if an order/job has to be allocated to one or several units in a group of affiliated companies/organisational units, e.g. by contractual guarantee given to a customer or to ensure customers’ acceptance, and the job receives a negative valuation from all of these units. In this case, efficient allocation implies assigning the job to the unit with the best (negative) valuation.
Proceeding from these considerations, the paper is organised as follows: In section 2 the characteristics of business planning situations are determined in order to classify planning problems that have to be solved by MAS using market-like coordination mechanisms. Using this classification, various coordination principles are discussed in section 3 and related to the classes already outlined. Finally, section 4 concludes the paper, presenting the main results and prospects for further research in this area.
2. CLASSIFICATION
The matching of various coordination mechanisms and different operational planning problems requires the identification of attributes relevant to these problems. Combinations of these attributes lead to different classes of planning problems and in this context it is necessary to distinguish between the following attributes:
• one job versus multiple jobs to be assigned
If there is only one job to assign to a single organisational unit, a simple assignment problem has to be solved. Assuming that there is more than one job, these can be assigned either successively or simultaneously.
This decision is determined by the complexity of the problem-solving process. The simultaneous assignment of jobs may improve the quality of the allocation if each organisational unit can identify its optimum plan for the whole set of orders, whereas successive assignment may result in inferior allocation (Sandholm 1996). The valuation of a job-package may differ significantly from the sum of the valuations of each job in the package, for example, if in transportation planning the package leads to a circular tour and reduces idle capacity costs. Hence, such packages have a higher valuation than the sum of the single valuations.
• decomposable jobs versus non-decomposable jobs
If a job is decomposable each part can be executed by different problem solvers. Decomposition may result in lower overall cost than the execution of the entire job by one organisational unit. Natural decompositions can be found, for example in the area of transportation, where orders can be divided by distance and/or freight. Further examples are dynamic load balancing
problems in distributed computer systems, scheduling problems in the area of production planning and portfolio planning in finance, where jobs with natural decompositions can also be found.
• fixed job-decomposition versus unknown job-decomposition
A fixed decomposition of the job(s) significantly reduces the possibilities of job assignments to organisational units. If the decomposition of a job is unknown, only a complete enumeration of all potential decompositions will ensure efficient allocation. In transportation planning the decomposition of air freight routes is determined by airports which are the only possible reloading points. Job shop scheduling in the domain of production can also be identified as an example of fixed job-decomposition. The decomposition of liquids or bulk material for transportation and the allocation of an investment (job) to different shares or bonds (units) in the area of portfolio planning represent planning problems with ex-ante unknown job decompositions.
• identical jobs/parts of a job versus different jobs/parts of a job
The last attribute for the classification of planning problems is whether the assignment is carried out for identical or different jobs or parts of a job. In the case of different jobs/parts there is a need to inform potential problem solvers about each individual part. In this situation each individual job and package has to be evaluated separately by the organisational units, whereas in the case of identical jobs or parts it is sufficient to value one job/part and its multiples. Examples of identical jobs and identical parts of a job respectively can be identified in container transport, flow shop scheduling and, in the case of single-instruction-single-data-problems, in load balancing.
Based on these four dimensions, table 1 surveys the planning problems to be considered.
Table 1: Categories of planning problems Decomposability non-decomposable decomposable Number
of jobs
decomposition given
decompositio n not given identical
parts
different parts
one job ¬ ¯ ° ³
identical jobs
different jobs
identical parts
different parts
Multiple jobs - ® ± ² ´
Given this classification scheme, in the next section we derive adequate coordination mechanisms for each category of planning problems in the table (shaded cells ¬ - ´).
3. MARKET-LIKE COORDINATION MECHANISMS FOR DECENTRALISED OPERATIONAL PLANNING
As discussed in section 1, we consider coordination mechanisms to determine an efficient allocation of jobs to organisational units. Therefore, a correct valuation of the jobs based on the individual contribution margins of the units is required, even if margins have negative values. The mechanisms have to be constructed in such a manner that strategic behaviour is excluded ex-ante, which means that for each organisational unit it is rational economic behaviour to reveal its true valuation. In the following, according to the categories ¬ - ´ appropriate market- like coordination mechanisms for MAS are derived which
(i) assure an efficient allocation of jobs to organisational units, because (ii) they prevent strategic behaviour
(iii) even in the case of negative contribution margins and (iv) perform with acceptable communication effort/cost.
¬ one job; non-decomposable
In this subsection common single auctions are investigated with regard to their ability to fulfil criteria (i) - (iv).
In the ENGLISH AUCTION, bidding is open to all participants and bids increase successively until only one bidder is left. The job is sold to him at the price of his last bid.
