Optimization Metrics

Clara and I have to take a written exam in order to transfer our driver’s licenses to Minnesota, so we’re studying the Minnesota driver’s manual. In the section on crosswalks I found this instruction:

The problem is that you can’t leave an intersection both as quickly as possible AND as safely as possible. To leave as quickly as possible would be to sprint the last few yards, raising the chances of tripping and falling. To leave as safely as possible would mean walking slowly and carefully, constantly scanning your surroundings for new dangers, which is not particularly speedy.

This curve represents the boundary of your possible choices for how to cross the street. Anywhere inside the curve, you can increase either your speed or your safety, or both. Combinations of speed and safety outside the curve are beyond your ability, such as trying to go both as fast as you can and as safely as you can, the way pedestrians in Minnesota are charged to do.

Of course, no one is really bothered by this in practice; what we actually do is something like “leave the intersection as quickly as possible while still staying reasonably safe.”


But this is only one way of solving the problem: you could decide instead that you need to leave reasonably quickly and then go as safely as you can at that speed. And either way, the choice of how safe is “reasonably safe” or how quickly is “reasonably quickly” is a little arbitrary.

Anyway, the reason I’m bringing this up is to try to make clear the idea that even if two people value the same things, they might disagree on where to allocate their efforts. This is especially relevant these days, as so many people seem to be talking past each other about what they want for our country. We all want everyone to be better off, but there are many ways to gauge the wellbeing of a population, and you can’t optimize them all at the same time. Here are some examples:

  • How good the best are. This is how we compare countries in the Olympics, for example.
  • How good the average are. This is what we are thinking of when we worry that U.S. students are falling behind those in other countries regarding their math scores, or when we compare different countries based on their GDP per capita.
  • How good the total is. If a life is valuable in itself, then all things being equal a larger population is preferable to a smaller one. Measures along these lines include total GDP; policy based on improving that might involve promoting birth rates so we have a larger workforce.
  • How bad the worst are. If we want to improve the minimum quality of life in the country, we should concentrate all our efforts on those people who need our aid the most.
  • How far apart the best and worst are. If we want everyone to have equal resources, then we should keep on robbing the rich to give to the poor until there’s no difference, regardless of where that middle point ends up being.
  • How many people fall below a certain threshold. If we’ve drawn a poverty line and only want to reduce the number of people below it, it’s better to adjust handouts so that everyone just makes it above the line, regardless of how much worse off those above the line end up being.

One of these may sound more like your preference than another, even though they’re all based on some way of trying to make things better for everyone. It’s impossible to optimize with respect to two different metrics at the same time, so you have to choose what to fix at “reasonable” and what you optimize given that, and the choice of “reasonable” is a little arbitrary. So next time you’re in an argument with someone, please remember that they might just be trying to optimize according to a slightly different metric, and just because they disagree with you doesn’t mean they don’t value the same things.


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