the Application of Nash Bargaining in Reverse Payment Cases in the Pharmaceutical Industry

Some applications of Nash bargaining in the antitrust analysis of reverse payment settlements estimate the bargaining power parameter from the split of the surplus in the actual settlement with alleged reverse payments. This estimated parameter is then used as the bargaining power parameter in the but‐for world where reverse payments are prohibited. We demonstrate that this approach is incorrect. Indeed, in the but‐for world there could theoretically be an alternative “no payment” settlement where reverse payments are prohibited.  However, the prohibition on reverse payments disproportionately affects the shape of the bargaining set in favor of the patent holder. With this asymmetry in the bargaining set, the estimated “bargaining power” in the actual world (with a settlement) no longer accurately reflects the split of the surplus that would have been agreed during the but‐for negotiation with no payments. Moreover, we find that keeping the split of the surplus constant between the actual and but‐for worlds is not innocuous because it overestimates the extent of alleged generic delay.


Introduction
Reverse payment settlements are agreements between branded drug manufactures, which are typically patent-holding monopolists, and potential generic entrants to settle patent infringement litigation or the threat of such litigation. 5 A reverse payment settlement usually involves an agreed-upon generic entry date and one or more transfers of value among the settling parties, including for example, cash payments or an agreement by the branded company not to launch an authorized generic for a specified period of time in certain circumstances.
Settlements of this nature could potentially harm consumers by delaying the entry of generic products that would compete against the brand (Edling et al., 2013). A common approach for evaluating the potential harm to consumers is estimating the extent of delay that was caused by the reverse payment settlement. In other words, if the parties agree to an entry date with reverse payments, the alleged harm is a function of how much earlier the entry date would have been absent those payments (this implicitly assumes that a "no-payment" settlement could have been reached instead of the actual settlement). As part of the process of estimating the purported damages, it is natural to consider and perhaps attempt to model the negotiation process among the parties that ultimately led to the observed reverse payment settlement.
The Nash bargaining solution (Nash, 1950) is an alternative that can be used to model the outcome of the parties' negotiations in reverse payment cases. The Nash bargaining solution is the most widely used axiomatic bargaining solution in economics. In general, an axiomatic bargaining problem has two elements: (1) a set of possible utility pairs that the parties can obtain through an agreement ("the Bargaining Set") and (2) a specific point that captures the level of utility the parties would obtain if they fail to reach an agreement("the disagreement point") (Muthoo, 1999). 6 Given these two elements, the Nash bargaining solution selects an outcome by satisfying a set of axioms that aim to represent the preferences of an "arbitrator who tries to distribute the gains from cooperation in a manner that reflects 'fairly' the bargaining strength of the different agents" (Mas-Colell et al., 2005, p. 838). 7 The Nash bargaining solution may yield asymmetric utilities for parties due to the asymmetry of the bargaining set or the asymmetry in the utilities from the disagreement point. In the generalized Nash bargaining solution, however, there is an additional parameter that represents the bargaining power of the bargaining parties originating from any remaining sources such as the case of "longer waiting time after one party's offer than after the other's offer and the case of different beliefs concerning the probability of breakdown in the risk model" (Binmore et al., 1986, p. 179). It is this latter exogenous parameter 6 We assume that this failure refers "to the payoffs that bargainers obtain by quitting the negotiation permanently." (Chiu, 1998) 7 These axioms are Invariance to Equivalent Utility Representations, Pareto Efficiency, Symmetry, and Independence of Irrelevant Alternatives. Symmetry is lifted in cases where an asymmetric bargaining power parameter is used (Muthoo, 1999). that we call the bargaining power or strength of the bargaining parties. 8 The generalized Nash bargaining framework can therefore be considered as a way to better understand the bargaining process in reverse payment settlements. A standard allegation in litigation in this context is that a cash payment from the incumbent pharmaceutical company to a potential entrant was made in exchange for the generic firm not entering the market or delaying entry. In general, monopoly profits are greater than the sum of duopoly profits. As a result, a settlement with reverse payments may be more profitable for both parties because there is a larger "pie" of profits to be split if the monopoly is indeed prolonged.
The Federal Trade Commission (FTC) initially sought to strongly deter reverse payment settlements (FTC, 2002). Indeed, even an appellate court held that these agreements were per se illegal. 9 The FTC's position was consistent with some of the early literature showing their potential for adverse effects on competition (Leffler and Leffler, 2004;Ecer and Higgins, 2004).
However, since 2005 the appellate courts started to uphold these agreements. 10 The FTC continued to express deep concerns, especially when the purported payments exceed expected litigation costs (FTC, 2010). Another strand in the literature shows that reverse payments higher than expected litigation costs may be justified based on risk aversion or asymmetric information (Shapiro, 2003), risk of under-compensation of the patent holder (Langenfeld and Li, 2003), ability to achieve welfare enhancing settlements (Willig and Bigelow, 2004), and increasing the incentives to develop new generic drugs and investment in patent challenges by generic manufacturers (Dickey and Rubinfeld, 2012). The thrust of this and related literature cautions against treating reverse payment patent settlements as per se illegal (Dickey, Orszag, and Tyson, 2010;Meunier and Padilla, 2015). Nonetheless, other papers propose a modified per se rule (Elhauge and Krueger, 2012). The legal treatment of reverse payments from an antitrust perspective was addressed by the Supreme Court in 2013, which concluded that rule of reason treatment should be the legal standard. 11 This decision led to series of papers to clarify the implementation and practical implications of this decision (Edlin et al., 2013;Harris et al., 2014).
In this paper, we contribute to the use of bargaining models and concepts in determining the entry date of the generic firm but-for the reverse payments (see, e.g., Ghili and Schmitt, 2017).
In reverse payments matters, we typically observe the actual generic entry date, but we do not observe the generic entry date in the absence of the reverse payment. In the textbook anticompetitive case, a delayed entry date brings lower expected profits to the entrant, and reverse payments are typically treated as a way to compensate the entrant for the lower profits.
When the delay in entry is considered an antitrust concern, the modeling of an entry date in the absence of reverse payments may include a prohibition on reverse cash payments. In such a butfor world, the bargaining analysis is less straightforward because the parties are explicitly prohibited from giving or receiving monetary transfers.
In this paper, we discuss the appropriate way to formulate the estimates of the patent holder's and the entrant's bargaining power in estimating the but-for entry date in reverse payment cases. We find that eliminating the possibility of cash payments in settlements disproportionately affects the shape of the bargaining set as the set is tilted in favor of the patent holder. This is because the entrant cannot access monopoly profits via transfers and must settle for lower competition profits. For the same reason, total profits are lower without reverse payments. Then, we show that if one applies the estimated bargaining power parameter from a settlement derived from the actual shares of the surplus with reverse payments to the but-for scenario where reverse payments are not allowed in the settlement, the estimated delay from the reverse payments would be overstated (potentially leading to higher damages for the entrant).
The following section presents the basic model. Sections 3 and 4 study the case where reverse payments are permitted and where such payments are not permitted, respectively.

