IJAHPArticle: Vargas/How to write a contract with the AHP International Journal of the Analytic Hierarchy Process 274 Vol. 9 Issue 2 2017 ISSN 1936-6744 https://doi.org/10.13033/ijahp.v9i2.490 HOW TO WRITE A CONTRACT WITH THE AHP Luis G. Vargas Professor of Business Analytics and Operations The Joseph M. Katz Graduate School of Business University of Pittsburgh lgvargas@pitt.edu ABSTRACT In this paper, we show how the Analytic Hierarchy Process could be used to develop a legal contract in the process of a negotiation. We illustrate the process with a well-known case used routinely in negotiation courses to illustrate that the AHP is particularly well suited for this type of application where most of the dimensions and criteria are intangibles, and the scales used to measure the gains and costs of parties involved in the negotiation do not always exist. Keywords: Negotiation; gain and loss ratios; value claim; value creation 1. Introduction The dictionary definition of “contract” is “a binding agreement between two or more persons or parties” or “a document describing the terms of a contract.” This implies that a contract has multiple dimensions and the parties must agree on each of the dimensions. For example, in the case of a recruiter trying to hire a candidate for a position in a company, the dimensions could be the signing bonus, salary, job assignment, company car, starting date, number of vacation days, percentage of moving expenses covered, the type of insurance coverage offered, and so on. Each dimension has a different impact on each of the parties. There are two types of outcome at work when two parties negotiate: Value claim, and Value creation (see Figure 1). Figure 1. Value Claims, Value Creation and the Pareto Frontier IJAHPArticle: Vargas/How to write a contract with the AHP International Journal of the Analytic Hierarchy Process 275 Vol. 9 Issue 2 2017 ISSN 1936-6744 https://doi.org/10.13033/ijahp.v9i2.490 Value claim occurs when one party can capture value from the other party during the negotiation process. This is most prevalent among those dimensions of the negotiation that are distributive (i.e., what one party gains, the other party experiences as a comparable loss). However, it can also manifest itself for integrative elements (i.e., when multiple factors are negotiated – some of which are more important to one of the parties, and some of which are more important to the other party). However, for both integrative and compatible dimensions (i.e., factors where the same element is perceived as a gain for both parties), there are also opportunities for exchange that leads to value creation. Thus, value creation takes place when both parties are made better off during the negotiation. When value creation occurs, the parties move closer towards the Pareto frontier – the point at which neither party can be made better off without the counterparty being made worse off. 2. A simple example We mentioned above that a contract has multiple dimensions and the parties must agree on each of the dimensions for the contract to be accepted by both parties. Thus, for a negotiation to arrive at a mutually agreed contract it needs to consider the gains and losses of the parties in each of the dimensions. For example, a recruiter is negotiating with a prospective employee for a position. They need to agree on the conditions of employment. The negotiation involves agreement on several dimensions. Each dimension can be considered a benefit or a cost. Table 1 shows an example of dimensions of a negotiation and their type. Table 1 Dimensions and their type In addition, within each type, the dimensions are not equally important. Table 2 shows the importance of the dimensions from both, the recruiter’s and the employee’s perspective. Dimensions Type SIGNING BONUS (SB) Benefit SALARY (S) Cost JOB ASSIGNMENT (JA) Cost COMPANY CAR (CC) Benefit STARTING DATE (SD) Benefit VACATION DAYS (VD) Benefit MOVING EXPENSES REIMB (MER) Benefit