\section{The Secretary}

\definition{The secretary problem}
Say we need to hire a secretary. We have exactly one position to fill,
and we must fill it with one of $n$ applicants. These $n$ applicants,
if put together, can be ranked unambiguously from \say{best} to \say{worst}.

\vspace{2mm}

We interview applicants in a random order, one at a time. \par
At the end of each interview, we either reject the applicant (and move on to the next one), \par
or select the applicant (which fills the position and ends the process).

\vspace{2mm}

Each applicant is interviewed at most once---we cannot return to an applicant we've rejected. \par
In addition, we cannot reject the final applicant, as doing so will leave us without a secretary.

\vspace{2mm}

For a given $n$, we would like to maximize our probability of selecting the best applicant. \par
This is the only metric we care about---we do not try to maximize the rank of our applicant. \par
Hiring the second-best applicant is no better than hiring the worst.

\problem{}
If $n = 1$, what is the best hiring strategy, and what is the probability that we hire the best applicant?

\vfill

\problem{}
If $n = 2$, what is the best hiring strategy, and what is the probability that we hire the best applicant? \par
Is this different than the probability of hiring the best applicant at random?

\vfill
\pagebreak


\problem{}
If $n = 3$, what is the probability of hiring the best applicant at random? \par
Come up with a strategy that produces better odds.

\vfill

\problem{}<bestyet>
Should we ever consider hiring a candidate that \textit{isn't} the best we've seen so far? \par
Why or why not? \hint{Read the problem again.}

\vfill


\remark{}
\ref{bestyet} implies that we should automatically reject any applicant that isn't
the best we've seen. We can take advantage of this fact to restrict the types of
strategies we consider.

\vspace{2mm}

We'll transform our sequence of $n$ secretaries into a sequence of $n$ random variables $I_1, I_2, ..., I_n$,
each producing values in $\{\texttt{0}, \texttt{1}\}$. Each $I_x$ will produce \texttt{1} if the $x^\text{th}$
secretary we interview is the best we've seen so far, and \texttt{0} otherwise.

\problem{}
What is the probability distribution of $I_1$? \par
That is, what are $\mathcal{P}(I_1 = \texttt{0})$ and $\mathcal{P}(I_1 = \texttt{1})$?

\vspace{1cm}


\problem{}
Convince yourself that the largest $x$ where $I_x = 1$ is the \par
position of the best candidate in our list of applicants.

\vspace{1cm}


\remark{}
Recall that we only know the \textit{relative} ranks of our applicants. \par
We have no absolute metric by which to judge each candidate.

\vspace{2mm}

Thus, all $I_x$ defined above are independent:
the outcome of any $I_a$ does not influence the probabilities of any other $I_b$.

\vspace{2mm}

We can therefore ignore any strategy that depends on the outcomes of
previous $I_x$. Since all random variables in this sequence are independent,
the results of past $I_x$ cannot possibly provide information about future $I_x$.

\vspace{2mm}

Given the above realizations, we are left with only one kind of strategy: \par
We blindly reject the first $k$ applicants, and select the first \say{best-seen} applicant we encounter afterwards.
All we need to do now is pick the optimal $k$.

\pagebreak

\problem{}
Consider the secretary problem with a given $n$. \par
What is the probability distribution of each of $I_1, I_2, ..., I_n$? \par
\note{That is, what are $\mathcal{P}(I_x = \texttt{0})$ and $\mathcal{P}(I_x = \texttt{1})$ for each $x$?}

\vfill

\problem{}
What is the probability that the $x^\text{th}$ applicant is \textit{not} the best-seen applicant? \par
In other words, what is the probability we reject the $x^\text{th}$ candidate? \par
\note{Assuming that $x > k$. Otherwise, the probability of rejection is 1!}

\vfill

\problem{}<phisubn>
Again, consider the secretary problem with a fixed $n$. \par
If we reject the first $k$ applicants and hire the first \say{best-yet} applicant we encounter, \par
what is the probability that we select the best candidate? \par
Call this function $\phi_n(k)$.

\vfill

\problem{}
Find $
	\underset{x \in \{0, ..., n\}}{\text{max}}
	\Bigl(\phi_n(k)\Bigr)
$ for all $n$ in $\{1, 2, 3, 4, 5\}$.

\vfill

\problem{}
Find $
	\underset{x \in \mathbb{R}}{\text{max}}
	\Bigl(~
		\underset{n \rightarrow \infty}{\text{lim}}
		\phi_n(k)
	~\Bigr)
$


\vfill