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les Nouvelles Article of the Month

The Nash Bargaining Solution

Doug Kidder
Doug Kidder

OSKR, LLC, Director, Emeryville, CA, USA

Vince O'Brien
Vince O'Brien

OSKR, LLC, Director, Emeryville, CA, USA


In patent litigation, the ruling against the 25% Rule has led some plaintiff's experts to search for a methodology to replace it. Some of these experts are putting forward claims based on a 50/50 split of profits by claiming that it is the result of the Nash Bargaining Solution. This is not only a misreading of Nash's work; the economic community has not accepted that Nash Bargaining is a good predictor of outcomes in the real world. The 50/50 split posited by Nash requires a set of assumptions that are

1. Introduction

Some patent damages experts are putting forward estimates that they claim are based on the Nobel Prize-winning work of John Nash.1 In their telling, the Nash Bargaining Solution (NBS) applied to the hypothetical negotiation leads to a royalty rate of 50 percent on the profits of the patented product. However, this is neither an actual application of the NBS nor is it supported by the outcome of actual negotiations.

2. The applications we've seen of the NBS (all in the context of patent litigation) are uniformly distortions of Nash's work. The NBS posits that, under idealized conditions when two parties can benefit by cooperation, they will split the surplus equally; i.e. "50/50." While plaintiff's experts have seized on the 50/50 split, they haven't been as quick to grasp what is to be split—and it's not the entirety of the profits from the patented product. Under Nash Bargaining the surplus to be split is not the total benefit from the sale of the patented product, it is the total benefit from the patented product minus the total benefit each side could receive if it chose not to enter the license ("cooperate"), but pursued a different course of action. For example, if I'm contemplating manufacturing a patented product that will earn me $10 and I have an unpatented product that I could manufacture that will earn me $9, then the amount to split is not $10, it is no more than $1. This understanding is crucial to applying the NBS. However, in the applications we've seen to date, damages experts have applied a 50/50 split to some level of profits from the accused product without subtracting anything for the infringer's alternative courses of action. This is methodologically indistinguishable from the 25% Rule, which the CAFC ruled is not an acceptable methodology for calculating a reasonable royalty.2

3. Even when the benefit is properly defined, the NBS doesn't work for patent damages because NBS relies on assumptions that are demonstrably false in the real world. The NBS and other solutions based on the notion that humans were entirely rational economic beings have been tested and found wanting. In fact, two economists were awarded the Noble Prize in Economics for developing the field of Behavioral Economics which challenged these rational solutions.3 In particular, the assumptions underlying the NBS do not hold for the Hypothetical Negotiation described in Georgia Pacific.4

2. What Is Nash Bargaining?

4. John Nash was attempting to devise a solution to a question that had been puzzling economists for decades: how do economic surpluses get divided up in the real world? Prior to Nash, the answer to this question was indeterminate. There was no economic theory that determined how much of the surplus each party to a negotiation would receive. If two parties were negotiating over $10, each side might receive any amount from $0 to $10. This is still true in the real world.

5. What Nash did was to show that, if both parties to the negotiation were perfectly rational with identical and linear preferences and perfect knowledge of each other, then—and only then—the surplus would be split equally. While much of the focus on Nash is on the equality of the split, in our opinion, the critical observation is the amount to be split. In the next sections we discuss how NBS works and then examine some of the assumptions it requires.

2.1 How NBS Works

6. The surplus to be divided under Nash Bargaining is not simply the benefit from cooperating (entering an agreement): it is the benefit from cooperating minus the sum of the "disagreement payoffs" of the parties. Disagreement payoffs are also called "reservation prices" or "threat points." The reservation price of a party is the amount that can be received by that party from pursuing a course of action other than cooperating.5

7. To illustrate the surplus to be divided by NBS, we will use a hypothetical example involving a manufacturer negotiating for an exclusive license from a patent owner. For our example assume that the benefit from manufacturing the patented good amounts to $10 per unit and that the manufacturer could obtain a benefit of $8 per unit from manufacturing an alternative product.6 Further, assume that the patent-owner had an offer to exclusively license the patent to a different manufacturer for $1 per unit.7 Thus, the disagreement payoffs are $8 for the manufacturer and $1 for the patent-owner. The manufacturer would not agree to any license that gave it less than $8 profit per unit and the patent-holder would not agree to any license that gave it less than
$1 per unit. Under NBS the surplus to be divided is the remaining $1; the $10 in benefits minus the $8 reservation price of the manufacturer minus the $1 reservation price of the patent-owner. In this way, the manufacturer and the patent-owner are negotiating over the amount that they can only receive by
cooperating. Nash then proposes that this $1 surplus will be split equally. For this example Nash Bargaining predicts that the manufacturer would get $8.50 per unit ($8 reservation price plus half of the $1 surplus) and the patent-holder would receive $1.50 per unit.

