References for particular definitions of risk and uncertainty 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?



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.
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