I think sometimes we’re too obsessed with optimisation. It’s a product of the industrial revolution, or something, everything can go faster, better and cheaper, we assume, except that we all know it can’t. You have to make compromises, obviously. In economics and engineering, the problem is referred to as Pareto optimality. Basically, if you can’t make something better in one respect without making it worse in another, it is Pareto optimal. A “Pareto improvement”, is a change that achieves what you want: making things better without making anything else worse, a change with no compromise. Policy makers know this (it is not an obscure theory) and are supposed to try and achieve Pareto improvements with the changes they make. The thing is, in a complex environment, getting a genuine Pareto improvement is, I suspect, almost certainly impossible.
This Guardian data blog thing is quite fun and full of numbers that an armchair statistician might want to play with. I found this set of poll-based ratings of which European countries are best at which things. I can’t quite believe they went to the effort to create this, but they did. So from this I can draw a fun if completely pointless chart:
The x scores are the percentage of people who think this country has the best food, the y scores are the percent who think it has the most attractive inhabitants. Lets assume the poll is “right” (instead of being completely silly), and you want to choose where to live to maximise these variables. You already live in Poland, you can’t eat anything nicer without moving to France or Spain, but if you do, the people will get uglier. If you must move, Spain is really the only choice, as it has the second best good looking people. Thus, Poland and Spain form the “Pareto front” of Pareto optimal countries based on these criteria, so I coloured them red. If you live in these countries already, you cannot make a Pareto improvement by moving.
The more you move to the right or upwards, the more likely you are to be on the Pareto front and thus already doing the best you can. This is step one of why I think you should be careful about maximising things: if you live in an industrialised country, most of the decisions you make will be about areas where you are already pretty much on the Pareto front because something has already been maximised. The fact is you almost certainly cannot get a faster car without it being more expensive, or less reliable, or uglier, or smaller (if you need space), or bigger (if you prefer it small). If you find a faster car than your current one, at least one of the other desirable characteristics will be made worse (though not necessarily all, of course).
This brings me to the second way in which you are probably on the Pareto front: as you add dimensions, the chances of being already there increases. The silly poll above asked more questions than just about food and how friendly people are. For example, Germany apparently has the best drivers, so if you live there, you can’t move somewhere for better food (such as Poland or Spain) without having to put up with worse drivers. So Germany is on the Pareto front as well now. But also, Poland and Spain are still on the Pareto front, because no matter what dimensions you add to the maximisation problem, the original logic about food and attractiveness will still put them there. Thus if you have a fixed number of options, as you increase the number of factors in your compromise, the more likely it is that you can’t find a better option than your current one without compromising on something.
A more serious multiple-objective optimisation problem would be health care spending. You can’t spend infinite money on health care, but you do (presumably) want to make people as healthy as possible. The OECD produces an amazingly useful set of data on health care. You can look at the above question by finding the Pareto front for the (admittedly rudimentary) measures of spending per capita and life expectancy at birth:
(I removed the USA from this chart as it was a pesky outlier and left the rest of the data squished up in the middle, in short, they spend way more than everyone else on health but it seems don’t get that much for it.)
Lets say that less spending (while maximising health) is better, so the Pareto front consists of:
Living in one of these countries, you couldn’t move to somewhere with better health outcomes that didn’t spend more to get it. You could however, make an informed choice that you consider one variable more important than the other, and move along the Pareto front in a particular direction. For example, if you live in Mexico, while you couldn’t find a higher life expectancy without spending more, it would seem that moving to Chile might be pretty sensible as they seem to have a much better life expectancy, whilst only spending a little more.
But health is very complex, there’s a lot of possible variables other than life expectancy and spending that might be important. In practise, you can probably only make a decision by deciding on the relative importance of each, and use all the information available to move to the appropriate point of the Pareto front. Trying to make an optimal change without any compromise is almost surely going to fail.
As a side note, it would be interesting to know how quickly the Pareto front grows as you add dimensions under various distributions of variables.
Update: I originally referred to Pareto “frontiers” everywhere, because that’s what Wikipedia said, but felt sure that Pareto front is what I heard before. I did a Google fight and Pareto front won, so Wikipedia is (amazingly!) just wrong (that article does look pretty bad, which is why I didn’t link to it before).