8. The outcome of the Nash Bargaining solution is very sensitive to the quantification of the reservation prices. Notice what happens if the manufacturer has an alternative product—one that is perhaps completely unrelated to the patented product—that can be produced for a benefit of $9 per unit if it isn't producing the patented product. Then the sum of the reservation prices would equal $10 and there is no surplus to be divided and hence no point to concluding a negotiation.

9. Thus, Nash Bargaining requires an analysis of each party's next best alternative assuming it doesn't enter the agreement. This goes significantly beyond the "invent around" discussion in patent cases and includes strategic as well as marketing alternatives. A completely unrelated product—not simply the allegedly-infringing product without the patented technology—can provide a reservation price. Without an examination of the parties' next-best alternatives, the analysis is not specific to the patent or the parties. In particular, applying a 50/50 split to any standard accounting profits without deducting the manufacturer's disagreement payoff will substantially overvalue the patented technology in the NBS framework.

2.2 Assumptions Critical To NBS

10. As Dr. Nash put it:

In general terms, we idealize the bargaining problem by assuming that the two individuals are highly rational, that each can accurately compare his desires for various things, that they are equal in bargaining skill, and that each has full knowledge of the tastes and preferences of the other.8

11. Nash's theory also relies on some additional assumptions:

  • The parties tastes and preferences, i.e. their utility functions, are linear; and
  • The outcome of the negotiation is independent of irrelevant alternatives.9

12. Nash also implicitly assumes that the negotiation is completely isolated from any other negotiation or strategic consideration. For example, in the NBS, there is no consideration of any relationship beyond the negotiation (e.g. whether the parties to the negotiation are competitors or inventor/promoter.) There is also no consideration of the results of previous negotiations or of any established licensing program. Under NBS each deal is unique and uninformed by past deals or the prospect of future deals. In the world of Nash Bargaining, a license for the same patent to 100 different companies would naturally yield 100 different royalty rates. This is methodologically opposite to a Georgia-Pacific analysis, which is grounded in an analysis of prior licenses.

13. A very important aspect of the NBS is that all of its assumptions and conditions are absolutely necessary to arrive at the conclusion of an equal division of the surplus. If any one of these assumptions is not true in any particular negotiation, then the outcome is indeterminate. In particular, one cannot use a Georgia-Pacific analysis to "adjust" the 50/50 split as that is an admission that one or more of the assumptions and conditions required to apply the NBS are invalid.

14. In general, Nash's theory relies on assumptions about human behavior that seem plausible, but are not valid. Specifically, people are assumed to be rational and utility maximizers. However, the growing evidence from behavioral economics indicates that people aren't as rational as required by NBS. As stated by the Nobel Laureate Daniel Kahneman writing with Amos Tversky:

We first sketch an analysis of the foundations of the theory of rational choice and then show that the most basic rules of the theory are commonly violated by decision makers. 10

15. One particular and explicit assumption underlying NBS that has been widely shown to be violated in the real world is the "Independence of Irrelevant Alternatives" or "IIA." The IIA assumes that the existence of a third choice that is worse than the first two choices will not affect the final decision. The IIA is illustrated by the following: if I am choosing between a peach and a pear and I prefer peaches, then I will choose the peach. And if I am choosing between a peach, a pear and an apple, then I will also choose the peach if I prefer pears to apples. My preferences are ranked peaches > pears > apples so the addition of an apple into my choice set is irrelevant to my selection. I will still choose the peach.

16. However much appeal the IIA has as a theoretical construct, it isn't actually true in the real world. People's choices are affected by irrelevant alternatives. There are multiple experiments demonstrating this. As described in a paper by Amos Tversky and Itamar Simonson:

One group (n = 106) was offered a choice between $6 and an elegant Cross pen. The pen was selected by 36 percent of the subjects and the remaining 64 percent chose the cash. A second group (n = 115) was given a choice among three options: $6 in cash, the same Cross pen, and a second less attractive pen. The second pen, we suggest, is dominated by the first pen but not by the cash. Indeed, only 2 percent of the subjects chose the less attractive pen, but its presence increased the percentage of subjects who chose the Cross pen from 36 percent to 46 percent...11

17. In this experiment, the addition of a lowquality pen to the choice between money and a high-quality pen—a third, supposedly irrelevant option—increased the percentage of people who chose the nice pen over the cash. In other words, approximately 10 percent of the people reversed their preference for cash over the nice pen because they were offered another option that was clearly inferior to the nice pen. This 10 percent originally ranked nice pen > cash, but when presented with an inferior pen they ranked cash > nice pen. The IIA states that the low-quality pen is the irrelevant option and thus should not affect the decisions of the players; but it does.