The Model
Consider an incumbent firm that operates as a monopolist in a market and realizes instantaneous constant profits denoted by 0 if it is able to maintain its monopoly through a patent of strength . Patent length is normalized to 1 and discounting is ignored throughout.
A potential entrant, denoted by (which stands for duopolist), can challenge the patent through litigation. When the entrant operates in the market, both firms obtain profits 0. We will assume that 2 . (1) If (1) did not hold, the patent-holder monopolist would always allow entry as long as it can set a minimum fee of to the entrant and restore its monopoly profits. In other words, enforcing the patent would not be profitable if (1) is violated. 12 Furthermore, we assume that if the entrant challenges the patent, litigation is resolved immediately in the beginning of the patent period at a cost of 0, which we assume to be equal for both firms. Hence, the entrant's expected litigation payoffs are 1 ; and the incumbent expected litigation payoffs are

.
To keep both litigation payoffs positive, it suffices to assume that 1 (litigation costs are sufficiently low relative to the duopoly profits and the weakness of the patent).
Instead of litigating, the parties can settle by agreeing on a specific generic entry date. To do so, they could negotiate two different types of settlements: one with or one without reverse payments, which we denote as ∈ , , defined to be from the monopolist to the entrant and assumed to be bounded. The reverse payment or transfer is therefore a payment made by the patent holder to the entrant if 0 or a payment from the entrant to the patent holder if 0. If 0, no payments are made by either party.
To complete the characterization of the possible settlements, let us call the time of entry under an agreement. Therefore, if reverse payments are permitted, the parties could negotiate over an entry date ∈ 0,1 and a transfer . If the parties are only allowed to negotiate over an entry date ∈ 0,1 , then the bargaining outcomes are only derived from profits on sales made in the marketplace.
In the Nash bargaining model, parties have ex ante exogenous bargaining strengths. We will denote the entrant's bargaining power as ∈ 0,1 , and the patent holder's bargaining power as 1 . The exogenous bargaining power parameter captures any advantages the entrant may have during the negotiations. For example, more skilled negotiators would allow the entrant to obtain a larger share of the surplus, ceteris paribus. More generally, bargaining tactics, the information structure, the procedures used to conduct negotiations, and the parties' discount rates all could potentially influence (Muthoo, 1999). 13