INSURANCE COVERAGE (IC) Benefit IJAHPArticle: Vargas/How to write a contract with the AHP International Journal of the Analytic Hierarchy Process 276 Vol. 9 Issue 2 2017 ISSN 1936-6744 https://doi.org/10.13033/ijahp.v9i2.490 Table 2 Recruiter/Employee Priorities for Benefits and Costs In many real-life contract negotiations, neither the dimensions of the contract nor the intensity scales may be known. To make tradeoffs we need to identify the dimensions and the intensity scales. Consider the dimensions of the recruiter-candidate example with scales as given in Table 3. Benefits Recruiter Employee SIGNING BONUS (SB) 0.270 0.270 COMPANY CAR (CC) 0.081 0.081 STARTING DATE (SD) 0.108 0.270 VACATION DAYS (VD) 0.270 0.108 MOVING EXPENSES (MER) 0.054 0.216 INSURANCE COVERAGE (IC) 0.216 0.054 Costs Recruiter Employee SALARY (S) 0.75 0.75 JOB ASSIGNMENT (JA) 0.25 0.25 Priorities Priorities IJAHPArticle: Vargas/How to write a contract with the AHP International Journal of the Analytic Hierarchy Process 277 Vol. 9 Issue 2 2017 ISSN 1936-6744 https://doi.org/10.13033/ijahp.v9i2.490 Table 3 Intensities and the benefits/costs accrued by the recruiter and the candidate These scales are not usually known and need to be constructed using relative measurement. For the moment, consider the intensity scales in Table 3 expressed in ideal terms (i.e., the elements are divided by the largest value) in Table 4. In this example, the scale values are all equispaced, i.e., they form a linear scale. However, in practice these values would be obtained through prioritization and they do not need to be linear.The negotiation process consists in finding out what value each dimension should take for the recruiter and the candidate so that the total amount they get (benefit/cost ratio) is maximized and satisfies the constraint that neither party gets more than the other, i.e., the contract is fair and equitable (Fisher & Ury, 1981). INTENSITY RECRUITER CANDIDATE SIGNING BONUS (SB) 10% 0 4000 8% 1000 3000 6% 2000 2000 4% 3000 1000 2% 4000 0 SALARY (S) 60,000.00$ -6000 0 58,000.00$ -4500 -1500 56,000.00$ -3000 -3000 54,000.00$ -1500 -4500 52,000.00$ 0 -6000 JOB ASSIGNMENT (JA) Division A 0 0 Division B -600 -600 Division C -1200 -1200 Division D -1800 -1800 Division E -2400 -2400 COMPANY CAR (CC) LUX EX2 1200 1200 MOD 250 900 900 RAND XTR 600 600 DE PAS 450 300 300 PALO LSR 0 0 STARTING DATE (SD) 1-Jun 1600 0 15-Jun 1200 1000 1-Jul 800 2000 15-Jul 400 3000 1-Aug 0 4000 VACATION DAYS (VD) 30 days 0 1600 25 days 1000 1200 20 days 2000 800 15 days 3000 400 10 days 4000 0 MOVING EXPENSES 100% 0 3200 REIMBURSEMENT (MER) 90% 200 2400 80% 400 1600 70% 600 800 60% 800 0 INSURANCE COVERAGE (IC) Allen Insurance 0 800 ABC Insurance 800 600 Good Health Insurance 1600 400 Best Insurance Co. 2400 200 Insure Alba 3200 0 IJAHPArticle: Vargas/How to write a contract with the AHP International Journal of the Analytic Hierarchy Process 278 Vol. 9 Issue 2 2017 ISSN 1936-6744 https://doi.org/10.13033/ijahp.v9i2.490 Table 4 Priorities of dimensions and relative scales for the recruiter – candidate case RECRUITER PRIORITIES CANDIDATE PRIORITIES Benefits' Priorities Costs' Priorities Benefits' Priorities Costs' Priorities RECRUITER CANDIDATE 0.2702701 0.27027 SIGNING BONUS Scale Ideal Scale Scale Ideal Scale 10% 0.01 0.00 4,000 1.00 8% 1,000 0.25 3,000 0.75 6% 2,000 0.50 2,000 0.50 4% 3,000 0.75 1000 0.25 2% 4,000 1.00 0.01 0.00 0.714285918 0.714285918 SALARY 10000.01 10000.01 $60,000 -6,000 1 0.01 0.00 $58,000 -4,500 0.75 -1,500 0.25 $56,000 -3,000 0.50 -3,000 0.50 $54,000 -1,500 0.25 -4,500 0.75 $52,000 0.01 0.00 -6,000 1.00 0.285714082 0.285714082 JOB ASSIGNMENT -14999.99 -14999.99 Division A 0.01 0.00 0.01 0.00 Division B -600 0.25 -600 0.25 Division C -1,200 0.50 -1,200 0.50 Division D -1,800 0.75 -1,800 0.75 Division E -2,400 1.00 -2,400 1.00 0.0810812 0.081081 COMPANY CAR -5999.99 -5999.99 LUX EX2 1200 1.00 1200 1.00 MOD 250 900 0.75 900 0.75 RAND XTR 600 0.50 600 0.50 