Applying the Nash Bargaining Solution for a Reasonable Royalty
AApplying the Nash Bargaining Solution for a Reasonable Royalty
David M. Kryskowski (Student) ∗ and David Kryskowski Wayne State University, Detroit, Michigan, USA UD Holdings, 2214 Yorktown Dr., Ann Arbor, Michigan, 48105, USA (Dated: May 21, 2020)There has been limited success applying the Nash Bargaining Solution (NBS) in assigningintellectual property damages due to the difficulty of applying it to the specific facts of the case.Because of this, parties are not taking advantage of
Georgia-Pacific factor fifteen. The intentof this paper is to bring clarity to the NBS so it can be applied to the facts of a case. Thispaper normalizes the NBS and provides a methodology for determining the bargaining weight inNash’s solution. Several examples are shown demonstrating the use of this normalized form, and anomograph is added for computational ease.JEL classification: K11; C78Keywords: Nash Bargaining Solution; Bargaining Power; Royalty; License
I. INTRODUCTION
In U.S. patent litigation, there are two predominateways to compensate a licensor when a firm infringes onthe licensor’s intellectual property. One way is to cal-culate the profit that was lost due to the infringement.The other way is to designate a reasonable royalty. Areasonable royalty is defined as a royalty that is assignedto the licensor for the use of their intellectual propertyby the licensee that is fair to both parties [1].Assigning a reasonable royalty is especially difficult ina dispute situation because of the difficulty of an arbiteror court to attribute a royalty that is perceived as fairfor both parties. A famous District Court case
Georgia-Pacific vs. United States Plywood Corp demonstratedthe complexity of assigning a reasonable royalty in litiga-tion involving patents. As a result of the case, the Dis-trict Court established fifteen guidelines for determininga reasonable royalty. However, guideline fifteen allowedfor the use of a hypothetical license negotiation at thetime the infringement began. This guideline implies thatthe NBS could be used as a justification for assigning areasonable royalty.In recent court cases, some judges have steered clearfrom using the NBS because parties often do not apply itto the specific facts of the case . This has caused judges ∗ Electronic address: [email protected] Georgia-Pacific Corp. v. U.S. Plywood Corp. , 318 F. Supp.1116, 1120 (S.D.N.Y.1970), mod. and aff’d, 446 F.2d 295 (2dCir. 1971), cert. denied, 404 U.S. 870 (1971) Notable cases include:
VirnetX, Inc. v. Cisco Systems, Inc. ,767 F.3d 1308 (Fed. Cir. 2014);
Oracle Am., Inc. v. GoogleInc. , 798 F. Supp. 2d 1111 N.D. Cal. 2011;
Suffolk Techs. LLCv. Aol Inc. , No. 1:12cv625, 2013 U.S. Dist. LEXIS 64630 (E.D.Va. Apr. 12, 2013);
Limelight Networks, Inc. v. Xo Communs.,LLC
Civil Action No. 3:15-CV-720-JAG, 2018 U.S. Dist. LEXIS17802 (E.D. Va. Feb. 2, 2018) to criticize the NBS solution when determining a rea-sonable royalty [2–4]. Because Nash’s solution is oftennot tailored to the specific facts of the case, parties arenot taking full advantage of the NBS. Another reasonfor critisim, is the NBS has not been easy to calculateand easy to interpret so that a court or jury can easilyunderstand it [5]. In order to demystify the NBS, we in-troduce certain normalizations that provide for a simplecalculation of damages. These normalizations make theNBS a powerful tool to value intellectual property andprovide guidance in assigning proper compensation.First, this paper applies Nash’s solution in a more busi-ness friendly manner by using terminology that is com-mon on corporate balance sheets. Additionally, this pa-per normalizes each term in the NBS by the operatingincome. By doing this, the parties can better interpretthe NBS and do not need to know exact dollar amountswhen determining a royalty. The Choi and Weinstein [6]Two Supplier World (TSW) model is the basis for ourmodifications.Second, Nash’s original solution assigns equal bargain-ing strength to each party. However, this equal bar-gaining strength assumption is, in general, not realis-tic [7]. This paper shows Nash’s solution with an arbi-trary bargaining weight to account for unequal bargain-ing strengths and presents a methodology for determiningthose strengths.Third, a nomograph of the NBS is supplied to make iteasy for parties to graphically obtain a reasonable royaltyusing a simple straight edge. Nomographs are useful to [5] gives two reasons why courts are reluctant to use the NBS:”First, damages experts often use the NBS improperly, failing toapply the specific facts of the case to their calculations.[internalcitation omitted] Second, damages experts typically fail to ad-equately explain the NBS to courts and juries.[internal citationomitted]” a r X i v : . [ q -f i n . GN ] M a y provide visualization so the NBS can be better explained.By taking these steps, parties can take advantage of Georgia-Pacific factor fifteen by allowing the NBS to betailored to the specific facts of the case. This paper isan attempt to bring clarity to the use of the NBS, sothe royalty that is assigned is both legally defensible andmutually beneficial.
