The present invention relates generally to methods for performing negotiations between parties and more particularly to a method for automatically performing a negotiation between multiple parties without necessarily disclosing the identity and positions of the parties to each other. The present invention also relates generally to methods and apparatuses for performing computerized trading, and more particularly to a method and apparatus for performing computerized trading of multiple securities in a single simultaneous transaction. This is known as "linked trading." In negotiations, parties often desire to negotiate a deal involving something of value to both parties or multiple parties, but are wary of disclosing their willingness to deal on any variable of the deal for fear of losing any negotiating strength. In fact, it is a well-known negotiating tactic to avoid being the first party to suggest an offer, since that sets the stage for the remaining offers. However, it is practically impossible to negotiate without a starting point.

In today's complex world, these negotiations now often involve multiple parties and multiple terms. Occasionally, the parties do not even know all of the other parties to the transaction, e.g., in large exchanges where trades are made between

millions of parties who may never even meet face to face. This presents significant difficulties in reaching an agreement even on a single term, much less multiple terms.

One particularly good example of this is securities trading, and in particular linked trading of securities. Many trading strategies in current markets involve simultaneous purchases and/or sales of multiple securities, where the combined transaction must satisfy a prescribed price objective. Simple examples of multiple securities trading include: 1) pairs trading, in which a party is interested in buying one security that is perceived to be undervalued using the proceeds from a simultaneous short sell of a correlated security that is perceived to be overvalued, in expectation that both securities will revert to their nominal values, allowing both positions to be closed out at a profit; and 2) buy-write trades that involve the simultaneous purchase of a stock and sale of a call option on the stock, wherein the call premium received partially hedges the risk of a decline in the value of the stock, at the cost of capping the potential profit of an upside move.

More complex examples of linked trades include extensions of pairs trading to more than two securities. One type of multiple securities trading is known as "basket trading," an example of which is index trading, wherein the objective is to buy or sell a defined set of securities (the basket) in a single transaction. Another example is combined equity-currency trades, in which one is purchasing foreign equity in the native currency and simultaneously effecting the required currency exchange trade. Yet another example are "swaps," wherein one income and/or payment stream is swapped for another. More general extensions can be envisioned, involving an arbitrarily complex set of linked trades across multiple securities types.

All current techniques and systems for implementation of linked trades are inefficient and carry some risk for the party desiring the trade for several reasons.

First, the practical requirements of carrying out the multiple trades in significant volume may require hundreds or even thousands of telephone calls and/or keyboard entries into an electronic terminal connected to individual centers of liquidity for the securities involved. Therefore, a common practice among parties to such trades is to employ an intermediary who is willing either to take the contra side of the

combined trade, acting as principal, or to negotiate and execute the simultaneous legs of the combined trade, acting as agent. Obviously, this service is not without cost to the original party, and in the case of a principal trade, the intermediary has now assumed the contra position in the corresponding securities, and in most cases, will want to unwind this position over time. In the case of an agency trade, the realized prices, and even the ability to execute the individual legs of the trade, are uncertain, which exposes the original party to financial risks that cannot be controlled in advance of trade implementation.

Second, the costs incurred in link trading are significantly high, especially where the contra position is negotiated with an intermediary acting as the principal in order to enable the trade to occur with certainty. As an example, an intermediary might charge as much as 3% of the total value of the trade on average, and more if the trade involves illiquid securities, to take the contra position.

Finally, index trading, which is a very prevalent example of multiple security linked trading, often incurs substantial "market impact" costs, in addition to the execution costs, as a result of the leakage of information into the marketplace.

Adverse price moves in the individual securities making up the index can be arranged to the benefit of other market participants (and the detriment of the party desiring the trade) if the other market participants have information about an impending index trade(s). Thus, the linked trader desires both anonymity of the fact that he is trading in linked securities and non-disclosure of his desired position in each of the securities to the market.

The present invention is therefore directed to the problem of developing a method and system for automatically negotiating an optimum agreement between parties desirous of obtaining such an agreement without necessarily identifying the parties and their positions to each other, which method and apparatus is suitable for performing linked trading in a way that optimizes the trade prices for the participants on both sides of the trades and does so without disclosing the identity of the trader and his position in the securities at issue.

SUMMARY OF THE INVENTION The present invention solves the above-identified problems by providing a method and system that calculates the mutual satisfaction between the negotiating parties and maximizes this mutual satisfaction over a range of decision variables and does so without requiring the parties to identify themselves and their positions to each other.

According to the present invention for automatically negotiating agreements between multiple parties, a computer accepts a satisfaction profile from an offering party who defines his degree of satisfaction to agree to a range of terms upon which the party is desirous of negotiating as a function of the relevant decision variables.

