References for particular definitions of risk and uncertainty

Question

I have some doubts about risk vs. uncertainty. I have read the thread "What is the difference between risk, uncertainty and ambiguity" and have skimmed through Knight's "Risk, Uncertainty, and Profit" Chapter VII (starting on p. 197). It seems there exist alternative definitions of these concepts, which is a bit confusing.

I have a particular example that implicitly defines risk and uncertainty. I would like to know if these implicit definitions are in line with any of the established ones.

Example: In a lottery, the probability to win is $p$ and the probability to lose is $1-p$. If I know $p$ and participate in the lottery, I am facing risk. If I do not know $p$ and participate in the lottery, I am facing uncertainty.

Does this match any existing definitions of risk vs. uncertainty?
Could I also get a reference to an academic paper or a textbook?

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Edit 1: User 123 notes correctly that there seems to be risk in both cases (whether I know the probabilities or not, I am still participating in the lottery which by itself is a source of risk). Hence, the term uncertainty seems to incorporate risk + the lack of knowledge of the probabilities. This may suggest the terms risk and uncertainty are not mutually exclusive; when we call something uncertain, it might also include an element of risk; but when we identify something as risky, we must know the probabilities, thus it is not uncertain. Yeah, it is convoluted...

Edit 2: Quoting Knight via Wikipedia, a measurable uncertainty, or 'risk' proper, as we shall use the term.... Thus it is enough that the probability be measurable in the above example and is does not have to be known precisely. So I understand that the implicit definitions in the example above are not in line with the Knightian ones.

Answer 1

Knight's 1921 essay was not written in formal mathematics (and trying to formulate a direct translation into modern mathematics may be quite problematic). Since Knight's time, a formal decision theory literature has developed which makes distinctions that are at least reminiscent of Knight's.

Lars P. Hansen (2012), writes "Motivated by the insights of Knight (1921), decision theorists use the terms uncertainty and ambiguity as distinguished from risk." Hansen references Gilboa et. al. (2008) who write:

In economics, Knight (1921) is typically credited with the distinction between situations of "risk” and of “uncertainty.” In his formulation, “risk” designates situations in which probabilities are known, or knowable in the sense that they can be estimated from past data and calculated using the laws of probability. By contrast, “uncertainty” refers to situations in which probabilities are neither known, nor can they be deduced, calculated, or estimated in an objective way.

Gilboa and Schmeidler (1989) introduced a max-min utility theory where instead of simply maximizing expected utility, agents solve a max-min problem where to model ambiguity aversion, the objective is minimized over different priors. This work has been expanded on by Hansen and Sargent in their work on robustness.

Sliding away from Knight and into the broader topic of decision theory, it would be mandatory to reference Leonard Savage's classic, Foundations of Statistics where he introduces the notion of subjective probability.

Your particular example (in more modern decision theory)

You have a Bernoulli likelihood function for winning the lottery.

• To a frequentist statistician, $p$ is a scalar value, a parameter (albeit unknown). There is only one possible outcome for $p$ and so there's no randomness.

• If you're a Bayesian in the spirit of Savage, you're willing to extend the tools of probability to model uncertainty in your own mind; you will treat $p$ as a random variable!

If we put a prior on $p$ (eg. the beta distribution is a conjugate prior to the Bernoulli likelihood), we can compute posterior probabilities and make decisions based upon standard expected utility. In terms of how we behave though, there's no difference between a 20% probability that's objective vs. subjective.

Under the ambiguity aversion, max-min model though, we may maximize our control variable over expected utility, taking into account that it will then be minimized over multiple priors: we choose control variable $x$ to maximize utility and then (after observing our choice $x$) a mean guy chooses the prior to minimize utility.

