WO2001090985A1

WO2001090985A1 - Method for conditional auctions

Info

Publication number: WO2001090985A1
Application number: PCT/SE2001/001093
Authority: WO
Inventors: Mattias Willman; Fredrik Ygge; Arne Andersson
Original assignee: Mattias Willman; Fredrik Ygge; Arne Andersson
Priority date: 2000-05-19
Filing date: 2001-05-17
Publication date: 2001-11-29
Also published as: AU2001258997A1

Abstract

A format and method for expressing bids in a computerized multi-commodity auction, and a method for managing auctions containing bids of the introduced format in a computationally efficient manner. The invention enables advanced auctions in which bidders can associate constraints, such as discounts and/or additional charges, with a set of bids.

Description

METHOD FOR CONDITIONAL AUCTIONS

This application claims the benefit of copending Provisional Application No. 60/205,403 filed on May 19, 2000.

Field of the Invention The present invention relates to a computerized multi- commodity auction, i.e. an auction in which several commodities are offered and/or requested simultaneously. The term "commodity" is used herein to include products as well as services or any other subject of a bid.

Background of the Invention

Multi-commodity auctions are, for example useful during purchase within the field of public service, such as transportation contracts, or a major-building project, wherein several bidders are invited to bid on one or more subparts of the building project. In a multi-commodity auction, the bidder may want to add conditional discounts and/or additional charges or make restrictions of some kind.

In some cases, such conditions may be handled using state of the art combinatorial auctions . A combinatorial auction is an auction in which bidders can express valuations not only on single items, but on bundles of items.

For example, example 1, the bid ^»ι offer the service A for $30/hour, the service B for $25/hour, and service C for $4θ/hour, but if I am selected to provide two or more of these services I will give a 10 % discount" is quite straightforward to express with today's combinatorial auctions (as will be described below) .

However, example 2, a bid such as "I am willing to purchase 20 units of commodity A for $30 each and 10 units of commodity B for $20 each, but if the difference between the number of acquired units of commodity A and commodity B differ more than 30%, then I demand a 20% discount" is, although easy to understand in itself, more complicated to compare with other bids having other conditions. This type of bid is impractical to express in present combinatorial auctions, such as eAuctionHouse part of eMediator, presently accessible at the Internet address htt : //ecom erce .cs . ustl . edu/e ediator/ .

Numerous patents have been published that deal with auctions in which computers are used to evaluate and/or organize the bids of an auction. Many of these patents also relate to auctions utilizing computer networks, such as the Internet. However, the inventors of the present invention are presently not aware of any patents relating to the type of multi-commodity auctions of the present invention.

Common to the presently available combinatorial auctions for computer administration is that the bids, when associated with certain conditions, such as discount offers or additional charges, have to be expressed in an indirect way. This means that at some stage during the bidding procedure, the bid has to be transferred to a set of bids covering the different possible outcomes of the offer. The number of such outcomes may be very large.

For example, to evaluate the bid of the example 1 above it has to be divided into the following mutually exclusive alternatives, in addition to the trivial case of not receiving any of the commodities :

Alternative 1) A for $30/hour

Alternative 2) B for $25/hour

Alternative 3) C for $40/hour Alternative 4) A for $27/hour and B for $22.5/hour

Alternative 5) A for $27/hour and C for $36/hour

Alternative 6) B for $22.5/hour and C for $36/hour

Alternative 7) A for $27/hour, B for $22.5/hour and C for $36/hour

This way to handle a bid of a multi-commodity auction, wherein the bid is associated with conditional discount offers or additional charge requests, is herein called "indirect bidding" which refers to the need for an enumerative representation of the original bid.

Summary of the Invention According to the present invention, a method is introduced for computerized multi-commodity auctions in which bidders are allowed to explicitly express discounts and additional charges associated with each bid. With the method of the invention, the bids of the auction are evaluated without the need to dissolve each bid into different outcome possibilities. The term "multi-commodity auctions with explicit discounts and/or additional charges" is herein defined as an auction in which each bidder has the possibility to combine a bid on one or several commodities with a specification of conditions for any discounts and/or additional charges.

The term multi-commodity auctions should also be understood to cover many variants of what is sometimes referred to as "multi- attribute auctions", which essentially are auctions in which bidders offer/request a single item with a number of different attributes, since for the purpose of this description the different attributes could be regarded as different commodities. That is auctions including bids like "I pay $1000 for a certain computer with some certain attributes, but request a discount of $100 if it comes without a DVD player" can be efficiently managed with the invention.

According to the method of the invention, in a first step the bidder is invited to describe a bid (or several bids) that include (s) conditional discounts and/or additional charges using a high-level language. This will enable the bidder to express the bid(s) in a more natural way than what is possible with present combinatorial auctions.

In a second step, the bids are transformed to a compact mathematical form referred to as a direct formulation, to be used in a third step of the invention. In said third step of the invention, the optimal selection of bid(s) is computed from the direct formulation. In this way, the invention provides means to determine a dinner, or a set of winners, using a computationally efficient management of the given bids.