II. ELEMENTS OF A LICENSING BARGAIN
The NBS is recast into a simple normalized form usingcommon terms found on a corporate balance sheet withthe intent to introduce common business terminology.
A. Operating Revenue
The operating revenue is the revenue generated fromthe use of the intellectual property and is denoted by O R .It does not include income from unusual events or incomethat is not primary due to the use of the intellectualproperty. B. Operating Cost
The operating cost is the expense associated with pro-ducing and selling the product incorporating the intellec-tual property. This is defined as O C and does not includecosts from non-primary sources or unusual events. C. Operating Income
The operating income, or profit, is determined by sub-tracting the operating cost from the operating revenue: O I = O R − O C . In formulating the asymmetric NBS, wedenote the operating income of licensor and licensee tobe π and π , respectively where the total profit in thesystem is O I . D. Operating Margin
The operating margin, O M , is operating income di-vided by operating revenue and is expressed as O M = O I /O R . E. Royalty
The royalty is what the licensee will pay the licensorfor the use of the intellectual property. There are twocommon ways to calculate a royalty. One way is assign-ing a royalty on each unit, and the other is obtaining a royalty based off a percentage of revenue [8] by multiply-ing the revenue with the royalty rate, r . In this paper,we focus solely on a royalty based off revenue. F. Disagreement Payoffs
A disagreement payoff is the opportunity cost of mak-ing the deal. In other words, disagreement payoffs areprofits that come from a hypothetical negotiation thatdid not happen but could have happened if the parties didnot agree to a deal. Disagreement payoffs are normallyexpressed as monetary amounts and are represented inthis paper by d and d for licensor and licensee respec-tively. However, for computational ease, we normalizethe disagreement payoffs to the operating income, andthese are expressed as d † and d † for the licensor and li-censee respectively. A normalized disagreement payoffequal to one implies a particular party is indifferent be-tween making the deal and not making the deal since theparty could earn the same amount of profit regardless.For example, if d † = 1 .
0, then the licensor is just as goodat making the deal than not making the deal. For empha-sis, a normalized disagreement payoff of d † = 0 . d † = (cid:16) d † , d † (cid:17) . G. Bargaining Weight
A bargaining weight quantifies the amount of influenceeach party has in the negotiation and determines howthe parties split the surplus from making the deal. Thebargaining weight for the licensor is α , and the bargainingweight for the licensee is 1- α , where the weight is betweenzero and one. Each party has a perception of how theyview their own and each other’s bargaining strengths in anegotiation. The bigger a party’s bargaining weight, themore influence that party has in the negotiation. Thismeans the party with the larger weight will obtain moresurplus from making the deal. When applying the NBS,it has been common practice to assign each party a weightequal to 1/2, which implies that each party has the sameinfluence in the negotiation [9]. III. THE ASYMMETRIC NASH BARGAININGSOLUTION
John Nash developed the NBS, which provides amethod for two parties who enter in a profit makingagreement to determine how to optimally share thoseprofits [10]. Nash’s solution determines how much morethe parties can make if they cooperate and equally dividethe surplus. The axioms that satisfy the classic NBS are:1.
Individual rationality:
No party will agree toaccept a payoff lower than the one guaranteed tohim under disagreement.2.
Pareto optimality:
None of the parties can bemade better off without making at least one partyworse off.3.
Symmetry:
If the parties are indistinguishable,the agreement should not discriminate betweenthem.4.
Affine transformation invariance:
An affinetransformation of the payoff and disagreementpoint should not alter the outcome of the bargain-ing process.5.
