Looking at the chart on Andy Gelman’s post about health-care expenses and outcomes, I wonder if there’s any way to put all of those data points in an order. You want to say that country A is better than country B in its health-care outcomes and expenses, and you want to be able to do that for all countries.
There’s an obvious *partial* ordering for all those countries: A’s health care is better than B’s if A’s health-care outcomes are better than B’s and if A spends less on health care. That is, if A is to the left of and above B, then A is better than B. But we’re unlikely to be so lucky that countries can be put into a line that slopes uniformly down and to the right.
If there were some widely accepted way to balance expenses and outcomes, then we could achieve a total ordering here. Let’s say, for instance, that we defined the “goodness” of a health-care system as 1/3 times its per-capita price, plus 2/3 times its health outcome. Then our two-dimensional chart would collapse into a one-dimensional line, and all countries would naturally be totally ordered. But unless I’m missing something, there’s no objective criterion for combining these two quantities.
What I’m asking, I think, mathematically, is whether there’s any natural total order on ordered pairs. Probably not, right?
__P.S.__: I wonder whether the ratio of quality to price has any claim to objectivity. One would expect, though, that the marginal gain in quality for every marginal dollar spent would decrease with the quantity of dollars. (Diminishing returns.) So if we’re not careful with this ratio, it will tend to reward those countries that spend hardly any money and have mediocre health outcomes. So I wonder whether the ratio of quality to price, limited to the set of countries with quality above a minimum threshold, would be an interesting metric. This does, however, start to get us into “how much money is an additional year of life worth?” territory, which is ethically contentious.
This particular ratio, too, depends on some possibly special features of the response function (i.e., the response of quality to increased cost). In particular, the response function probably has a positive first derivative (every extra dollar buys you *some* increase in quality) and a negative second derivative (…but the amount of extra quality attained for every dollar is decreasing). This is somewhat specialized, but decreasing returns of this sort are fairly common.
__P.P.S.__: Even without this specialization, it seems fair to say that country A’s health index is less than country B’s if they spend the same amount of money but A has lower quality.