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

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.

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