Bargaining with Reverse Payments
In this section, we solve the bargaining problem between the firms in the case where they can use reverse payments in the bargaining process. We assume that this setting corresponds to the actual world. Let and be the (expected) utilities for the entrant and the incumbent, respectively. Following Muthoo (1999), we can define a decreasing function ≡ that characterizes how one firm loses (gains) as the other firms gains (loses). Thus, the function .
is the frontier of the bargaining set, which is where a typical efficient solution resides, including the Nash and generalized Nash bargaining solutions.
Given that we allow for cash transfers as part of the settlement, we can show that it is optimal for the parties to always agree on 1 to maximize total surplus, and then bargain over the split of profits net of litigation cost. 14 To see this result, note that for any entry date the total rents generated by both parties is 2 1 , where the inequality holds by (1). Therefore, the parties always elect to bargain over the split of the monopoly profits using the reverse payment. Hence the bargaining frontier has a negative 45 ∘ slope when monetary transfers are possible: 13 See Dagranska et al. (2010) for an empirical application. 14 If the parties expect an antitrust scrutiny of such settlement, we might constrain to account for the maximum acceptable entry date. Our primary conclusions are unchanged after such a modification. bargaining solution is the solution to the following problem max , . ( Plugging (2) into (3) and taking the first order condition of leads to the solution 15 ′ .
By (2), we know that ′ 1, then 1 1 With simple manipulation, we can rewrite the bargaining outcome , as: 1 and .
Noting that , we have 1 and . (5') In Figure 1, we plot the entrant's payoffs as on the x-axis and the patent holder's payoffs as on the y-axis. The line segment ( , ) represents the frontier of the bargaining set in the actual case where reverse payments are allowed. The curve represents the "Generalized Nash product" (the utility function representing the preferences of the "imaginary arbitrator"), which is maximized at the point of tangency ( , ). Hence, the solution of the bargaining problem in this case is given by , , which for both players exceeds the utility of the litigation payoffs , .
At (5), the expression represents the surplus at the solution point and as can be seen from (5') this surplus is constant at the efficient frontier. Note that the share of the surplus obtained by the entrant is given by the exogenous bargaining power at (5) and 15 The solution is proven in Muthoo (1999) Note that in the case with reverse payments, as demonstrated above. However, this equality between the exogenous bargaining power and the share of the surplus does not hold in general.
We next demonstrate why it is erroneous to assume, when using the Nash bargaining solution, that the observed or estimated split of the surplus in a settlement is always equal to the exogenous bargaining power of the two parties, including in a hypothetical negotiation for which reverse payments are not allowed. Indeed, as we show below, this equality fails when reverse payments are not allowed. 16 By (1) we can show that both payoffs are positive. The corner solution when 1 (which gives the entrant all the bargaining power) requires that the patent holder realizes payoffs and the entrant, the remainder, so . Furthermore, we can check that the greater the entrant's bargaining power is, the greater its payoffs are, and the reverse is true for the patent holder. After developing the expression for the litigation payoffs we get 1 1 1 2 1 and 1 2 1 1 . Remark that when the patent strength increases, the patent holder obtains greater payoffs, whereas the entrant gets lower payoffs. The reason for this effect is that an increase in the patent strength increases the patent holder's litigation payoffs. Additionally, if the litigation costs increase, the party with less bargaining power will see its payoffs decrease and the other party earns more. Finally, given that 1, the entirety of the entrant's payoffs comes from the transfer , hence .