The term "high-level language" is here understood as a machine-readable language that allows for compact description of relevant parameters and expressions. The language should also make it possible for programmers and others skilled in the art to interpret different data written in the language. The term "direct formulation", used above for the second step of the invention, is defined as any, explicitly mathematically expressed, conditional discount offer or additional charge request. The term is used herein as a distinction with respect to the indirect formulation, wherein the discount offers or additional charges are defined by an enumeration of different possibilities, as described by the indirect bidding above.

There are no assumptions whether the bids are sealed or open, or whether the auction is one-shot or iterated. It is also possible to apply the method according to the present invention on commodities of non-discrete entities, for example where the amount of traded goods is expressed in kilograms and where the bid is specified over a continuous interval, like 3000 - 5000 kg.

Further scope of applicability of the present invention will become apparent from the detailed description given hereinafter. However, it should be understood that the detailed description and specific examples, while indicating preferred embodiments of the invention are given by way of illustration only. Various changes and modifications within the spirit and scope of the invention will become apparent to those skilled in the art from this detailed description.

The present invention will become more fully understood from the detailed description given herein, including the accompanying drawings which are given by way of illustration only, and thus are not limiting the present invention.

Brief Description of the Drawings

Fig. 1 is a block diagram showing the essential steps of a multi-commodity auction according to the present invention,

Fig. 2 is an example of a user interface, according to the invention, allowing for an explicit discount offer,

Fig. 3 is a second example of a user interface, according to the invention, allowing for an explicit discount offer wherein the bidder can bid on different combinations of goods, and

Fig. 4 is a block diagram depicting a system, according to the invention, for administrating multi-commodity auctions with explicit discount and/or additional charge bids.

Detailed Description of Preferred Embodiments Generally, the method according to the present invention is outlined in Fig. 1, although there are a number of variations. For example, in the case of an auction with iterative bidding, some steps may be repeated a number of times. The method of the invention is implemented in a computer software, run on a stand- alone computer or preferably on a (number of) server (s) connected to a suitable computer network, such as the Internet.

With reference to Fig. 1, the rules of the auction, e.g. if the auction is open or closed, the number of commodities of the auction, the possible extent of discounts or additional charges, parameters to be observed when evaluating the bids etc., are determined by the party setting up the auction, such as a seller or a dedicated auctioneer and are made public to the bidders 101. The auction rules are implemented in the dedicated computer program, as will be described below (indicated with dashed arrows in Fig. 1) . The bidders are invited to express their bids by some high- level language or graphical user-interface 102, such as a printed form or a form displayed on a monitor.

The bids may be entered into the computer system in several ways, where the main ones are (i) the bids are entered by a human bidder through a graphical user-interface, and (ii) the bids are generated by a software program (here referred to as a software agent) representing some real world entity. The descriptions below will focus on (i) above for readability reasons. With the invention, a simplified user-interface, compared to the previously known methods, is provided as described below. The bids could be input by personnel provided by the auctioneer, but preferably the bids are input directly by the bidder, for example via an open computer network such as the Internet . The bids are translated by the computer program into a format suitable for the winner determination algorithm. According to the invention, an objective function 103, together with a number of constraints 104, are determined. The objective function and the constraints are formed, i.e. generated, from the rules selected for the auction 101 (indicated with dashed arrows) , together with the input bids 102.

The best bid or bids are determined by solving a winner determination problem using a- suitable algorithm 105. As will be more fully discussed below, this can be done in a variety of ways depending on how the bids were translated. When the best bid(s) have been determined, the bidders may be informed and the commodities reallocated according to the winning bid(s) . Optionally the bidders may be allowed to enter new bids (or even withdraw bids) based on the received information.

A multi-commodity auction with explicit discounts and/or additional charges according to the invention enables users to give bids in a natural and easy-to-understand way, and allows software agents to express bids in a compact manner.

The exact appearance of a user interface is of course dependent of the software application used, as well as the artistic freedom of the programmer, but generally the user interface should comprise separate fields for input of all the characteristics of a bid in a predetermined way. The characteristics of a bid include such entities as amount, price, conditions regarding discounts and/or additional charges, restraints with respect to other bids, etc. Examples of such graphical interfaces are shown in Figs. 2 and 3. The data acquired in the user interface, or generated by a software should, via the bid translating steps 103, 104, be coordinated with the winner determining step 105 in such a way that during the translation step the input data are translated to an objective function and a number of constraints from which the winner determining software is able to compute the most optimal bid(s) .

In the case of bidders giving the bids through a user- interface, the user interface guides the bidder (or the operator responsible for entering the bids) by clearly appointing the fields wherein specified information is to be input, as is shown in the user interface examples of Figs. 2 and 3.

