What is Risk

How the New Yorker's Atul Gawande Caused the debacle at UVA

"What's the difference between a risk and a gamble? You can rescue yourself from a risk. If a gamble goes wrong, the consequences are irrevocable. I first learned this distinction from a battalion commander who had served several tours in Afghanistan. I interviewed him for a series of articles I wrote about risk."

That quote is from John Dickerson of Slate discussing the firing of UVA President Theresa Sullivan.  Apparently the plot to unseat Sullivan was hatched by UVA rector  Helen Dragas.  While trying to convince other decision makers to back her coup attempt she provided the transcript of a commencement address by Atul Gawande.  In his address to Williams College graduates Gawande had discussed risk, failure, and recovering from failure.  Dickerson suggests that perhaps Dragas had missed the point of the Gawande's speech - the importance of being able too rescue oneself when a risk goes wrong.  Dragas's plan did go wrong (as protests broke out until Sullivan was reinstated) and Dragas and her co-conspirators had not planned for that eventuality.  While discussing Dragas et als mistake Dickerson makes the above comparison between risk and gamble.

I am going to show my ignorance here - I am a risk manager by profession and I have never heard this definition of risk versus gamble before and frankly it sounds strange to me.  However when I started thinking about why it was wrong I realized that "risk" is a hard word to define - possibly because the word is used in multiple ways.

Let's see what Websters has to say

"risk noun \'risk\

1 possibility of loss or injury: PERIL

2 someone or something that creates or suggests a hazard

3a: the chance of loss or the perils to the subject matter of an insurance contract; also: the degree of probability of such a loss

3b: a person or thing that is a specified hazard to an insurer

3c: an insurance hazard from a specified cause or source <war risk>

4: the chance that an investment (as a stock or commodity) will lose value"

 I don't think that any of the other definitions really express the word risk in a way that I know it.

Let's see what Wikipedia has to say

"Risk is the potential that a chosen action or activity (including the choice of inaction) will lead to a loss (an undesirable outcome). The notion implies that a choice having an influence on the outcome exists (or existed). Potential losses themselves may also be called "risks". Almost any human endeavor carries some risk, but some are much more risky than others."  

Wikipedia also provides a number of alternative definitions of "risk" prefaced by  "The many inconsistent and ambiguous meanings attached to "risk" lead to widespread confusion and also mean that very different approaches to risk management are taken in different fields."  Ok so that may explain why I have trouble defining "risk".  It appears that there are multiple manners in which the word "risk" is used. 

First "risk" can be used as a synonym for probability or hazard.  As in "what is the risk of getting hit by a truck?"   In this case we assume that the implication of the event occurring is very negative and the word "risk" just denotes the probability of the event occurring.  Furthermore when used in this context there are usually phrased as there being two distinct possibilities - getting hit by a truck or not getting hit by the truck.

Second "risk" can refer to a listing of possible bad states without assigning probabilities to them.  An example would be "What is the risk if you were to contract the flu?"  In this case the reference is not to the probability of the event occurring but rather to what the possible outcomes are.   If you are healthy then the "risk" is that you feel crappy, or be in bed for a week, whereas if you have a compromised immune system the "risk" of catching a flu is that it could be fatal. I guess one could attach probabilities to each possible outcome conditional on the initial health status of the person in question but that would be an atypical response to the question.

But how do the above uses of the word "risk" apply to this statement "what is the risk of holding one share of IBM stock for a week".  Well obviously the "risk" is that the price of the stock could drop but that is pretty uninformative.  Someone who asks that question wants to know how much the price of IBM could fall and what probability you would assign to each of those falls.  Just saying how much the price of IBM could fall is nearly tautological.  It "could" fall to zero but that not likely.  So this use of the word "risk" asks for both the possible outcomes as well as the probability of each occurring.  

The case of getting hit by a truck is really just a special case of this third meaning of "risk".  And it is one where the "risk" is easy to understand since there are only two outcomes.  But once we start assigning probabilities to multiple outcomes then "risk" becomes difficult to define.  Is "risk" the probability of the worst possible case? or the probability of being significantly worse off? or the probability of being any any worse off?  should every outcome get the same weighting?  We could present the "risk" as a distribution over possible outcomes - for example the left hand tail of a t-distribution with mean of 0.5% and standard deviation of 5% and 3 degrees of freedom.  That fully enumerates the probabilities of different negative outcomes but does it tell us what the "risk" is?  That tells us what the probabilities are.

I think of risk as having three distinct parts (1) a set of possible outcomes (2) a probability distribution over those outcomes (3) and a valuation or utility measure which assigns a value to each possible outcome.  Assuming the outcomes are distinct we can then construct a single measure of the "risk" of any situation by summing over the probability weighted valuations of the outcomes - similar to expected utility.

For example one possible valuation could be assign 0 to any outcome except the worst and assign 1 to the worst possible outcome.  Weighting each valuation by its respective probability and summing over our measure of risk would then be the probability of the worst case outcome.  There are literally an infinite number of possible weighting schemes one could come up with for different outcomes.

Now back to Dickerson's statement.  "What's the difference between a risk and a gamble? You can rescue yourself from a risk. If a gamble goes wrong, the consequences are irrevocable."  I don't see anything in the definition of risk which alludes to the ability of one to rescue oneself or not.  The point of his article may be correct but that is not the definition of risk.