References

Hansen, Lars P., 2012, "Challenges in Identifying and Measuring Systemic Risk", NBER

Hansen, Lars P. and Thomas Sargent, 2001, "Robust Control and Model Uncertainty," American Economic Review

Gilboa, Itzhak and David Schmeidler, 1989, "Maxmin Expected Utility with Non-unique Prior," Journal of Mathematical Economics

Gilboa, Itzhak, Andrew W. Postlewaite, and David Schmeidler, 2008, "Probability and Uncertainty in Economic Modeling", Journal of Economic Perspectives

Savage, Leonard Jimmie, 1954, Foundations of Statistics

Answer 2

At the risk of being a bit repetitive from my earlier comments, I believe there are a few notable caveats and assumptions made in my answer. Wherever possible, I’ll try to highlight the assumptions made, and how they impact my opinions. That said, any clarification you can provide regarding what exactly your project’s scope and goals are, the more I can refine some of the literature suggestions and my takes (for what that’s worth- admittedly, not a whole lot haha).

The first major point to make is that there really is a lot of variation in terminology used inside the field of decision theory. That’s partially driven by the different underlying axioms different models of utility are based on. Using a relatively well-known example, Savage’s axioms don’t assume objective probabilities need to exist for most events. This contrasts with Expected Utility theory, for example, which does directly require some notion of objective probability (whether it’s known or not) of an event’s likelihood of occurring. Therefore, if your model/paper is assuming Savage’s axioms, definitions of uncertainty and risk won’t necessarily be tied to the nuances of how well known a probability is (it allows a bit more flexibility at the real ground level). At the other extreme, Expected Utility Theory, because of the strict reliance on probabilities from the Independence axiom, assumes there’s no real difference between a known and unknown probability- through reduction, any multi-stage uncertainty collapses to a single dimension of uncertainty.

I don’t mean to get too far removed from the subject at hand, but I feel it’s worth keeping in mind the high level of context dependence the definitions of these terms often have- “risk” can mean something to Savage, but something slightly different to von Neumann, and something else to Kahneman and Tversky.

The next related caveat on this terminology is in regards to the fluidity of definitions over time. Even beyond the different implications the terms have for different utility structures, different authors have used these terms in a variety of related, but subtly different ways. I think your definitions fall reasonably well inside the umbrella definitions for these terms. As long as you explicitly define how you’re going to use words like uncertainty, risk, and ambiguity, I think you should be fine.

**EDIT: TO CLARIFY A BIT MORE ON THIS POINT: Historically, several works have directly conflated ambiguity and uncertainty, defining them as the same concept, contrasted with risk. While this has fallen a bit out of favor it seems (using the ambiguity/risk phrasing more commonly), it would still be a historically correct definition. Hence, your definitional example you provide in the question is correct, but perhaps a bit dated. That said, here's an example from 2012 that does include the duality explicitly in their definition:

In real life, of course, most important decisions present a mixture of risk and uncertainty. Since Keynes (1921), Knight (1921) and Ellsberg (1961) distinguished between risk (known probabilities) and uncertainty/ambiguity (unknown probabilities), there have been many studies of the differences between risk and ambiguity attitudes.

Though again, this paper is a bit more outside the central decision theory core (applying the field to healthcare and medicine). **

The last major source of terminology variation I’ve noticed is regarding what the format of the project is. For example, elite econ journal articles published recently (a few examples provided below) tend to hew more closely with the emerging consensus definition in recent decision theory work in risk and ambiguity attitudes, and the theoretical implications. On the other hand, if your project is for lecture notes for a class, or a book chapter, then you get much more leeway in how you delineate between these similar concepts- again as long as you’re clear at the beginning. If you’re writing for a journal article in a related field, like finance or management science, then there very well be conventions in those journals that don’t conform to my experience. I speak from a pretty pure economic theory background, and while I haven’t read everything out there on the subject clearly, I’ve spent a fair amount of time working with a decent subset.

So What are the proper definitions in my experience? Traditionally, you can consider one way of partitioning the set of all decision models into: decisions with uncertainty as problems where not all outcomes are necessarily known, and decisions with no uncertainty where all outcomes are deterministically known (and there are no “economic lotteries” in any stage). While there are of course many additional ways of partitioning these two subsets of problems, you can consider problems with uncertainty as being composed of:

i)problems where probabilities of each outcome is known to the agent as models of risk, and

ii)problems where some probabilities are not point identified for the agent, which makes up models with ambiguity.

