Many advisors tell us that they have a hard time finding and hiring good team members.

This seems to be a bigger problem for small advisors than for larger firms. Based on our observations, we believe that hiring is harder for small firms for several reasons. These reasons mostly have to do with economies of scale. This post addresses some aspects of the hiring problem that might shed some useful light on it for the next time you need to hire.

Analytical Tools

Hiring is far from a science. We do not have a magic formula. However, that does not mean there are no formulas, or at least rules of thumb, that might be useful.

We can derive some informative rules of thumb derives from an area of Operations Research[1] or applied math called optimal stopping problems.

A famous problem in the literature of optimal stopping is called the Secretary Problem. The secretary problem, which may apply in many hiring situations, is basically this:

Suppose you need to hire a secretary (or executive assistant, or nanny, or bookkeeper – the job title doesn’t matter), and you are able to look at candidates sequentially. For each candidate, you must either: 1) hire, or 2) pass and not hire that candidate. You cannot go back after you pass (the assumption is that the candidate moved on). Also, for simplicity, the problem statement assumes that a candidate will accept an offer. Of course, in the real world that is not always the case.

Further suppose that you don’t have any real idea of what a good candidate will look like. You need to get some idea of “what’s out there?” before you can really know. If you have only one candidate, then by definition if you hire that candidate, you’ve hired the best one. But you’ve also hired the worst one. This is not a very interesting situation.

If you have only two candidates, if you hire the first one, you have a 50% chance of hiring the best one. And if you pass on the first and hire the second, you will know whether that second candidate is better, but you still have a 50% chance of hiring the best candidate. With two candidates, you have a choice, but you have no ability to make a choice that isn’t random.

With three candidates, if you hire at random, you would have a 1/3 chance of hiring the best, and also a 1/3 chance of hiring the worst. Can we do better? If you hire the first candidate, you have only a 1/3 or 33% chance of the hiring the best.

If, instead, you meet the first candidate and pass, you now have some information. You can then meet the second candidate, and hire candidate number 2 if the second is better than the first. If the second is not better than the first, you pass on the second and hire the third.

What is the probability of hiring the best candidate out of the three? The answer is 50%. Here’s an explanation.

Let’s rank the candidates using letter grades A, B, and C, with A being best and C being the worst. Because the order they appear is random, there are six possible orders.[2]

The table below shows those six possibilities, and shows which candidate will be selected under each.

For example, in the first scenario, we pass on A, then do not select B because B is not better than A. In the third scenario, we pass on the first candidate B, but select A because it’s better than the first candidate.

Order	Candidate Selected	Did We Select the Best?
ABC	C	No
ACB	B	No
BAC	A	Yes
BCA	A	Yes
CAB	A	Yes
CBA	B	No

Notice that we have a 50% probability of selecting the best, but only a 1/6 (or 16.66%) probability of selecting the worst. That’s important. And almost all employers would think this was a much better strategy than making a random selection from the three candidates.

What if we have 4 candidates? With 4 candidates, there are 24 possible orders in which the candidates could be met. And we’d have to evaluate two possible selection rules: pass on the first candidate and take the next best one, or pass on the first two and then take the next best one.

You can see that this would get complicated quickly. As the number of candidates grows, the complexity of the problem explodes.[3] It would be great if there were a simpler rule to answer the question: how many candidates do you meet, and not hire no matter how good they are, just for the purpose of learning what the candidate pool looks like?

The answer to that question depends on what you’re trying to achieve. Here we’ll look at how to maximize the chance of selecting the best candidate.

Optimization

As a society, we are obsessed with finding “the best.” In a moment, we’ll come back to the wisdom of that obsession. But for now, let’s look at the strategy that, given the assumptions of the hiring problem described above, gives you the maximum chance of selecting the absolute best candidate out of a pool of n people.

We’ll spare you the math, but to maximize your chances of finding the best candidate, you should meet, write down the value of the candidate, and then pass on that candidate until you’ve found the values of the first 37% candidates. You then continue to review each subsequent candidate, and when you find one with a higher value than the highest of the first 37%, you hire that candidate.

Instead of a proof, below we show the results of simulating the decision problem assuming that the pool contains 1000 candidates. Here’s how that graph looks:

Along the vertical axis is the probability of selecting the highest ranking of all the candidates; and along the horizontal axis is the number of candidates we look at and skip, before selecting the next candidate who exceeds the rank of all the candidates examined so far.

This simulation shows that the maximum probability of selecting the highest ranked candidate occurs when we skip the first approximately 350 to 400. We rely on the mathematicians to tell us that the actual optimum occurs when we skip the first 1/e (where e is Euler’s number, approximately 2.718), or about 36.79%, which we round to 37%.

Is Optimal Best?

This approach is almost certainly better than randomly selecting a candidate (which you probably weren’t going to do anyway!). If there are n candidates, and we select randomly, the probability of selecting the best is 1/n.

For example, if there were 10 candidates, a random selection would have a 1 in 10, or 10%, probability of selecting the best. As n grows, the probability of randomly selecting the best falls. With 20 candidates, it would be just 5%, with 100, just 1%.

Compared to that, 37% is pretty darn good.

But the probability of hiring the best candidate is still only about 37%. That’s because the best one could either be among the first 37% you passed on, or that candidate could be in the part of the pool that you would see only after the candidate you did hire.

Even with the optimal algorithm, when you try to find the best, you have only a 37% probability of doing so.

But it’s worse. Because if the best candidate did happen to be in the 37% that you skipped, you’ll end up with the last candidate. And that candidate will, on average, just be average.

For most practical problems, there are ways to do much better.

If you found this interesting, and would like to see more, please email, comment, or call Connor at 703 437 9720 to let us know, and we’ll do more on the topic.

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[1] Operations research is a branch of optimization theory that was developed originally in response to the unprecedented logistical problems faced by the US armed forces in fighting two wars (the European and Pacific theaters of WWII) at the same time on opposite sides of the globe, thousands of miles away from the continental US. One of the early successes was the development of linear programming.

[2] In general, with n candidates, there will be n! (“n factorial”) possible orders.

[3] For the computer science folks, the Big O complexity of the problem, as you no doubt already know, is n!.