This means that the design of the user interface should include a limited (and optional) number of different manners in which the bid may be given, and the fields should be provided in accordance therewith. Thus, a pre-selected set of discount offer and/or additional charge request alternatives are provided by the auctioneer, and are implemented both in the user interface and the computer software that is used to evaluate the bids. For example, the example of a user interface shown in Fig. 2 allows the bidder to give a bid on one or more of three contracts, and the bidder may offer a discount (in percentage) for a selected number of contracts .

In the user interface example of Fig. 3, the bidder is allowed to give up to four different bids, each bid defining how many items each of five commodities the bidder intends to buy for an offered payment. In addition to this, the bidder may request a discount if he buys for more than a certain sum. Thus, the bidder may have one or more winning bids (or none) .

These examples illustrate one general idea according to the invention, namely that the user interface is designed to direct the bidder to input his bid(s) in a way that allows an unambiguous translation of the bid(s) into a mathematical formulation for the calculation of the winning bid(s) . Of course, more than one way of adding offers or constraints to a bid could be provided with the user interface.

Thus, the rules of the auction should be built into the user interface in an unambiguous way.

In the case of an input from a software agent, the input should be correspondingly formatted. A suitable high-level language is used for communication between the bidder software agent and the auction. The high-level language is then translated using the translation rules given below into a form suitable for a winner determination algorithm.

The winner determination problem is here defined as the problem of finding the most preferred combination of bids.

In the implementation description below, the most preferred combination is the combination that generates the highest surplus in terms of some numeraire commodity, typically money. To illustrate the surplus in monetary units: The surplus for the combination "I buy 2 units of commodity 1 for $12", "1 sell 1 unit of commodity 1 for $2", and "I sell 1 unit of commodity 1 for $3" is $12-$2-$3 = $7.

However, it should be noted that the invention is not limited to the case where the most preferred combination is the combination with the highest surplus in monetary units. For example, in an auction wherein a public company is sold, it could be in the interest of the seller to extend the ownership to as many shareholders as possible. In such a case, the most preferred combination is not necessarily the one with the highest surplus in monetary units.

In some cases finding the optimal solution is impractical even with highly optimized algorithms. Instead, the chosen combination may, for example, be taken as the best combination found after a given amount of calculating time. According to the invention, the winner determination problem for multi-commodity auctions with explicit discounts and/or additional charges is managed by the use of a formulation which directly describes the discounts/additional charges in a format suited for an efficient winner determination. In the following, this will mainly be demonstrated with integer linear programming formulations, but the invention is not limited to this case. Other formulations and software, such as mixed linear integer programming, constraint satisfaction, hybrid solvers etc. are also possible without deviating from the scope and spirit of the invention. Below it will be described how the winner determination problem for the multi-commodity auction can be translated into linear programming terms by presenting a number of translation rules. From these rules, and the accompanying discussions, anyone skilled in the art will understand how to apply the present method to very general multi-commodity auctions with discounts and/or additional charges for different solvers, optionally by adding further types of translation rules without departing from the spirit or the scope of the invention. The general formulation used herein for the winner determination is to maximize the surplus, subject to a number of constraints. The function describing the degree of how preferred a combination is (i.e. here how much surplus it generates) is referred to as the objective function . Common for all winner determination problems is that there are some feasibility constraints saying that it must be possible to

(re) allocate the commodities following the selected combination of bids. Optional relevant constraints are constraints related to some mutual exclusion of different bids and constraints describing discounts/ additional charges.

An illustration of a basic winner determination problem (without any discounts or additional charges) is given by the following example. Buyer A bids $5 for one unit of commodity 1 and one unit of 2, buyer B bids $3 for one unit of commodity 3, buyer C bids $6 for one unit of commodity 2 and one unit of commodity 3, seller D offers one unit of commodity 1 for $2, seller E offers one unit of commodity 2 and one unit of commodity 3 for $ 1, and finally seller F offers one unit of commodity 2 for $1. This gives the following winner determination problem:

max 5A+ 3B+ 6C- 2D- IE- IF (1) subject to A-D=0 (2)

A+ C- E- F= 0 (3)

B+ C- E= O (4)

The variables A - F, representing the bids of the six bidders, are binary and a value equal to 0 denotes that the bid is not selected and 1 that it is selected. Eq. (1) is the objective function. Eqs. (2) -(4) are the feasibility constraints. Here we have assumed that supply must match demand. Sometimes it is sufficient that supply is at least as high as the demand for each commodity. This is referred to as free disposal. If free disposal is assumed, then "=" should be changed to "≤" in Eqs. (2) -(4) . The optimal solution here is A = C = D = E = F = 1 and B = O.

When discounts and additional charges are added they will both contribute to the objective function and add different constraints as described by the translation rules described below. In the translation rules below, the following notation is used:

There are k commodities in the market. There are n variables, b_, b₂, .... b_n, representing the bids (without discounts or additional charges) of the bidder under observation, and b = {b₁, b₂, .... b_n} . Furthermore, let there be a function, v (b) , giving the value of a specific assignment, and k functions, q_ (b) , giving

the traded quantity of commodity i. Finally, let .

