A good heuristic I keep handy to avoid looking like a fool is to be cautious when answering what seems like an obvious question when posed by someone clearly smarter than me. Here, we've got a few minor semantic issues and at least one problem of application. As I attempt to clarify, please bear in mind that I'm combining a brief survey of what others have done with what I think is useful. Out of an abundance of caution, I'll use a bit of technical jargon from statistics, but I'll do my best to illustrate what I mean with metaphors. If you're a casual reader, please don't let mentions of probability density functions throw you.
For the difference between risk and uncertainty, I turn to Frank Knight (who was the professor of James Buchanan, sort of making him my intellectual great-grandfather). In chapter VII of Risk, Uncertainty, and Profit (alas, I can find no electronic copy worth a pig's knuckle—even my paper copy is a photocopied, semi-legible disaster), Knight distinguishes risk arising from a defined, known probability distribution from uncertainty arising from an inscrutable, unknown probability distribution. Importantly, he takes a Cartesian, empirical approach to the question. Since he's an economist looking at real world phenomena—he mentions life and fire hazard insurance—he's leery of a priori probability reckoning. From p.225:
It must be emphasized that any high degree of confidence that the proportions found in the past will hold in the future is still based on an a priori judgment of indeterminateness. Two complications are to be kept separate: first, the impossibility of eliminating all factors not really indeterminate; and, second, the impossibility of enumerating the equally probable alternatives involved determining their mode of combination so as to evaluate the probability by a priori calculation.Knight is saying here that careful analysis must not ignore what Nassim Taleb calls "fat-tail" events, or "black swans", to be alert for the financial profession's maxim "past performance does not guarantee future returns." The rapid summary is found in the opening sentence to the next chapter (p.233): "[t]o preserve the distinction which has been drawn in the last chapter between the measurable uncertainty and an unmeasurable one we may use the term "risk" to designate the former and the term "uncertainty" for the latter." That's more than 140 characters, but it's about as pithy as one might hope. Risk is governed by a defined, known probability distribution. Uncertainty is not.
For ergodicity (and the lack thereof), I turn to Douglass North. In Understanding the Process of Economic Change, Chapter 2, Section II (p.19) opens with a dictionary definition of ergocity:
"involving or relating to the probability that any state will recur, especially having zero probability that any state will never recur." Therefore, "an ergodic stochastic process simply means that averages calculated from past observations cannot be persistently different from the time average of future outcomes." For Samuelson the ergodic hypothesis was essential for a scientific economics.Obviously, Knight disagreed with Samuelson. For Knight, the challenge faced by firms was to deal with uncertainty and the role of the economist was to catalog and observe rather than to predict and manage. For more on these (and more) intellectual traditions in economics, I strongly recommend Pete Boettke's Living Economics. At any rate, North continues:
To an economic historian surveying the ten millennia of human history from the onset of the Neolithic revolution, however, the ergodic hypothesis is a-historical. Further, the extraordinary changes in every facet of present-day society are evident all around us; and it is evident that we have been and are creating societies that are unique in comparison to anything in the past.Non-ergodicity concerns novelty. Adding North to the Schumpeter-Knight-Kirzner synthesis isn't a matter of managing shocks within industries, it's how institutions (and to sprinkle a dash of McCloskey on top, rhetoric and moral sentiment as well) respond to system-wide disruptions. The commercial viability of the steam engine sounded the death knell of the stagecoach industry. Ditto the reversion of the purse strings to parliament in the Orange Revolution, ditto the moldboard plough, cooking meat over a fire, or fore-and-aft rigging on sailboats. Some developments kill entire industries, entire ways of life. These are not problems of the firm, but of society.
So to answer Lynne's question: both Northian non-ergodicity and Knightian uncertainty deal with inscrutable outcomes that arise from processes that we can only roughly and imprecisely model using historical evidence. But the way I understand uncertainty from Knight is that outcomes are still basically in the same ballpark as before: we've still got the same basic rule of law, the same basic patterns of production and exchange, the same bourgeois rhetoric, and the same types of relationships. The way I understand non-ergodicity from North is that when something big hits, all bets are off. Whole social orders, entire nation-states, entrenched ways of life, broad political coalitions are threatened by non-ergodic developments. The two are related in type, but not in scale.
To close, picture three urns, labeled "risk", "uncertainty", and "non-ergodicity".
In the risk urn, there are 50 white balls and 50 black balls. You know a priori the odds of what you're going to draw.
In the uncertainty urn, there are 50 white balls and some other balls. You know a priori that you could draw a white ball, but you can't calculate the odds beyond what you've seen other people do in the past.
In the non-ergodicity urn, there are 50 white balls and some other things. They could be balls, but as far as you know, they could be snapping turtles or vinyl copies of Fleetwood Mac's classic album Rumours. You can't calculate the odds, and you might not even have a very good idea of what the downside risks could be.
To recap, this is simply my understanding of the differences. Much like the "political kayfabe" meme, distinguishing the three is a little personal pet project. As you might imagine, the boundaries between risk, uncertainty, and non-ergodicity are hazy. And the implications for ensuring an environment where peaceful, mutually-beneficial voluntary exchange aren't always perfectly clear. But I am of the opinion that distinguishing these three helps clarify how and why some societies seem to be robust against minor change and major upheaval while others collapse. I hope this helps.