The computer then accepts input from all other parties regarding their degree of satisfaction to agree to each of the terms as a function of a particular relevant decision variable. The computer then calculates a satisfaction profile for each of these terms based on all of the individual inputs. Next, the computer calculates a joint satisfaction profile for all of the terms as a function of the particular relevant decision variables, and then calculates the mutual satisfaction function for the offering party and the other parties, also as a function of the particular relevant decision variables. Finally, the computer calculates the decision variable values yielding the maximum mutual satisfaction and provides this output to the parties.

According to the present invention, a method for automatically trading linked securities includes the steps of receiving a satisfaction function from an offering party who defines its degree of satisfaction to trade multiple securities simultaneously based on the overall cost of the transaction, receiving a satisfaction profile from other traders who indicate their degree of satisfaction to trade particular securities as a function of price/volume, creating a composite satisfaction function for each of the individual securities from the input of all of the individual traders, creating a joint satisfaction function for all of the securities from all of the satisfaction density functions, creating a mutual satisfaction function from the joint satisfaction function and the satisfaction density profile entered by the offering party, maximizing the mutual satisfaction function, which establishes a set of prices, volumes and parties for trading each of the individual securities, and executing a

trade in the multiple securities simultaneously with the identified parties at the prices and volumes established by the maximum mutual satisfaction.

According to the present invention, a system for automatically trading linked securities includes a linked trading workstation at which a linked trader can input a desired linked trade and a satisfaction function that defines its degree of satisfaction to trade multiple securities simultaneously based on the overall cost of the transaction, a plurality of trader workstations at which individual securities traders can enter a satisfaction density profile that indicates their degree of satisfaction to trade particular securities as a function of price/volume, a central control engine determining: (1) a composite satisfaction function for each ofthe individual securities based on the input of the traders, (2) determining a joint satisfaction function for all of the securities from these satisfaction functions, (3) determining a mutual satisfaction function from the joint satisfaction function and the satisfaction function entered by the linked trader, (4) maximizing the mutual satisfaction function, which establishes a set of prices, volumes and parties for trading each of the individual securities, and executing simultaneously a trade among the identified parties for the multiple securities at the established prices and volumes.

BRIEF DESCRIPTION OF THE DRAWINGS FIG 1 depicts a satisfaction contour plot of SL(C) for the case of buying one unit of X, and buying two units of X2, respectively.

FIG 2 depicts a satisfaction contour plot of SL(C) for the case of buying one unit of X, and selling two units ofX2, respectively.

FIG 3 depicts a satisfaction contour plot of SL(C) for the case of selling one unit of X, and buying two units of X2, respectively.

FIG 4 depicts a satisfaction contour plot of SL(C) for the case of selling one unit of X, and selling two units of X2, respectively.

FIG 5 depicts a contra-side satisfaction contour plot of Sc(P,, P2J for the case of selling one unit of X, and selling two units of X2, respectively.

FIG 6 depicts a contra-side satisfaction contour plot of SL(C) for the case of selling one unit of X, and buying two units ofX2, respectively.

FIG 7 depicts a contra-side satisfaction contour plot of Sc(Pi, P2) for the case of buying one unit of X, and selling two units of X2, respectively.

FIG 8 depicts a contra-side satisfaction contour plot Of SL(C) for the case of buying one unit of X, and buying two units of X2, respectively.

FIG 9 depicts a contour plot ofMS(P1, P2) for the case of buying one unit of X, and buying two units of X2, respectively.

FIG 10 depicts the segment , representing the locus of points of tangency between satisfaction contours of SJc) and Sc(P,, P2).

FIG 11 depicts one embodiment of the system of the present invention for implementing the method of the present invention.

DETAILED DESCRIPTION The present invention provides a novel method for enabling parties to negotiate with each other both quickly and easily without disclosing their willingness to negotiate on specific terms and in fact without necessarily disclosing their identities. In its broadest sense, the present invention is a tool that can discover areas of overlap in negotiating positions between parties having a mutual desire to enter into some type of agreement. The applications to which the present invention may be applied are as varied as the imagination of the users. Some examples are disclosed below, however, these are merely indicative of the broad range of possibilities. While the present invention is described mainly herein in the context of securities trading, it is equally applicable to other linked trading applications involving non-securities goods and services, such as commodities, etc.

One possible example is as follows.

