## Median Utilitarianism

Posted by Michael Dickens on February 17, 2010

Recently I’ve been thinking about different sorts of averages. The mean average is the most common: add up all the values, and divide by how many values there are. But another useful average is the median average: put all the values in a line, and take the one in the middle. This is useful for some types of averages. For example, if you want to find the average person’s income, the mean would not be very accurate since people like Bill Gates would push the average up. But a median average would be more reasonable.

This got me to thinking about Utilitarianism. On one particular axis, the two types of Utilitarianism are what could be called Average Utilitarianism and Total Utilitarianism. They may have more proper names, but I think that those are descriptive enough. The basic idea there is that Total Utilitarianism seeks to maximize happiness and minimize suffering overall, while Average Utilitarianism seeks to maximize happiness and minimize suffering only for the average person. This is when I started thinking about what kind of average we’re talking about here. Mean average is the type that people usually talk about. But using a median average would possibly be more advantageous. For example, it would be the solution to problems such as the Mere Addition Paradox, also known as the Repugnant Conclusion. The basic idea is that, according to Total Utilitarianism, a great massive population filled with people whose lives are barely worth living is more valuable than a small population filled with people whose lives are rich and enjoyable. Median Utilitarianism solves this problem (so does Mean Utilitarianism).

Another problem solved by Median Utilitarianism is the problem of the Utility Monster. The best description I can find on the web is from a blog:

The “utility monster” was one of philosopher Robert Nozick’s objections to utilitarian theory.

Nozick postulated a creature who received 100 units of utility (pleasure, happiness) per unit of resource consumption, in a universe where everybody else received 1 unit of utility per unit of resource consumption. In this type of universe, Nozick argued, utilitarian would require that all of the people who got lesser utility be sacrificed (give up any and all resources) to the utility monster. This moral demand for sacrifice, however, is absurd. Therefore, basic utilitarianism is defeated by means of a reduction to absurdity.

Median Utilitarianism fixes this problem on an intuitive level. By Median Utilitarianism, making more and more people unhappy to support the happiness of one being is not actually a good thing, since it reduces median happiness — but notice that it still increases total and mean happiness.

Median Utilitarianism probably has problems of its own, but it resolves these two objections to Utilitarian moral theory.

## phynnboi said

I’m working on a separate problem, but it ties so closely into this post of yours that it immediately reminded me of it, and I wanted to run the idea by you.

Okay, the basic idea is, intuitively (at least, to me), we want to maximize total utility in the system, but we

alsowant maximize thebalanceof utility in the system. I think this may be what you were getting at with the “median utility” idea.Well, I think there’s an easy mathematical way of looking at it. I may be off on this. The idea just popped into mind, so I may be overlooking a huge flaw. It may be totally stupid. But, here goes.

Say your system includes only two people, A and B. Higher utility ratings are better. Say that A has 100 utility and B has 0 utility. The sum utility there is 100. Same as if A and 0 utility and B has 100 utility. Well, what about when A has 50 utility and B has 50 utility? That’s still a sum utility of 100. In other words, total utility alone does not differentiate between one person getting completely shafted and both people having an equal share. My hunch is, our genetic wiring tells most of us that the situation in which A has 50 and B has 50 is preferable to the other two situations. (At least, that’s how I feel.) However, maximizing for sum of utility isn’t a total loss–it at least ensures that the system has the highest possible utility, even as it doesn’t ensure an even distribution of utility.

Say that we instead take the sum of

squaresof utilities. In this case, we actually want thelowestsum. So, if A has 100 and B has 0, that’s a sum-of-squares of 10,000. Likewise if A has 0 and B has 100. If A has 50 and B has 50, that’s a sum-of-squares of 5,000. We want the lowest sum of squares, so we prefer the perfect balance over one of the two people getting perfectly shafted. However, since we are aiming for the lowest sum, the obvious way of optimizing the system is to remove all utility. Clearly, we don’t want that.How about combining the two? We seek to maximize the sum of utility (so we have the most utility in our system) while at the same time minimizing the sum-of-squares of utility (so we have the best balance of utility in our system).

(What’s the gain? Well, mean utility is just scaled sum of utility, and thus, has the same problems as sum of utility. Median utility is a partial solution, but still allows for the “it’s okay for the majority to totally shaft the minority” problem, since it’ll be a member of the majority that determines the median. Maybe there’s still a way of gaming the system I proposed, but I think it would be a lot harder.)

I’m not sure how to turn the system I proposed into something useful in day-to-day decision-making. It’s just something that flashed into mind while working on something else. However, I thought you might find it interesting.

(Geeky aside: Minimizing the sum-of-squares of utility could be thought of as “minimizing the ‘error’ in the distribution of utility.”)

## Michael Dickens said

I posted a reply to this comment as a blog post.

## Phil Goetz said

Dammit, I just conceived of least-square utilitarianism today, and googled to see if anybody else had thought of it. You did; and also Francis Ysidro Edgeworth, a famous economist from the 19th century.

It’s shocking that economists who are familiar with the least-squares method, and have been for a century, keep using the same simplistic and hopelessly-flawed utility-combining functions of Bentham (average or total utility), Nash (greatest product), and Rawls (greatest minimum).

## Phil Goetz said

Francis Y. Edgeworth (1883). The method of least squares. Philosophical Magazine Series 5, 1941-5990, Volume 16, Issue 101, Pages 360 – 375. The publisher has this behind a paywall even though its copyright has expired; but Google provides it for free here.