In the DUTCH AUCTION the auctioneer decreases an initial price gradually until the first bidder stops the procedure and is awarded the job at the current price.
In the FIRST-PRICE-SEALED-BID AUCTION all participants make one sealed bid. The highest bidder is awarded the object at the price he has offered.
The Second-price-sealed-bid auction or VICKREY AUCTION (Vickrey 1961) differs from the FIRST-PRICE-SEALED-BID AUCTION merely by splitting the award and the settlement of the price. Again, the highest bidder is awarded the object but at the price of the second-highest bid.
Each auction yields the same expected value of the price: the second-highest valuation of all participants (McAfee/McMillan 1987, p. 711). An efficient allocation (criterion (i)), i.e. the bidder with the highest valuation receives the job, is not ensured, if the auction mechanism does not prevent strategic behaviour.
In the DUTCH AUCTION and the FIRST-PRICE-SEALED-BID AUCTION bidders anticipate their competitors’ valuations for the job because they want to obtain the award and want to maximise their profit, i.e. the difference between their own valuation and the price they have to pay. Therefore, they need to gather information about the other bidders in order to make strategic bids. In the ENGLISH AUCTION the individual valuations of participants are revealed during open bidding, consequently there is no need for counterspeculation and an efficient allocation is assured. For the VICKREY AUCTION it can be shown (Vickrey 1961, p. 20) that it is a dominant strategy to bid one’s true valuation (criterion (ii)). 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. The individual profit of the bidder who receives the job is the difference between his own bid and the price. Hence, the ENGLISH AUCTION and the VICKREY AUCTION are candidates for coordination mechanisms in MAS.
With regard to criterion (iii) the ENGLISH AUCTION will not lead to an efficient allocation if the valuations of all the agents are ≤ 0. In this case, there will be no bid and the job can only be assigned hierarchically.
Valuations ≤ 0 lead to a special case for the analysis of the VICKREY AUCTION. The Vickrey principle is still valid: the highest bidder receives the job, but instead of paying the second-highest bid he receives the amount of the second-highest bid as a payment. This is a sufficient incentive for each agent to bid for a job with a negative valuation, because the winner is sure to receive a payment that is higher than his own (negative) valuation. As in the case of positive valuations, it can be shown that the bidders will also offer their true valuations.
In considering suitable coordination mechanisms for MAS, communication costs (criterion (iv)) cannot be neglected. It is obvious, that on average a sealed- bid auction generates lower communication costs than any open auction, where each bid must be transmitted to all participants/agents.
Therefore, the VICKREY AUCTION is an appropriate coordination mechanism in MAS for the assignment of a single job which is not decomposable (see also Hubermann/Clearwater 1995, Rosenschein/Zlotkin 1994).
- multiple jobs; non-decomposable; jobs are identical
If n identical jobs have to be assigned simultaneously, the data relating to this job must first be transmitted to all agents by broadcasting or by the use of a black-board. Agents then calculate their individual contribution margins for 1 to n jobs. In general, the valuation of n jobs will not be n times the valuation of one job, because there is often neither a linear nor a monotonous relation between the valuation of one job and that of its multiples. Agents next transmit their bids to the auctioneer, who determines an efficient allocation by setting up a matrix with the number of jobs in the columns and bidders in the rows. The cells of the matrix represent agents’ bids for each number of jobs.
An efficient allocation (criterion (i)) can be found by evaluating this matrix using an assignment algorithm that has to take into account that the maximum of assignments in each row equals one. Moreover the sum of the assignments in each column multiplied by the column number has to add up to n at most.
Table 2 shows an example consisting of four identical jobs and five organisational units:
Table 2: Matrix for the allocation of multiple identical jobs number
of jobs 1 2 3 4
bidders
A 55 70 80 110
B 30 80 -10 impossible
C 25 100 impossible impossible
D -10 30 40 70
E 20 45 60 80
The shaded cells represent the optimum assignment for the example in table 2. The settlement of the prices is again based upon the Vickrey principle: for each number of jobs belonging to the optimum allocation the highest bidder receives the job and has to pay the second-highest bid (here: C has to pay 80 for two jobs). If there is more than one shaded cell in a chosen column the mechanism Vickrey describes for m identical objects (multiple auction) is used (Vickrey 1961, p. 24): m objects (here: jobs) are sold to the m highest bidders at the price of the
(m+1)st highest bid (here: A and B both have to pay 25 for the job). This mechanism prevents strategic behaviour (criterion (ii)): each bidder has an incentive to offer his correct valuation for the job(s) because his bid only determines whether he belongs to the m highest bidders and it does not affect the price he has to pay. As in the case of the arguments for the single auction (see subsection ¬), it is not rational for a bidder to make a bid above or below his true valuation. In addition, this mechanism can handle negative contribution margins (criterion (iii)) in the same way as described in ¬.
