# Dietary Debates among the Fruit Gnomes

_Somewhat experimental._

Among the Fruit Gnomes, there were two groups that were involved in an intense and annoying debate: the No-Banana-Eaters against the All-Fruiters. It was unclear who had started this kind of argument. But now it usually went as follows.

The All-Fruiters pointed out that excluding bananas from your diet was completely unnatural and obviously unhealthy. Of course, they said, some Fruit Gnomes should reduce their banana consumption. But bananas still contained many important nutrients that were hard to replace. The proponents of the banana taboo, in turn, pointed out that usually the all-fruiters were less healthy than those who did not eat bananas. And most of the all-fruiters basically ate way too many bananas. And that was bad for the Fruit-Gnome metabolism.

One day, an annoyed Vegetable Dwarf listened to the fruit gnome debate and said: "Come on people, it is not possible that each of your groups has a healthier diet than the other." So what was the problem?

With some reflection, the parties of the debate realized that they should apply two rules to their debate:

1.  Do not compare an optimized all-fruit diet to a typical no-banana diet, except if the typical no-banana diet is an optimized no-banana diet.
2.  Do not compare an optimized no-banana diet to a typical all-fruits diet, except if banning bananas makes the typical all-fruiter choose the optimized no-banana diet.

The first rule follows from a simple insight; if you choose a diet such that it makes you as healthy as possible, and this optimal diet contains bananas, then taking the bananas away will make you less healthy (that is, the optimized value of an objective function must decrease if you add a binding constraint). But if you want to have a fair comparison, then you should compare the resulting diet to the diet of a no-banana-eater who has also optimized her diet to be as healthy as possible, except for the bananas. Maybe the lack of bananas is not that much of a problem in the end.

The second rule is the mirror-image of the first; if you have someone who has chosen his no-banana diet in a way that makes him as healthy as possible, then comparing to a typical all-fruiter is not meaningful. The exception is for simple cases where people who must not eat bananas will, by themselves, choose the most healthy no-banana diet. (There are some reasons why this could be the case: maybe this just reflects their preferences, or maybe banning bananas makes you talk to people who tell you how to choose a healthy diet.)

Applying these rules to their debate, the Fruit Gnomes saw that you could eat healthy without bananas, though maybe you might need some supplements. However, they also saw that an actually healthy all-fruits diet did not include many bananas, and that the typical All-Fruiter got healthier by reducing banana consumption, though she might also find it very costly (in terms of investing time to learn new recipes and changing old habits) to choose a sufficiently healthy banana-free diet.

When the dust from the old debates had settled, the Fruit Gnomes saw clearer that many participants in these debates had never chosen their diet primarily for health reasons. Instead, most members of the No-Banana group had wanted to avoid taking the bananas that otherwise would be eaten by the banana birds; these birds could not eat anything else. To the All-Fruiters, banana birds did not matter enough to change their ways and forgo the delicious taste of bananas. Of course, these motivations had been visible before, but with confusing debate rules, you sometimes do not see clearly. For some, the banana question was really about health, but hardly anybody of them had taken part in the debates. For the others, the healthy-diet topic became a proxy fight. So the Fruit Gnomes decided to augment the new debating rules:

3\. Do not present secondary reasons to choose a diet as though they were the primary ones.

And suddenly, it was possible to talk about banana birds (in fairy tales, this works). The debates had not become friendlier, but at least they were a bit less confused, and a bit more honest.