Hence, the contribution (without discounts or additional charges) of the bidder under investigation to the objective function is v(b) (cf. Eq. (1)), and the contribution (without discounts or additional charges) to the feasibility constraint of commodity i is q_ (b) (cf. Eqs. (2) -(4)). For the examples below, v and q_t are assumed to be linear functions. An example: Let there be three commodities, k = 3 , and let the bidder under observation place two bids stating

Bid h_ : $4 is bid for 7 units of commodity 2, and Bid b₂ : 2 units of commodity 1 and 3 units of commodity 3 are offered for $6. Then we define the bids by two variables, b_ and b₂ (giving n = 2). Furthermore v(b) , q₁ (b) , q₂ (b) , and q₃ (b) , are determined by v(b) = 4b, - 6b₂ q,(b) = -2b₂ q₂ (b) = lb_ , and q₃(b) = -3b₂ (5)

According to the invention, the objective function and the related constraints are generated using translation rules. A number of presently preferred translation rules will be described below. First, preferred examples of how to express the conditions for when the discounts/additional charges can be offered or are requested will be described (translation rules 1 to 7) . Then the issue of how to give the discounts/charge the additional charges will be described later (translation rules 8 to 9) .

Translation rule 1.

The first translation rule is preferred when a discount offer condition, defined in terms of value, is translated to a linear constraint; if a seller can sell for more than a specific value V, a specific discount is offered. Linear formulation :

Let d be a binary variable which is 0 if the discount is not used, and 1 if it is used. Then the following constraint is added to the winner determination problem, saying that if the total value of what is sold is above V, then d may be set to one (the discount may be accepted) :

-v(b) ≤ Vd. (6)

It should be noted that V is a negative number as it denotes a sale bid, cf. the example described above in association with Eqs. (1) - (4) .

Translation rule 2.

The second translation rule is preferred when a discount request condition, defined in terms of value, is translated to a linear constraint; if a buyer buys for more than a specific value V, a specific discount is requested. Linear formulation:

Let d be a binary variable which is 0 if the discount is not used, and 1 if it is used. Then the following constraint is added to the winner determination problem, saying that if the total value of what is sold is above V, then d must be set (the discount is requested) : v(b) ≤ Cd + V, (7) where C is a (just) sufficiently large positive constant. A very high C always works, but setting it as low as possible often increases the computational efficiency of the algorithm used. If, for example, it is known that the value can never exceed V , C should be set to V - V.

Translation rule 3.

The third translation rule is preferred when a discount offer condition defined in terms of quantity, is translated to a linear constraint; if a seller gets to sell N or more items, a certain discount is offered. Herein, the interpretation is the net amount sold. That is, if the seller sells two units and buys one the net amount sold is one unit . Linear formulation:

Let d be a binary variable which is 0 if the discount is not used, and 1 if it is used. The following constraint is added to the winner determination problem, saying that if N or more units are sold, d can be set:

-q(b) ≥ Nd. (8)

Translation rule 4. The fourth translation rule is preferred when an additional charge condition, defined in terms of quantity, is translated to a linear constraint; if a seller gets to sell N units or more, a certain additional charge is requested.

Linear formulation : Let a be a binary variable which is 0 if the additional charge is not enforced, and 1 if it is enforced.

Then the following constraint is added to the winner determination problem, saying that if N or more units are sold, a must be one : -q(b) < Ca + N, (9) where C is a (just) sufficiently large positive constant.

Translation rule 5.

The fifth translation rule is preferred when an additional charge condition defined in terms of minimal quantity, is translated to a linear constraint; if a seller gets to sell less than N units, a certain additional charge is requested. (If no units are sold, no additional charge can be claimed.)

Formulation: This condition is translated into a discount offer corresponding to the third translation rule above. For example, if an additional charge of V is requested if N or less units are sold then, instead of defining an additional charge variable a, some conditions for it, and adding -aV to the objective function (cf . Translation rule 9) , we introduce a discount variable, d, which can be set if N or more units are sold, cf. Translation rule 3. (We then add dV to the objective function.)

Translation rule 6. The sixth translation rule is preferred with an example of an additional charge condition based on specific properties: "I offer to deliver a package for a certain fee, but for each package I can deliver to the same area, I give a certain discount". Assume that commodities 1 to k ' correspond to packages to one area and that commodities k ' + 1 to k correspond to packages to another area, k ' < k.

Linear formulation .

Let e be an integer variable representing the number of discounts. Let e_ and e₂ be variables representing discounts of the respective areas. Then the conditions are: -q_ (b) ≥ 2e_x -q₂ (b) ≥ 2e₂ e ≤e, + e, Translation rule 7

The seventh translation rule is preferred when an additional charge condition is based on difference between numbers of units sold of different commodities, for example: "I offer to sell n_ units of commodity 1 and n₂ units of commodity 2 at some specific prices, but if the difference between the number of purchased units for the respective commodities is higher than 30% of all commodities sold, I require a specific additional charge" . Such a condition could occur in a case where one of the products is a by-product of the other.

