A recent conversation with an old colleague has prompted me to write some thoughts about commercial decision making under uncertainty. I have written about oil and gas prospect and portfolio analysis recently, and because I have some strong opinions about the topic more generally, I thought it would be worth sharing them here.
Having strong opinions doesn’t mean I’m right about anything, of course, but take it as evidence that I’ve wrestled with the subject enough to develop them. If the subject interests you enough to continue reading, see how they map onto your own.
To get it out of the way up front…
What do I mean by uncertainty?
The oil and gas industry, where my career began, has a particularly strange set of definitions for the terms risk and uncertainty, that I think have damaged our ability to think clearly on this subject.
In oil and gas, risk loosely refers to “the chance of something happening” and uncertainty refers to “the range of outcomes we expect”.
This maps (somewhat clumsily) onto the notions of discrete and continuous uncertainty, noting that risk semantics as used in oil and gas have a hard time handling multiple categories, caught in a binary “success or failure” frame.
Watch a room erupt into argument at what the Chance of Success of an exploration prospect targeting 3 reservoirs should be, and you’ll start getting suspicious that something has gone very wrong. The word games being played impede logical thinking, so I would prefer that the terminology is thrown out altogether.
So with that in mind—cutting through the jargon and warfronts of strange, esoteric definitions of words—when I say uncertainty, I just mean
“I’m not sure…”
That’s it.
Heads or tails? Not sure? You’re uncertain. How tall am I? Not sure? Uncertainty. What happens next? Not sure? Uncertainty. What is the average porosity of the reservoir? Not sure? Uncertainty.
If I use the word risk I usually mean exposure to a bad thing, contrasting with the term opportunity, which I use to mean exposure to a good thing. This is closer to the semantics used in finance, and I think closer to popular usage as well.
Ultimately, I really don’t care what words are used outside of their role in communicating technical concepts clearly. The use of risk and uncertainty in the oil and gas context make things more confusing, so I avoid using them where possible.
How does it relate to decision making?
Thinking about decision making under uncertainty is mostly just figuring out what we do know, thinking through the consequences of what we don’t, and deciding what to do despite our lack of knowledge, given how we feel about those consequences.
That involves wondering about all the things that could happen, and the math of describing those possibilities.
Sometimes this involves discrete possibilities (tossing a coin, rolling a die), sometimes it involves a continuous range of possibilities (how much volume, how much time), and very often it involves both; but the concept itself should not be confusing because we live every moment of our lives with it.
We’re not sure. So… what should we do?
Well, a good first step—apart from agreeing on what the hell we’re talking about when we use words—is to be honest about what we know.
EMV is not enough
Expected Monetary Value (EMV) is often used as a metric for evaluating commercial opportunities, where positive means “good” and negative means “bad”.
It is calculated as:
\[ \mathrm{EMV} = \sum_{i=1}^{n} p_i x_i \]
This is the sum of each outcome multiplied by its probability. In binary business decision making terms it is often simplified to success and failure outcomes:
\[ \mathrm{EMV} = p_s V_s + p_f V_f \]
Where the \(f\) and \(s\) subscripts denote the failure and success outcomes, respectively, and \(V\) is the value (often in NPV) of the outcome.
Drill a well with a 50% chance of discovering a field worth $200MM, or else fail and eat the $10MM CAPEX:
\[ \begin{aligned} \mathrm{EMV} &= p_s V_s + p_f V_f \\ &= 0.5(\$200\mathrm{MM}) + 0.5(-\$10\mathrm{MM}) \\ &= \$100\mathrm{MM} - \$5\mathrm{MM} \\ &= \$95\mathrm{MM} \end{aligned} \]
It is neat, tidy, convenient to calculate, and easy to communicate.
But, I would argue it is disastrously misleading when conveyed to the people that most need to understand the context: the decision makers.
Example
To illustrate why EMV is insufficient for rational decision making, let’s start with a simple coin toss game.
On heads, I pay you $2, on tails you pay me $1.
In EMV terms, one play is:
\[ \begin{aligned} \mathrm{EMV}_{\text{small bet}} &= 0.5(\$2.00) + 0.5(-\$1.00) \\ &= \$0.50 \end{aligned} \]
Now compare that with a superficially similar choice:
\[ \begin{aligned} \mathrm{EMV}_{\text{large bet}} &= 0.5(\$1{,}000{,}000) + 0.5(-\$999{,}999) \\ &= \$500{,}000 - \$499{,}999.50 \\ &= \$0.50 \end{aligned} \]
Both choices have the same EMV:
\[ \mathrm{EMV}_{\text{small bet}} = \mathrm{EMV}_{\text{large bet}} = \$0.50 \]
But they are obviously not the same decision. The first game should be played as many times as you can get away with, but the second should be avoided unless you are Elon Musk. You need exceedingly deep pockets to profit from the second game.
