### Why Inequality Can't Matter

A famous experiment of Hsee's asks people how much they would pay for two different sets of dishware:

Set ASet B
Dinner plates: 8, all in good condition 8, all in good condition
Soup/salad bowls: 8, all in good condition 8, all in good condition
Dessert plates: 8, all in good condition 8, all in good condition
Cups: 8, 2 of them are broken
Saucers: 8, 7 of them are broken

Note that Set A is a Pareto improvement over Set B - it has everything in Set B and some additional items as well. Therefore, people should be willing to pay at least as much for A as they are for B.

Nonetheless, people are willing to pay almost 50% more for B than for A. The explanation for this "less is better" result is that the "hard" question of finding the absolute value of the set is subconsciously replaced with the "easier" question of finding the relative value of each item in the set.

A similar phenomenon occurs in population ethics. Consider two populations:

Population APopulation B
Investment Bankers:100, very well off100, very well off
Secretaries:100, moderately well off

My guess is that Population A would raise more ire than Population B, even though A is a Pareto improvement over B. Suppose we require our population ethics to follow what is sometimes called "Dominance" or "Pareto Dominance":

If Population A and Population B differ by only one person, and that person is better off in A than in B, then A is better than B.

Note that this is a pretty weak condition: in real life, there will almost always be winners and losers to any policy change, so it's rare to be able to decide things based solely on the Pareto Dominance principle.

Despite being a weak condition, it rules out population ethics that value equality, diversity etc.

Consider an extreme example: we only care about inequality (as measured by say the Gini index). In the example above, Population A had more inequality (higher Gini index) and so it would be worse. But A was a Pareto improvement over B, so a contradiction arises; hence, the Gini index can't be the way we compare populations.

A more general version of this is true:

Suppose $(G,+)$ is a population ethics that obeys the group axioms and Pareto Dominance. Let's say there is also some function $f$ whereby if $pop_a$ and $pop_b$ differ by only one person $\Delta$ then $pop_a > pop_b$ if and only if $\Delta > f(pop_b)$, i.e. $f$ defines the minimum welfare needed for a person to "improve" the total value of the population.

Then $f$ is constant. Specifically, $f(x)=0$ for all $x$, where 0 is the identity of $G$.

In some ways, this is not a very surprising result - it just says that whether your life is good is independent of whether my life is good. But it seems to contradict a lot of things we believe as a society.

Proof: Arbitrarily choose some population $pop$ and consider $pop+f(pop)$, i.e. adding a person right on the "margin". There are two possibilities: $pop+f(pop) < pop$ (adding this person is a bad idea), or $pop+f(pop)=pop$ (adding the person doesn't matter).

Suppose that $pop+f(pop) < pop$. We know that there is some element $0$ such that $pop+0=pop$. If $0 < f(pop)$ then $pop+f(pop)$ is a Pareto improvement over $pop+0$, so $pop+0 < pop+f(pop) < pop$, which is a contradiction because $pop+0 = pop$. If $0 > f(pop)$ then by the definition of $f$, $pop+0 > pop$, another contradiction. Therefore $0=f(pop)$, proving the theorem in the first case.

Alternatively, suppose that $pop+f(pop)=pop$. This means that $f(pop)$ is an identity of $G$, and since identities in a group are unique, $f(pop)$ must be $0$.

Since $pop$ was chosen arbitrarily, we have shown this is true for all populations. QED.