Litigation and settlement in patent infringement cases.

(i) If [G.sup.i.sub.t]([bar]x) [greater than or equal to] 0, entry is a dominant strategy and the patentholder spends [x*.sub.t], which can be a value between zero and [bar]x, as well as zero or [bar]x if this is the best way for the patentholder to maximize his expected reward from a trial.

(ii) Conversely, if [G.sup.i.sub.t](0) [less than or equal to] 0, entry would be a bad decision for any level of monitoring effort. Facing the no-entry dominant strategy, the patentholder decides to spend zero.

(iii) The problem is more complicated when [G.sup.i.sub.t](0) > 0 > [G.sup.i.sub.t]([bar]x). Suppose first that [x*.sub.t] < [x.sub.t]. It results that [G.sup.i.sub.t] [x*.sub.t] ) > 0, so we obtain an equilibrium with entry despite the monitoring effort of the incumbent. In other words, from the best response functions

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we conclude that the Nash equilibrium is [x.sup.n] = [x*.sub.t], and [e.sup.n] = 1.

Consider on the contrary the case where [x*.sub.t] > [x.sub.t], as in Figure 1. Since [G.sup.i.sub.t]([x*.sub.t]) < [G.sup.i.sub.t]([x.sub.t]) = 0, the best response of the challenger to the effort [x*.sub.t] is to stay out (point A). But when the challenger is out, [x.sub.t] = 0 is the best choice of the incumbent (point D). And since we are in the case where [G.sup.i.sub.t](0) > 0, no monitoring encourages entry (point C), which in turn triggers a positive monitoring effort [x*.sub.t] (point B). Consequently, it appears that there is no Nash equilibrium in pure strategies.

But we can determine equilibria in mixed strategies. For example, if the patentholder spends [x.sub.t] (which can be viewed as a degenerate mixed strategy), the challenger is indifferent between entering and not entering. If she chooses to enter with probability [[1 + d[G.sup.h.sub.t]([x.sub.t])/dx].sup.-1], it is easy to check that [x.sub.t] is the best choice for the patentholder.

Another equilibrium is such that the patentholder draws randomly between [x.sub.t] = 0 and [x.sub.t] = [x*.sub.t]. In this equilibrium, (i) the infringer enters with probability [x*.sub.t]/[[G.sup.h.sub.t]([x*.sub.t]) + [x*.sub.t] - [G.sup.h.sub.t](0)] and (ii) the patentholder decides a zero level of enquiry with probability [G.sup.i.sub.t]([x*.sub.t])/[[G.sup.i.sub.t] ([x*.sub.t]) - [G.sup.i.sub.t](0)] and spends [x*.sub.t] with the complementary probability. Whatever the exact value of the probability of entry, the only thing that matters is that neither entry nor no-entry occurs with certainty.

These results are summarized in Proposition 3.

Proposition 3. In the simultaneous game, when the postentry solution is to sue the infringer at law,

(i) if [G.sup.i.sub.t]([bar]x) [greater than or equal to] 0, there exists a Nash equilibrium in pure strategies [x.sup.n] = [x*.sub.t], [e.sup.n] = 1,

(ii) if [G.sup.i.sub.t](0) [less than or equal to] 0, there exists a Nash equilibrium in pure strategies [x.sup.n] = 0, [e.sup.n] = 0,

(iii) if [G.sup.i.sub.t](0) > 0 > [G.sup.i.sub.t]([bar]x), either [x*.sub.t] [less than or equal to][x.sub.t] and there exists a Nash equilibrium in pure strategies [x.sup.n] = [x*.sub.t], [e.sup.n] = 1, or [x*.sub.t] > [x.sub.t] and there exist Nash equilibria in mixed strategies where the patentholder monitors his market with a probability smaller or equal to one and the infringer enters with a probability strictly smaller than one.

Settlement solution. In the "settlement outcome" the net profits are respectively

(8) [G.sup.h.sub.s](x) = p(x)([[PI].sup.h] - [F.sub.s]) + (1 - p(x))[[pi].sup.h.sub.d] - x

for the patentholder and

(9) [G.sup.i.sub.s](x) = p(x)[[PI].sup.i] + (1 - p(x))[[pi].sup.i.sub.d]

for the entrant, where [[PI].sup.h] and[[PI].sup.i] are defined in (2) and (3) respectively.