Again, of course, these two subsets don’t always make intuitive sense- if your agents all follow expected utility theory, then there’s no difference between (i) and (ii) from a utility perspective. Any uncertainty over the probabilities (i.e. ambiguity) the agent faces collapses into just a compound lottery (i.e. a model of risk). This, for example, is the issue Ellsburg Paradox “exploits” to show an intuitive failure of expected utility theory.

The “modern” definitions draw from Ellsburg’s 1961 landmark paper (starting on page 657), though it should be noted that here the notions of “ambiguity” and “risk” aren’t particularly clearly delineated, they’re defined as I described above. Other utility models started considering “ambiguity” as it’s own, independent feature of preferences largely starting in the late 1980’s. The definition of “ambiguity” becomes a bit clearer in Segal 1987, starting on page 176, then continuing to 177

Ambiguous probabilities (i.e., situations where decision makers do not know the exact values of the probabilities) has some clear economic relevance… Other situations where ambiguity of the probabilities may play a significant role occur in search problems or in optimal investment problems. In all these cases, decision makers have some information about the objective probabilities, but they do not know their exact values. This paper suggests that the ambiguous lottery (x, S; 0, S) (ambiguous in the sense that the decision maker does not know the probability of S) should be considered a two-stage lottery, where the first, imaginary, stage is over the possible values of the probability of S.

(And is then defined in far more precise detail starting on page 183.)

Another prominent early example of adding this sensitivity to ambiguity conceptually into the modeling framework can be found in Segal 1988 , which implicitly uses the definitions for “ambiguity” and “risk” as I defined above.

More recent papers traditionally shorten their definitions to a common theme. Using a relatively random smattering of papers I’ve seen recently, this 2011 working paper defines the terms:

developments in the theory of decision making under ambiguity (i.e., subjective uncertainty about probabilities) recognize that ambiguity is not always treated the same as a known risk

This 2014 Handbook chapter similarly defines decision making under risk and ambiguity:

In many decisions under uncertainty, the decision maker has only vague information about the probabilities of potential outcomes of her actions. Following Ellsberg (1961), such situations with unknown or uncertain probabilities are often called ambiguous, to distinguish them from situations with objectively known probabilities, which are typically called risky.

Recent experimental papers, attempting to measure attitudes (usually in different specific settings) toward risk and ambiguity, have also followed in this general definition trend. For example, this work in Econometrica in 2017 measuring the effects of different types of ambiguity on observed decision making highlights the difference between “compound risk” (just a set of two lotteries with known probabilities of winning in a row) and “ambiguity” (where one stage led to uncertainty over the true probability of winning in that stage, which varies exogenously and unknown to the individual). For example, an example of a lottery with compound risk is:

Stage 1: flip a fair coin. If it’s heads, then go to Stage 2a. If it’s tails, go to step 2b

Stage 2a. Draw a card from a perfectly shuffled standard deck of 32 cards. If the card is not a spade, then win \$100. Else, win \$0

Stage 2b. Draw a card from a perfectly shuffled standard 52 card deck. If the card is a spade, then win \$100. Else, win \$0.

As you can see, while the probabilities might be dependent on previous draws, the objective likelihood is known at all stages. An example of an “ambiguous lottery,” then could be:

Stage 0: Draw a number $n$ randomly between $0$ and $50$, but do not show the individual playing the lottery. Then define the number of red balls in an urn as $25+n$, and the number of yellow balls in the urn as $75-n$.

Stage 1: Draw a ball randomly from the urn and flip a fair coin. If the result is either $(red, heads)$ or $(yellow, tails)$, then win \$100. Else, win \$0.

Note that, of course, in both cases the overall probability of winning is 50/50 in either lottery. However, sometimes we find that individuals will prefer one lottery over the other, suggesting preferences are partially formed along some other dimension.

Sorry this has dragged on. I’ll get back to summarize and add a few more references in a bit, but hopefully this can start the conversation going a bit. If you have any details about the project you have in mind, then I can try and tailor what I add!