® multiple jobs; non-decomposable; jobs are not identical
Again we consider n jobs that are not decomposable. In contrast to -, jobs that are not identical have to be allocated, which increases the amount of data transmitted to agents. Agents calculate their valuation for each potential combination of jobs (2n-1 combinations altogether). As in the case of -, the auctioneer sets up a matrix to identify the optimum allocation. The algorithm for this version of the assignment problem has to take into consideration that the maximum of assignments in each row equals one. Beyond this, columns/combinations of jobs that have any job in common must not be selected jointly.
The shaded cells of table 3 show the optimum allocation for an example with three different jobs and four organisational units:
Table 3: Matrix for the allocation of multiple different jobs combina
tion {1} {2} {3} {1, 2} {2, 3} {1, 3} {1, 2, 3}
bidders
A 10 30 40 60 10 -20 -60
B 5 -10 30 -30 40 70 80
C -10 70 60 40 -20 40 impossible
D 5 40 30 40 60 -30 50
In each selected column the Vickrey principle is used to determine the purchase price. This ensures that the agents offer their true individual valuations (criterion (ii)). Again the principle can cope with negative valuations (criterion (iii)). Multiple auctions do not occur because of the restrictions mentioned above.
¯ one job; decomposable; decomposition given; identical parts
Identical parts, resulting from a given decomposition of one job can be handled in the same way as multiple identical jobs that are not decomposable.
Consequently, the coordination mechanism discussed in subsection - can be applied to this kind of allocation problem.
A further analogy can be revealed when we consider a coordination mechanism for
° one job; decomposable; decomposition given; parts are not identical
This allocation problem is characterised by the same attributes as the problem of multiple different jobs that are not decomposable. Hence, the coordination mechanism of subsection ® is a suitable mechanism for the allocation of different parts of one job.
± multiple jobs; decomposable; decomposition given; identical parts
The extension of the allocation problem from one job (see ¯ and -) to multiple decomposable jobs with identical parts does not affect the coordination mechanism itself, but merely leads to an increase in the number of elements to be assigned. If job i (i = 1 to n) is decomposed into ki parts, the number of identical parts/number of columns in the allocation matrix (see table 2) equals i ki
n
∑= 1 .
² multiple jobs; decomposable; decomposition given; different parts
Considering n jobs with a fixed decomposition into ki different parts, we can refer to the coordination mechanism in subsections ° and ®. The number of parts and consequently the number of combinations (columns in the allocation matrix of table 3) increases to 2i 1ki 1
n
∑=
− .
Summing up, in subsections - - ² we have distinguished two coordination mechanisms, one for identical jobs/parts of a job (-,¯,±) and another for different jobs/parts (®,°,²). Although the mechanisms use different assignment algorithms to determine an efficient allocation, they both apply the Vickrey principle and take a matrix as a basis for the allocation. In the following both mechanisms are therefore called MATRIX AUCTIONS.
³ one job; decomposable; decomposition not given
In contrast to cases ¬ - ², the decomposition of the jobs is now arbitrary and ex-ante unknown. For this assignment problem neither the VICKREY AUCTION nor MATRIX AUCTIONS result in an efficient allocation: the VICKREY AUCTION
only ensures an efficient allocation of one job to a single organisational unit. For the representation of arbitrary and unknown decompositions in a matrix all imaginable - in the worst case continuously - decomposable parts and their combinations have to be listed in the columns.
An allocation mechanism for jobs with unknown decompositions should ensure the use of synergies by cooperation among agents, i.e. coalitions of organisational units. In this case, besides individual agents, coalitions of agents participate in the bidding. Ex-ante a single organisational unit is opposed to the following trade-off: it does not know whether joining a coalition will be superior to making a bid on its own, i.e. it is not clear to an agent whether its share in the profits of a coalition will exceed the profit of a bid made on its own. Thus, the mechanism must guarantee that each organisational unit has an incentive to join a coalition, i.e. a single bidder joining a coalition has to be assured a minimum margin that equals the margin he receives when bidding on his own.
The Multistage Extended VICKREY AUCTION (MEVA) was developed and investigated in detail (Gomber et al. 1996, pp. 299-307) for solving such problems.
In this coordination mechanism jobs are auctioned off in an iterative bidding process, whose number of iterations equals the number of participating organisational units (l). In each iteration i a VICKREY AUCTION is carried out and coalitions with i participants are called to make a bid.