Airline Negotiations Example If a party, such as an airline, wishes to negotiate with three unions, such as the airline pilots, the flight attendants and the mechanics unions, the party might employ the present invention. Assuming the airline had a fixed cost (i.e., salaries, benefits, etc.) in mind that it could pay to all employees, and with some constraints, it did not care which unions were paid more than the others, the airline would

employ the present invention as follows. First, the party would determine its satisfaction function based on its understanding of its industry and needs and quantify its desired terms and satisfaction for agreeing to these terms. The method for determining this satisfaction function is different for each industry and participant, and hence is beyond the scope of this invention, but is well understood to those who are responsible for managing companies in these industries. The airline would then enter this data into a computer and transmit that data to the control engine. Each of the three unions would be provided templates for entering their degree of satisfaction for agreeing to the terms, which templates correspond to the arrangement of data of the airline. However, some terms might not necessarily apply to all participants. By enabling the unions and the airline to enter their degree of satisfaction without disclosing it to the other, the present invention permits the parties to be more aggressive in specifying terms upon which they might agree. For example, knowing that an earlier agreement might get them more of the total funds available one of the unions might agree to lower wages initially then they might otherwise.

Once all of the data is entered into the system, the system calculates the satisfaction function for each individual term, and then calculates the joint satisfaction function for all of the terms. Finally, the system calculates the mutual satisfaction by determining the area of overlap between the joint satisfaction function and the offering party's satisfaction function. The maximum point of this mutual satisfaction is then determined, which establishes the terms for each of the parties and the overall agreement is then specified. The maximum value of the mutual satisfaction may not lie exactly on a discrete term, but may be located between two terms, in which case the nearest term is used. Furthermore, once a point within some tolerable degree is located, then finding the absolute maximum is not necessary and the search can then end.

Securities Trading Example A more concrete example can be found in the securities trading arena. U.S.

Patent Application No. 08/430,212, which has been incorporated by reference

above describes the concepts of satisfaction-based representations of mutual trading desire and optimization of trading parameters on the basis of mutual satisfaction.

The same concepts are used in the present invention for the individual securities traders.

With regard to the present invention, inter alia, this patent application discloses a method that a linked trader uses to enter its orders to buy or sell multiple securities simultaneously in the form of a satisfaction function. The present invention differs from this prior patent application in that it matches the multitude of individual orders with the linked trade order to perform the linked trade and maximizes the satisfaction of all traders involved.

Theory We shall describe the theory behind the present invention in the context of securities trading. The same theory applies equally to the above described situation, and others (such as broadcast time trading, commodities trading, or any agreement where multiple parties and/or multiple terms are involved and the parties would rather not disclose their willingness to enter into the agreement on certain terms), however, the theory is significantly more abstract in these instances.

Assume a market participant ("linked trader") desires to execute a set of N linked trades involving the simultaneous purchase of V, units of k different securities, Xj, i = 1 ... k, and sale of Vj units of N - k different securities X;-, where i = k +1 ... N. Any indifferent set of price outcomes from this set of trades can be expressed as a hyperplane in N-dimensional price space: where Pj is the price of security X, Vj is positive for purchases and negative for sales, and c is the net cost of the trade. Positive values of C represent a net cash outflow from the composite trade, while negative values represent a net cash inflow.

In other words, the linked trader is merely interested in trading certain volumes of securities in a linked trade (i.e., all simultaneously) and does not care about the price obtained for each individual security, but only cares about the total cost of the transaction. Obviously, there is a definite relationship between the costs of the total transaction and the price of an individual security, but the satisfaction function is defined as a function of the cost of the overall transaction rather than the price in the individual securities, as in the prior patent application referenced above.

The linked trader therefore specifies each individual security he desires to trade and a volume associated with that security. Once all of the individual securities are specified, the linked trader then specifies a range of costs for the whole transaction (positive or negative depending whether it is a buy or sell overall) along with a degree of satisfaction for each cost within the range.

For example, Table 1 is indicative of the type of data entered by the linked trader. The table indicates the individual securities and the volume of each desired to be bought/sold.

Table 1 Security (Xj) Volume (V,) IBM 500,000 Xerox -100,000 Microsoft 200,000 Intel 400,000 AT&T 1,000,000 Lucent -500,000 Viacom 1,000,000 Citicorp -200,000 Conrail -400,000 Continuing the theoretical discussion, the user desiring the linked trade can then define an associated scalar satisfaction function SL(C) for the composite linked trade, which always has the property that SL(C) equals unity for sufficiently small

values of c (which may be negative in the case where net cash inflow is required, as in a sale) and, over some range of C values, transitions monotonically to zero as c increases. This satisfaction function expresses the degree of willingness to perform the prescribed set of linked trades, as a function of the net cost c.

Table 2 indicates the degree of satisfaction the linked trader would have in effecting the overall transaction; ranging from full satisfaction to pay $3.5 million, to unwillingness to pay $5.4 million or more.