Linear formulation :

Let a be a binary variable which is 0 if the additional charge is not enforced, and 1 if it is enforced.

First we need the standard conditions -q_ (b) ≤n_ and -q₂ (b) <n₂. Then we introduce two variables a_{χ l} and a₂ and the constraint (s) -a₂ ≤ - q_ (b) + q₂ (b) ≤ a_ (11) saying that if the number of sold units of commodity 1 is larger than the number of sold units of commodity 2, then a₁ is (at least) the difference between the two, else if the number of sold units of commodity 2 is larger than the number of sold units of commodity 1, then a₂ is (at least) the difference between the two.

Then, finally, we can introduce the constraints a_ ≤ 0. 3 (-q₁ (h) - q₂ (b ) ) + 0 . 7 n_xa, and (12) a₂ ≤ 0 . 3 (-q_x (b) - q₂ (b) ) + 0 . 7n₂a . Stating that a must be one if the difference, represented by a_x and a₂, is more than 30% of the sold volume.

Comments on translation rules 1-7

Discounts and additional charges may, but need not, be mutually exclusive or be restricted by other constraints. For example, if a seller offers at most n out of m discounts, d_, we can introduce

a constraint 7 , d, < n

In terms of discounts, combining the conditions is normally not problematic, but combining additional charge discounts requires some more care. Assume, for example, that the condition of translation rule 4 above is extended with another required additional charge if N' (N' > N) or more units are sold, and that the constraints are mutually exclusive. Then the straightforward formulation -q (b) < C_a₁ + N, -q (b) < C₂a₂ + N' , and a_I + a₂ ≤ 1 , would exclude valid solutions, since if -q (b) > N' , then both a_ and a₂ must be one according to the first two inequalities, giving a contradiction with the third inequality. Instead a formulation of the form -q (b) < C_a_ + C₂₁a₂ + N, -q (b) < C₂₂a₂ + N' , and a_ + a₂ ≤ 1 or similar, is required.

It has been described above how to express the conditions for when the discounts/additional charges can be offered, or are requested, and the consequent issue of how to give the discounts/charge the additional charges will now be described.

Translation rule 8.

The eighth translation rule is preferred when a discount/additional charge condition results in a discount offer defined in relative terms; if some condition is fulfilled, an x% discount is offered. Linear formulation :

Let d be a variable which is 0 if the discount/additional charge is not used, and 1 if it is used. Furthermore, let c_, 1 ≤ i ≤ n be discount variables with the interpretation shown below, and let c =

[c_ , c₂,...,c_n] . The following three inequalities ensures that if d = 1 then c_± = _{i t} else c = 0 :

- Cd ≤ c_ ≤ Cd c_ ≤ b_ + C (l -d) (13) c_ ≥ b_ + C (d - 1) , where C is a (just) sufficiently large constant.

If d represents a discount, it adds the following term to the sellers contribution to the objective function to be maximized, cf . Eq. (1) ,

-x V(C) (14)

100 -X and if d represents an additional charge, v(c) is instead

100

added.

Note that Eq. (13) often can be significantly simplified. For example, if there is an offered discount and it is known that b_± is binary, then it is sufficient with a condition saying that if d = 1 and b_L = 1, then c_ (which then also is a binary) may be 1, i.e.

2c_ ≤ b_ + d (15)

Similarly if d represents that an additional charge is required, and b_ is binary, it is sufficient to say that if d = 1 and b_ = 1 , then c_ (which then also is a binary) must be 1, i.e. c_ + 1 ≥ b_ + d . (16)

Quadratic formulation:

For this type of discount/additional charge, the number of variables can be reduced significantly by using a quadratic formulation. If we again let d denote the discount, the contribution to the objective function is

-d (x/l 00) v (b) , (17) and correspondingly for an additional charge.

Translation rule 9. The ninth translation rule is preferred when a discount defined in absolute terms is translated to a linear constraint; if some condition is fulfilled, a seller offers a discount of value V. Linear formulation:

Let d, be a variable which is 0 if the discount is not used, and 1 if it is used. This discount adds the following term to the seller's contribution to the objective function to be maximized, cf . Eq. (1) ,

Vd (18)

This rule is also directly applicable for an additional charge, but then -Vd is added instead.

It should be noted that the translation rules above are typical for the actual translations but, at the same time, they are examples only.

The examples of translation rules presented above represents mainly linear formulations. A winner determination problem expressed only in linear terms is well adapted to be solved by integer programming algorithms implemented in computer program. Actually, the number of commodities and bids need not be very high before it is practically impossible to perform the calculations without the aid of computers .

Though the above translation rules focused on linear formulations, the invention is not limited to this case. Other types of solvers, such as quadratic or hybrid solvers may be applicable to different multi-commodity auctions with explicit discounts and/or additional charges. Then the translation rules above need to be modified (and/or new ones may need to be added) , to suit the selected solver. Similarly new types of discounts and/or additional charges may be added and the above ones may be modified by establishing new translation rules (and/or modify the above ones) . However, such modifications will be obvious for anyone skilled in the art. The main principles as described above will still be the same.