This is an extreme and contrived example, but the point is that the uncertainty matters. Without explicit communication of the possible outcomes, an EMV metric can hide absurdity.
So if someone asks “would you take a $0.50 EMV positive bet?” the rational response is very clearly not yes, but I don’t have enough information to decide.
With this in mind, how many commercial decisions in your business are riding on EMV calculations? Are opportunities being ranked using it?
Uncertainty all the way down
In the above example the uncertainty is just “heads or tails”; but outside of casino games where odds and payoffs are known, this simplification covers up a (crucially important) mess.
In the real world, the outcome probabilities and their values are almost always unknowns, we just pretend we know them so the spreadsheets work. The real world isn’t business metric friendly.1
Take the $200MM field example: that success case NPV is hiding tens of millions of dollars of uncertainty, and you would be kidding yourself if you thought $200MM was actually the “true” middle of the distribution, because the true distribution isn’t real.
The probability going into that EMV calculation is based on a subjective risk assessment from the subsurface team, which they may or may not even believe themselves, but regardless is also unknowable.
It’s uncertainty all the way down. So ending up with a single number should make you uncomfortable, especially when a lot of money is on the line.
But does it actually matter?
Why uncertainty matters
You might think that, because at the end of the day we need to decide what to do, and we’ll likely decide based on some number being above some other number, all this nerdy talk about uncertainty is a waste of time.
The reason that is wrong is because of exposure. It isn’t possible to make a rational decision without considering exposure to consequences, and that exposure is not defined outside of the context of uncertainty. To ignore the uncertainty is to cover it up!
Take the coin toss example. The game of small bets exposes you to losing a dollar, or winning 2 dollars each time you play. You don’t know what will happen (you are uncertain), but the consequence of your uncertainty is acceptable, so you play.
Compare that with the game of big bets. The EMV is the same, but each time you play you are exposed to a loss of a million dollars. How many losses can you survive? The consequence of your uncertainty (not knowing if you will win or lose the game) is basically existential. So you don’t play!
I made the EMVs equal to make the point, but even if you stood to gain 10 million dollars on a win, you still shouldn’t play if a single loss would wipe you out.
Uncertainty matters! A win for the nerds.
I feel like all of this is totally intuitive to us in normal circumstances, but in business we get seduced by abstractions, calculating precise numbers while losing sight of the obvious.
Business contexts are a lot less intuitive, though, so it is worth formalizing this stuff. Here is one approach that I like.
Utility
We can avoid many of these limitations by mapping dollars to “utility” with an agreed upon function. I think this provides a fairly simple to communicate path to better decision theory in business.
The idea is dead simple: commercial outcomes are estimated in dollars, our utility is some function of those dollars.
We may respond linearly to opportunity—$100 provides utility \(u\) and $1000 provides utility \(10u\)—but our response to risk is quite different, because risk can kill us… so, you know… fair enough.
One way of capturing the asymmetry here is with a function like this. In plain English, it treats upside linearly, but increasingly punishes losses as they approach the point where the company can no longer survive.
\[ U(x) = \begin{cases} x, & x \ge 0 \\[0.8em] \dfrac{T x}{T + x}, & -T < x < 0 \\[0.8em] U_{\min}, & x \le -T \end{cases} \]
where:
- \(x\) is the commercial outcome, measured in dollars
- \(U(x)\) is the utility-adjusted value of that outcome
- \(T > 0\) is the maximum survivable loss, so bankruptcy occurs at \(x = -T\)
- \(U_{\min}\) is a finite utility floor used to represent outcomes at or beyond ruin
The important behavior is that the function is linear for positive outcomes, but becomes hyperbolic for losses as they approach a survival threshold:
\[ \lim_{x \to -T} \frac{T x}{T + x} = -\infty \]
It helps to cap that divergence at some finite floor \(U_{\min}\), because once an outcome is catastrophic enough to kill the company, making the plotted penalty even more negative stops being useful. Negative infinite utility is something like damning all sentient creatures in the universe to eternal torture, and—unless you run an AI company—is probably beyond the scope of your impact.
