Problem Statement
Ann and Bob are negotiating the sale of an asset. Bob owns the asset currently but has no value for it. Ann has value somewhere in the set $\{1,2,3,4\}$. She knows her value, but Bob does not. He thinks that the probabilities are as follows:
| Value | Probability |
|---|---|
| $1$ | $0.4$ |
| $2$ | $0.15$ |
| $3$ | $0.25$ |
| $4$ | $0.2$ |
Assume that Bob can make only one take-it-or-leave-it offer (i.e., if Ann rejects that offer, Bob cannot improve it, and if she accepts it, he cannot change his mind). Assume that if Ann is indifferent (i.e., if her value is $4$ and the price is $4$) she buys the object.
a. For each possible offer $1,2,3,4$, what is the probability that Ann accepts the offer?
b. What is Bob's expected profit as a function of the offer he makes?
c. What is the optimal offer?
My Attempt
a.
$1 : 1 \\2 : 0.6 \\ 3 : 0.45 \\4 : 0.20 \\$
b.
$EP(p) = \bigg(1-\frac p4 \bigg)\cdot p$ with $EP$ being expected payoff and $p$ being the bid.
c.
Put the derivative of answer $b$ to zero to work out for $p$.
I get $p=2$.