Lawsuits may sometimes be signaling games. Here is one example. In 2003, AT&T filed suit against eBay, alleging that its

[answerhappygod]

Lawsuits may sometimes be signaling games. Here is one example. In 2003, AT&T filed suit against eBay, alleging that its Billpoint and PayPal electronic-payment systems infringed on AT&T’s 1994 patent on “mediation of transactions by a communications system.” Let’s consider

this situation from the point in the time when the suit was filed. In response to this suit, as in most patent-infringement suits, eBay can offer to settle with AT&T without going to court. If AT&T accepts eBay’s settlement offer, there will be no trial. If AT&T rejects, the outcome will be determined by the court. The amount of damages claimed by AT&T is not publicly available.

Let’s assume that AT&T is suing for $300 million. In addition, let’s assume that if the case goes to trial, the two parties will incur court costs (paying lawyers and consultants) of $10 million each. Because eBay is actually in the business of processing electronic payments, we might think that eBay knows more than AT&T does about its probability of winning the trial. For simplicity, let’s assume that eBay knows for sure whether it will be found innocent (i) or guilty (g) of patent infringement. From AT&T’s point of view, there is a 25% chance that eBay is guilty (g) and a 75% chance that eBay is innocent (i). Let’s also suppose that eBay has two possible actions: a generous settlement offer (G) of $200 million or a stingy settlement offer (S) of $20 million. If eBay offers a generous settlement, assume that AT&T will accept, thus avoiding a costly trial. If eBay offers a stingy settlement, then AT&T must decide whether to accept (A) and avoid a trial or reject and take the case to court (C). In the trial, if eBay is guilty, it must pay AT&T $300 million in addition to paying all the court costs for both parties. If eBay is found innocent, it will pay AT&T nothing, and AT&T will pay all the court costs for both parties.

(i) Draw a game tree to describe this game. (Let the first payoff number be eBay’s and the second be AT&T’s.) What is eBay’s optimal strategy when it is innocent?

(ii) Does this game have a separating equilibrium with perfect signaling where eBay makes the generous settlement offer when it’s guilty and the stingy offer when it’s innocent?

Explain your answer.

(iii) Does this game have a pooling equilibrium where eBay always makes the stingy offer?

Explain your answer.

*(iv) Calculate the bluffing equilibrium where eBay, when it is guilty, makes the generous offer with probability x and the stingy offer with probability 1 − x, and AT&T, when facing the stingy offer, accepts with probability k and rejects with probability 1-k.