Maths for Economics 6: Probability and Economics Topic 1 - Uncertainty

Question 1

Describe the advance of probability theory over the years

Answer 1

c. 1650 - First systematic discussion, by Fermat, Pascal, and
Huygens. Applied to: gambling, theology.
1650-1850 - Extensive mathematical development
c. 1850 - Applied to physics, statistics, social sciences
c. 1900 - ‘Random’ - unpredictable - nature of stock market
fluctuations realized.
c. 1945 - Uncertainty comes to be understood as a central
feature of economic life.

Question 2

List different areas of economics & finance in which risk & uncertainty is relevant

Answer 2

 Financial markets
 Insurance markets, gambling industry (could not exist if there
were no uncertainty)
 Investment banking
 Portfolio selection
 Forecasting
 Choice under uncertainty (Decision Theory)
 Statistics, and its applications, such as:
 Econometrics

Question 3

Describe & explain ‘Generality and Security vs. Richness and Power’ and its link to probability & uncertainty

Answer 3

A common situation in mathematics and its applications:
Should we prefer a general theory, or a special theory?
General theories
 Make relatively few assumptions (and are therefore
relatively ‘secure’)
 apply to a relatively wide range of cases
 draw relatively cautious (even unexciting) conclusions.
Special theories
 are special cases of general theories. They make additional
assumptions and are therefore ‘riskier’.
 apply to a relatively narrow range of cases
 but draw relatively rich and powerful conclusions
In our case, the relatively general theory is the Theory of Uncertainty.
The relatively special theory is the Theory of Probability.
By ‘theory’ I just mean ‘subject’, not that either subject can be proved
or disproved.

Question 4

Describe the relationship between the ‘Theory of Probability’ and ‘Theory of Uncertainty’

Answer 4

The Theory of Probability:
 Is a special case of the Theory of Uncertainty
 It makes relatively demanding assumptions. (The existence
of probability numbers.)
 applies to a relatively narrow (but still huge!) range of cases
 but is far richer and more powerful in conclusions than the
general theory of uncertainty.

Question 5

Describe & explain Frank Knight’s distinction between ‘uncertainty’ and ‘probability’

Also, when did he do this

Answer 5

Frank Knight’s distinction (1921) between ‘uncertainty’ and
‘probability’ (sometimes called ‘risk’, in Knight’s honour).
If we can attach probability numbers to the possible outcomes, for
instance P(Heads) = 0.5, we have a situation of probability (or risk).
If we are not willing to do this, there is a situation of uncertainty, also
known as radical uncertainty (or Knightean uncertainty).
For instance, I might be unwilling to assign a meaningful probability
number to the event that humans will walk on Mars before 2050. For
me, this would then be a radically uncertain event.
We shall see that there are far more uncertain events than that

Question 6

Describe & explain a common error students make with radical uncertainty

Answer 6

Student Y interprets ‘radically uncertain’ as meaning ‘no knowledge at
all’, which may be correct …
… but then goes on to interpret this as ‘all outcomes have equal
probabilities’ …
… which (for instance in the case of a six-sided die) allows us to
attach probability numbers (1/6) to the outcomes …
… so violating the definition of ‘radically uncertain’.
The key point about radical uncertainty is that no numbers at all will
do.
Those who claim that events are radically uncertain can sometimes be
challenged by offering them bets about whether the event will occur,
or not – an important idea in (Bayesian) decision theory.

Question 7

Describe & explain Rumsfeld’s ‘unknown unknowns’

Answer 7

In what we called uncertainty so far, we were at least able to make a
complete list of things that might happen.
But some future events - for instance, revolutionary scientific
breakthroughs and revolutionary inventions, such as the invention of
the Internet, cannot even be thought of and named, by most of us,
before they take place. The first farmers, around 10000BCE, did not
speculate on when the wheel might be invented.
Rumsefld’s ‘Unknown unknowns’ (very radical uncertainty) are events, thoughts, or discoveries of which,
before they took place, no-one had any inkling. If they are very
dramatic and influential, they are called (by Nassim Nicholas Taleb)
‘black swans’. For instance, the discovery of IT and the Internet, and
perhaps now ChatGPT and similar resources.