Table 2 Overall Cost ($M) Cost ($M) 3.5 1 3.9 0.9 4.2 0.75 4.5 0.5 4.8 0.25 5.4 0 Two dimensional examples Of SL(C) are shown in the contour plots of FIGs 1-4, wherein the independent variables are P, (horizontal axis) and P2 (vertical axis). These satisfaction profiles cover the four cases of buying X, and buying X2, buying X, and selling X2, selling X, and buying X2, and selling X, and selling X2, respectively, with each case involving the purchase or sale of one unit of X, and two units ofX2.

In FIG 1, for example, region 5 indicates the region of prices where the trader would be fully satisfied to performed the linked trade. The diagonal lines (i.e., hyperplanes) indicate contour levels of intermediate satisfaction, which decreases approaching the black region. Along any of these diagonal lines, the net cost of the linked trade (and the corresponding satisfaction) is constant, and the trader is therefore indifferent to any price pair lying on a given diagonal line.

Region 1 indicates the region of prices where the trader would be unwilling to perform the linked trade. Moving in the direction of the arrow from region 1 to region 5, one passes through regions of increasing intermediate

satisfaction. In other words, region 2 represents a region of price combinations in which the trader is at least somewhat satisfied to trade. Region 3 represents a region of price combinations where the trader is more satisfied than region 2, but less satisfied than region 4. Region 4 represents a region of price combinations where the trader is more satisfied than region 3 but less than completely satisfied, as in region 5. Finally, region 5 indicates the region of price combinations where the trader would be completely satisfied to trade.

The same is true for FIG 2, however, the whole figure has been rotated clockwise ninety degrees from FIG 1. In FIG 2, for example, region 10 indicates the region of prices where the trader would be fully satisfied to performed the linked trade. Region 6 indicates the region of prices where the trader would be unwilling to perform the linked trade. Moving in the direction of the arrow from region 6 to region 10, one passes through regions of increasing intermediate satisfaction. In other words, region 7 represents a region of price combinations in which the trader is at least somewhat satisfied to trade. Region 8 represents a region of price combinations where the trader is more satisfied than region 7, but less satisfied than region 9. Region 9 represents a region of price combinations where the trader is more satisfied than region 8 but less than completely satisfied, as in region 10. Finally, region 10 indicates the region of price combinations where the trader would be completely satisfied to trade. As before, along any of these diagonal lines, the net cost of the linked trade (and the corresponding satisfaction) is constant, and therefore the trader is indifferent to any price pair lying on a given line.

As with the previous figure, FIG 3 is similar to FIG 1 but has been rotated ninety degrees counterclockwise from FIG 1. In FIG 3, for example, region 15 indicates the region of prices where the trader would be fully satisfied to performed the linked trade. Region 11 indicates the region of prices where the trader would be unwilling to perform the linked trade. Moving in the direction of the arrow from region 11 to region 15, one passes through regions of increasing intermediate satisfaction. In other words, region 12 represents a region of price combinations in which the trader is at least somewhat satisfied to trade. Region

13 represents a region of price combinations where the trader is more satisfied than region 12, but less satisfied than region 14. Region 14 represents a region of price combinations where the trader is more satisfied than region 13 but less than completely satisfied, as in region 15. Finally, region 15 indicates the region of price combinations where the trader would be completely satisfied to trade. As before, along any of these diagonal lines, the net cost of the linked trade (and the corresponding satisfaction) is constant, and therefore the trader is indifferent to any price pair lying on a given line.

Finally, FIG 4 is similar to FIGs 1-3, but has been rotated ninety degree counterclockwise from FIG 3. In FIG 3, for example, region 20 indicates the region of prices where the trader would be fully satisfied to performed the linked trade. Region 16 indicates the region of prices where the trader would be unwilling to perform the linked trade. Moving in the direction of the arrow from region 16 to region 20, one passes through regions of increasing intermediate satisfaction. In other words, region 17 represents a region of price combinations in which the trader is at least somewhat satisfied to trade. Region 18 represents a region of price combinations where the trader is more satisfied than region 17, but less satisfied than region 19. Region 19 represents a region of price combinations where the trader is more satisfied than region 18 but less than completely satisfied, as in region 20. Finally, region 20 indicates the region of price combinations where the trader would be completely satisfied to trade. As before, along any of these diagonal lines, the net cost ofthe linked trade (and the corresponding satisfaction) is constant, and therefore the trader is indifferent to any price pair lying on a given line. This same concept extends to an arbitrary number of dimensions, but cannot be depicted graphically beyond three dimensions.