The mechanism is now explained by an example where a maximum of two participants constitute a coalition. In the first iteration each single agent makes a bid for the job. The auctioneer stores all bids and their respective bidders, but does not announce them. In the second iteration the auctioneer calls on coalitions with two participants to make their bids. In this case bilateral negotiations lead to a joint bid, if it exceeds the maximum of both individual bids.
Again, the auctioneer stores all bids and their respective coalitions, but does not announce them. In our example no coalitions will constitute in iterations 3 to l.
Hence, the auctioneer does not receive any further bids.
The organisational unit/coalition with the highest bid of all iterations is awarded the job. The price corresponds to the second highest bid of all iterations. In the determination of the price bids made in the first iteration are not taken into account if their bidders join a coalition in the second iteration. The reason for this is that no organisational unit would stick to its single offer if it joined a coalition. In order to achieve incentive compatibility the auctioneer has to determine a minimum margin for the following situation: a coalition has made the highest bid of the two iterations and one of both coalition partners made the highest bid in the first iteration and the highest bid of the first iteration exceeds the second highest bid of the second iteration. The auctioneer announces the minimum margin to both participants in the coalition. The agent with the highest bid in the first iteration receives the minimum margin and a share in the profits of the coalition. The share is negotiated among the coalition partners. This implies that the MEVA mechanism ensures a compelling incentive (criterion (ii)) to join a coalition and therefore guarantees an efficient allocation (criterion (i)).
Considering criterion (iii), the award follows the Vickrey principle and the mechanism can be applied to negative contribution margins.
´ multiple jobs; decomposable; decomposition not given
The extension of the planning problem described in ³ from one job to multiple jobs with an unknown decomposition does not effect the selection of the coordination mechanism. The determination of an efficient allocation for each job separately based on the MEVA does not guarantee a globally efficient allocation of all jobs. The MEVA has to be applied to the whole set of jobs, because an efficient allocation can only be found as a result of multilateral negotiations among the organisational units on all imaginable parts of the jobs.
As in the case of ³, the MEVA leads to an efficient allocation (criterion (i)) by ensuring that each unit has a compelling incentive (criterion (ii)) to join coalitions. As the Vickrey principle is valid, criterion (iii) is again fulfilled.
Finally, the coordination mechanisms discussed in this section are investigated with regard to communication effort/cost (criterion (iv)). As mentioned in ¬, sealed bid auctions are superior to all other auctions because the mean communication costs are lower. Since - to ´ are based on the VICKREY AUCTION as a sealed bid auction, its characteristics with regard to communication costs will also apply to the mechanisms identified in these subsections.
Beyond this, communication costs increase in ³ and ´ due to negotiations that are inherent in the MEVA. For all conceivable coalitions the number of (bi-, tri-, ... multilateral) negotiations and consequently communications costs increase rapidly. This, however, is not a problem that is associated with the MEVA. Any coordination mechanism allowing cooperation has to handle this kind of problem.
4. CONCLUSION
The quality of problem-solving competence in MAS primarily depends on the coordination within the group of participating agents. The main object of this paper is to disclose relations between different kinds of problems to be solved on the one hand and adequate coordination mechanisms on the other.
Starting with the identification of relevant attributes for planning problems in a business environment, a classification scheme based on these characteristics is derived. We then point out essential criteria for market-like problem solving (criteria (i) - (iv)) and assign suitable coordination mechanisms to all categories in the classification scheme (table 4).
Table 4: Categories of planning problems and suitable coordination mechanisms
decomposability non-decomposable decomposable number
of jobs
decomposition given
decomposition not given identical
parts
different parts
one job VICKREY
AUCTION
MATRIX AUCTIONS
MEVA
identical jobs
different jobs
identical parts
different parts
multiple jobs MATRIX AUCTIONS MEVA
The results of these considerations are as follows:
• The VICKREY AUCTION is identified as the most suitable mechanism for the allocation of one non-decomposable job.
Proceeding from this, we develop
• MATRIX AUCTIONS for the allocation of one or multiple decomposable job(s) with a given decompostion and
• the MEVA to allocate one or multiple decomposable jobs with an unknown decomposition.
These three mechanisms share two essential elements: sealed bidding to ensure acceptable communication costs and separation of the award (highest bid) and the price settlement (second-highest bid) in order to avoid strategic behaviour.
In the context of our project the main focus of future research should be (bi- or multilateral) negotiations among coalition partners from a theoretical, economic point of view. In addition, with regard to the implementation of the coordination mechanisms developed, adequate representations and architectures of MAS have to be realised for handling the planning process (Lohmann et al.
1996).
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