# An Introduction to the Concepts of Risk, Uncertainty, and Ambiguity

Zedekiah G. Higgs

Risk, uncertainty, and ambiguity are closely related concepts in economics. Depending on who you ask, different definitions may be provided, but the following are common definitions for each of the three terms.

Both risk and uncertainty refer to situations where some future outcome has multiple possibilities. The distinction made between the two is that risk is used to describe situations where the possible future states occur with *known probabilities*, while uncertainty refers to situations where the probabilities associated with each possible state are *unknown* (and the set of all possible outcomes may itself not be known). For example, gambling on the outcome of a flip of a "fair" coin would be considered to entail *risk* because the probabilities of winning and losing are known, while gambling on a horse race would entail *uncertainty* because you presumably have no way of identifying the probabilities associated with each outcome.

In the case of uncertainty (such as gambling on a horse race), you may still be able to make some judgments, such as, "Horse A is *more likely* to win than Horse B," despite not being able to identify exact probabilities. That is, you're not necessarily completely in the dark regarding the probability of each outcome. In fact, depending on the assumptions you're willing to make, it is possible to use an individual's preferences over uncertain gambles to derive their *subjective probabilities* for each outcome (see L.J. Savage's *The Foundations of Statistics (1954)*). That is, theoretically you can present an individual with different possible gambles on the horse race and, using their preferred choice in each of the gambles, back out the probability they implicitly assign to each outcome. In the extreme scenario in which an individual is completely agnostic regarding the relative likelihood of all events, they will always prefer the bet that pays more (irrespective of which event the payout is contingent upon), and their preferences over gambles will imply a uniform distribution over all possible outcomes.

It can be argued that, in the real world, *all* probabilities are *subjective probabilities*. For instance, in the example of the "fair" coin, the probabilities are only "known" because the coin is *assumed* to be "fair," which is equivalent to assuming that the odds of flipping heads or tails is 50-50. In reality, no coin will actually provide *exactly* 50-50 odds (beyond unavoidable imperfections in the construction of the coin, the odds will also depend on the method used to flip it, etc.). Thus, the concept of "known probabilities" is really only a theoretical idealization. And as a result, the distinction made between *risk* and *uncertainty* is really only a matter of degree. That is, risk refers to situations where we are relatively *confident* about the probabilities associated with each possible outcome (such as the flip of a coin), and uncertainty refers to situations where we are relatively *unconfident* about the probabilities associated with each possible outcome (such as a horse race).

In this formulation of risk and uncertainty all gambles exist on a spectrum, with *certain probabilities* on one end (representing risk) and completely *uncertain probabilities* on the other end (representing extreme uncertainty). Where we are on this spectrum is sometimes referred to as the level of *ambiguity*. That is, ambiguity measures the amount of uncertainty with respect to the true probabilities associated with each outcome. For a gamble on the flip of a coin, while we may never know the *exact* odds, we can be pretty confident that the odds are close to 50-50. Thus, there is little ambiguity associated with a gamble on the flip of a coin. However, for a gamble on a horse race, while we may have a general idea about which horses are most likely to win, we will typically have little confidence that our estimated probabilities accurately reflect the true probabilities (assuming we are even able to derive estimated probabilities). Therefore, a gamble on a horse race will embody much more ambiguity.

The extent to which an individual seeks to avoid ambiguity is sometimes referred to as *ambiguity aversion*.