picking opportunities

In this article I want to explore the processes involved in choosing an opportunity.

In keeping with the ongoing theme, I will reduce this to two binary approaches. There is more mathematics in this post than previous, but I will somewhat gloss over it, as I am attempting to simplify the matter. For the mathematically adept I will point to some references, for others I will ask you to accept the findings of some well-trodden bodies of study.

Evaluating opportunities generally means determining if you should take an opportunity, or leave things as they are.

It is usually a Type A decision. To truly embrace an opportunity, you cannot have a bet each way. If you can, then that is good, but experience tells me that adopting an opportunity is almost irreversible, or the cost of reversing the decision is very high.

I suggest that for evaluating opportunities there are three categories, of which two share a similar approach. The first and simplest case is what I will call the “singleton” situation. This is where you have only the one opportunity, other than doing nothing, to consider. The next case, where all the alternatives are present at the same time, is a “parallel” situation. The third case, and the far more problematic, is the “serial” situation, where the alternatives appear one after another.

Considering a merger or acquisition? That is a singleton situation. Want to buy a car? That is usually a parallel situation. Want to buy a house? That is a serial situation.

In the singleton case, you will have one opportunity to consider, and your decision is straightforward. You will analyse the opportunity, apply risk, confidence limits, and probability to the outcomes and benefits. Then you take your decision to adopt this opportunity or not. A straightforward and well-understood issue.

The parallel situation has you considering several alternative opportunities, and selecting the best amongst them. You apply the same reasoning as the singleton case, but you optimise for the alternative that offers the best outcome. Now, that raises a question – what is the best outcome? In a later post I will discuss solving the right problem, rather than the obvious one. For the purposes of this discussion, let us assume that “best” is clear and well understood.

This parallel scenario is often called a “beauty parade” or “dog show”, for obvious reasons. All the contestants are present at the same time, and you can pick and choose, revisit, and play one against the other. You can make direct comparisons, and, like the singleton situation, you have the information necessary to choose the best alternative, given your desired outcome. Your procurement department loves these; your sales department hates them.

For this discussion I am classifying singleton and parallel as fundamentally identical.

The serial scenario, however, is very different, and is the hardest of the decision situations.

The serial scenario sees opportunities presented one after another, and you usually cannot progress until you have accepted or rejected the current opportunity. Further, taking one precludes you from taking another. You know nothing about the opportunities you have yet to examine. Typically, these scenarios also have a constraint that prevents revisiting a discarded opportunity.

Conventional wisdom advises us against selecting too soon, because something better might turn up. Equally, you could go through many alternatives without seeing anything better than one of the early ones.

This is a well-known problem, and much been written on this topic.

It is known in statistics and probability literature as “the secretary problem”. The task is to interview candidates one by one for a secretarial position, and have to make a decision at the end of each interview. There are related problems, such as the “Sultan’s Dowry” that occupy similar problem spaces.

line of people waiting to enter a building
line of people

The takeaway point from the literature is this: there is no known way of ensuring you get the best result.

All you can do is adjust the odds a little. Clearly, if you can remove the constraint about no revisitation, then it becomes a far more tractable problem, and is just a special case of the parallel situation. However, this revisitation constraint often accurately reflects the real world.

There several strategies that you can use to make decisions of this type, and the most common, and most successful, is called the optimal stopping strategy. Put in its simplest terms, you examine some candidates, accepting none. Once you have examined these candidates, you then choose the first one who is better than all you seen before.

What you are doing with the first candidates is calibration. You are determining what “good” looks like. How many candidates do you need for calibration? This is where busloads of mathematicians come your aid, because over the fifty years or so that they have studied this problem, they have shown that the answer is roughly 37%.

As it happens, roughly thirty-seven percent is one of the universe’s magic numbers. It is up there with the gravitational constant, the square root of two, and pi. Thirty-seven percent is 1/e. I will revisit this number in subsequent posts.

So, in the secretary problem, if you have 100 candidates, you will use the first 37 to set your expectations, then you will choose the next candidate who is better that anyone you have seen before. Your probability of selecting the best candidate is a little over 37%. A word of warning – do not read anything into this apparent symmetry around 37%, it is just a mathematical artefact. The real relationship between stopping value and the probability of selecting the best is much more complex than this suggests.

This will almost certainly prompt a question in your mind. What would happen if I reject all candidates after the first 37, and am forced to accept the last one? Surely, the chance of that candidate being the best is 1 in 100, or 1%. The answer is no, it is higher than that, but is certainly not 37%. The mathematics to prove this are beyond the scope of this article, but I would ask you for now to accept that answer.

The other almost certain question is whether a probability of 37% is good enough? This probability could be expressed in another way – you have a better than even chance of NOT finding the best candidate with this method.

The answer to this comes from an intriguing part of the mathematics; what you now do is relax the constraint of “best” to “suitable”.

You rank the first 37 candidates, and set your threshold of “suitable” to, say, the 80th percentile. You select the next person who falls into that range. Probability of getting a “suitable” candidate? Around 85%, or nearly a certainty. Probability of getting the best? A bit below 10%.

What you are doing is switching from a “stopping strategy” to a “threshold strategy”. It reduces the probability of finding the best to be almost negligible, but, depending on the width of your acceptable band, gives you this 85% chance of finding a suitable candidate.

To be clear – this means you almost certainly won’t find the best, but will almost certainly find someone or something good enough.

This 37% magic number also applies to the house-buying problem, mentioned briefly before. This problem is different because you do not know how many opportunities there are. Much of the statistical reasoning related to the serial problem assumes a known number of alternatives, even if you haven’t explored them. The house-buying scenario introduces a new dimension to the problem, and assumes instead that there is known rate at which new alternatives appear (say, two per month). Intriguingly, the outcome and strategy end up in a very similar place.

Instead of looking at 37% of a number of candidates, in this scenario you set a time limit, and spend 37% of the time on calibration. From there, similar logic and outcomes apply. You can apply the pure optimal stopping strategy, and may never find the house better than one of the early ones you saw, or you can switch to a threshold strategy, and settle for something good enough.

Which introduces the question of why settle for something that is only good enough? Why not wait for something better to turn up?

Effectively, the serial scenario proposes a finite approach to waiting for something better to turn up. If you make it infinite, then most of what I have said goes out the door. Further, if you wait for something better when faced with a singleton or parallel situation you have just moved that decision into the serial category, with its attendant uncertainties and deep compromises.

This is a well-understood psychological matter. It is hard for people to decide between options when there may be no clear winner. It is much, much easier to make decisions in favour of a clear winner than to make compromises between close alternatives. As a result, the closer the opportunities, the greater the risk of no decision – or procrastination.

All these approaches have assumed the decision-making process has no cost. Real-world analysis shows something different. It reveals that whilst the cost of the decision-making process may be low, the cost of inaction may not be.

My next post will discuss this cost of procrastination. And you already know the answer – it is high.

A closing digression on the topic of dog shows or beauty parades. In my career I have participated in many, many of these dog shows, principally as a contestant.

The process goes like this. The procurement department issues a “request for information”, and a number of people respond. The procurement department then goes into a huddle with these responses and emerges with a “request for proposal”. What the huddle has produced is a set of requirements that has taken the best parts from all the RFI responses. They have also set a budget that is the cheapest price from all the RFI responses. This is the measure they use on you, the contestant. It is a superset of the best – which doesn’t exist – with an impossibly low price. Every time you admit to not meeting a requirement, they tell you do not have the same value as this non-existent cheap one, and therefore should be reducing your price to compensate. I hate dog shows.

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