18. The IIA assumption matters greatly to the Nash Bargaining Solution. Without a neat ordering of preferences, utility functions become not just non-linear, but essentially unmanageable. Much as we'd like to be able to simplify actual utility functions, they are not the neat, linear constructs needed to arrive at a 50/50 split of the profits from agreement.

19. The NBS also requires that each side know with certainty the other sides' disagreement payoffs and that both sides have identical utility functions. These assumptions are clearly not true in real negotiations. The NBS 50/50 split is elegant, but it requires a set of idealizations that render it useless for predicting the outcome of a real negotiation.

20. As one game theorist wrote:

… I have never heard an economist seriously claim that the Nash bargaining solution is a good predictor of bargaining in real markets … 12

21. The economic community has never accepted the NBS as an accurate predictor of actual outcomes. As such, it clearly fails the Daubert requirement that it be an established methodology.

3. The NBS Can't Be Used With Georgia-Pacific

22. The Georgia-Pacific Factors can't be combined with the NBS because they are methodologically opposed. The hypothetical negotiation described in Georgia Pacific is designed to replicate a real world negotiation and its outcome. In contrast, the NBS is described by Nash as an "idealized" bargaining situation.

23. In the hypothetical negotiation, the parties do not know each other's "tastes and preferences" or "utility functions." Moreover, their utility functions are certainly not linear or identical.

24. The hypothetical license is usually for a multiple years and, by implication from Georgia Pacific Factor 7, for the life of the patent. The NBS does even not address the long term consequences of cooperation between the parties. The negotiation is assumed to take place and benefits obtained instantaneously.

25. The hypothetical negotiation described in Georgia-Pacific incorporates past licensing history in Factors 1, 2 and 12, and strategic considerations in Factors 4, 5, and 6. The NBS does not. All of the broader implications of the bargain are not considered by the NBS.

26. The NBS also requires certainty and symmetry between the parties—and without that certainty the split of any surplus is indeterminate. Under NBS, there is no room for the calculation of the surplus to be uncertain—as it is in the real world. In the real world, not only are the benefits of the bargain uncertain (How many units will be sold? What price will be acceptable to consumers? How will competitors respond? What will manufacturing costs be in two years? Etc.) but the disagreement payoffs are also uncertain. Thus, there is considerable uncertainty about the surplus and consequently significant risk in the outcome. The NBS does not address this.

27. Uncertainty about the surplus is a defining characteristic of the hypothetical negotiation. The Hypothetical Negotiation is required to be set at the date infringement begins. This is a date before the benefits of cooperation are known. An appeal to the Book of Wisdom to incorporate future knowledge into the date of the hypothetical reduces the uncertainty about the benefits from the patented product, but it does not reduce the uncertainty about the amount of the disagreement payoffs.

28. The NBS does not consider the relative risks of the parties. Under NBS, there is no way for either party to lose money from cooperating, yet, in the real world, that is a possible outcome. A manufacturer could pay more to access a patented technology than he will obtain in benefits from selling that patented technology.13 Thus, one consideration in the Hypothetical Negotiation is the assumption of risk. Under a running royalty license: the risk of sales volume being higher or lower than expected is shared, the risk of profitability is entirely absorbed by the manufacturer, and the risk that the manufacturer will stop using the patented technology is entirely absorbed by the licensor. In a lump sum license, all risk is absorbed by the manufacturer—in exchange for a discount to the present value of an expected running royalty. Under Nash Bargaining, there is no consideration of these tradeoffs for risk and division of any surplus.

29. Nonetheless, we do believe that the NBS does contribute to patent damages calculations. While the 50/50 split rests on idealized assumptions that are not valid in the real world, the definition of the surplus under the NBS is useful when considering patent damages. The calculation of the surplus takes into account each party's alternatives, reservation prices and costs of capital. However, using the 50/50 split posited by NBS as a check or supplement to a Georgia-Pacific is wrong as it is an admission that the NBS assumptions and conditions are violated.

4. Discussion Of Cases

30. The NBS has been met with mixed success in the courts. When it was used as the sole basis for the royalty rate, it has been rejected. When used as "a check" on or supplement to a Georgia-Pacific analysis, it has met with some acceptance. A discussion of these cases is below with the most recent first.

