A regional government has created a Venture Fund (VF) to support early-stage start-up companies that could benefit the local economy. Those are risky, but when start-ups succeed, VF can recoup its investments, sometimes with substantial profits, and re-invest in new budding companies.

VF is considering a $2 million investment in, a start-up company that has patented a novel technology for efficient water purification or desalination. Considering the uncertainties surrounding whether the new technology will lead to commercialization, VF managers estimate that investing $2m in H2O would lead to one of three equally likely outcomes (1/3 chance each):

1) Success: the start-up thrives, and VF will be able to sell their share in it for $7m (in present value), that is, they will earn a net payoff of $5m.

2) "Living dead": the start-up manages to bring a product to market but with mediocre results; VF will be able to just recoup their $2m investment, i.e., a breakeven.

3) Failure: the patented technology does not yield a viable product and the start-up fails, the business is worth nothing; VF will not recoup anything, i.e., a net loss of $2m.

a) Construct a decision tree to help evaluate this investment opportunity. What is its expected value? What should VF do if they make decisions based on expected value?

Suppose a diagnostic study could determine with 100% reliability whether the technology can be developed into a viable product or not. This would provide perfect information that predicts just whether the start-up will fail or not; in case of no failure, it would not predict the market outcome (Success or Living-dead).

b) Add a branch to your decision tree from Part (a) to show the alternative, described above, of seeking perfect information on failure. Based on expected value, what is the most VF would be willing to pay for that information?