DE PAS 450 300 0.25 300 0.25 PALO LSR 0.01 0.00 0.01 0.00 0.1081082 0.27027 STARTING DATE 3000.01 3000.01 1-Jun 1,600 1.00 0.01 0.00 15-Jun 1,200 0.75 1,000 0.25 1-Jul 800 0.50 2,000 0.50 15-Jul 400 0.25 3,000 0.75 1-Aug 0.01 0.00 4,000 1.00 0.2702701 0.108108 VACATION DAYS 4000.01 10000.01 30 days 0.01 0.00 1,600 1.00 25 days 1,000 0.25 1,200 0.75 20 days 2,000 0.50 800 0.50 15 days 3,000 0.75 400 0.25 10 days 4,000 1.00 0.01 0.00 MOVING EXPENSES 10000.01 4000.01 0.0540542 0.216216 REIMBURSEMENT 100% 0.01 0.00 3,200 1.00 90% 200 0.25 2,400 0.75 80% 400 0.50 1,600 0.50 70% 600 0.75 800 0.25 60% 800 1.00 0.01 0.00 0.2162161 0.054054 INSURANCE COVERAGE 2000.01 8000.01 Allen Insurance 0.01 0.00 800 1.00 ABC Insurance 800 0.25 600 0.75 Good Health Insurance 1,600 0.50 400 0.50 Best Insurance Co. 2,400 0.75 200 0.25 Insure Alba 3,200 1.00 0.01 0.00 8000.01 2000.01 IJAHPArticle: Vargas/How to write a contract with the AHP International Journal of the Analytic Hierarchy Process 279 Vol. 9 Issue 2 2017 ISSN 1936-6744 https://doi.org/10.13033/ijahp.v9i2.490 3. The trading model To find the solution of this problem we model it with integer programming. A solution is represented by an 8-by-5 matrix  ijx of 0’s and 1’s. Each row corresponds to a dimension and each column corresponds to an intensity of the scale corresponding to that dimension (see Table 5). Table 5 A solution with the Benefit/Cost Ratios The benefits and costs are obtained from Table 4. For example, the benefit for the recruiter of selecting a 6% Signing Bonus (0.135135) is obtained by multiplying the weight of Signing Bonus (0.27027) by the scale intensity 3 (0.50). Thus, 1 ij x  if the i th dimensions takes the j th intensity value. Let ( ) R C ij ij b b and ( ) R C ij ij c c the benefit and cost corresponding to the j th intensity of the i th dimension for the recruiter (candidate). The benefits/costs ratios of the recruiter and the candidate are given by benefits ( ) costs R R i ij ij i j R R R i ij ij i j w x b r x v x c       and benefits ( ) costs C C i ij ij i j C C C i ij ij i j w x b r x v x c       , respectively. The objective is to find a solution *x such that the parties gain as much as possible, ( *) ( *) {Min{ ( ), ( )}} S R C A B x X r x r x Max r x r x    , where S X is the solution space defined as the set of matrices ( ) ij x that satisfy the conditions 5 1 1 ij j x   , for all i, 0,1ijx  , for all i and j, and the two parties gain the same, i.e., their ratios are equal. A Solution Intensities 1 2 3 4 5 Benefits Costs Benefits Costs SB 0 0 1 0 0 0.135135 0 0.135135 0 S 0 1 0 0 0 0 0.535714 0 0.178571 JA 0 1 0 0 0 0 0.071429 0 0.071429 CC 1 0 0 0 0 0.081081 0 0.081081 0 SD 0 0 1 0 0 0.054054 0 0.135135 0 VD 0 1 0 0 0 0.067568 0 0.081081 0 MER 0 0 1 0 0 0.027027 0 0.108108 0 IC 0 0 1 0 0 0.108108 0 0.027027 0 B/C Ratio 0.7790 2.2703 Recruiter Candidate IJAHPArticle: Vargas/How to write a contract with the AHP International Journal of the Analytic Hierarchy Process 280 Vol. 9 Issue 2 2017 ISSN 1936-6744 https://doi.org/10.13033/ijahp.v9i2.490 A given solution has benefits/costs ratios that are different for the parties. For example, in Table 5 we give a solution. In this solution, the recruiter has a lower benefits/costs ratio than the candidate, so the recruiter will try to change to another solution where he will get a greater benefits/costs ratio. Table 6 shows the solution in matrix form. Table 6 Optimal solution Translated into the original scale values of the dimensions we have Table 7. Note that now both the recruiter and the candidate gain the same. Table 7 Terms of the contract Obviously, the scales within each dimension do not have to be linear. For example, if the recruiter and the candidate have relative intensities as given in Table 8, the solution (Table 9) would not be the same as the one in Table 7. The solutions in Table 9 are within 3.125% of each other. No other closer solutions exist. 