HW# The Hiring Problem Mathematically
The problem. Searching for a new hire and interviewing potential candidates. When is the
candidate good enough? What\u2019s the stopping criteria?
Formalize an abstract problem. Let us consider each candidate as an integer, the integer
representing a ranking criterion. For example: nine candidates whose rank = {1,3,7,5,8,3,1,9,4}.
This problem would be trivial, just pick the element with maximum value, if it weren\u2019t for two
properties.
\uf0b7 There is no look-ahead. When I\u2019m selecting any one candidate, you are unable to look
forward into the future and consider who you will select in the future. No crystal ball.
\uf0b7 There is no undo. If you select a candidate and after a while decide to fire them in a
misguided attempt to find someone better, there\u2019s a good chance this person will be
unavailable in the future gone working for a competitor.
We can think of it visually as a machine which is fed a tape of integers. It has two actions:
\uf0b7 it can either stop; or
\uf0b7 it can consider the next integer.
The machine\u2019s objective is to stop on the highest integer.
Real world problem. At the heart of the hiring problem is conflict. Do I reject the current
possibility in hopes of landing something better if I keep looking, or do I stick with what I have?
Can be applied to almost anything a selection choice decision is required. Take it or leave it?
Solving the hiring problem analytically.
Random selection. Choose the 7th element in the list. No reason just 7th element in the list.
The probability, then, of picking the best element from an integer sequence of length N with
this random pick rule is (1 divided by N).
To improve on this random selection strategy, search for a while, gain some insight, determine
your options, and then choose the next best element that presents itself.
In terms of the hiring problem, such a strategy would be to scan through the first r integers and
then choose the first option that is greater than any of the integers in [1,r].
How does this new strategy compare to random selection? The above image is a prop to help
understand the discussion that follows. Assume that i, is the greatest integer, occurs at n+1.
In order for this strategy to return the maximum integer, two conditions must hold:<\/p>\n\n\n\n
To calculate the probability for some r, P\u00ae, not for arbitrary n, but for everything, we need
to sum over n\u2265r:
This is a Riemann1
approximation of an integral so we can rewrite it. By letting
Now, we can find the optimal r by solving for P\u2032\u00ae=0.
By plugging roptimal back into P\u00ae we will find the probability of success.
What The Math Says
Well, the optimal solution is for us to estimate how many people we believe we might
reasonably interview in the future, say 20. We plug this into the equation (N divided by e),
where N=20, (20 divided by e) \u22487.
This result says that, if we want to maximize our probability of ending up with the best possible
candite, we should interview 7 candidates and then, choose the next candidate who is better
than all of those candidates.
1 Gary L. Miller. Riemann\u2019s hypothesis and tests for primality. Journal of Computer and System Sciences, 13(3):300\u2013317, 1976.
However, the typical hiring problem maximizes the chances of landing the best candidate and
considers all other outcomes equally bad. Most on the choosers are not thinking this way, they
want to maximize the probability that they end up with a pretty good candidate. It is not all or
nothing.
Maximizing the Probability of a Good Outcome
Fear not, there\u2019s a modification of the hiring problem that maximizes the probability of finding
a high-value candidate. Suffice it to say, the strategy is the same except we use a cutoff of \u221aN
rather than (N divided by e).
What Sort of Optimal?
At the end of the day, the secretary problem is a mathematical abstraction and fails to take into
account much of complexity of, you know, reality.
The solution to the secretary problem suggests that the optimal hiring strategy is to estimate
the maximum number of people you are willing to interview, N, and then
interview \u221aN people and hire the next person who is better than all of those. In laboratory
experiments, people often stop searching too soon when solving searching problems. This
suggests that the average person doesn\u2019t search enough candidates prior to choosing.
At the end of the day, the hiring problem is a mathematical abstraction and there is more to
finding the \u201cright\u201d candidate than interviewing a certain number of people.
Question: Who would you choose? Quantity thirty elements. Each with a ranking criterion.
3 8 3 9 2 3 8 5 1 4 2 7 9 2 4 4 9 7 1 0 1 7 4 2 1 8 1 9 5 1<\/p>\n","protected":false},"excerpt":{"rendered":"
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