A comparative example

An illustrative example will now be given in order to exemplify the basic principles, as well as to make a comparison to basic combinatorial auctions. The objective of this example is to illustrate advantages with the invention.

The example is as follows: A company has four contracts, for example four tasks that need to be performed or four products that need to be delivered. There are three entrepreneurs bidding on the contracts. Two of them are small; they would prefer to get only a small number of contracts . The third entrepreneur is large and would like to get many contracts. More specifically, they would like to express the following offers:

Entrepreneur A: "I can take any of the first three contracts for $500, but if I get more than one contract, I need to charge $100 extra." (The entrepreneur has an increasing cost with scale.) Entrepreneur B: "I can take any contract for $550, but I cannot take more than two contracts, and if either contract 1 or contract 2 (or both) are accepted, I need to charge $50 extra."

Entrepreneur C: "I can take any of the contracts for $600, and I give 10% discount if I get more than one contract." (The entrepreneur has decreasing cost with scale . )

The solution with the highest surplus is to let Entrepreneur A sign up for contract 1, and let Entrepreneur C sign up for contracts 2, 3 , and 4.

Prior art method

With a . traditional combinatorial auction, all combinations that each bidder could possibly accept have to be explicitly enumerated. As seen below this gives a large number of bids per entrepreneur.

Entrepreneur A: Bid A_ expresses the case when this entrepreneur gets all the three first contracts, bids A₂, A₃, and A₄ the case when it gets two out of the three first contracts, etc. As all the bidders of this example, bidder A also has to declare that he only accepts one of the enumerated bids (a so called XOR constraint is used) .

Entrepreneur B: All combinations wherein the entrepreneur gets one or two contracts of the four possible contracts have to be enumerated .

Entrepreneur C: All combinations wherein the entrepreneur gets at least one contract have to be expressed with the corresponding costs.

The large number of bids generates a large number of variables in the optimization problem. In this section, this is illustrated by giving the corresponding integer linear formulation, but a similar result is obtained • with any conventional combinatorial auction winner determination algorithm.

The objective function to maximize, according to this prior art example, for solving the winner determination problem (finding the solution with the lowest cost) with one variable for each bid is, (cf . Eq. (1) ) :

-1600A₁-1100A₂-1100A₃-1100A₄-500A₅-500A₆-500A₇-1150B₁-1150B₂- 1150B₃-1150B₄-1150B₅-1100B₆-600B₇-600B₈-550B₉-550B₁₀-2160C₁- 1620C₂-1620C₃-1620C₄-1620C_S-1080C₆-1080C₇-1080C₈-1080C₉-1080C₁₀- lO^βOCu-^βOOCia-SOOC^-βOOC^-βOOC_jg. (19)

The constraints of the winner determination problem are as given below. (Some explanations are given after the constraints.)

A₁+A₂+A₃+A₅

+B₁+B₂+B₃+B₇ +C₁₊C₂+C₃+C₄+C₆+C₇+C_a+C₁₂ = 1 (20)

^Al^+A2^+A4^+A6

+B₁+B₄+B_s+B_a +C₁+C₂+C₃+C₅+C_S+C₉+C₁₀+C₁₃ = 1 (21) A₁+A₃+A₄+A₇ +B₂+B₄+B₆+B₉ +C₁+C₂+C₄+C_S+C₇+C₉+C₁₁+C₁₄ = 1 (22) B₃+B₅+B₆+B₁₀

+C₁+C₃+C₄+C₅+C₈+C₁₀+C₁₁+C₁₅ = 1 (23)

A₁+A₂+A₃+A₄+A₅+A₆+A₇ ≤ 1 (24)

B₁+B₂+B₃+B₄+B₅+B_s+B₇+B₈+B₉+B₁₀ ≤ 1 (25)

C₁+C₂+C₃+C₄+C₅+C₆+C₇ +C₈+C₉+C₁₀+C₁₁+C₁₂+C₁₃+C₁₄+C₁₅ < 1 (26)

Explanations: Eq. (20) states that one entrepreneur gets contract 1. Eqs. (21) -(23) state the same thing for contracts 2, 3, and 4, (cf. Eqs. (2) -(4)) . One can also use "≥" rather than "=" in Eqs. (20) -(23) , as that would mean that it could be acceptable that the optimal combination includes a solution when more than one entrepreneur has committed to the contract. This can typically always be resolved by relieving an entrepreneur from a contract. Eqs. (24), (25), and (26) state that at most one bid per entrepreneur is accepted.

In terms of the variables of this formulation, the optimal solution is A_s = C₅ = 1, and all other variables are equal to zero.