Having posed the satisfaction function SL(C) for the linked trade, it remains <BR> <BR> <BR> to describe the joint contra satisfaction function Sc(Pi P PN; VI , VN), V, which is a 2N-dimensional function of price and volume in each of the N securities involved in the linked trade. For the moment, let us assume that adequate contra-side liquidity exists in all of these securities to enable the purchase and/or sale of the desired volume Vi in each security. We shall address the situation when such

liquidity does not exist later.

For security Xi, we can then construct a contra-side satisfaction function S,(P, V) by determining, for each price, the volume-weighted contra satisfaction to trade V shares at that price. Given our assumption of adequate liquidity, this may be expressed as a scalar function of price alone, Scj (PJ, for the implicit volume Vi required for the linked trade.

For example, a sell contra satisfaction function may be constructed by determining the minimum price for which the volume Vi can be purchased, along with the corresponding volume-weighted satisfaction at that price, and then incrementing the price upward, calculating volume-weighted satisfaction at each price increment until a volume-weighted satisfaction value of unity is obtained.

Under the assumption of adequate liquidity, the joint contra satisfaction function is then defined as the product of these individual satisfaction functions: This contra-side satisfaction function is defined over a "fuzzy hypercube" in N- dimensional price space, in the sense that contours of constant satisfaction, which is a fuzzy variable, correspond to nested hypercubes, with rounded corners at intermediate satisfaction values, lying in the positive quadrant Of RN. Again employing the two-dimensional cases used in FIGs 1-4 for illustration, we show contour plots of corresponding contra-side satisfaction functions SL(C) in FIGs 5- 8, respectively.

For example, in FIG 5, the region 27 indicates the region of prices where the contra side sellers jointly have unity satisfaction (i.e., are completely satisfied ) to trade at any pair of prices within this region. The contour lines stepping down to region 22 indicate intermediate levels ofjoint satisfaction. the traders would be less satisfied to trade at prices in region 26 than region 27, but more satisfied than in region 25. The same is true for regions 24 and 23. Region 22 represents the region where no sellers would be willing to trade.

In the degenerate case, where all the individual contra side satisfaction functions are step functions that transition abruptly from unity to zero (i.e., corresponding to regular limit orders) the joint contra satisfaction function is unity over the interior of a single hypercube and zero elsewhere. In this case, with reference to FIGS 5-8, there would be an abrupt transition from the black region to farthest most white region without any transition regions.

In FIG 6, the region 28 indicates the region of prices where the contra side buyers/sellers jointly have unity satisfaction (i.e., are completely satisfied ) to trade at any pair of prices within this region. The contour lines stepping down to region 33 indicate intermediate levels ofjoint satisfaction. In other words, the traders would be less satisfied to trade at prices in region 29 than region 28, but more satisfied than in region 30. The same is true for regions 31 and 32. Region 33 represents the region where the contra side would be unwilling to trade.

In FIG 7, region 34 indicates the region of prices where the contra side buyers/sellers jointly have unity satisfaction (i.e., are completely satisfied ) to trade at any pair of prices within this region. The contour lines stepping down to region 39 indicate intermediate levels ofjoint satisfaction. In other words, the traders would be less satisfied to trade at prices in region 35 than region 34, but more satisfied than in region 36. The same is true for regions 37 and 38. Region 39 represents the region where the contra side would be unwilling to trade.

In FIG 8, the region 45 indicates the region of prices where the contra side buyers jointly have unity satisfaction (i.e., are completely satisfied ) to trade at any pair of prices within this region. The contour lines stepping down to region 40 indicate intermediate levels ofjoint satisfaction. In other words, the traders would be less satisfied to trade at prices in region 44 than region 45, but more satisfied than in region 43. The same is true for regions 42 and 41. Region 40 represents the region where the contra side would be unwilling to trade.

Note by comparison between corresponding cases (e.g., FIG 1 versus FIG 5, etc.) that the support region of SL(C), i.e., the region of price values where the linked-trade satisfaction function has non-zero values, lies opposite, and potentially intersects, the interior vertex of the fuzzy hypercube.(i.e., corner) of

the support regions of the corresponding contra-side satisfaction function Sc(P1,...PN). When these regions intersect, the hyperplane contours of SL(C) literally slice the corner of the fuzzy hypercube support region of Sc(P1, w PN).

This feature generalizes to the N-dimensional case as well, although it cannot be visualized beyond three dimensions.