Bargaining Without Reverse Payments
Now we consider a situation where the parties cannot use payments as a negotiation tool.
Compared to the previous case, the bargaining set and hence the function . needs to change to reflect the incumbent's opportunity cost of ceding profits to the entrant (it is no longer one for one in the case of a pure cash transfer). We next determine the new split of the surplus generated through the Nash Bargaining solution as a function of the exogenous bargaining power . In this case, the maximization of the Nash product as in (3) yields where ′ is the slope of the frontier of the (but-for) bargaining set evaluated at the optimal solution when reverse payments are prohibited. We show that the split of the surplus does not equal in this case (NR). Indeed, the entrant's ex post share of the surplus, , henceforth simply , is given by 17 , or, with some manipulation, by, .
Inserting (9) in (8)  Finally, to characterize the bargaining outcomes we need to calculate the function . We do so with the aid of Figure 1. When 0, both parties realize duopoly profits .
And when 1, the entrant makes zero profits, while the incumbent obtains monopoly profits throughout the whole period. We assume that the entrant and the patent holder face constant trade-offs as a function of , leading to a linear frontier of the bargaining set, so where | ′ | 1 by (1). Since | ′| 1, for any ∈ 0,1 , it must be that . In addition, we can write ≡ .
The bargaining set in this case is therefore the trapezoid , , . 18 Now, recall that we assumed 1 . In addition, in order for a settlement without reverse payments to be individually rational relative to the disagreement point represented by the expected litigation payoffs , , remark that (consistent with the display in Figure 1) and . 19 Hence, the solution is given by the point of tangency of the same Nash product (function of the exogenous bargaining power ) on the frontier, , .
Let us now denote the incremental monopoly gains with respect to duopoly (or, equivalently, the absolute value of the slope of the bargaining set) with Δ ≡ 1. In Figure   2, we plot the split of the surplus against the exogenous bargaining power , for various values of Δ. Note that, as Δ increases, the difference between and increases. This means that as the monopoly is more profitable, the error of using bargaining power as a prediction for the split of the surplus when transfers are not allowed also increases. Conversely, if 2 , the error is small. 18 Note that the points , and -, are not feasible without reserve payments. 19 It follows immediately that so that a settlement with reverse payments is possible.

Figure 2: Split of the surplus for different values of
To compute , accurately, we plug in (9) and (10)  Remark that in the case where 0, the model yields * , since that is the only feasible point in the frontier of the bargaining set where both parties get at least their litigation payoffs (In Figure 1,  Intuitively, when litigation is free, the parties will not agree on a distribution of rents different than what they would achieve during litigation. When is different than zero, its effect on * is mediated by the factor . Then, for instance, if the entrant had considerable exogeneous bargaining power ( ), an increase in the litigation costs would favor the entrant as the parties would agree on an earlier entry date. 22 Furthermore, as , the exogenous bargaining power of the entrant, increases, the but-for entry date is earlier , which can be explained by the fact that the party with more bargaining power can obtain a more favorable outcome.

Keeping the Split of the Surplus Constant Overestimates Delay
In this subsection, we calculate the entry delay implied by the Nash bargaining solution when the split of the surplus (i.e., the bargaining power parameter) is kept constant across both types of settlements (with and without reverse payments). Denote as the entry date under this erroneous assumption. By Equation (6), we know that if the bargaining power parameter is also the split of the surplus in the but-for world, the solution is given by: , where min max 0, , 1 .
And given this result, it can be verified that * and hence 1 * 1 .
Since the entry delay is defined as 1 , the erroneous method overestimates delay (see Figure 3). The intuition for this result is that when reverse payments are not permitted the correct split of the surplus is always smaller than the exogenous bargaining power parameter (see Figure   2). Hence, the correct but-for entry date * occurs later to account for a split of the surplus that rewards the entrant less. Conversely, the incorrect entry date occurs earlier since it is based on a split of the surplus that is more favorable to the entrant.

Conclusion
In this paper, we first show that the parameter in Nash bargaining problems that exogenously characterize the bargaining power of the parties does not necessarily correspond to the split of the surplus in the case where reverse payments are not allowed. Then, we show that ignoring the distinction between the bargaining power and split of the surplus leads to an overestimation of entry delay in pharmaceutical reverse payment cases, which can in turn lead to an overestimation of damages. Practitioners, therefore, should be wary of this difference when applying the Nash bargaining solution to problems where the structural conditions of the negotiation change. In particular, an estimate of bargaining power in the actual world cannot be used to calculate the split of the surplus in a but-for world where reverse payments are not permitted.