# What is the difference between risk, uncertainty and ambiguity

I am trying to pin down the difference between risk, uncertainty and ambiguity.

As I understand, when behavioral economists talk about choice under uncertainty, they mean choice when agents face risk (known probability distribution over a range of outcomes) versus ambiguity (unknown probability distribution). So uncertainty is a blanket concept that can be broken down into risk and ambiguity. See for example Dannenberg et al (2014).

I also understand there is a debate on the meaning of these terms going back to Knight (1921) and Ellsberg (1961). Are there any competing definitions to the one above?

• Do you want to get literature argument on each of the terms you mentioned ? Or you are confuse about redefinition of the terms by individual economist? – mootmoot Dec 22 '16 at 16:21
• @mootmoot Regarding your first question, I am interested in hearing other formal definitions of the relationships between uncertainty, risk and ambiguity. The example provided implies uncertain outcomes can be risky (known probabilities) or ambiguous (unknown probabilities). Regarding your second question, sorry, it is incomprehensible. – invictus Dec 22 '16 at 16:28

Here is a decision-theoretic formalization of your definitions.

The usual framework to talk about objective risk is the situation where a decision-maker expresses preferences over objective lotteries. Formally, if $X$ is a prize space, objective lotteries are defined as elements of the space $\Delta(X)$ of probability distributions (usually with finite support) over $X$. For instance, the decision-maker might be asked to form preferences between the lottery that offers her/him an apple with probability 0.3 and an orange with probability 0.7, and the lottery that offers her/him an apple with probability 0.5 and an orange with probability 0.5. The standard result in that area (von Neumann-Morgenstern theorem) delivers a representation that identifies the agent's attitude towards objective risk (her/his utility function), while the probabilities are given as a primitive of the model.

The usual framework to talk about ambiguity is the situation where a decision-maker expresses preferences over uncertain acts. Formally, if $X$ is a prize space and $S$ is a state space, acts are mappings $f:S \rightarrow X$ from $S$ to $X$. For instance, the decision-maker might be asked to form preferences between the act that offers her/him an apple if Novak Djokovic wins the 2017 Australian Open and an orange otherwise, and the lottery that offers her/him an apple if Andy Murray wins the 2017 Australian Open and an orange otherwise. The standard result in that area (von Neumann-Morgenstern theorem) delivers a representation that identifies both the agent's probabilistic beliefs regarding the states and her/his attitude towards risk (her/his utility function).

There is a third widely used concept, usually called Anscombe-Aumann acts or horse races, which associates both objective lotteries and uncertain acts. Formally, given a prize space $X$, an Anscombe-Aumann act is a mapping $f:S \rightarrow \Delta(X)$ that associates an objective lottery to any state in $S$.

Notice that the definitions of objective risk and ambiguity are to some extent subjective. The fact that risk is called "objective" relies very much on the assumption that the decision-maker agrees with the underlying probability model. For instance, if you observe the outcome of a coin toss, you might believe that heads happen with an objective probability of 0.5. It is implicitly imposed in the theory that the decision-maker agrees with this statement.

Regarding ambiguity, you might yourself believe that the act "receive an apple if Novak Djokovic wins the 2017 Australian Open" is very ambiguous because you have no idea on how to compute a subjective probability for this event. That said, another decision-maker might very confidently believe that Djokovic has a 74% chance of winning the tournament, in which case she/he does not perceive this act as ambiguous at all. Ambiguity is a subjective notion, which is given by people's preferences and behavior and not by the choice situation itself.

• I wonder if you could take a look at this new question of mine. Thank you! – Richard Hardy Aug 10 '17 at 14:52

So far you have got no answer to your last question, about Knight and others' view on risk and uncertainty. In fact, there is quite a radical distinction between the view of uncertainty in Knight (and Keynes) and that presented in Oliv's answer.

In brief, according to Knight (1921) risk refers to situations where the classification of states, events or alternatives is objective and known, and their probabilities can be objectively determined. For example, in the context of a house insurance policy, events could be "house burns down" or "house doesn't burn down", which probabilities can more or less be asserted objectively, based on house/environment/individual characteristics.

Conversely, for Knight uncertainty arises from "the impossibility of exhaustive classification of states" (Langlois and Cosgel 1993, p.459). Thus, regardless of whether probabilities of events are objective or subjective, nature/economy can be so complex that all possible states are simply not known. As such, any categorisation of events used to predict the future is based on intution and judgment, thereby being subjective.

This view of uncertainty is shared by Keynes, and it is essential in his theorizing of the "animal spirits". Precisely because the future is unknown - not only in the probability of events but also on the range of events that could happen, expectations in investment decisions is not just a matter of mathematical calculation. As Keynes said, decisions to invest are taken “as a result of animal spirits – of a spontaneous urge to action rather than inaction, and not as the outcome of a weighted average of quantitative benefits multiplied by quantitative probabilities” (Keynes 2008 [1936]: 144, emphasis is mine). Interestingly, the approach highlighted in italics is what Oliv describes in his answer, and the foundation of much of neoclassical investment theory under uncertainty.