Independence of irrelevant alternatives:
Allthreats the parties might make have been ac-counted for in the disagreement point.The introduction of a bargaining weight into the NBSaffords the opportunity for the parties to be distinguish-able when d † = d † (potentially violating symmetry), andthis is known as the asymmetric NBS [11]. The bargain-ing weight can be influenced by other forces or tacticsemployed by the parties which can be independent of thedisagreement payoffs. These forces should be accountedfor because they ultimately affect how the surplus is di-vided .The asymmetric NBS is formed from the constrainedmaximization problem:max π ,π ( π − d ) α ( π − d ) − α (1)Subject to the following conditions: π ≥ d (2) [11] states: ”However, the outcome of a bargaining situation maybe influenced by other forces (or, variables), such as the tacticsemployed by the bargainers, the procedure through which negoti-ations are conducted, the information structure and the players’discount rates. However, none of these forces seem to affect thetwo objects upon which the NBS is defined [the disagreementpayoffs], and yet it seems reasonable not to rule out the pos-sibility that such forces may have a significant impact on thebargaining outcome.” π ≥ d (3) π + π ≤ O I (4)Maximum occurs when:(1 − α ) ( π ∗ − d ) = α ( π ∗ − d ) (5) π ∗ + π ∗ = O I (6)Solving for optimal partition of the profits gives thefinal result: π ∗ = d + α ( O I − d − d ) (7a) π ∗ = d + (1 − α ) ( O I − d − d ) (7b)The interpretation of Eq. (7) is that the parties firstagree to give each other their respective disagreementpayoffs, and split the remaining profit according to theirbargaining strength. A. Normalized Royalty Model
In order to make TSW model more practical, we modi-fied Eq. (7) to introduce a royalty based on a percentageof revenue. Moreover, by simple algebraic manipulation,we can re-cast Eq. (7) where every term is normalizedby the operating income and varies between zero andone. Having each term normalized is powerful becausethe parties do not need to think in terms of specific dol-lar amounts. Instead, the parties can think in terms offractions of profit.We refer to the licensor as party 1 and the licensee asparty 2. Under these assumptions, we derive the payoffsfor party 1 and 2 as: π ∗ O I = r O R O I = rO M (8) π ∗ O I = O R − O C − rO R O I = 1 − rO M (9)Additionally defining: d † = d O I ≤ d † ≤ d † = d O I ≤ d † ≤ rO M = d † + α (cid:16) − d † − d † (cid:17) (12)Where: 0 ≤ d † + d † ≤ ∂r∂d † > ∂r∂d † < IV. ESTIMATION OF THE BARGAININGWEIGHT
The bargaining weight, α , represents how the partiesperceive their own bargaining strength and how they seethe other’s bargaining strength. To account for all theperceptions of bargaining strength, we introduce the pa-rameter, P m,n , and it is defined as party m’s bargain-ing strength as perceived by party n. For example, P , is how the licensee perceives the licensor’s bargainingstrength.Making the simple assumption that the bargainingstrength of each party is the average of their own per-ception and the perception of the other party, we candefine the following mathematical ansatz using two dif-ferent equations to describe the bargaining strength ofparty 1: α = 12 [ P , + P , ] (15a) α = 1 −
12 [ P , + P , ] (15b)Averaging Eqs. (15a)–(15b), we obtain the complete ex-pression for the bargaining weight of party 1: α ≡
12 [ α + α ]= 12 + 14 [ P , + P , − P , − P , ] 0 ≤ P m,n ≤ FIG. 1: Family of Nash Bargaining Solutions Given EqualBargaining Power
Eq. (16) is critically important because we now havea simple procedure in defining the bargaining weight ofparty 1. By formally defining the bargaining weight, wecan incorporate the bargaining strengths of each party tofit the particular facts of a case.There are three basic approaches when calculating abargaining weight. One approach is to treat α as functionthat is independent of the disagreement payoffs. Thesecond approach is to make the bargaining weight strictlya function of the disagreement payoffs. The third is amixture of the first two approaches. A. The Classic Nash Bargaining Solution
When P , + P , = P , + P , in Eq. (16), then α =1 / rO M = 12 (cid:16) d † − d † (cid:17) (17)Fig. 1 presents the family of solutions of Eq. (17).Note that the lines of equal d † are linear and equidistantfrom each other. Also note that the lines are not thesame length due to the constraint of Eq. (13). V. DISCUSSION
In this section we present some hypothetical situationsto demonstrate the use of the NBS. Since the assignmentof a party’s perception of bargaining strength to a partic-ular P m,n can be somewhat arbitrary, the examples givenin this section are for illustration only. In the end, it isthe job of the parties to provide a careful assessment ofeach of their perceptions and incorporate them properlyinto Eq. (16). By choosing these perceptions, the NBScan be applied to the specific facts of the case. A. Estimation of Bargaining StrengthsIndependent of the Disagreement Payoffs
We demonstrate the use of Eq. (16) by a simple hypo-thetical negotiation involving bargaining strengths inde-pendent of the disagreement payoffs.