As an illustration, FIG 9 shows the mutual satisfaction support region resulting from the product of SL(P1, P) in FIG 1 and 5c (P" P) in FIG 5, although only three regions for each are depicted for simplicity sake. The highest mutual satisfaction occurs in region 55, where the regions of complete satisfaction overlap, i.e., regions 5 (FIG 1) and 27 (FIG 5). Next, region 54 represents a lower value of mutual satisfaction, as it is the intersection of region 27 (FIG 5) with region 4 (FIG 1). Region 52 represents the overlap of regions 26 (FIG 5) and region 5 FIG 1). As one moves farther away from region 55, the mutual satisfaction decreases. The arrows in FIG 9 represent gradients of decreasing mutual satisfaction. Thus, region 49 has higher mutual satisfaction than region 48 and region 47, etc. The price coordinates ofthe region 55 indicate the optimum set of prices for this two-security linked trade.

Returning to the theory, the problem now reduces to finding the optimum price point in N-dimensional price space that results in the maximum mutual satisfaction MS(P" ..., PN), as defined by MS(P19 9 PN) = SL(C) SC(P1 ..., PN) (3) where c is a function ofP,, Vp i =1... Nas defined in equation (1).

Optimization Approach Since none of the Vi in equation (1) are zero, it is straightforward to show that satisfaction contours of SL(C) are not parallel to any axis in N-dimensional price space, and thus are not parallel to any side of the SC(P" ..., P hypercube support region. Thus, the negative gradient of SL(C) with respect to c along any hypercube contour where 0 <SL(C) < 1 is a vector that points to the interior of the

support region of Sc(P1, , PA), i.e., decreasing values of SL(C) yield increasing (or at least non-decreasing) values of Sc(P1, ..., PN).

We first consider the category of cases where the intersection between the support regions of SL(C) and Sc(P1, ..., PA) is such that the values of either SL(C) or ScP1, --, PA) (or both) are strictly less than unity over the region of intersection, yileldingMS(P1, ..., PJ values less than unity in (3). The implication ofthe gradient geometry is that the optimum mutual satisfaction will occur at a point of tangency between a hyperplane contour of SL(C) and a convex satisfaction contour of SC(PI, --, PA), since the point of tangency achieves the maximum value of Sc(P1, ..., PA) for a given value of SL(C).

Since the individual contra-side satisfaction functions Sc(Pi) in equation (2) are strictly monotonic over their respective regions of transition from unity to zero values, the locus of the points of tangency between hyperplane contours of SL(C) and satisfaction contours ofSc(P1, ..., PA) traces out a one-dimensional curvilinear segment T in N-dimensional price space. One end of the segment T is the point of tangency corresponding to the zero satisfaction boundary of SL(C), if Sc(P1, ..., PA) < 1 at this point, or to the hypercube vertex point corresponding to the unity satisfaction boundary contour of Sc(P,, ..., PA) otherwise. The other end of the segment 'P is the point of tangency corresponding to the unity satisfaction boundary contour of SL(C), if Sc(P1, ..., PA) > O at this point, or to the hypercube vertex point corresponding to the zero satisfaction boundary contour of Sc(P1, ..., PA) otherwise.

Along this segment #, the values ofSc(P1, ..., PN) decrease strictly monotonically from the former endpoint to the latter, due to the strict monotonicity of the Sci(Pi) functions. Conversely, the values of SL(C) decrease strictly monotonically proceeding from the latter endpoint of # to the former.

Thus there exists at least one optimum solution (#1, ..., #N) that maximizes MS(P" ..., PA) at some point along , which can be determined via a one- dimensional search along the segment #. As an illustration, FIG 10 shows the mutual satisfaction surface resulting from the product of SL(P1, P) in FIG 1 and 5c(Pi, P) in FIG 5, although only three regions for each are depicted for simplicity

sake. The segment # is shown. The maximum mutual satisfaction occurs somewhere on this line segment as discussed above. The price coordinates of the peak indicate the optimum set of prices for this two-security linked trade.

We now describe a constructive method for evaluating MS(P1, --, PA) along the segment #, and thereby determining the optimum set of prices that maximizes MS(P1, ..., PN). Les #CO = (P1(0), ..., PN(0)) be the set boundary prices at which the respective individual SCi(Pi) satisfaction functions just attain their zero values, and let #C1 = (P,O), ..., PN(1)) be the set of boundary prices at which the individual SCi(Pi) satisfaction functions just attain their unity values. Let PRO be the point oftangency corresponding to the intersection ofthe zero satisfaction boundary hyperplane of SL(C) with a 0 <SC(P1, ..., PA) < 1 contour of SC(P1, ..., PA), where such exists, and let #L1 be the point of tangency corresponding to the intersection of the unity satisfaction boundary hyperplane of SL(c) with a 0<SC(P1, ..., PN) < 1 contous of SC(P1, ..., PN), where such exists.

