In my last post I talked about statistical decision making. Decision making when the probabilities — and the costs and payoffs of the alternatives are known — is easy in principle, if often hard in practice.
How about when one or more of those — the outcomes that are possible, or the probabilities of those outcomes occurring — are not known? Then we move out of the realm of statistical decision-making and into decision-theory, a branch of science that spans multiple disciplines.
Take the landing page example I used in my last post. The data that I started with allowed us to model the distribution of possible conversion rates for each version of the page. We could also presume that the company knows the value of each conversion, and I implicitly assumed that pretty much nothing else would change. Again: easy in principle, if hard in practice.
But many decisions in the real world are much more complex: we may not even know what outcomes are possible, and even if we do, we may not know which ones are better for us (what their payoffs are), and we may not be able to estimate the probabilities of them occurring.
Okay — what the hell did I mean by that?
When you are playing roulette and you bet on black, you know two things:
The probability that it will land on black is 47.4%
The payoff if it lands on black is the size of your bet
In this particular case, the logical move is to not bet at all, but imagine that the terms were more attractive. The question that we’re exploring here is how to make decisions when either the probabilities of the possible outcomes are unknown, and/or the payoffs are unknown.
When the probabilities are unknown, it’d be like knowing that the ball can land on red, black, zero, or double-zero, but having no idea what the probabilities were of each outcome.
When the payoffs are unknown, it’d be like knowing that that red and black each had 47.4% probabilities, zero and double zero at 2.6%, but having no idea how much each bet pays off.
In the world of decision-science, it’s common to lay out the four possible combinations as a two-by-two grid:
On the lower left — the case where you know the payoffs but not the probabilities — is often called “Knightian uncertainty”, after the economist Frank Knight, who detailed how this problem was fundamentally different from the case of “risk”, where both probabilities and payoffs are known.
Other scholars later added a new dimension — knowledge about the outcomes — that fills in the rest of the table. Maybe you know what could happen, but you don’t know which outcome is better for you. Think of an election where you’re undecided — you know who could win the election, and you may even know what the probabilities are of each candidate winning — but you don’t know which candidate would be better for you.
Or maybe you just don’t know shit. The term for this is called, appropriately, “ignorance”.
It may be dawning on you that “risk” — and what we typically think of as making “risky” decisions — is actually the easiest type of decision to make, because making the best decision is possible, at least in principle. Once you venture out of “risk” into the other cells, you’re in much murkier waters. But, as I’ll explain below, you’d much rather give up your knowledge of probabilities than your knowledge of outcomes.
In the realm of “uncertainty”, there is a common framework that can lead to better decision-making, if not perfectly optimal. And that framework is called “minimax regret”, so called because you want to minimize your maximum regret. Basically, this means picking the outcome that has the smallest “regret” — which is a term of art I need to define for you. It’s easier with an example:
Pull lever A: you could get $100, or you could lose $20
Pull lever B: you could get $10, or you could lose $10
“Regret” is the amount that you regret your decision. If you pull A and get $100 for it, you don’t regret your decision at all! So your regret is $0. But if you pull A, lose $20, and B would have given you $10, you’re down $30 relative to where you would have been if you had pulled B. So your regret is $30.
To figure out which decision minimizes the maximum amount you’d regret your decision, lay out all the possibilities in a table:
If you choose B, the maximum amount you’d regret your decision is $110, when A would have returned $100, and choosing B costs you $10. But if you choose A, the maximum regret is only $30. Choosing A minimizes your maximum regret; so in this framework, it is the right choice.
So in the realm of “uncertainty”, where the possible outcomes are known but the probabilities of each outcome happening are unknown, you can at least follow the minimax regret principle.
And, really, are the probabilities ever totally unknown? In Knight’s time, it may have been forgivable to think of one-off events as not having a probability attached to them, but I am reminded of this anecdote about David Blackwell’s conversion to Bayesianism: (you may have to click on the picture to open the tweet to be able to read the small text)
And as Philip Tetlock’s research into superforecasting has shown, even quite complex events can get probabilities attached to them. You can see some of the outcomes that his firm, Good Judgment, has attached probabilities to here.
So, with “uncertainty” you have two options: use minimax regret if you can’t get the probabilities, or use tools like superforecasting or Bayesian reasoning to attach probabilities to the outcomes.
What about venturing into the other type of missing knowledge? Oh man, does it get more complex. Take this graphic (from Andy Stirling), which lays out the ontology of incertitude — ⅚ of the graphic is dedicated to the types of situations you can find yourself in if the outcomes are underdetermined.
I’m exhausted just looking at that thing, and I would be lying if I told you that I knew all the permutations. To keep it simple:
It’s bad to know what outcomes are possible, but to not know what their payoffs are
It’s worse to not know what outcomes are even possible, let alone what their payoffs are
It’s worse still if you can’t know what outcomes are possible, because the system you’re playing in is simply too complex
It’s worst of all if you can’t put bounds your incertitude1
Most of us — thankfully — don’t have to deal with decisions where phrases like “hermeneutic ignorance” or “deterministic chaos” come to mind. But, those on the world stage often do. Questions like “what will happen if we invade {country}?” actually do require that someone, somewhere, think of the fact that they maybe actually can’t know what outcomes are possible. Not simply that they don’t know, but it’s not possible to know.
Don’t worry — I’m not going any further here. But it would be irresponsible of me to not conclude with perhaps the most famous quote on this issue — whose very existence is the height of irony, because it elucidates, ever so clearly, why the reasoning that led to the quote itself was so flawed. And that quote, of course, is the famous “known knowns” from Donald Rumsfeld, Secretary of Defense during the run up to and the beginning of the Iraq War:
Reports that say that something hasn't happened are always interesting to me, because as we know, there are known knowns; there are things we know we know. We also know there are known unknowns; that is to say we know there are some things we do not know. But there are also unknown unknowns—the ones we don't know we don't know. And if one looks throughout the history of our country and other free countries, it is the latter category that tends to be the difficult ones
I’m using this weird word to disambiguate from the situation of “uncertainty”, where possibilities are well-defined but probabilities are unknown
Going beyond probabilities
That slideshare by Andy Sterling is terrific, I wonder if there's a recording of him giving the presentation.