1. Number of Competitors as Strength
The bargaining strength of party 1 is dependent onthe relation between the hypothetical number of licen-sors and licensees in the market [13]. This is because ifparty 1 has a wide range of options to sell its intellec-tual property, then party 1 is presumably less worriedabout making a deal with party 2 because it can crediblywalk away. Therefore, if party 1 can sell its intellectualproperty to multiple licensees, we expect party 1 to havemore bargaining strength. Conversely, if party 2 can li-cense an acceptable substitute, then party 1’s bargainingstrength will diminish. The following equation is drivenby the ratio of the number of licensors to the number oflicensees in the relevant market [13]. The component ofparty 1’s bargaining strength as derived from the numberof licensors and licensees in the market is: P L ,n = 1 − min (cid:20) , LicensorsLicensees (cid:21) (18)The perception is assigned as P ,n because either partymay perceive (18) as a component of party 1’s bargainingstrength.
2. Market Share as Strength
In business, market share is regarded as the essentialelement of dominance [14]. As a result, valuing a com-ponent of party 1’s bargaining strength by the amountof market share, s , is attractive as opposed to a mea-surement of potential profits. Using potential profits asa measurement of bargaining strength may not be ap-pealing because profits are highly variable from year toyear while market share is relatively constant over longerperiods of time. Additionally, courts often measure thedominance of firms by market share rather than prof-its [15]. Therefore, another measurement of bargainingstrength is to determine how much market share party2 would gain as a result of the deal. The component of party 1’s bargaining strength as derived from marketshare is: P S ,n = sS ≤ s ≤ S (19)In Eq. (19), S denotes that fraction of the total marketparty 2 realistically desires.
3. Life of the Patent as Strength
Another perception of strength can be the time left un-til the patent expires. Presumably, party 1 is in a strongbargaining position when the patent has been recently is-sued but is in a weak bargaining position when the patentis about to expire. Let the life of the patent be denotedby T and the time that has elapsed since issue by t . Thecomponent of party 1’s bargaining strength as derivedfrom patent life is: P T ,n = 1 − tT ≤ t ≤ T (20)
4. Example
In this hypothetical example, party 1 perceives its bar-gaining strengths, with equal weight, to be the lack ofacceptable substitutes for its patent and the potentialmarket share that the patent can bring to party 2. Party2 perceives party 1’s bargaining strength as only the lifeof the patent. Party 2 has a unique manufacturing basethat can take full advantage of party 1’s patent and per-ceives its own bargaining strength as P , = 2 /
3. Party 1is well aware of party 2’s unique manufacturing capabil-ities but only perceives party 2’s strength as P , = 1 / α = 12 + 14 (cid:34) P L , + P S , P T , − − (cid:35) (21)Eq. (21) can now be substituted into Eq. (12) toobtain the royalty for party 1. B. Estimation of Bargaining Strengths UsingDisagreement Payoffs
Disagreement payoffs can be a reasonable measureof bargaining strength because the parties can poten-tially walk away from the negotiation and make a dealsomewhere else based on the disagreement payoffs alone.Therefore, α can be a function of each party’s disagree-ment payoff. This approach requires the least amount FIG. 2: Family of Nash Bargaining Solutions for Table I Case1 of information but requires the parties to determine afunctional form of α (cid:16) d † , d † (cid:17) that adequately representsthe negotiation. For a standard of fairness, we stipulatethat when d † = d † , the parties should split the profitsequally, which implies that we are reintroducing symme-try. It is possible to construct an α (cid:16) d † , d † (cid:17) that reintro-duces symmetry and yet provides for variability in thebargaining weight.Cases 1-3 in Table I are examples of symmetric bar-gaining weights driven by the parties’ disagreement pay-offs.
