In this lecture we had Theorem 3.1. It had an easy but nontrivial proof. It is important because it states that $F(x)$ satisfies a DP equation (3.7). It holds under the assumption of D (discounted), N (negative) or P (positive) programming. I explained what these three assumptions mean.
I gave an example, $c(x,u) = x$, $a(x,u) = –x$, $\beta = 1$, for which $F(x)$ does not exist. Note that a problem with this data fails to satisfy D, N or P.
Notice that the asset selling example in Section 3.5 (selling a tulip bulb collection) is very much like the secretary problem in Section 2.3. The difference is that now (i) we observe values (not just relative ranks), (ii) wish to maximize the expected value of the selected candidate (rather than probability of choosing the best), and (iii) the number of offers is infinite, but with a discount factor $\beta$.
The tulip bulb example illustrates that one way in which discounting by a factor beta can naturally arise is when a catastrophe can occur, with probability $1–\beta$, bringing the problem to an end.
Can you figure out the answer to the assess selling problem if past offers for the tulip bulbs remain open (so long as the market has not collapsed)? The state $x_t$ is now be the best of the first $t$ offers.
Can you figure out the answer to the assess selling problem if past offers for the tulip bulbs remain open (so long as the market has not collapsed)? The state $x_t$ is now be the best of the first $t$ offers.