Thus the endpoints of the segment # are given by #0=#0 if SL(#0)<1, or #0 = #L1 otherwise; and (4) E1 = #1 if SL(#1)> O, or W1 = PLO otherwise, (5) respectively.

The task now is to evaluate MS(pj, ..., PA) at points along the segment # to determine the set of prices (#1, ,#N) that maximize this quantity. In certain mathematically tractable cases of the functions SL(c) and SC(P1, ..., PN), a rigorous optimization may be performed using the calculus of variations. In other

instances, a number of standard numerical optimization approaches may be used.

For the latter cases, interior points of the segment # may be determined by finding, for any value of c whose corresponding hyperplane satisfaction contour of SL(c) defined by (1) lies between the points #0 and #1, the point of tangency Py(c) (in N-dimensional price space) between that hyperplane contour and its corresponding satisfaction contour of SC(P1, ..., PN). As described previously, this is equivalent to finding, for the given value of c, the unique set of prices (constrained by (1)) that maximize Sop,, ..., PA) and thus maximize MS(P1, ..., PN) (the product of SL(C) and SC(PI, ..., PN)) for that value of c. This ycalculation may be performed for incremental values of C lying between those values corresponding to the endpoints of the segment , and the optimum P#(c) is selected that results in the maximum value ofMS(P1, ..., PN).

In practice, satisfaction values of the individual SCi(Pi) functions typically are specified at discrete price points (e.g., multiples of eighths of a dollar), and each SCi(Pi) function is assumed to be continuous and strictly monotonic with respect to its respective price variable Pi between these points. The same holds true for the linked trade satisfaction function S,(c). In between these points of specification of these functions, we assume the satisfaction functions are piecewise linear. For each SCi(Pi), let {Pij, j = 0 ... Ki} be the set of discrete price points at which SCi(Pi) is defined for values of satisfaction Sijbetween zero and unity, viz., SCi(Pi,j) = Si,j, (6) where Note that by this definition, if SCi (Pj) corresponds to a buyer satisfaction function, then the values Of Pj y decrease with increasing values of the indexj, and vice versa

if SCi(Pi) corresponds to a seller satisfaction function. In this fashion, SCi(Pi) always increases strictly monotonically with increasing values of the indexj.

Similarly, let {ck, k = 0 ... K0} be the set of discrete values of net cost at which SL(C) is defined, viz., SL(ck) = SL,k, where SL,0 = 0 at c0 and SL,K0 = 1 at ck.0 (8) Assuming piecewise linearity of the individual satisfaction functions Sci(Pj) and the linked satisfaction function SL(C) between their specified values above, we can then write: SCi(Pi) = ri,j, (Pi-Pi,j) + Si,j; Pi,j, # Pi # Pi,j+1, (9) j = 0 ... where and similarly for SL(c): SL(C) = sk(c Ck) + ck #c # ck+1, (11) k = 0 ... Ko-1 where

k = 0...Ko-1 Outside the price and/or net cost ranges defined above, the satisfaction functions take on unity or zero values, as appropriate.

With the above construction, we can use standard numerical optimization techniques to perform an exhaustive evaluation of MS(Pb, ..., PA) over all c, typically involving, at most, a few dozen values) to determine the optimum prices (#1, ...,#N), which amounts to a very modest computational burden.

In the event of multiple optima along the segment , generally accepted notions of fair trading practice would give price improvement to the contra-side participants (since the linked trade represents a contingent order and thus has lower priority or standing with respect to non-contingent orders) and thus we would select the optimum price corresponding to the maximum value of C.

However, this rule can be modified to suit individual market conventions.

In the second category of cases, where the unity satisfaction regions of SL(C) and Sc(PIs ..., PN) intersect, the mutual satisfaction MS(PZ, ..., PA) is unity over some region of price space. While any point in this region of price values over which Me,, ..., PA) is unity could be chosen to execute the linked trade, generally accepted principles of fairness would again suggest that price improvement should be evenly distributed among the contra-side participants.

This can be achieved by selecting the unique centroid point in the hyperplane section bounding the region of unity values of MS(P1, ..., PN), which hyperplane corresponds to the maximum value of C for which SL(C) has unity value, thus yielding the most favorable prices to the contra-side parties in the region of unity values ofMS(P1 ..., PJ. For the two dimensional case as in the continuing example, one can see this in FIG 9, where the region 55 contains unity values of MS(P1, PA). The centroid of region 55 is the point yielding the most favorable prices to the contra-side parties. This hyperplane is defined by

where c1 is the maximum value of C in equation (1) for which SL(C) has unity value.