1. Case 1
In Case 1 of Table I, each party assumes that itsbargaining strength is equal to its disagreement payoff.Moreover, each party agrees that the other party’s bar-gaining strength is its own disagreement payoff. Substi-tuting Case 1 of Table I into Eq. (12), we obtain: rO M = d † − d † + 2 (cid:16) d † − d † (cid:17) + 12 (22)Eq. (22) shows a quadratic dependence on both d † and d † , and this dependence is illustrated in Fig. 2. Note thata party is penalized to a much greater extent for havinga weak disagreement payoff position over the classic NBSof Fig. 1. FIG. 3: Family of Nash Bargaining Solutions for Table I Case2
2. Case 2
In Case 2 of Table I, each party assumes that its bar-gaining strength is equal to its fraction of the total dis-agreement payoff position d † + d † . Moreover, each partyagrees that the other party’s bargaining strength is itsown fraction of the total disagreement payoff position.Substituting Case 2 of Table I into Eq. (12), we obtain: rO M = d † d † + d † (23)Interestingly, the payoff for each party is the party’sown bargaining weight. Moreover, the solution is inde-pendent of O I , which makes this a non-cooperative bar-gain and is equivalent to a limiting case of the Rubinsteinmodel , where the parties take turns in making an offeruntil agreement is secured. In [11] the Subgame Perfect Equilibrium solution, where the timelimt between offers ∆ →
0, is presented in terms of discountrates ( r A , r B ) where d † /d † = r B /r A . The payoff pair obtainedthrough perpetural disagrement, the Impass Point, is ( I A , I B ) = (cid:16) d † , d † (cid:17) . See Corollary 3.1 and Definition 3.1. [11] discusses the Rubinstein model where the parties take turnsin making an offer until agreement is secured. ”...Another in-sight is that a party’s bargaining power depends on the relativemagnitude of the parties’ respective costs of haggling, with theabsolute magnitudes of these costs being irrelevant to the bar-gaining outcome. ...In a boxing match, the winner is the rela- TABLE I: Three Cases of Symmetric Disagreement Payoff Driven Bargaining Weights
Case P P P P α (cid:16) d † , d † (cid:17) d † d † d † d † + d † − d † d † d † + d † d † d † + d † d † d † + d † d † d † + d † d † d † + d † d † d † + d † d † d † + d † − d † − d † − d † − d † − d † − d † d † + (cid:16) d † − (cid:17) d † + d † − d † (cid:16) d † + d † (cid:17)(cid:16) − d † + d † (cid:17) Fig. 3 shows the family of solutions for Eq. (23). Notethe rapid collapse to zero of party 1’s royalty for anyconstant d † as d † approaches zero.
3. Case 3
In Case 3, we present an example where a party’s bar-gaining strength (party 2) depends on the weakness ofhis opponent. As in the previous examples, all partiesagree on each other’s bargaining strength. SubstitutingCase 3 of Table I into Eq. (12), we obtain: rO M = d † − d † − d † + d † + 12 − d † − d † (24)Fig. 4 shows the family of solutions for Eq. (24). Thefigure shows the same quadratic dependence as Case 1Fig. 2 where the lines of constant d † get closer togetheras d † becomes dominant. Party 1’s bargaining advantagehas increased from Case 2 for small d † because party 2’sstrength is derived from party 1’s weakness and not itsown strength as in Case 2. C. Estimation of Bargaining Strength UsingCombinations
We can combine perceptions using both constants andfunctions involving the disagreement payoffs. This im-plies that the parties’ perceptions can be both indepen-dent or dependent on the disagreement payoffs. However,there are cases when combinations of perceptions are notPareto efficient, and we examine this next.
