## Mathematical models and the box we should think outside of

The phrase
thinking outside the box is of course a cliche.
But the notion that the implicit assumptions of a model provide invisible boundaries for the validity of the model's predictions is a useful notion, and the "box" metaphor is as good as any.
Let me give two familiar examples and then one less familiar one.
#### A freshman statistics example.

A typical use of the Binomial distribution is to predict that,
amongst 4-child families, about 6/16 should have 2 boys and 2 girls.
In an actual freshman course I take 5 minutes to ask the class how this prediction might be inaccurate;
They give some predictable suggestions; identical twins, sex ratio not excatly 50-50,
"boys run in certain families", etc.
But a more interesting possibility is to imagine
a hypothetical society in which parents continue
having children until having at least one boy and one girl, then stop.
In such a society there would be **no**
4-child families with 2 boys and 2 girls.
The point, of course, is that to use the Binomial distribution requires
"n fixed in advance", but this requirement is often not emphasized in textbooks.
it's an invisible box.

#### Domination in decisions under uncertainty

Let's think abstractly about a "decision under uncertainty" setting.
You can choose between action A and action B.
The outcome (quantitative benefit to you) depends on your choice of action and
on some external variable you can't control.
Say *A dominates B* if,
regardless of the external variable, action A provides more benefit
to you than action B.

Common sense says

- (*)
if action A dominates action B then you should always do action A instead of B.

And indeed many philosophical-style discussions of axiomatic foundations of
Bayesian statistics, decision theory etc start out by declaring some similar
principle as a self-evident axiom.
But where's the box that delimits applicability of rule (*)?
Well, one type of box is well known from game theory.
The whole point of
Prisoner's dilemma type games
is that, if both players follow rule (*), then they both get less benefit from action A
than they would have gotten from action B.
But can you think of other boxes?

#### A venture capital maxim.

Well, another box is suggested by the maxim
*better to fail quickly than to fail slowly*
which I'm attributing to venture capitalists but also applies in everyday life.
Better to have a marriage fail after 7 months rather than 7 years;
better to decide you don't really want to do a Ph.D. after 7 months
rather than 7 years.
For the entrepreneur, there is some unknown function f(t)
giving profit at future time t.
Rule (*) says that if f_A(t) > f_B(t) regardless of external variables, then
do A.
What's the box?
If you were forced to remain in business up to a fixed time T
then (*) is reasonable.
But you're not.
That's why actions that turn slow-failure outcomes into fast-failure outcomes
may be better; you can stop what you're doing and try something completely different,
outside the context of the model.