Method accordin^g; to the present invention With the method of the present invention, bidders are able to express their bids in a very convenient and natural way. For example, an interface for input of the bids could typically and schematically look like the following, wherein the exemplifying bids per above are inserted:

Entrepreneur A:

ADDITIONAL CHARGE: more than [_____] contract implies | 100 1 extra

Entrepreneur B :

(LIMITATION : at most [Tl contract are accepted

ADDITIONAL CHARGE : accepting contract [T] or [T] implies |^~5O extra

Entrepreneur C :

DI SCOUNT : If I get more than [Tj contract , I give [ 10% | discount

With the present invention, the integer programming formulation contains comparatively few variables. The objective function of the direct formulation according to the invention is, cf. Eq. (1) -500A₁₀₁-500A₁₀₂-500A₁₀₃-550B₁₀₁-550B₁₀₂-550B₁₀₃-550B₁₀₄-60OC₁₀₁- 600C₁₀₂-600C₁₀₃-600C₁₀₄-100a-50b+60c₁+60c₂+60c₃+60c₄ (27) Apart from the objective function, there are a number of inequalities stating the constraints under which the objective function should be maximized:

^■"202 ^+B101 ⁺ (~ιoi = 1 (28)

^102^+B102 ^{+ (}~-102 = 1 (29) t 03 ^+B103 ⁺ ~103 = 1 (30)

Bl04^{÷ <}~₁₀₄ = 1 (31)

^A 01 ^+J^202 ⁺Α 03 ≤ 2a+l (32)

^B101 ^+B102 ^+B103 ^+B104 ≤ 2 (33)

^B101^+B102 ≤ 2b (34)

^-101 ^{+ (}~102 ⁺ ^103 ⁺ ^104 ≥ 2d (35)

2c_ ≤ C +d (36)

2C₂ ≤ C₂+d (37)

2c₃ ≤ C₃+d (38) 2c_ ≤ C₄+d (39)

Explanation: Eq. (28) states that only one entrepreneur can get contract 1, Eqs. (29) - (31) state the same thing for contracts 2-4, Eq. (32) states that the variable a should be 1 if entrepreneur A gets more than one contract, Eq. (33) states that B should get 2 contracts at most, Eq. (34) states that the variable b must be set if B₁₀₁ or B₁₀₂ are set, Eq. (35) states that variable d must be 0 if C gets less than two contracts, Eq. (36) states that the discount variable c_ can only be 1 if entrepreneur C gets contract 1 and the variable d is 1, Eqs. (37) - (39) express the same thing for contracts 2-4.

In terms of the variables of this formulation, the optimal solution is A₁₀₁=C₁₀₂=C₁₀₃=C₁₀₄=d=c₂=c₃=c₄=l, and all other variables equal to zero.

In this example, for arriving at the above direct formulation of the winner determination problem, translation rules according to the invention, and as described above, have been applied as follows:

For entrepreneur A, the value v = -500A₁₀₁-500A₁₀₂-500A₁₀₃, and the quantities q_1# q₂, and q₃ are -A₁₀₁, -A₁₀₂, and -A₁₀₃, respectively. The corresponding holds for the other entrepreneurs. The terms -100a and -50b in Eq. (27) were obtained from translation rule 9. The terms 60c_x in Eq. (27) resulted from translation rule 8. Translation rule 4 generated Eq. (32) and a simple variant of the same rule generated Eq. (34) . Eq. (35) was obtained from translation rule 3, and finally Eqs. (36) - (39) were obtained from translation rule 8.

The example described above illustrates a significant difference both in terms of simplicity of the bids and in terms of number of variables in the resulting winner determination problem between a conventional approach and the utilization of the method according to the present invention.

The example was selected for simplicity and readability, and it should be emphasized that the differences become much more evident with larger and/or more complicated examples. For example, even a simple construction like "I am willing to purchase 200 units of commodity A for $30 each and 100 units of commodity B for $20 each, but if the number of acquired units of commodity A and commodity B differ by more than 30%, I demand a 20% discount", becomes impractical by use of a conventional method, as this requires the enumeration of some 17,000 combinations.

An example system, according to the invention, for administrating multi-commodity auctions with explicit discount and/or additional charge bids is schematically shown in Fig. 4. Typically, the auction is administered via a computer server 110 connected to a number of bidders 121, 122, 123 via a computer network, such as the Internet. The server 110 includes a software 113 designed to perform the steps of the method as described above, a memory 112 to store data during the auction process, an I/O unit to connect the server with the bidders 121, 122, 123, and for communication with any device 115 for presenting the result of the auction, such as a printer, a monitor or a link to the bidders. A CPU 114 controls the server 110.

When the bidders are human, they are preferably presented with a user-friendly graphical interface, such as in the examples of Figs. 2 and 3. Alternatively, the bid could be automatically generated by a software agent representing the bidder, in which case the agent should provide the bids in a high-level language suitable for the method of the invention.