1. Solutions That Violate Pareto Efficiency
When α is a function of the disagreement payoffs, therecan be combinations of perceptions that violate Pareto tively stronger of the two boxers; the absolute strengths of theboxers are irrelevant to the outcome.” FIG. 4: Family of Nash Bargaining Solutions for Table I Case3 efficiency in a part of the solution space, and Fig. 5 is onesuch example. Substituting the following hypothetical α into Eq. (12), we obtain Fig. 5: α (cid:16) d † , d † (cid:17) = 12 + 14 (cid:34) d † + 13 − d † d † + d † − (cid:16) − d † (cid:17)(cid:35) (25)From Fig. 5, we can see that the solution space is notPareto efficient everywhere because when both d † and d † are small, party 1 will receive a lower royalty for a smallincrease in d † , which is counter-intuitive.It is easily shown that the royalty in Fig. 5 violatesEq. (14) when d † is small. The reason for this violationis that the specification of P , causes party 2’s strengthas perceived by party 1 to be lower as party 1’s disagree-ment payoff lowers. This influences a small section of thesolution space to violate Pareto efficiency. FIG. 5: Family of Nash Bargaining Solutions With RegionsThat Violate Pareto Efficiency
VI. NOMOGRAPHS
In order to make it easy to compute a royalty usingthe asymmetric NBS, a nomograph was constructed (seeFig. 6) with PyNomo [16, 17]. A nomograph is a dia-gram that is a graphical representation of a mathemati-cal function. It allows for quick computation without theneed to substitute numbers into a formula. Nomographsalso provide visualization as to how the function behavesso the NBS can be easily explained.To use the nomograph, pick any three variables on thegraph and draw a straight line to get the fourth vari-able. For example, suppose that the normalized dis-agreement payoffs are d † = 0 .
20 and d † = 0 .
30. Ad-ditionally, suppose α = 0 .
40. Using a straight edge, aline is drawn from α = 0 .
40 to a point on the grid where(d † , d † ) = (0 . , . VII. CONCLUSION
In this model of the asymmetric NBS, there are threeessential variables needed to obtain a royalty. They arethe disagreement payoffs of both party 1 and party 2,and the bargaining weight. At a minimum, the parties Type 9 General Determinant was used.
FIG. 6: The use of the nomograph is demonstrated with d † =0 . d † = 0 .
30, and α = 0 .
40 to solve for r/O M = 0.40. should have a good understanding of the licensed prod-uct’s operating margin if a royalty rate is to be com-puted along with the need to take educated guesses onthe disagreement payoffs of both parties. Various ex-amples were given to demonstrate how the bargainingstrengths of each party can be incorporated into the bar-gaining weight, and these individual bargaining strengthscan be used to apply the NBS to the specific facts of thecase. Although Georgia-Pacific factor fifteen is the ba-sis for this analysis, the other fourteen factors could alsobe used to obtain the normalized disagreement payoffsand to choose the bargaining strengths. Finally, we haveproduced a nomograph for the parties to make a quickcalculation of the asymmetric NBS to solve for a reason-able royalty.
Acknowledgments
One of the authors (D.M. Kryskowski) would liketo thank Professors Li Way Lee and Vitor Kamada ofWayne State University for their encouragement in pur-suing this topic and to Professor J.J. Prescott of the Uni-versity of Michigan Law School for sparking the author’sinterest in Law & Economics. [1] N. J. Linck and B. P. Golob, IDEA , 13 (1994).[2] J. G. Sidak, Stan. Tech. L. Rev. , 1 (2015).[3] J. C. Jarosz and M. J. Chapman, Stan. Tech. L. Rev. ,769 (2012).[4] Z. Yang, Berkeley Tech. LJ , 647 (2014).[5] L. Wyatt, Santa Clara Computer & High Tech. LJ ,427 (2014).[6] W. Choi and R. Weinstein, Idea , 49 (2001).[7] R. Higgins and J. Klenk, Fed. Cir. BJ , 125 (2015).[8] R. Goldscheider and A. Gordon, Licensing Best Prac-tices: Strategic, Territorial, and Technology Issues (JohnWiley & Sons, 2006), ISBN 9780471740674.[9] M. A. Lemley and C. Shapiro, Tex. L. Rev. , 1991(2006).[10] J. F. Nash Jr, Econometrica: Journal of the EconometricSociety pp. 155–162 (1950).[11] A. Muthoo, Bargaining theory with applications (Cam-bridge University Press, 1999).[12] L. W. Lee, The American Economic Review , 848 (1980).[13] S. Zimmeck, Alb. LJ Sci. & Tech. , 357 (2011).[14] L. W. Lee, Industrial Organization: Minds, Bodies, andEpidemics (Palgrave Macmillan, Switzerland, 2019).[15] D. Cameron and M. Glick, Managerial and Decision Eco-nomics , 193 (1996), ISSN 01436570, 10991468.[16] L. Roschier, PyNomo - Nomographs with Python (2016), URL .[17] R. Doerfler,
Creating Nomograms with thePyNomo Software (2009), URL http://myreckonings.com/wordpress/2009/07/31/creating-nomograms-with-the-pynomo-software/ . Appendix A: Blank Nomograph0