Answer 3

I finally found a reference that defines the terms risk and uncertainty the way I do. Sven Ove Hansson "Decision Theory: A Brief Introduction" (1994) writes on p. 27-28:

In one of the most influential textbooks in decision theory, the terms are defined as follows:

"We shall say that we are in the realm of decision making under:
(a) Certainty if each action is known to lead invariably to a specific outcome (the words prospect, stimulus, alternative, etc., are also used).
(b) Risk if each action leads to one of a set of possible specific outcomes, each outcome occurring with a known probability. The probabilities are assumed to be known to the decision maker. For example, an action might lead to this risky outcome: a reward of \$10 if a 'fair' coin comes up heads, and a loss of \$5 if it comes up tails. Of course, certainty is a degenerate case of risk where the probabilities are 0 and 1.
(c) Uncertainty if either action or both has as its consequence a set of possible specific outcomes, but where the probabilities of these outcomes are completely unknown or are not even meaningful."
(Luce and Raiffa 1957, p. 13)

These three alternatives are not exhaustive. Many – perhaps most – decision problems fall between the categories of risk and uncertainty, as defined by Luce and Raiffa. Take, for instance, my decision this morning not to bring an umbrella. I did not know the probability of rain, so it was not a decision under risk. On the other hand, the probability of rain was not completely unknown to me. I knew, for instance, that the probability was more than 5 per cent and less than 99 per cent. It is common to use the term "uncertainty" to cover, as well, such situations with partial knowledge of the probabilities. This practice will be followed here. The more strict uncertainty referred to by Luce and Raiffa will, as is also common, be called "ignorance". (Cf. Alexander 1975, p. 365) We then have the following scale of knowledge situations in decision problems:

  certainty      deterministic knowledge 
  risk           complete probabilistic knowledge 
**uncertainty**  partial probabilistic knowledge
  ignorance      no probabilistic knowledge

It us common to divide decisions into these categories, decisions "under risk", "under uncertainty", etc. These categories will be used in the following chapters.

(Emphases and asterisks around uncertainty are mine.)

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Update: From personal communication with the author:

The two definitions [i.e. those of risk and uncertainty above] that you quote <...> are intended to describe the established usage. <...> I am not sure it would be meaningful to propose new terms for these concepts; these terms are well established.

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References

• Alexander, E. R. (1975). The limits of uncertainty: A note. Theory and Decision, 6(3), 363-370.
• Hansson, S.O. (1994). Decision Theory: A Brief Introduction.
• Luce, R.D., and Raiffa, H. (1957). Games and Decisions: Introduction and Critical Survey.

Answer 4

At the risk of spamming this question- this answer focuses more closely on the papers I've come across for models of uncertainty, risk, and ambiguity in the broader finance landscape. Note- this isn't my primary area of study, so this almost certainly isn't a complete, or even necessarily representative sample of the work there. Similarly, these papers are more likely to at least be somewhat tied to the economics literature, but then again, so is the broader field of decision theory.

EXPERIMENTAL PAPERS

Here, we propose a behavioral framework, ambiguity aversion, to help better understand the cause of equity market home bias. Simply put, we argue that ambiguity aversion inhibits people from investing in unfamiliar companies. Unlike previous studies, we use an experimental design with real world assets and test for ambiguity aversion instead of using fictitious assets or simply showing home bias without an explanation... Decision theorists have defined and modeled ambiguity in several ways. The most intuitive way of defining ambiguity is that the individual is uncertain about the distribution of the risk (Knight 1921). More uncertain the individual is about the distribution implies a higher level of ambiguity. [Emphasis mine]

From Ambiguity Aversion in Asset Market: Experimental Study of Home Bias (2009) (Using ambiguity aversion to explain the home-bias puzzle in finance)

Ambiguity preferences are important to explain human decision-making in many areas in economics and finance... In many circumstances of everyday life, people take decisions in uncertain environments. In most situations, the probabilities of the possible outcomes are only vaguely known to decision makers, if at all. Since the seminal works of Knight (1921) and Ellsberg (1961), the absence of precise information on probabilities is referred to as ambiguity, and has been recognized as a form of uncertainty distinct from the standard notion of risk. Preferences towards ambiguity – and ambiguity aversion in particular – have shown to be an important determinant of individual decision-making.1 Incorporating ambiguity preferences in economic models helps to explain a variety of phenomena in economics and finance that cannot be attributed to risk aversion alone...

