# Beware of Trolls (Part 2)

Read the first part of this post, with an introduction into the business model of patent trolls, here!

Let’s get into the economics. Think of two companies, call them “Pear” and “Samson”, who are both active in the smart phone business and each making a profit of $\pi_P = \pi_S = \pi$. But in their production process both firms risk to infringe a patent held by a troll with probability $\theta$. Choi and Gerlach assume that when a troll files a law suit and actually wins in court, he can extract a licensing fee of $\pi/2$, which results from bargaining with symmetric bargaining power (Nash solution).

First, consider the case when the troll is dealing with one firm in isolation. If he faces a litigation cost of $L$, he has a credible threat to go to court if

$\theta \frac{\pi}{2} > L$.

In words, the probability to win an infringement litigation times the money a troll can gain has to be higher than the costs of litigation. Otherwise the business is just not profitable for the troll and firms could ignore the letters in their mailbox as cheap talk. If the litigation threat is credible, however, the firm and the troll decide to settle out of court in order to save litigation costs (assuming that both the plaintiff and the defendant pay costs when a case goes to court).

But what happens when Pear and Samson produce very similar phones? If a court were to find that Pear is infringing on a troll’s patent, Samson is very likely to infringe as well. This can be conceptualized by a correlation coefficient in litigation outcomes of $\rho$. In the most extreme case of perfect correlation ($\rho=1$), the troll can extract royalty payments from both firms with (essentially) a single law suit. He then has a credible threat to litigate in twice as many cases

$\theta \frac{2*\pi}{2} > L$.

Now, let’s consider the case of imperfect correlation. Then, even if Pear wins in court against a troll (i.e., the infringement case is dropped), Samson can still not be sure that it isn’t infringing itself. But since Samson knows the correlation of litigation outcomes $\rho$, a dropped case against Pear still bears information. Technically speaking, Samson does a Bayesian updating of its beliefs. If Pear doesn’t infringe the troll’s patent, then Samson’s probability to infringe is equal to

$(1 - \rho) \theta$

which is smaller than $\theta$. This captures the information externality which arises in repeated litigation.

Who benefits from this information externality, the operating firms or the troll? Interestingly, it goes both ways. Assume a sequential game; the troll starts with Pear and afterwards deals with Samson. The model is solved backwards. In the first case, assume that litigation costs are so high that the troll would not have a credible threat against Samson in isolation

$L > \theta \frac{\pi_S}{2}$.

But if the troll wins against Pear in court, updated infringement probabilities would make the threat credible (remember, if Pear infringes its more likely for Samson to infringe as well, leaving the troll with a higher probability to win). In this case, the information externality is in favor of the troll. The troll is more likely to litigate Pear because he might benefit from the litigation outcome when dealing with Samson. Pear foresees that the odds are not in its favor and settles early.

The story changes when litigation costs are low and the troll would, seen in isolation, always want to litigate against Samson (the second firm he visits), i.e.,

$L < \theta \frac{\pi_S}{2}$.

The  odds of a litigation of Pear are now in favor of Samson because if Pear survives in court, the troll could lose the credible threat against Samson. Thus, the troll now has something to lose in litigation, namely his credibility against Samson. Whereas, in isolation (without information externalities), the troll would always be credible. Again, Pear foresees the troll’s risk of losing credibility and is now willing to settle in fewer cases.

That’s a pretty cool result, in my view. In the case of bottom-feeder trolls (remember part 1 of this post), who target small firms, litigation costs should be quite high compared to the money that can be made from settlement. The above model predicts that the possibility to target several firms sequentially allows the troll to settle more easily and thus collecting more money from small businesses — making the problem even worse.