4. General contract model In many contract negotiations, the parties do not always act in good faith or share information with the other party. In this case, one should also consider the perceptions of the parties about the benefits and costs of the tradeoffs. For example, in a merger transaction, the buyer (A) and the seller (B) may not always agree as to the terms of the merger, and hence the transaction may fail. The steps to make tradeoffs in this more general situation are as follows: Optimal Solution Intensities 1 2 3 4 5 Benefits Costs Benefits Costs SB 0 0 1 0 0 0.135135 0 0.135135 0 S 0 1 0 0 0 0.357143 0 0.357143 JA 1 0 0 0 0 0 0 0 0 CC 1 0 0 0 0 0.081081 0 0.081081 0 SD 0 0 0 0 1 0 0 0.27027 0 VD 0 0 0 0 1 0.27027 0 0 0 MER 1 0 0 0 0 0 0 0.216216 0 IC 0 0 0 0 1 0.216216 0 0 0 B/C Ratio 1.9676 1.9676 Recruiter Candidate SB S JA CC SD VD MER IC Total 6% 56,000.00$ Division A LUX EX2 1-Aug 10 days 100% Insure Alba Points Recruiter 2000 -3000 0 1200 0 4000 0 3200 7400 Candidate 2000 -3000 0 1200 4000 0 3200 0 7400 IJAHPArticle: Vargas/How to write a contract with the AHP International Journal of the Analytic Hierarchy Process 281 Vol. 9 Issue 2 2017 ISSN 1936-6744 https://doi.org/10.13033/ijahp.v9i2.490 1. Identify the dimensions of the problem 2. Identify the tradeoffs of each party within the dimensions 3. Identify the benefits accrued by a party from the other party’s tradeoffs 4. Identify the costs incurred by a party from its own tradeoffs 5. Identify the perceived benefits that the other party received from your tradeoffs 6. Identify the perceived costs incurred by the other party from their tradeoffs 7. Find out what tradeoff each party must make to maximize the total minimum gain they obtain, ensuring that the gains of a party are as close as possible to the other party gains. This is what makes the final contract fair, equitable and balanced. Table 8 Intensities with Non-Linear Relative Scales Relative Scales Dimensions RECRUITER CANDIDATE RECRUITER CANDIDATE SIGNING BONUS 10% 0.01 4,000 1E-06 1 8% 1,000 3,000 0.1 0.75 6% 2,000 2,000 0.5 0.5 4% 3,000 1000 0.9 0.1 2% 4,000 0.01 1 1E-06 SALARY 10000.01 10000.01 $60,000 -6,000 0.01 1 1E-06 $58,000 -4,500 -1,500 0.75 0.1 $56,000 -3,000 -3,000 0.5 0.5 $54,000 -1,500 -4,500 0.1 0.9 $52,000 0.01 -6,000 1E-06 1 JOB ASSIGNMENT -14999.99 -14999.99 Division A 0.01 0.01 1E-06 1E-06 Division B -600 -600 0.1 0.1 Division C -1,200 -1,200 0.5 0.5 Division D -1,800 -1,800 0.9 0.9 Division E -2,400 -2,400 1 1 COMPANY CAR -5999.99 -5999.99 LUX EX2 1200 1200 1 1 MOD 250 900 900 0.75 0.75 RAND XTR 600 600 0.5 0.5 DE PAS 450 300 300 0.1 0.1 PALO LSR 0.01 0.01 1E-06 1E-06 STARTING DATE 3000.01 3000.01 1-Jun 1,600 0.01 1 1E-06 15-Jun 1,200 1,000 0.75 0.1 1-Jul 800 2,000 0.5 0.5 15-Jul 400 3,000 0.1 0.9 1-Aug 0.01 4,000 1E-06 1 VACATION DAYS 4000.01 10000.01 30 days 0 1,600 0 1 25 days 1,000 1,200 0.1 0.75 20 days 2,000 800 0.5 0.5 15 days 3,000 400 0.9 0.1 10 days 4,000 0.01 1 1E-06 MOVING EXPENSES 10000 4000.01 REIMBURSEMENT 100% 0.01 3,200 1E-06 1 90% 200 2,400 0.1 0.75 80% 400 1,600 0.5 0.5 70% 600 800 0.9 0.1 60% 800 0.01 1 1E-06 INSURANCE COVERAGE 2000.01 8000.01 Allen Insurance 0.01 800 1E-06 1 ABC Insurance 800 600 0.1 0.75 Good Health Insurance 1,600 400 0.5 0.5 Best Insurance Co. 2,400 200 0.9 0.1 Insure Alba 3,200 0.01 1 1E-06 8000.01 2000.01 IJAHPArticle: Vargas/How to write a contract with the AHP International Journal of the Analytic Hierarchy Process 282 Vol. 9 Issue 2 2017 ISSN 1936-6744 https://doi.org/10.13033/ijahp.v9i2.490 Table 9 Terms of the contract for the Non-Linear Intensity case The mathematical model that helps identify the proper contract is given below. Let ( ) k X x the scale of the kth dimension. The parties will negotiate on the value of that scale according to their preferences. The realized value of the scale is determined by the benefit, the cost, the perceived benefits and the perceived cost that the value has for each party. Let ( ) i k B x be the benefits accrued by party i from the