That cap doesn’t need to be arbitrary. It could represent the finite, but very large, cost of corporate ruin: the value of the firm, plus distress costs and any additional penalty assigned to the disruption imposed on employees, creditors, partners, and future opportunities. One rough way to express it:
\[ U_{\min} = -\left( V_{\text{firm}} + C_{\text{distress}} + C_{\text{stakeholder disruption}} + C_{\text{strategic optionality}} \right) \]
where \(V_{\text{firm}}\) might be enterprise value, market capitalization, or the present value of future distributable cash flows, depending on the organization and context. The important point is that this number is not infinity. It is just very large, because destroying a company destroys more than the cash immediately lost on a bad bet. It also doesn’t need to be perfect (good luck estimating the dollar value of distress), just agreed upon ahead of time.
In the example plot above, the company is very small, closer to an individual household: it can survive a $1MM loss, but anything worse than that pushes it into ruin. Positive outcomes are still valued linearly, so making $2MM is twice as good as making $1MM. But losses close to the survival threshold are punished much more severely than the same-sized gains are rewarded. A $900k loss is not merely the opposite of a $900k gain if the company only has around $1MM of effective loss-bearing capacity.
This is the intuition I want the function to capture. A small company with limited cash reserves, limited access to credit, and no easy way to raise emergency capital should not evaluate a large bet using the same utility function as a much larger organization with deep cash reserves and diversified risk. The same project-level NPV distribution can be acceptable for one company and existential for another.
Same NPV distribution, different utility function.
So there is some math here, but I think it maps so well to our intuitions that it becomes quite powerful for communicating risk to decision makers. Agree upon a company utility function, then map the uncertainty of commercial project outcomes to it.
We can still communicate in dollars and NPV—with a small addendum of “utility adjusted”—and for smaller projects there may be very little difference between raw and utility-adjusted NPV.
Example
To show this in action, here are two projects that are loosely analogous to the “small” and “big” bet coin toss game, but here we have a known cost to play (e.g. the CAPEX of each project) and an uncertain commercial outcome.
Imagine we’ve got a million bucks in the bank, and we’re trying to decide what to do next.
The following plots show the outcome uncertainties of each project, in raw and utility-adjusted NPV terms. The means—which can be taken as EMV or Expected Utility (EU)—are shown as points along each ECDF.
Project A is a low-risk, low-reward opportunity with a clear positive EMV. It will cost half a million dollars to complete, and we estimate the gross commercial payoff is mostly in the $0.5–$1.2MM range, so it is very likely to make returns.
Here the NPV and utility-adjusted NPV essentially overlap, because we are not exposed to a high chance of ruin.
Project B is much larger, with a more uncertain payoff. We estimate the gross commercial payoff is mostly in the $10–$15MM range, but it will cost $11MM to complete. Ignoring the fact that we will need to go into debt to fund the project at all, there is a real chance it will lose money. We also estimate there is about an 11% chance that it will bankrupt the company.
But then again… the EMV is very healthy, and much larger than the EMV of Project A.
Obviously, because we’re not insane, we will go with Project A. The benefit of the utility function approach is the sane thing is also reflected in the maths. The utility-adjusted EMV is much lower, and negative in this example. The function is helping to guide us back to sane decision making (or just behaving consistently with our sanity, which is also good).
I should also note here that the important information really isn’t that Project B has negative Expected Utility, but rather that it exposes the company to ruin. The expectation isn’t the point, it’s just a shorthand, the uncertainty and its consequences are always the point.
Hopefully this example illustrates the usefulness of this approach.
I’ve never seen it used in oil and gas contexts, but it wouldn’t surprise me if it was employed in other industries that aren’t spending as much time arguing about what the word risk means.
Closing thoughts
EMV is a useful shorthand, but it is insufficient for good decision-making under uncertainty. The full distribution of outcomes is important, especially when there is exposure to large consequences (like bankruptcy), and an expected value calculation fails to capture this.
The use of company utility functions is a nice way of making an organization’s actual capacity to absorb loss explicit, and avoids the pitfalls of a naive EMV metric, especially when uncertainty in commercial outcomes is preserved. In this framing, a project can be perfectly rational for one company and existentially risky for another, even under the same commercial assumptions.
I’m not advocating for replacing human judgment with a shinier decision-making calculus, I am arguing that the consequences of uncertainty should take center stage when presenting projects to decision makers, rather than abstracted away with expected value calculations. The uncertainty really matters, so it should be honestly communicated while we try to figure out what to do next in a messy and complicated world.
Thanks for reading.
Footnotes
Unless you run a casino, in which case, shut it down as soon as possible, thanks.↩︎