From Measuring Ambiguity Preferences: A New Ambiguity Preference Module, 2016

GENERAL EMPIRICAL PAPERS:

Households must consider both risk and ambiguity when making investment decisions. Risk refers to events for which the probabilities of the future outcomes are known; ambiguity refers to events for which the probabilities of the future outcomes are unknown. Ellsberg (1961) argues that most people are ambiguity-averse, that is, they prefer a lottery with known probabilities to a similar lottery with unknown probabilities, and numerous theoretical studies explore the implications of ambiguity for economic behavior. In particular, a large body of theory suggests that ambiguity aversion can explain several household portfolio choice puzzles.

From Ambiguity aversion and household portfolio choice puzzles: Empirical evidence (2016)

We introduce a simple, easy to implement instrument for jointly eliciting risk and ambiguity attitudes. Using this instrument, we structurally estimate a two-parameter model of preferences. Our findings indicate that ambiguity aversion is significantly overstated when risk neutrality is assumed. This highlights the interplay between risk and ambiguity attitudes as well as the importance of joint estimation... Individuals face uncertainty daily. People must evaluate the likelihood of uncertain future outcomes, such as whether a business venture will succeed or fail, the future performance of a stock, or whether their vacation will be ruined by rain. In cases where the outcomes are not accompanied by objective probabilities, the normative approach of Subjective Expected Utility (SEU) theory introduced by (Savage, 1954) has traditionally been used. In this framework, an individual behaves as though she holds a single (subjective) prior over all states of the world and maximizes the expected value of utility given this prior. Ellsberg (1961), however, proposed that most individuals treat ambiguous uncertainty differently than objective risk.1 In particular, he argued that people exhibit a significant degree of ambiguity aversion, placing a premium on outcomes for which probabilities are known.

From Estimating Individual Ambiguity Aversion: A Simple Approach, 2015

We introduce a tractable method for measuring ambiguity attitudes, which requires only three observations and five minutes per subject, and we apply this method in a large representative sample. In addition to ambiguity aversion, we confirm a-insensitivity, a new ambiguity component recently found in laboratory studies. A-insensitivity means that people insufficiently discriminate between different levels of likelihood, often treating all likelihoods as fifty-fifty, which results in the overweighting of extreme events. Our ambiguity measurements can predict real economic decisions of the subjects; specifically a-insensitivity has a negative relation with stock market participation and private business ownership. Surprisingly, ambiguity aversion is not significantly related to stock market participation, except under high ambiguity perception

From Ambiguity Attitudes in a Large Representative Sample, 2015

We match administrative panel data on portfolio choices with survey data on preferences over ambiguity. We show that ambiguity averse investors bear more risk, due to a lack of diversiÖcation. In particular, they exhibit a form of home bias that leads to higher exposure to the domestic relative to the international stock market. While more sensitive to market factors, their returns are on average higher, suggesting that ambiguity averse investors need not be driven out of the market for risky assets... Ambiguity has been widely studied both theoretically and experimentally in the past decades...

From Ambiguity Preferences and Portfolio Choices: Evidence from the Field (2017)

THEORY PAPERS:

We present a new approach that enables investors to seek a reasonably robust policy for portfolio selection in the presence of rare but high-impact realization of moment uncertainty. In practice, portfolio managers face difficulty in seeking a balance between relying on their knowledge of a reference financial model and taking into account possible ambiguity of the model. Based on the concept of Distributionally Robust Optimization (DRO), we introduce a new penalty framework that provides investors flexibility to define prior reference models using moment information and accounts for model ambiguity in terms of “extreme” moment uncertainty.... Thus, the need arises to take into account additional levels of uncertainty: model uncertainty, also known as model “ambiguity”. Ellsberg [14] has also found that decision makers in fact hold aversion attitudes toward the ambiguity of models. As a classical example, even with lower expected return, investors have higher preference for investments that are geographically closer due to their better understanding of the return distribution.

