# Why is the expected value conditional on the trade taking place in an adverse selection problem?

I have been asked the following question, and I don't understand why the expected value of the firm is conditioned on the trade taking place.

"Suppose that a firm owns a business unit that it wants to sell. Potential buyers know that the seller values the unit at either $100m,$110m, $120m, . . . ,$190m, each value equally likely. The seller knows the precise value, but the buyer only knows the distribution. The buyer expects to gain from synergies with its existing businesses, so that its value is equal to seller’s value plus $10m. (In other words, there are gains from trade.) Finally, the buyer must make a take-it-or-leave-it offer at some price p. How much should the buyer offer?" The answer is the $$(10 + E[value|trade takes place] - (Price Paid for business)) \times Prob(trade takes place|Price paid)$$ I don't understand why$E[value|trade takes place]$is conditioned on the trade, I would have though that it would just be$E[value]\$.

Can anyone shed light on why this is the case?

• Have you looked into the "winner's curse" (see, for example, en.wikipedia.org/wiki/Winner%27s_curse#Explanation ) phenomena yet for common value auctions? If you're still unsure of the reason post back and I can try to explain! – AndrewC Sep 10 '18 at 6:48