31. Suffolk Tech. LLC v. AOL Inc. and Google Inc., Case No. 1:12-cv-625 (Doc. No. 518), Eastern District of Virginia, Judge Ellis, April 12, 2013. N S rejected as being no different than the 25% Rule.

32. VirnetX Inc. v. Cisco Systems, Inc., Case No. 6:10-cv-00417-LED (Doc. No. 745), Eastern District of Texas, Tyler Division; March 1, 2013; Judge Leonard Davis. NBS allowed. NBS accepted as "… the traditional 50 percent—50 percent profit split" and applied to "gross profit." While we don't know the specific details that led to a 50/50 split being characterized as being "traditional," the court appears to have gotten this wrong. NBS is not traditional and does not involve splitting gross profit.

33. VirnetX Inc. v, Apple Inc., Case No. 6:10-cv417) E.D. of Texas, Tyler Division; February 26, 2013; Judge Leonard Davis. NBS allowed although most of the dispute was over whether the expert had correctly measured the profits due to the infringed technology.

34. Gen-Probe Inc. v. Becton Dickinson & Co., Case No. 09-CV-2319 BEN NLS and 10-CV-0602 BEN NLS, Southern District of California, November 26, 2012. NBS allowed in addition to the expert's real world observations.

35. Mformation Techs., Inc. v. RIM, No. C 08-04990, Northern District of California, Judge Ware, March 29, 2012. NBS allowed as a "check" on a reasonable royalty derived through a Georgia-Pacific analysis.

36. Solvay, S.A. v Honeywell Specialty Materials LLC et al., 06-557-SLR, District of Delaware, Judge Sue L. Robinson, September 8, 2011. NBS allowed because the expert testified that a 50/50 split is also supported by her "review of thousands of agreements over 33 years of [her] career." Interestingly, the expert challenged was retained by the Defendant and the Plaintiff was challenging the use of the NBS. Also, it's hard to see how a review of license agreements would reveal anything about the parties' split of profits. About all one can extract is that the royalty rates or lump sum payments tend to be small relative to sales. This would suggest that either the incremental profit is small or that the split favors the licensee.

37. Oracle America, Inc. v. Google Inc., No. C 1003561, Northern District of California, Judge Alsup, July 22, 2011. NBS firmly rejected with comments like:

"It is no wonder that a patent plaintiff would love the Nash bargaining solution …"

"The Nash bargaining solution would invite a miscarriage of justice by clothing a fifty-percent assumption in an impenetrable facade of mathematics..."

  1. In particular, see: Weinstein, R., Romig, K., Stabile, F., "Taming Complex Intellectual Property Compensation Problems," TTI Vanguard Conference "Taming Complexity," October 4-5, 2011.
  2. Uniloc USA v. Microsoft, US CAFC 2010-1035, -105, January 4, 2011. For a more detailed discussion of the multitude of problems with the 25% Rule, see Kidder, D., O'Brien, V., "Simply Wrong: The 25% Rule Examined," les Nouvelles Journal of the Licensing Executives Society, December, 2011.
  3. For example: Daniel Kahneman and Vernon Smith in 2002.
  4. Georgia-Pacific Corp., v. United States Plywood Corp., 318 F. Supp. 1116 S.D.N.Y. (1970); aff'd, 446 F.2d 225 (1971).
  5. The use of terms "benefit" and "payoff" as opposed to "profit" or "incremental profit" is intentional. The latter are accounting concepts that do not take into account the alternatives nor include a charge for the use of capital. These are important in determining the surplus that the parties would be willing to divide.
  6. For the purposes of this example, assume that the quantities that would be manufactured and sold of the patented and unpatented items are identical.
  7. Assume that the alternative licensor would produce a volume equal to the hypothetical manufacturer.
  8. Nash, J., "The Bargaining Problem," Econometrica, Vol. 18, No. 2 (Apr. 1950), pp. 155-162.
  9. We explain this rather abstract term later in the paper.
  10. Tversky, A., Kahneman, D., "Rational Choice and the Framing of Decisions," The Journal of Business, Vol. 59, No. 4, Part 2: The Behavioral Foundations of Economic Theory. (Oct., 1986), pp. S251-S278.
  11. Tversky, A., Simonson, I., "Context-Dependent Preferences," Management Science, Vol. 39, No. 10, pp. 1179-1189.
  12. Rubinstein, A., "John Nash: The Master of Economic Modeling," Scand. J. o/Economics, 97(1), 9-13,1995.
  13. This is even before a consideration of alternative products that could have been produced that would have yielded greater total benefits.