The vertices vi of the hyperplane section bounding the region of unity values of Me,, ..., PA) are the points of intersection of (13) with the Nunity-satisfaction contour line segments of SC)P1, ..., PA), given by the equations The coordinates of the vertices vi thus satisfy the equations from which we can solve for the Pi coordinate of the ith vertex vi: Thus the set of vertices of the hyperplane section bounding the region of unity values of MS(P1, ..., PN) is expressed by:

i.e., the ith coordinate ofthe ith vertex is Pi as determined by (16) and thejth coordinates ofthe ith vertex forj + i are equal to p(11, the coordinates of the unity-satisfaction boundary interior vertex of SC (P" ..., P. The optimum centroid point PC of the hyperplane section is simply the arithmetic average of the vertices in (17), viz., which can be shown by substitution to lie on the hyperplane defined by (14).

Summary Thus we have constructed a feasible computational procedure for finding the optimum set of prices to execute the linked trade in all cases of intersection between SL(C) and SC(Pj, ..., PA), which maximizes the joint mutual satisfaction as defined by (3).

In the above, we assumed adequate liquidity to provide the required volume in each security involved in the linked trade. There may be instances where adequate contra volume does not exist in one or more securities. Such cases may be handled by a number of strategies, including: . Determine the minimum percentage of desired volume that is available in any of the involved securities. Adjust all volumes, and the satisfaction functions SL(C) and Sc(P , ..., PA), to reflect this downsized trade. This option amounts to executing a scaled-down version of the desired trade on a pro-rata basis for each security, preserving the relative volume mix among the individual securities.

Determine the minimum volume available in each involved security, not to exceed the desired volume. Adjust those volumes that are less than the desired volume, and the satisfaction functions SL(C) and Sc (P" ..., PA), to reflect the reduced trade. This option amounts to an eclectic execution of

the desired trade, preserving as much of the original desired volume in each security as possible.

Using the above eclectic approach, adjust upward the volume in securities where adequate liquidity is available in order to keep the same SL(C), while adjusting Sc(P" ..., PA) to reflect the new volumes. This option amounts to executing a trade of similar overall net cost to the original desired trade, but with a different mix of volumes of the requisite securities.

It will be apparent that numerous other strategies may be considered for dealing with cases of inadequate liquidity, all of them falling within the scope of the basic invention.

Entry of the Linked Trade The party desiring to perform a linked trade can access the system using a specially designated terminal or the same trading terminal with a menu selection for selecting a linked trading profile entry. The linked trading profile includes fields for each security and the desired volume. Once all of the securities have been entered, the screen includes a series of fields for entering the costs for the linked trade and each associated satisfaction. Price is not entered for the individual securities as the costs for the overall linked trade essentially accounts for price in the individual securities.

System Referring to FIG 11, the system 60 to implement the present invention includes a central matching controller 61, i.e., a computer, such as an IBM RS6000/SP, a plurality of trader terminals 62-67 (e.g., PC based workstations), and a linked trader terminal 68. While only one linked trader terminal 68 is shown, the system could employ a large number of these terminals, and in fact, these terminals can be identical to the trader terminals for individual securities markets. During each transaction, however, only one linked trade can be processed at a time. Thus, the linked trades will be processed according to a priority, such as first in time.

The traders A, through An represent traders in one market, traders B, through B, represent traders in another market, and traders ZZ through Zk represent traders in yet another market. A large number of markets could be represented within the system of the present invention, thus enabling transactions in a variety of securities, commodities and equities. For example, traders A, through An could represent traders from the Pacific Exchange, while traders B through B, represent traders in the foreign currency market in London, and traders Z, through Zk could represent traders in the commodities market in Tokyo.

Optionally, each trader terminal could have the capability for performing linked trades by selecting an option from a menu of choices.

A central database stores all of the information received from the traders for later processing by the control engine. A database extraction routine can be used to extract certain types of information. For example, some traders may enter information regarding specific securities that are not being traded by any other traders, hence this data may not be needed for some of the calculations. In this case, the extraction routine may only extract data regarding securities that are part of the linked trade.

This extraction routine operates under control of the main control engine, which ultimately receives all of the data from the plurality of trader terminals and performs the necessary calculations discussed above. Once the identities of all of the parties and the prices and the volumes have all been determined, the control engine then executes the trade in each of the relevant markets, and informs the parties of the results of the trades. Clearing is performed in accordance with known rules and regulations.

In the above system, the workstations can be any type of user terminal, from the low power end to the high power end. For example, the workstation could simply be a dumb terminal that is operated by the central control engine, or a high performance workstation, such as a Sun workstation, or a RISC-based workstation, or the latest version of a personal computer. In addition, the workstation can employ any of the following operating systems: UNIX, DOS, Windows-95, and Macintosh.