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The advantage multiple versus selection formula. Let us simplify and leave risk aversion and money management matters aside. Further posit, following BS, that you are able to make a credible take-it-or-leave-it offer of 1. The value of the asset to him is v, an unknown quantity. The value to you is av, where a is your
absolute advantage. Your subjective prior probability distribution on v is f(v). 30The mean value of your prior is m < 1. In a stripped-down model, three parameters describe this situation: your advantage multiple, a; the probability that the other side is informed, p; and the selection factor against you, s, if the other 31side is informed. Thus s is the fraction of expected value that will apply, on average, if the other side is informed, and therefore only sells when the asset has low value to her. Of course, given the UU situation, you do not know s, but you should rely on your mean value of your subjective distribution for that parameter. If you knew p = 0, that the other side knew no more than you, you would simply make the offer if am > 1. If you knew there were selection, i.e., p = 1, you would invest if your multiple more than compensated for selection, namely if ams
> 1. The general formula is that your return will be:
am[ps + (1-p)1] . (1)
Maxim E: A significant absolute advantage offers some protection against potential selection. You should invest in a UU world if your advantage multiple is great, unless the probability is high the other side is informed and if, in addition, the expected selection factor is severe.
Following Maxim E, you should make your offer when the expression in (1) exceeds 1. In practice, you will have a choice of offer, t. Thus, s will vary with t, i.e., 32s(t). The payoff for any t will be
am[ps(t) + (1-p)1] - t. (2)
If at the optimal offer t*, this quantity is positive, you should offer t*
相关注释：
30It is important that m < 1. Otherwise the seller would refuse your offer if he were uninformed.
31 In health care, this process is called adverse selection, with sicker people tending to enroll in more generous health plans.
32 Let v be the conditional mean of x < v. The value of s will be constant if v/v = positive k for all v. This will be the case if f(v) is homogeneous, i.e., f(kv) = k f(v), as with the uniform or
triangular distribution starting at 0.
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