Imperfect Uncertainty
Choosing without probabilities
The subject of how to make choices and decisions in the absence of full information is pervasive, critically import, and terribly complex. It is also burdened with a jungle of conflicting terminology, academic (not to say baroque) nuance, and mathematical and philosophical “hair”. My plan beginning with this post is to explore various facets of the subject, leaving the crossing of the “t”s and dotting of the “i”s to other venues. Monographs and even treatises are in order, just not here where I aim to communicate core ideas in a manner that is as jargon-free and comprehensible as I can manage.1 Comments and suggestions are most welcome.
Consider two decision-making contexts or problems:
Deciding what to believe. (“the belief problem” What should I believe?) Will there be a functioning democracy in the United States in 2050? When if ever will the manufacturing costs of electric vehicles fall to less than 80% of the cost of internal combustion vehicles? Which of an ensemble of hypotheses should one believe? And so on.
Choosing what to take. (“the choice problem” What should I choose?) Or accept. Given a choice between a safe streets program and subsidies for rooftop solar PV, which policy should be implemented (in the relevant time and place)? Which house to purchase? Which car to buy? And so on.
A natural and attractive way to answer to belief problems is to choose a belief that on average has the highest probability of being true. Similarly, a natural and attractive way to answer to choice problems is to choose alternatives that on average have the highest average value to the decision maker. What I mean by highest average value to the decision maker is best illustrated with an example, kept as simple as possible.
Imagine that the choice is between two houses contemplated for purchase. There are many factors to consider. Price, taxes, and insurance are more or less certain. Much less certain are neighborhood developments over the next 5—10 years. These are many-faceted and each facet has its own uncertainty. If you could associate a probability with each relevant facet and associate a value to you with its outcomes, then you can compute (or estimate roughly) the average value of the outcomes and compare alternatives on their average values. Choose an alternative with the highest average value.
In sum, belief problems and choice problems, while quite distinct (let us assume) have in common a comforting core reliance on probabilities to answer to them. Belief: believe an alternative with the highest average probability. Choice: choose an alternative with the highest average value.
There are very many problems with this tidy picture. The rabbit hole lurks. Instead, I want to focus on just one such problem with the account just given. The problem is that the required probabilities are often, even most of the time, unavailable to us.2 This can happen for any number of reasons, but it is entirely common.
So, what is the alternative to using probabilities? How can we choose without them? The short answer is that there are in fact many accessible approaches to these decision problems, approaches that do not require inaccessible probabilities. The full story necessitates lengthy navigation of the rabbit hole. Instead, I want to focus on the choice problem and say something in particular about it. To wit for the choice problem:
Choice uncertainty—choice problems in which probabilities are effectively unavailable—is real, contra the pure Bayesians.
In the case of strong uncertainty (alias ignorance, pure uncertainty, etc.), when there is not warrant for assuming relevant probabilities, the main decision principle is, or should be, robustness. This is a widely-agreed point, e.g., within the DMDU community.
Practicable working notions of robustness are evident in real practice today. Nevertheless, much work needs to be done to articulate and refine the concept. We can get along with intuitive concepts, but would do better to investigate the matter further.
These points hold with respect to weaker notions of uncertainty, including what Kay and King call radical uncertainty: contexts in which the relevant probabilities are changing too rapidly to be at all useful.3
Thes points hold with respect to contexts in which there is limited information available regarding relevant probabilities. I call these contexts of imperfect uncertainty. Discussion of how this works with robustness will come in a future note.
While robustness is a main decision rule in these contexts, other factors are typically relevant. Warranted choice may often be supported on an MCDM (multiple criterion decision making) basis, with robustness one of several evaluation criteria.
These claims should be seen as (here, lightly) warranted principles, to be investigated in detail by application and testing. Indeed, there lies the main action.
I do plan to annotate this and subsequent postings, likely on a continuing basis after they originally appear. My hope is that the bodies of the works are clear and comprehensible. Let me know.
Some philosophers, at least since Hume, would argue that the probabilities are never available to us. Others, including subjectivists (Bayesians) hold that the probabilities are always available to us. Both views, IMHO, are mistaken. I’m happy to deal with that, but in another venue. It requires navigating the rabbit hole. Let practice be our guide and in practice the unavailability of relevant probabilities is recognized as quite real. The DMDU (decision making under deep uncertainty) community is commendably articulate on this. See DMDU society. See also Kimbrough, S. O. (1982). Circumventing the Problems of Induction: A Theory of Rational Hypothesis Choice [Ph.D. thesis]. University of Wisconsin, Madison.
Kay, J., & King, M. (2020). Radical Uncertainty: Decision-Making beyond the Numbers. W.W Norton & Company.