Claims

1. A method for selecting a winning bid(s) of a multi-commodity auction, said auction including a possibility to include in a bid a conditional discount and/or a conditional additional charge, comprising the steps of providing a computer interface for input of a bid to the auction, said interface allowing the bid to be explicitly input in a high-level language in accordance with a pre-selected set of discount and/or additional charge alternatives; using a computer software to generate an objective function and a number of constraints, based on said preselected set of discount and/or additional charge alternatives and on the bids of the auction, said objective function representing a measure of how preferred a combination of bids is; and using a computer software to maximize said objective function with respect to said constraints to obtain the winning bid(s) .

2. The method according to claim 1, wherein said computer interface comprises a graphical user interface for human input of the bids, said graphical user interface including specific fields for explicit input of a basic bid as well as for input of any conditional offers or requests included in the basic bid.

3. The method according to claim 1, wherein said computer interface comprises an interface with formats for input of the bids, as well as any conditional offers or requests included in the bid, from a software agent.

4. The method according to claim 1, wherein a direct formulation of said objective function and said constraints are generated, and said objective function is solved, using integer linear programming.

5. The method according to claim 1, wherein a direct formulation of said objective function and said constraints are generated, and said objective function is solved, using mixed integer linear programming.

6. The method according to claim 1, wherein a direct formulation of said objective function and said constraints are generated, and said objective function is solved, using constraint satisfaction formulations.

7. The method according to claim 1, wherein a direct formulation of said objective function and said constraints are generated, and said objective function is solved, using formulations for hybrid solvers .

8. The method according to claim 1, wherein a direct formulation of said objective function is a measure of a total reported economical surplus of all bids.

9. The method according to claim 1, wherein, in the case of a discount offer defined in terms of value under a condition of a minimum sale value, the condition is translated to the constraint -v (b) ≤ Vd, wherein v (b) is a function giving the value of a specific assignment

(wherein b= {b_, b₂, . . . .b_n) , and b_1# b₂, ....b_n are n variables representing the bids) , V is the specific value for the discount offer, and d is a binary variable which is 0 if the discount is not used, and 1 if it is used.

10. The method according to claim 1, wherein, in the case of a discount request defined in terms of value under a condition of a minimum procurement value, the condition is translated to the constraint v (b) ≤Cd+V, wherein v (b) is a function giving the value of a specific assignment (wherein b= {b_{l f} b₂, ... , b_nJ , and b_{l t} b₂, . . . .b_n are n variables representing the bids) , C is a positive constant, V is the specific value for the discount request, and d is a binary variable which is 0 if the discount is not used, and 1 if it is used.

11. The method according to claim 1, wherein, in the case of a discount offer defined in terms of quantity under a condition of a minimum sold quantity, the condition is translated to the constraint

-q (b) ≥Nd, wherein q (b) is a function giving the bought quantity (i.e. sold quantities are negative) of commodity i (wherein b= {b_x, b₂, ....b_n} , and b_ , b₂,....b„ ae π variables representing the bids) ,

N is the minimum net amount sold, and d is a binary variable which is 0 if a discount offer condition defined in terms of quantity is not used, and 1 if it is used.

12. The method according to claim 1, wherein, in the case of an additional charge request defined in terms of quantity under a condition of a minimum sale, the condition is translated to the constraint -q (b) ≤ Ca + N, wherein q (b) is a function giving the traded quantity of commodity i (wherein b= {b_{l r} b₂, . . . , b_nj , and b_{l f} b₂, ....b_n are n variables representing the bids) , C is a positive constant, a is a binary variable which is 0 if an additional charge defined in terms of quantity is not enforced, and 1 if it is enforced, and N is the minimum net amount sold.

13. The method according to claim 1, wherein, in the case of a discount offer defined in terms of quantity under a condition of a minimum number of items delivered to the same location

(or other categorizations) , the condition is translated to the constraint

-q_ (b) ≥ 2e₂

-g₂(JbJ^2e₂ e^e_I+e₂ wherein q_ (b) is a function giving the traded quantity of commodity i

(wherein b= {b₁, b₂, ... , b_n} and b_, b₂, .... b_n are n variables representing the bids) , e_ and e₂ are variables representing discounts of each one of two different locations, and e is an integer variable representing the number of discounts.

14. A computer server for managing a multi-commodity auction, wherein the server includes an I/O unit for communication with at least one bidder, comprising a computer software for performing at least one of the tasks selected from the group consisting of: a) providing a communication interface for input of a pre- selected set of discount/additional charge offer/request alternatives ; b) forming an objective function, based on the pre-selected set of offer and/or request alternatives, as well as on bids input by the at least one bidder; c) defining constraints to said objective function based on the pre-selected set of offer and/or request alternatives as well as on the bids; and d) maximizing said objective function with respect to said constraints to obtain the winning bid(s) .

15. The computer server according to claim 14, wherein said computer software provides a human bidder with a user interface allowing a bid to be explicitly input in a high- level language in accordance with said pre-selected set of offer and/or request alternatives.

6. The computer server according to claim 14, wherein said computer software provides a software agent with a high-level language interface allowing a bid to be explicitly input in accordance with said pre-selected set of offer and/or request alternatives .