Because the problem is similar to the preceding one, using obvious notations we can directly assert the following:

Proposition 4. In the simultaneous game, when the postentry solution is to come to arrangement with the infringer,

(i) if [G.sup.i.sub.s]([bar]x) [greater than or equal to] 0, there exists a Nash equilibrium in pure strategies [x.sup.n] = [x*.sub.s], [e.sup.n] =1,

(ii) if [G.sup.i.sub.s](0) [less than or equal to] 0, there exists a Nash equilibrium in pure strategies [x.sup.n] = 0, [e.sup.n] = 0,

(iii) if [G.sup.i.sub.s](0) > 0 > [G.sup.i.sub.s]([bar]x), either [x*.sub.s] [less than or equal to] [x.sub.s] and there exists a Nash equilibrium in pure strategies [x.sub.n] = [x*.sub.s], [e.sup.n]= 1, or [x*.sub.s] > [x.sub.s] and there exist Nash equilibria in mixed strategies where the patentholder monitors his market with a probability smaller than or equal to one and the infringer enters with a probability strictly smaller than one.

Whether the patentholder decides to go to court or to reach an agreement, we find different kinds of equilibria depending on the expected payoff of the imitator. In type-(i) equilibria, it is always worthwhile for the imitator to enter the market, either because the expected fine she will have to pay if her identity is discovered is not too high or because the probability of being identified is low. The patentholder monitors the market but cannot prevent entry. The probability of being identified will be low if the infringed patent is in an area far away from the infringer's product.

The opposite arises in type-(ii) equilibria. Here, it is not worthwhile at all for the imitator to enter the market. This is true essentially because the imitator knows that if she is discovered (and the probability of being discovered is very high), the expected penalty will be harmful. The patentholder does not need to monitor. This would be the case of a small infringer against a big tough patentholder. We can also think of pharmaceutical patents in Western countries. (20) In this industry, patents protect innovations that have been very costly to obtain and that promise high returns. Because going through clinical trials and administrative approval is lengthy and can be verified easily, pure imitation is easy to detect and to prove. It results that patent challenges on drugs are less frequent than in other areas (two trials for 100 patents against six for 100 in all areas (Lanjouw, 1993)).

In case (iii), the potential imitator has no dominant strategy. When the patent owner is not ready to spend much on monitoring ([x*.sub.j] < [x.sub.j], j = s or t), the equilibrium path is the same as in case (i). In the opposite case when both firms play randomly, any of the four nonequilibrium outcomes depicted as points A, B, C, and D in Figure 1 can occur. This does not facilitate the task of interpreting data on patent cases.

Graphical illustration. The various equilibria of the former section can be charted with respect to several sets of parameters. We have chosen a presentation in terms of the following pair of parameters: the fine charged to the infringer in case of a finding of infringement [F.sub.t], and the cost to launch on a settlement round [F.sub.s]. This pair is one of the "four key determinants of the likelihood of observing a filed case" (Lanjouw and Schankerman, 1998). The three others are the likelihood of a potentially litigious situation, a divergence in the parties' expectations about their chances of prevailing at trial, and the expectation of the stakes.

Depending on the values of [F.sub.t] and [F.sub.s], we analyze the equilibria of Propositions 2, 3, and 4 within each of the three areas of Figure 2 described in Section 3.

(R) In the left part of the figure, where the final outcome is "no reaction by the patent owner," we have the simplest type of equilibrium: equilibrium in dominant strategies where the incumbent spends nothing to monitor his market and the imitator enters. This pair of decisions needs no strategic reasoning by players: it is based only on the very low observed or expected value of the fines in case of infringement [F.sub.t]. The obvious question is why the innovator has paid for a patent if he is not ready to defend it against violators. This question is to be asked before our story begins. Actually, even though some recent econometric studies show that an increasing number of innovators worry about litigation cost before patenting (Lerner, 1995), we observe as a stylized fact that many innovators apply for a patent without exploring the future costs of legal protection. This is especially true for small firms that are short of cash and are unable to sue infringers. It can also apply in the case of big firms that do not have a rigorous patent policy. "Xerox owned some 8,000 patents (in 1997) ... [S]ome of its patented technologies were being illegally copied by other companies, [but] no steps were taken to detect and stop such patent infringement." (Rivette and Kline, 2000, p. 59).

(T) Consider now the trial zone, that is, the upper part on the right of Figure 2. By Proposition 3, we have to distinguish three equilibria.

(i) When [G.sup.i.sub.t]([bar]x) [greater than or equal to] 0, that is, by (6) when [F.sub.t] [less than or equal to] [[pi].sup.i.sub.d], entry is a dominant strategy for the imitator, and the incumbent does not have a dominant strategy but, knowing the challenger's decision, his best choice is to spend [x*.sub.t] on monitoring.