From Portfolio selection under model uncertainty: a penalized moment-based optimization approach (2013)

In this paper, we consider the problem of finding optimal portfolios in cases when the underlying probability model is not perfectly known. For the sake of robustness, a maximin approach is applied which uses a ”confidence set” for the probability distribution. The approach shows the tradeoff between return, risk and robustness in view of the model ambiguity... In his 1921 book Knight [1921], the American economist Frank Knight made a famous distinction between ”risk” and ”uncertainty”. In Knight’s view, ”risk” refers to situations where the decision-maker can assign mathematical probabilities to the randomness, which he is faced with. In contrast, Knight’s ”uncertainty” refers to situations when this randomness cannot be expressed in terms of specific mathematical probabilities. Since the days of Knight, the terms have changed. As introduced by Ellsberg [1961], we refer today to the ambiguity problem if the probability model is unknown and to the uncertainty problem, if the model is known, but the realizations of the random variables are unknown. [Bolding mine, perhaps the most direct to your question]

From Ambiguity in portfolio selection

The most familiar model of choice under uncertainty follows Savage (1954) in positing that agents maximize expected utility according to subjective priors. However, Knight (1939), Ellsberg (1961) and others argue that agents distinguish between risk (known probabilities) and ambiguity (unknown probabilities), and may display aversion to ambiguity, just as they display aversion to risk.1 The financial literature, while admitting the possibility that some individuals might be averse to ambiguity, has largely ignored the implications for financial markets.

From Ambiguity in Asset Markets: Theory and Experiment

The Ellsberg paradox suggests that people behave differently in risky situations — when they are given objective probabilities — than in ambiguous situations when they are not told the odds (as is typical in financial markets). Such behavior is inconsistent with subjective expected utility theory (SEU), the standard model of choice under uncertainty in financial economics. This article reviews models of ambiguity aversion. It shows that such models — in particular, the multiple-priors model of Gilboa and Schmeidler — have implications for portfolio choice and asset pricing that are very different from those of SEU and that help to explain otherwise puzzling features of the data.

From AMBIGUITY AND ASSET MARKETS, 2010

In contrast to the Bayesian approach to estimation error, where there is typically a single prior and the investor is neutral to ambiguity, we consider the case where the investor has multiple priors and is averse to ambiguity... We illustrate how to use the multi-prior model with ambiguity aversion by considering the portfolio problem of a fund manager allocating wealth across eight international equity indices; our empirical analysis suggests that portfolios that incorporate aversion to parameter and model uncertainty tend to over-weight the risk-free asset, are more stable over time, and deliver a higher out-of sample Sharpe ratio than the portfolios from both classical and Bayesian models... The Bayesian decision-maker, however, is assumed to have only a single prior or, equivalently, to be neutral to uncertainty in the sense of Knight (1921). Given the difficulty in estimating moments of asset returns, the sensitivity of portfolio weights to the choice of a particular prior, and the substantial evidence from experiments that agents are not neutral to ambiguity (Ellsberg (1961)), it is important to consider investors with multiple-priors who are averse to this ambiguity, and hence, desire robust portfolio rules that work well for a set of possible models.

From Portfolio Selection with Parameter and Model Uncertainty: A Multi-Prior Approach, 2007

CONCLUSION

Sorry there's just a lot of text there- I tried to pull out early sections that might give you a flavor of their definitions for the different terms, and how they were applying them. If there are any more big ones, I'll make sure to update it! Hope this helps aid in your project!!

Answer 5

It is incorrect to say that uncertainty is an extension of risk, because (among other possibilities) if there is uncertainty, there remains the case of risk being nil but you just don't know it.

"Risk" involves a known payoff matrix with specified probabilities for specified outcomes. "Uncertainty" can involve a payoff matrix that is incompletely defined or undefined, with respect to probabilities of outcomes. In the first case, you can precisely calculate an expected value, in the second case you cannot.

Running through some of the simpler exercises from one or more textbooks (e.g., recent editions on the shelf at the library) could help demonstrate this.