other party tradeoffs in dimension k. Let ( ) i k C x be the costs incurred by party i from its own tradeoffs in dimension k. Let ( ) i k PB x be the benefits party i perceives the other party receives from its tradeoffs in dimension k, and let ( ) i k PC x be the costs the other party perceives that party i incurs from its tradeoffs in dimension k. Thus, for a given dimension k, the gain of party i is given by the benefits it accrues from the tradeoffs of the other party in that dimension times the costs it perceives the other party incurs in that dimension, i.e., ( ) ( ) i k i k B x PC x . Similarly, the loss in each dimension k is given by ( ) ( ) i k i k C x PB x . Thus, the gain to loss ratio for a party for a given dimension k is given by: ( ) ( ) ( ) ( ) i k i k i k i k B x PC x C x PB x and the total gain-to-loss ratio for a party is given by ( ) ( ) ( ) ( ) i k i k i all k i k i k B x PC x r C x PB x   . This formulation was suggested by Saaty (1988) to handle retributive conflicts. Let ( ) k x s be a binary variable, where ( ) 1 k x s  if the parties agree on selecting the intensity s of the kth dimension as the best decision for both. The problem now consists in finding values of s for each dimension that maximizes the smallest gain-to-loss ratio of both parties, i.e., [ ( )] [ ( )][ ( )] [ ( )] ( ) , ( ) [ ( )] [ ( )] [ ( )] [ ( )] j k j ki k i k i j s all k i k i k j k j k B x s PC x sB x s PC x s Max Min r s r s C x s PB x s C x s PB x s                SB S JA CC SD VD MER IC Total 10% 56,000.00$ Division A LUX EX2 1-Jul 10 days 90% Insure Alba Points Recruiter 0 -3000 0 1200 800 4000 200 3200 6400 Candidate 4000 -3000 0 1200 2000 0 2400 0 6600 IJAHPArticle: Vargas/How to write a contract with the AHP International Journal of the Analytic Hierarchy Process 283 Vol. 9 Issue 2 2017 ISSN 1936-6744 https://doi.org/10.13033/ijahp.v9i2.490 Subject to ( ) 1 k s x s  and ( ) 1 ( ) i j r s r s   , where  is the tolerance that measures how far the two parties are in terms of their total gain-to-loss ratio. 5. Conclusions In this paper, we assume that a multidimensional contract between two parties is an agreement among the parties about the values of the dimensions that gives each party a fair and equitable gain. The values of the dimensions can be estimated through relative measurement when no scales exist. This approach was first used by Saaty (1988) to address the conflict in South Africa. The main difference between the approach in this paper and that used in the analysis of the Palestinian-Israeli conflict is that in the later the tradeoffs were analyzed in pairs (Saaty & Zoffer, 2011; 2013). In the case of a contract, the scales in which the dimensions are measured makes it impossible to analyze all possible pairs of tradeoffs. For example, in the simple case given above, the number of tradeoffs is 8 4 4, 294, 962 7, 296   . Here we used a non-linear integer optimization formulation to derive the solutions shown using a genetic algorithm. IJAHPArticle: Vargas/How to write a contract with the AHP International Journal of the Analytic Hierarchy Process 284 Vol. 9 Issue 2 2017 ISSN 1936-6744 https://doi.org/10.13033/ijahp.v9i2.490 REFERENCES Fisher, R. and W. Ury (1981). Getting to YES: Negotiating agreement without giving in. New York: Penguin Books. Doi: 10.2307/40202101 Saaty, T. L. (1988). The negotiation and resolution of the confilct in South Africa: The AHP. Orion, 4(1), 3-25. Doi: http://dx.doi.org/10.5784/4-1-488. Saaty, T. L. and H. J. Zoffer (2011). Negotiating the Israeli Palestinian controversy from a new perspective. International Journal of Information Technology and Decision Making, 10(1), 5-64. Doi: https://doi.org/10.1142/S021962201100421X Saaty, T. L. and H. J. Zoffer (2013). Principles for implementing a potential solution to the Middle East conflict. Notices of the American Mathematical Society ,60(10), 1300- 1322. Doi: http://dx.doi.org/10.1090/noti1053 https://doi.org/10.1142/S021962201100421X