Among Post-Keynesian circles, this uncertainty is called "radical, fundamental, or ontological uncertainty", in contrast with the neoclassical view of uncertainty described in Oliv's answer, which is sometimes called "epistemological uncertainty". For example, see here.

Finally, regarding ambiguity, to my knowledge that concept is not used in the early literature on this matter.

Oliv i think that what you are referring to in this paragrapgh:

The usual framework to talk about ambiguity is the situation where a decision-maker expresses preferences over uncertain acts. Formally, if is a prize space and is a state space, acts are mappings from to . For instance, the decision-maker might be asked to form preferences between the act that offers her/him an apple if Novak Djokovic wins the 2017 Australian Open and an orange otherwise, and the lottery that offers her/him an apple if Andy Murray wins the 2017 Australian Open and an orange otherwise. The standard result in that area (von Neumann-Morgenstern theorem) delivers a representation that identifies both the agent's probabilistic beliefs regarding the states and her/his attitude towards risk (her/his utility function).

Is the pure subjective world of Savage, as presented in Savage "Foundations of Statistics".

My interpretation of the terms are:

Risk: decision making with given/objective probabilities. Notice that the two terms arent equal. Im not going to expand on this because it is not the topic under discussion, but given probabilities $\Rightarrow$ objective probabilities. The reference for this is Gilboa "Theory of Decision Under Uncertainty". This means that the primitives in the problem are both probabilities and preferences. The standard model is VnM's Expected Utility.

Uncertainty: decision making with subjective probabilities. This means that given an event, two people might have different probabilistic beliefs over it, where none can convince the other of the superiority of their own probabilistic assessment. In this case the primitive of the decision problems is only the preference relation (beliefs are derived from it). The standard model is Savages Subjective Expected Utility

Mixture of both: here is where Anscombe Aumann enters the scene. They axiomatize a preference functional where both subjective and objective probabilities are present. In their representation the decision maker takes a double expectation (over lotteries and over states of the world)

Ambiguity: now, ambiguous scenarios are the ones where the decision maker does not hace sufficient information to be completely sure that his (unique) belief is the correct one. Quoting Cerreia Vioglio et.al "Ambiguity and Robust Statiatics",

Ambiguity refers to the case in which a DM does not have sufficient information to quantify through a single probability distribution the stochastic nature of the problem he is facing"

Thus, a natural way to model ambiguity is through sets of priors, where de DM is not forced to say "the probability of event E is x%" (as in savage world), an cam say "the probability of event E is between $[ x\%, y\%]$

Notice that by definition, ambiguity scenarios need to have subjective frameworks thus the natural way to model in this case is by having preferences over Savage acts $f:S \rightarrow X$ or Anscombe Aumann acts $f:S \rightarrow \Delta (X)$

• Given the downvote, I offer a survey paper by Gilboa and Marinacci "Ambiguity and the Bayesian Paradigm", which is in line with my answer. tau.ac.il/~igilboa/pdf/… – thekiciminister Dec 23 '16 at 14:25
• Hope it helps to clarify the terms, and by the way, OP, this paper is an excellent first read as an introduction to ambiguity – thekiciminister Dec 23 '16 at 14:26
• I wonder if you could take a look at this new question of mine. Would the example therein be in line with Savage's view? Thank you! – Richard Hardy Aug 10 '17 at 14:28
• Risk - the probability of being exposed to a loss or damage
• Uncertainty - the state of being unsure of something
• Ambiguity - unclearness by virtue of having more than one meaning
• This is not very informative in the formal modelling context that the OP asks about. – Giskard Jan 19 '17 at 10:08

Ambiguity is lack of explicit context. A cook are are given flours, eggs, sugar, salt, spices. If you ask the cook to "make food out of it", this is ambiguity.

Uncertainty is random outcomes cause by ambiguous instruction/direction. E.g. Use above ambiguous instruction, the results can be a pasta, cakes, biscuit, charcoal, etc. The outcome is "uncertain".

Risk is all possible tangible accessible "bad" outcome/events by probabilities associate to specific activities, depends on context.
E.g. When you lies the flat ground, your chances of falling is low. But, that doesn't mean you can't fall : what if the earth below you break apart due to earthquake. (winks)

• I'm pretty sure the OP wasn't asking for an explanation in layman's terms. They were looking for formal definitions that are present in the research literature. – Theoretical Economist Dec 22 '16 at 15:48
• @TheoreticalEconomist : First line, the OP say : I am trying to pin down the difference between risk, uncertainty and ambiguity. . I choose not to complicate the matter. You are welcome to post your answer. – mootmoot Dec 22 '16 at 16:02
• Yes. I understand that. Can you outline the differences between the models of von Neumann/Morgenstern, Anscombe/Aumann, and Gilboa/Schmeidler in terms of how they treat the concepts of risk, uncertainty, and ambiguity? That answer will be a lot closer to what the OP is asking, rather than what you give here. – Theoretical Economist Dec 22 '16 at 16:07
• @TheoreticalEconomist : let me ask the OP he is looking for those. – mootmoot Dec 22 '16 at 16:19
• @mootmoot Thanks for your answer. Unfortunately it is does not sharpen the distinction between the terms in any meaningful way. – invictus Dec 22 '16 at 16:25