Suppose that Beth Harmon is considering buying DeepMind Technologies, an artificial intelligence research company that is currently a wholly owned subsidiary of Alphabet Inc. Due to antitrust concerns unrelated to DeepMind's operations, Alphabet is considering selling the company. As it turns out, Harmon is the only potential buyer for DeepMind before it is eventually liquidated.

Harmon can make a single take-it-or-leave-it offer to Alphabet, and she judges that there are three possible scenarios for the technology DeepMind is currently developing. She believes that each scenario is equally likely. Moreover, suppose that (for now, at least) there is nothing Harmon can do to distinguish between the three possible scenarios.

(in billions)

Alphabet's valuation

Harmon's valuation

Cutting-edge technology

$2.0

$4.0

Efficient technology

$1.2

$1.6

Antiquated technology

$0.5

$0.2

You can assume that Alphabet's management would accept an offer equal to their internal valuation of DeepMind, and that they know what the state of DeepMind's technology is.

(a)[10 points] Suppose Harmon offers a price of $0.5 billion. What is the probability that this offer is accepted? And what are Harmon's expected profits (taking into account the possibility of rejection) from this offer?

(b)[10 points] Suppose Harmon offers a price of $1.2 billion. What is the probability that this offer is accepted? And what are Harmon's expected profits (taking into account the possibility of rejection) from this offer?

(c)[10 points] What price would you advise Harmon to offer, assuming that you have no more information than she does? What would her expected profits be? Be sure to fully explain your reasoning.

(d)[10 points] A "consulting firm" offers to conduct some corporate espionage and determine the state of DeepMind's technology. Suppose that Harmon trusts that the hackers' information will reveal the true state of DeepMind's technology, but Harmon has to pay for the information in advance (before learning which state will be revealed). What price would Harmon be willing to pay for this information? Explain.