Have I got a deal for you! I've got this great used car, and I might be willing to sell. The actual value of the car depends on how well it has been maintained, and this is known only to me; expressed in terms of the car's value to me, you believe it equally likely to be worth any amount between $0 and $500. You, who would utilize the car to a greater extent than I, would derive 50% more value from its ownership (e.g., if it's worth $300 to me, then it's worth $450 to you). How much are you willing to offer me? (I'll interpret your offer as "take-it-or-leave-it.")
This example was discussed in class in order to introduce the general notion of adverse selection. To reprise the key points:
You are subject to adverse selection whenever
In the example, (1) you made me an offer, (2) I knew how well the car had been maintained, and this was of relevance in determining the value of the car to you, and (3) I would accept your offer only if the car was worth less than that to me, i.e., relatively poorly maintained, and reject your offer if the car was worth more to me., i.e., was relatively well-maintained.
When you are subject to adverse selection, your expected benefit from making any particular offer is
and you must take into account that the latter conditional expectation is typically less than
The acceptance of your offer conveys information to you. In the used-car example, your offer of $x would yield you an expected ultimate benefit of
The critical observation was that, if I'd only accept your offer if it was at least as great as the value of the car to me, then the expected value of the car to you given that I say "yes" is only 1.5 · (x/2) , and therefore your expected benefit if the offer is accepted is negative for any positive offer.
To see how we could analyze the same problem in a more general setting, in which your uncertainty about the value of the car to me is normally distributed, look at this spreadsheet.
The transaction being proposed is "Approve our patent application, and we'll enter the Dentosite market." The patent clerk can approve or deny the application. The clerk holds information not yet available to National, i.e., whether Ware has already filed for a patent on the same process or not. If Ware has already filed, this is good news for National concerning the likely future potential of the market, since Ware would only have entered the race if the market potential was above $18 million. (Of course, the fact that Ware has already filed is bad news for National overall, since National's patent application will be denied.) Consequently, if Ware has not already filed (and the clerk says "yes" to National's application), this must be relatively bad news for National.
If we assume that the future annual sales potential was initially equally likely to be anywhere between $15 and $20 million, and that Ware's entry "cutpoint" was set at $18 million, then National must face the fact that either Ware was "in", the process was feasible, and Ware lost the race (an original chance of 40% (Ware is in) · 50% (the process is feasible) · 50% (Ware loses the race) = 10%), or Ware was "out" and the process was feasible, in which case National would surely win the race (an original chance of 60% · 50% · 100% = 30%). Hence the relative chance that Ware was in the race (which would lead National to raise its estimate of future annual sales to $(18+20)/2 = $19 million) is 10/(10+30) = 25%, and that Ware was out of the race (which would lead National to lower its estimate of future annual sales to $(15+18)/2 = $16.5 million) is 30/(10+30) = 75%. So National, after receiving a "yes" to its application, must revise its expectation of future annual sales potential down from $17.5 million to 25% · $19 + 75% · $16.5 = $17.125 million.